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abstract: 'Assume $G$ is a finite abstract simplicial complex with $f$-vector $(v_0,v_1, \dots)$, and generating function $f(x) = \sum_{k=1}^{\infty} v_{k-1} x^k=v_0 x + v_1 x^2+ v_2 x^3 + \cdots$, the Euler characteristic of $G$ can be written as $\chi(G)=f(0)-f(-1)$. We study here the functional $f_1''(0)-f_1''(-1)$, where $f_1''$ is the derivative of the generating function $f_1$ of $G_1$. The Barycentric refinement $G_1$ of $G$ is the Whitney complex of the finite simple graph for which the faces of $G$ are the vertices and where two faces are connected if one is a subset of the other. Let $L$ is the connection Laplacian of $G$, which is $L=1+A$, where $A$ is the adjacency matrix of the connection graph $G''$, which has the same vertex set than $G_1$ but where two faces are connected they intersect. We have $f_1''(0)={\rm tr}(L)$ and for the Green function $g=L^{-1}$ also $f_1''(-1)={\rm tr}(g)$ so that $\eta_1(G) = f_1''(0)-f_1''(-1)$ is equal to $\eta(G)={\rm tr}(L-L^{-1})$. The established formula ${\rm tr}(g)=f_1''(-1)$ for the generating function of $G_1$ complements the determinant expression ${\rm det}(L)={\rm det}(g)=\zeta(-1)$ for the Bowen-Lanford zeta function $\zeta(z)=1/{\rm det}(1-z A)$ of the connection graph $G''$ of $G$. We also establish a Gauss-Bonnet formula $\eta_1(G) = \sum_{x \in V(G_1)} \chi(S(x))$, where $S(x)$ is the unit sphere of $x$ the graph generated by all vertices in $G_1$ directly connected to $x$. Finally, we point out that the functional $\eta_0(G) = \sum_{x \in V(G)} \chi(S(x))$ on graphs takes arbitrary small and arbitrary large values on every homotopy type of graphs.'
address: |
Department of Mathematics\
Harvard University\
Cambridge, MA, 02138
author:
- Oliver Knill
date: 'May 29, 2017'
title: 'On a Dehn-Sommerville functional for simplicial complexes'
---
Setup
=====
####
A [**finite abstract simplicial complex**]{} $G$ is a finite set of non-empty sets with the property that any non-empty subset of a set in $G$ is in $G$. The elements in $G$ are called [**faces**]{} or [**simplices**]{}. Every such complex defines two finite simple graphs $G_1$ and $G'$, which both have the same vertex set $V(G_1)=V(G')=G$. For the graph $G_1$, two vertices are connected if one is a subset of the other; in the graph $G'$, two faces are connected, if they intersect. The graph $G_1$ is called the [**Barycentric refinement**]{} of $G$; the graph $G'$ is the [**connection graph**]{} of $G$. The graph $G_1$ is a subgraph of $G'$ which shares the same topological features of $G$. On the other hand, the connection graph is fatter and be of different topological type: already the Euler characteristic $\chi(G)$ and $\chi(G')$ can differ. Both graphs $G_1$ and $G$ are interesting on their own but they are linked in various ways as we hope to illustrate here. Terminology in this area of combinatorics is rich. One could stay within simplicial complexes for example and deal with “flag complexes", complexes which is a Whitney complex of its $1$-skeleton graphs. The complexes $G_1$ and $G'$ are by definition of this type. We prefer in that case to use terminology of graph theory.
####
Let $A$ be the adjacency matrix of the connection graph $G'$. Its Fredholm matrix $L=1+A$ is called the [**connection Laplacian**]{} of $G$. We know that $L$ is unimodular [@Unimodularity] so that the [**Green function operator**]{} $g=L^{-1}$ has integer entries. This is the [**unimodularity theorem**]{} [@Helmholtz]. The Bowen-Lanford zeta function of the graph $G'$ is defined as $\zeta(s) = {\rm det}((1-sA)^{-1})$. As $\zeta(-1)$ is either $1$ or $-1$, we can see the determinant of $L$ as the value of the zeta function at $s=-1$. We could call $H=L-L^{-1}$ the [**hydrogen operator**]{} of $G$. The reason is that classically, if $L=-\Delta$ is the Laplacian in $R^3$, then $L^{-1}$ is an integral operator with entries $g(x,y) = 1/|x-y|$. Now, $H \psi(y) = (L \psi)(y) - \psi(y)/|x-y|$ is the Hamiltonian of a Hydrogen atom located at $x$, so that $H$ is a sum of a kinetic and potential part, where the potential is determined by the inverse of $L$. When replacing the multiplication operation with a convolution operation, then $L^{-1}$ takes the role of the potential energy. Anyway, we will see that the trace of $H$ is an interesting variational problem.
####
There are various variational problems in combinatorial topology or in graph theory. For the later, see [@BollobasExtremal]. An example in polyhedral combinatorics is the upper bound theorem, which characterizes the maxima of the discrete volume among all convex polytopes of a given dimension and number of vertices [@Stanley1996]. An other example problem is to maximize the Betti number $b(G)=\sum_{i=0} b_i$ which is bounded below by $\chi(G)=\sum_{i=0} (-1)^i b_i $ which we know to grow exponentially in general in the number of elements in $G$ and for which upper bounds are known too [@Adamaszek]. We have looked at various variational problems in [@KnillFunctional] and at higher order Euler characteristics in [@valuation]. Besides extremizing functionals on geometries, one can also define functionals on the on the set of unit vectors of the Hilbert space $H^n$ generated by the geometry. An example is the free energy $(\psi,L\psi) - T S(|\psi|^2)$ which uses also entropy $S$ and temperature variable $T$ [@Helmholtz].
####
Especially interesting are functionals which characterize geometries. An example is a necessary and sufficient condition for a $f$-vector of a simplicial d-polytope to be the $f$-vector of a simplicial complex polytope, conjectured 1971 and proven in 1980 [@BilleraLee; @Stanley1980]. Are there variational conditions which filter out discrete manifolds? We mean with a discrete manifold a connected finite abstract simplicial complex $G$ for which every unit sphere $S(x)$ in $G_1$ is a sphere. The notion of sphere has been defined combinatorially in discrete Morse approaches using critical points [@forman95] or discrete homotopy [@I94a]. A $2$-complex for example is a discrete $2$-dimensional surface. In a 2-complex, we ask that every unit sphere in $G_1$ is a circular graph of length larger than $3$. For a 2-complex the $f$-vector of $G_1$ obviously satisfies $2 v_1-3v_2=0$ as we can count the number of edges twice by adding up 3 times the number of triangles. The relation $2v_1-3v_2=0$ is one of the simplest Dehn-Sommerville relations. It also can be seen as a zero curvature condition for $3$-graphs [@cherngaussbonnet] or then related to eigenvectors to Barycentric refinement operations [@valuation; @KnillBarycentric2]. Dehn-Sommerville relations can be seen as zero curvature conditions for Dehn-Sommerville invariants in a higher dimensional complex.
####
One can wonder for example whether a condition like $\eta(G) = 2v_1-3v_2=0$ for the $f$-vector $(v_0,v_1,v_2)$ of the Barycentric refinements $G_1$ of a general $2$-dimensional abstract finite simplicial complex $G$ forces the graph $G_1$ to have all unit spheres to be finite unions of circular graphs. For this particular functional, this is not the case. There are examples of discretizations of varieties with $1$-dimensional set of singular points for which $2v_1-3v_2$ is negative. An example is $C_n \times F_8$, the Cartesian product of a circular graph with a figure $8$ graph. An other example is a $k$-fold suspension of a circle $G=C_n + P_k$, where $C_n$ is the circular graph, $P_k$ the $k$ vertex graph without edges and $+$ is the Zykov join which takes the disjoint union of the graphs and connects additionally any two vertices from different graphs. In that case, $\eta_0(G)=n(2-k)$ which is zero only in the discrete manifold case $k=2$ where we have a discrete 2-sphere, the suspension of a discrete circle.
####
Our main result here links a spectral property with a combinatorial property. It builds on previous work on the connection operator $L$ and its inverse $g=L^{-1}$. We will see that $\eta(G_1)={\rm tr}(L-L^{-1})$, where $L$ is the connection Laplacian of $G$, which remarkably is always invertible. If $G$ has $n$ faces=simplices=sets in $G$, the matrix $L$ is a $n \times n$ matrix for which $L_{xy}=1$ if $x$ and $y$ intersect and where $L_{xy}=0$ if $x \cap y$ is empty. We establish that the [**combinatorial functional**]{} $\eta(G_1)=2v_1-3v_2 +4v_3 -5 v_4 + \dots$ which is also the [**analytic functional**]{} $f'(0)-f'(-1)$ for a generating function $f(x)=\sum_{k=1}^{\infty} v_{k-1} x^k$ is the same than the [**algebraic functional**]{} ${\rm tr}(L-L^{-1})$ and also equal to the [**geometric functional**]{} $\eta_1(G) = \sum_{x \in V(G_1)} \chi(S(x))$. The later is a Gauss-Bonnet formula which in general exists for any linear or multi-linear valuation [@valuation].
####
The functional $\eta_1$ is a valuation like the Euler characteristic $\chi(G) = v_0-v_1+v_2- \dots$ whose combinatorial definition can also be written as $f(0)-f(-1)$ or as a Gauss-Bonnet formula $\sum_x K(x)$ or then as the super trace of a heat kernel ${\rm tr}(e^{-t L})$ by McKean-Singer [@knillmckeansinger]. The Euler curvature $K(x)=\sum_{k=0}^{\infty} (-1)^k v_{k-1}(S(x))/(k+1)$ [@cherngaussbonnet] could now be written as $K(x)=F(0)-F(-1)$, where $F(t)=\int_0^t f(s) \; ds = \sum_{k=0}^{\infty} v_{k-1} x^{k+1}/(k+1)$ is the anti-derivative of the [**reduced generating function**]{} $1-f$ of $S(x)$. We see a common theme that $F(0)-F(-1), f(0)-f(-1), f'(0)-f'(-1)$ all appear to be interesting.
####
Euler characteristic is definitely the most fundamental valuation as it is related to the unique eigenvector of the eigenvalue $1$ of the Barycentric refinement operator. It also has by Euler-Poincaré a cohomological description $b_0-b_1+b_2- \dots$ in terms of Betti numbers. The minima of the functional $G \to \chi(G)$ however appear difficult to compute [@eveneuler]. From the expectation formula ${\rm E}_{n,p}[\chi] = \sum_{k=1}^n (-1)^{k+1} \B{n}{k} p^{\B{k}{2}}$ [@randomgraph] of $\chi$ on Erdös-Renyi spaces we know that unexpectedly large or small values of $\chi(G)$ can occur, even so we can not construct them directly. As the expectation of $\chi = b_0-b_1+b_2 - \dots$ grows exponentially with the number of vertices. Also the total sum of Betti numbers grows therefore exponentially even so the probabilistic argument gives no construction. We have no idea to construct a complex with 10000 simplices for which the total Betti number is larger than say $10^{100}$ even so we know that it exists as there exists a complex $G$ for which $\chi(G)$ is larger than $10^{100}$. Such a complex must be a messy very high-dimensional Swiss cheese.
####
After having done some experiments, we first felt that $\eta(G)$ must be non-negative. But this is false. In order to have negative Euler characteristic for a unit sphere of a two-dimensional complex, we need already to have some vertex for which $S(x)$ is a bouquet of spheres. A small example with $\eta(G)<0$ is obtained by taking a sphere, then glue in a disc into the inside which is bound by the equator. This produces a geometry $G$ with Betti vector $(1,0,2)$ and Euler characteristic $3$. It satisfies $\eta(G)=-8$ as every of the $8$ vertices at the equator of the Barycentric refinement of $G_1$ has curvature $\chi(S(x))=-1$ and for all the other vertices have $\chi(S(x))=0$.
####
This example shows that $\eta(G)$ can become arbitrarily small even for two-dimensional complexes. But what happens in this example there is a one dimensional singular set. It is the circle along which the disk has been glued between three spheres. We have not yet found an example of a complex $G$ for which $G_1$ has a discrete set of singularities (vertices where the unit sphere is not a sphere.) In the special case where $G$ is the union a finite set of geometric graphs with boundary in such a way that the intersection set is a discrete set, then $\eta(G) \geq 0$.
Old results
===========
####
Given a face $x$ in $G$, it is also a vertex in $G_1$. The [**dimension**]{} ${\rm dim}(x) = |x|-1$ with cardinality $|x|$ now defines a function on the vertex set of $G_1$. It is locally injective and so a [**coloring**]{}. We know already $g(x,x) = 1-\chi(S(x))$ [@Spheregeometry] and that $V(x) = \sum_y g(x,y) = (-1)^{{\rm dim}(x)} g(x,x)$ is [**curvature**]{}: $\sum_x V(x)=\chi(G)$. It is dual to the curvature $\omega(x) = (-1)^{{\rm dim}(x)}$ for which Gauss-Bonnet $\sum_x \omega(x)$ is the definition of Euler characteristic. Both of these formulas are just Poincaré-Hopf for the gradient field defined by the function ${\rm dim}$. The Gauss-Bonnet formula $\sum_x V(x)= \chi(G)$ can be rewritten as $\sum_{x,y} g(x,y) = \chi(G)$. We call this the [**energy theorem**]{}. It tells that the total potential energy of a simplicial complex is the Euler characteristic of $G$. By the way, $\sum_{x,y} L(x,y) = |V(G')|+2 |E(G')|$ by Euler handshake.
####
If $v_k$ counts the number of $k$-dimensional faces of $G$, then $f(x)=\sum_{k=1}^{\infty} v_{k-1} x^k =
v_0 x + v_1 x^2 + \dots $ is a [**generating function**]{} for the $f$-vector $(v_0,v_1, \dots)$ of $G$. We can rewrite the Euler characteristic of $G$ as $\chi(G)=-f(-1)=f(0)-f(-1)$. If $G$ is a graph, we assume it to be equipped with the [**Whitney complex**]{}, the finite abstract simplicial complex consisting of the vertex sets of the complete subgraphs of $G$. This in particular applies for the graph $G_1$. The $f$-vector of $G_1$ is obtained from the $f$-vector of $G$ by applying the matrix $S_{ij} = i! S(j,i)$, where $S(j,i)$ are [**Stirling numbers**]{} of the second kind. Since the transpose $S^T$ has the eigenvector $(1,-1,1,-1, \dots)$, the Euler characteristic is invariant under the process of taking Barycentric refinement. Actually, as $S$ has simple spectrum, it is up to a constant the unique valuation of this kind. Quantities which do not change under Barycentric refinements are called [**combinatorial invariants**]{}.
####
The matrices $A,L,g$ act on a finite dimensional Hilbert space whose dimension is the number of faces in $G$ which is the number of vertices of $G_1$ or $G'$. Beside the usual trace ${\rm tr}$ there is now a [**super trace**]{} ${\rm str}$ defined as ${\rm str}(L)= \sum_x \omega(x) L(x,x)$ with $\omega(x) = (-1)^{{\rm dim}(x)}$. The definition $\chi(G) = \sum_x \omega(x)$ can now be written as ${\rm str}(1)= \chi(G)$. Since $L$ has $1$’s in the diagonal, we also have ${\rm str}(L)= \chi(G)$. A bit less obvious is $\chi(G) = {\rm str}(g)$ which follows from the Gauss-Bonnet analysis leading to the energy theorem. It follows that the Hydrogen operator $H$ satisfies ${\rm str}(H)=0$, the super trace of $H$ is zero. This leads naturally to the question about the trace of $H$. By the way, the super trace of the Hodge Laplacian $L=(d+d^*)^2$ where $d$ is the exterior derivative is always zero by Mc-Kean Singer (see [@knillmckeansinger] for the discrete case).
####
The Barycentric refinement graph $G_1$ and the connection graph $G'$ have appeared also in a number theoretical setup. If $G$ is the countable complex consisting of all finite subsets of prime numbers, then the finite prime graph $G_1(n) \subset G_1$ has as vertices all square-free integers in $V(n)=\{2,3,4 \dots ,n\}$, connecting two if one divides the other. The prime connection graph $G'(n)$ has the same vertices than $G_1(n)$ but connects two integers if they have a common factor larger than $1$. This picture interprets sets of integers as simplicial complexes and sees counting as a Morse theoretical process [@CountingAndCohomology]. Indeed $\chi(G_1(n)) = 1-M(n)$, where $M(n)$ is the Mertens function. If the vertex $n$ has been added, then $i(n)=1-\chi(S(n))=-\mu(n)$ with Möbius function $\mu$ is a Poincaré-Hopf index and $\sum_x i(x)=\chi(G_1(x))$ is a Poincaré-Hopf formula. In combinatorics, $-i(G)=\chi(G)-1$ is called the [**reduced Euler characteristic**]{} [@Stanley86]. The counting function $f(x)=x$ is now a discrete Morse function, and each vertex is a critical point. When attaching a new vertex $x$, a handle of dimension $m(x)={\rm dim}(S(x))+1$ is added. Like for critical points of Morse functions in topology, the index takes values in $\{-1,1\}$ and $i(x)=(-1)^{m(x)}$. For the connection Laplacian adding a vertex has the effect that the determinant gets multiplied by $i(x)$. Indeed, ${\rm det}(L)=\prod_x \omega(x)$ in general while $\chi(G) = \sum_x \omega(x)$, if $\omega(x) = (-1)^{{\rm dim}(x)}$.
The functional
==============
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Define the functional $$\eta(G) = {\rm tr}(H) = {\rm tr}(L - L^{-1})\; .$$ Due to lack of a better name, we call it the [**hydrogen trace**]{}. We can rewrite this functional in various ways. For example $\eta(G) = \sum_k \lambda_k -1/\lambda_k$, where $\lambda_k$ are the eigenvalues of $L$. We can also write $\eta(G) = \sum_k \mu_k \frac{2+\mu_k}{1+\mu_k}$ where $\mu_k$ are the eigenvalues of the adjacency matrix $A$ of the connection graph $G'$. It becomes interesting however as we will be able to link $\eta$ explicitly with the $f$-vector of the complex $G_1$ or even with the $f$-vector of the complex $G$ itself.
####
We will see below that also the [**Green trace functional**]{} ${\rm tr}(g)$ is interesting as $g=L^{-1}$ is the Green function of the complex. It is bit curious that there are analogies and similarities between the Hodge Laplacian $H = (d+d^*)^2$ of a complex and the connection Laplacian $L$. Both matrices have the same size. As they work on a space of simplices, where the dimension functional defines a parity, one can also look at the [**super trace**]{} ${\rm str}(L) = \sum_{{\rm dim}(x)>0} L_{xx} - \sum_{{\rm dim}(x)<0} L_{xx}$ It is a consequence of Mc-Kean Singer super symmetry that ${\rm str}(H) = 0$ which compares with the definition ${\rm str}(L)=\chi(G)$ and leads to the McKean Singer relation ${\rm str}(e^{-tH})=\chi(G)$. We have seen however the Gauss-Bonnet relation ${\rm str}(g) = \chi(G)$ which implies the energy theorem $\sum_{x,y} g_{xy} = \chi(G)$. It also implies ${\rm str}(L-g) = 0$.
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The invertibility of the connection Laplacian is interesting and lead to topological relations complementing the topological relations of the Hodge Laplacian to topology like the Hodge theorem telling that the kernel of the $k$’th block of $H$ is isomorphic the $k$’th cohomology of $G$. Both $L$ and $H$ have deficits: we can not read off cohomology of $L$ but we can not invert $H$, the reason for the later is exactly cohomology as harmonic forms are in the kernel of $H$. So, there are some complementary benefits of both $L$ and $H$. And then there are similarities like ${\rm str}(H) = {\rm str}(L-L^{-1})=0$ and ${\rm str}(e^{-H}) = {\rm str}(L^{-1}) = \chi(G)$.
Gauss-Bonnet
============
####
The following [**Gauss-Bonnet**]{} theorem for $\eta$ shows that its curvature at a face $x$ is the Euler characteristic of the unit sphere $S(x)$ in the Barycentric refinement $G_1$. We use the notation $\eta_0(G) = \sum_{x \in V(G)} \chi(S(x))$, $\eta_1(G)=\eta_0(G_1)$ and $\eta(G)={\rm tr}(L-L^{-1})$.
Let $G$ be an arbitrary abstract finite simplicial complex. Then $${\rm tr}(L-L^{-1}) = \eta(G) = \eta_1(G) = \eta_0(G_1) = \sum_{x \in V(G_1)} \chi(S(x)) \; .$$
The diagonal elements of $g=L^{-1}$ has entries $(1-\chi(S(x))$. We therefore have have ${\rm tr}(g) = \sum_x (1-\chi(S(x))$. We also have ${\rm tr}(G) = \sum_x 1$.
[**Examples.**]{}\
[**1)**]{} If $G=C_n$, then $G_1=C_{2n}$. Now, $\chi(S(x))=2$ for all vertices $x \in G_1$. We see that $\eta(C_n) = 4n$.\
[**2)**]{} For a discrete two-dimensional graph $G$, a graph for which every unit sphere is a circular graph, we have $\eta(G) = 0$.\
[**3)**]{} For a discrete three-dimensional graph $G$, a graph for which every unit sphere is a two dimensional sphere, we have $\eta(G) = 2 V(G_1) = 2 \sum_{k=0}^{\infty} v_k(G)$. For example, for the 3-sphere, the suspension of the octahedron, which can be written as $G=3 P_2 = P_2+P_2+P_2$, we have $\eta(G) = 160$ because the $f$-vector of $G$ is $\vec{v} = (8, 24, 32, 16)$.\
[**4)**]{} For a graph without triangles, we have $\eta(G) = \sum_{x \in V(G_1)} {\rm deg}(x)$ which is by handshaking $2 v_1(G_1)$. Since Barycentric refinement doubles the edges, we have $\eta(G)=4 v_1(G)$. This generalizes the circular case discussed above.\
[**5)**]{} For $G=K_n$ we have $\eta(G) = 2^{n}$ if $n$ is even and $2^n-2$ if $n$ is odd. The numbers start as following: $\eta(K_1)=0$ $\eta(K_2)=4$, $\eta(K_3)=6$, $\eta(K_4)=16$, $\eta(K_5)=30$ etc.
Generating function
===================
####
Let $f_{G_1}(x) = 1+v_1 x + v_2 x^2 + \dots = 1+\sum_{k=1}^{\infty} v_{k-1} x^k$ be the (reduced) generating function for the Barycentric refinement $G_1$ of $G$. The [**Zykov join**]{} of two graphs $G_1 + H_1$ is defined as the graph with vertex set $V(G_1) \cup V(H_1)$ for which two vertices $a,b$ are connected if they were connected in $G_1$ or $H_1$ or if $a,b$ belong to different graphs. The generating function of the sum $G_1+H_1$ is the product of the generating functions of $G_1$ and $H_1$.\
Since the Euler characteristic satisfies $\chi(G) = f(0)-f(-1) = \chi(G_1) = f_1(0)-f_1(-1)$, the following again shows that the functional $\eta$ appears natural;
$\eta(G) = f_1'(0)-f_1'(-1)$.
####
To prove this, we rewrite the Gauss-Bonnet result as a Gauss-Bonnet result for the second Barycentric refinement $G_2$. Define for a vertex $x$ in $G_2$ the curvature $$k(x) = (-1)^{1+{\rm dim}(x)} (1+{\rm dim}(x)) \; .$$
$\eta(G) = \sum_{x \in V(G_2)} k(x)$, where the sum is over all vertices $x$ in $G_2$ which have positive dimension.
This is a handshake type argument. We start with $\eta(G) = \sum_x \chi(S(x))$. Since every $d$-dimensional simplex in $S(x)$ defines a $(d+1)$-dimensional simplex containing $x$, super summing over all simplices of $S(x)$ gives a super sum over simplices in $G_2$ where each simplex such $y$ appears ${\rm dim}(y)+1$ times.
[Remark.]{} This gives us an upper bound on the functional $\eta$ in terms of the number of vertices in $G_2$ and the maximal dimension of $G$: $\eta(G) \leq |V(G_2)| (1+d)$. If $G_1$ has $n$ elements, then $G_2$ has $\leq (d+1)! n$ elements. We see:
$\eta(G)$ is bounded above by $C_d |V(G_1)|$, where $C_d$ only depends on the maximal dimension of $G$.
####
Now we can prove the result:
As $f_1(x) = v_0 x + v_1 x^2 + v_2 x^3 + \dots$, we have $f_1'(x) = v_0 + 2 v_1 x + 3 v_2 x^2 + \dots$ and $f_1'(0)-h_1'(-1) = 2 v_1 - 3 v_2 + 4 v_3$ which is the same than $\sum_{x, {\rm dim}(x)>0} (-1)^{1+{\rm dim}(x)} (1+{\rm dim}(x))$.
As an application we can get a formula for $\eta_0(G_1+H_1)$, where $G_1+H_1$ is the Zykov sum of $G_1$ and $H_1$. The Zykov sum shares the properties of the classical join operation in the continuum. The Grothendieck argument produces from the monoid a group which can be augmented to become a ring [@Spheregeometry].
On the set of complexes with zero Euler characteristic, we have $\eta_0(G_1+H_1) = \eta_0(G_1) + \eta_0(H_1)$.
We have $f_{G_1 + H_1} = f_{G_1} f_{H_1}$. Now $\eta_0(G_1) = f_{G_1}'(0)-f_{G_1}'(-1)$ and $\eta_0(H_1) = f_{H_1}(0)-f_{H_1}'(-1)$. By the product rule, $f_{G_1+H_1}' = (f_{G_1} f_{H_1})' = f_{G_1}' f_{H_1} + f_{G_1} f_{H_1}'$. we have now $f_{G_1+H_1}'(0) = f_{G_1}'(0) + f_{H_1}'(0)$ and $f_{G_1+H_1}'(-1) = f_{G_1}'(-1) (1-\chi(H_1)) + (1-\chi(G_1)) f_{H_1}'(-1)$ so that $\eta_0(G_1+H_1) = \eta_(G_1) + \eta_(G_2) + f_{G_1}'(-1) \chi(H_1) + f_{H_1}'(-1) \chi(G_1)$.
Geometric graphs
================
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We will see in this section that for graphs which discretized manifolds or varieties which have all singularities isolated and split into such discrete manifolds, the functional $\eta$ is non-negative. A typical example is a bouquet of spheres, glued together at a point.
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A [**$d$-graph**]{} is a finite simple graph for which every unit sphere $S(x)$ is a $(d-1)$-graph which is a $(d-1)$-sphere. A [**$d$-sphere**]{} is a $d$-graph which becomes collapsible if a single vertex is removed. The inductive definitions of $d$-graph and $d$-sphere start with the assumption that the empty graph is a $-1$-sphere and $-1$ graph and that the $1$ point graph $K_1$ is collapsible. A graph $G$ is [**collapsible**]{} if there exists a vertex $x$ such that both $G \setminus x$ and $S(x)$ are collapsible. It follows by induction that $d$-sphere has Euler characteristic $1+(-1)^{d} \in \{ 0,2\}$.
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A simplicial complex is called a $d$-complex if its refinement $G_1$ is a $d$-graph. We now see that for even-dimensional $d$-complexes, the functional $\eta$ is zero. A graph is a [**$d$-graph with boundary**]{} if every unit sphere is either a sphere or contractible and such that the [**boundary**]{}, the set of vertices for which the unit sphere $S(x)$ is contractible is a $d-1$-graph. An example is the wheel graph $G$ for which the boundary $\delta G$ is a circular graph.
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Since for an even dimensional $d$-graph with boundary the Euler characteristic of the unit spheres in the interior is zero, Gauss-Bonnet implies $\eta(G) = |V(\delta G)|$. In the case of an odd-dimensional $d$-graph with boundary, it leads to $\eta(G)=2 |{\rm int}(G)| + |\delta G|$. This leads to the observation:
If $G$ is a $d$-graph with boundary, then $\eta(G) \geq 0$. Equality holds if and only if $G$ is an even dimensional graph without boundary.
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This can be generalized: if $G$ is a union of finitely many $d_k$-graphs $G_k$ such that the set of vertices which belong to at least $2$ graphs is isolated in the sense that the intersection of any two $G_k$ does not contain any edge, then $\delta(G) \geq 0$. Equality holds if $G$ is a finite union of even dimensional graphs without boundary touching at a discrete set of points. The reason is that the unit spheres are again either spheres or then finite union of spheres of various dimension. Since the Euler characteristic of of a sphere is non-negative and the Euler characteristic of a disjoint sum is the sum of the Euler characteristics, the non-negativity of $\eta$ follows. We will ask below whether more general singularities are allowed and still have $\eta(G) \geq 0$.\
####
Maybe in some physical context, one would be interested especially in the case $d=4$ and note that among all $4$-dimensional simplicial complexes with boundary the complexes without boundary minimize the functional $\eta$. In the even dimensional case, the curvature of $\eta$ is supported on the boundary of $G$. If we think of the curvature as a kind of charge, this is natural in a potential theoretic setup. Indeed, one should think of $V_x(y) = g(x,y)$ as a potential [@Helmholtz]. In the case of an odd dimensional complex, there is a constant curvature present all over the interior and an additional constant curvature at the boundary. Again, also in the odd dimensional case, the absence of a boundary minimizes the functional $\eta$.\
####
We should in this context also mention the [**Wu characteristic**]{} for which we proved in [@valuation] that for $d$ graphs with boundary, the formula $\omega(G) = \chi(G)-\chi(\delta G))$ holds. The Wu characteristic $\omega$ was defined as $\omega(G) = \sum_{x \cap y \neq \emptyset} \omega(x) \omega(y)$ with $\sigma(x)=(-1)^{{\rm \omega}(x)}$. The Wu characteristic fits into the topic of connection calculus as $\omega(G) = {\rm tr}(L J)$, where $J$ is the [**checkerboard matrix**]{} $J_{xy} = (-1)^{{\rm dim}(i)+{\rm dim}(j)} = \omega^T \cdot \omega$ so that $J/n$ is a projection matrix [@Helmholtz]. Actually, in the eyes of Max Born, one could see $\omega(G)/n = (\omega,L \omega )$ as the expectation of the [**state**]{} $\Omega=\omega/\sqrt{n}$.
The sum of the sphere Euler characteristic
==========================================
####
We look now a bit closer at the functional $$\eta_0(G) = \sum_{x \in V(G)} \chi(S(x)) \;$$ on graphs. It appears to be positive or zero for most Erdös-Renyi graphs but it can take arbitrary large or small values. We have seen that $\eta(G) = \eta_0(G_1) = {\rm tr}(L-L^{-1})$. But now, we look at graphs $G$ which are not necessarily the Barycentric refinement of a complex.\
[**Examples.**]{}\
[**1)**]{} For a complete graph $K_{n}$ we have $\eta_0(G)=n$.\
[**2)**]{} For a complete bipartite graph $K_{n,m}$ we have $\eta_0(G) = 2 n m$ and $\eta(G)=4 n m$.\
[**3)**]{} For an even dimensional $d$-graph $G$, we have $\eta_0(G)=0$.\
[**4)**]{} For an odd dimensional $d$-graph $G$ we have $\eta_0(G) = 2|V(G)|$.\
[**5)**]{} For the product $G$ of linear graph $L_m$ of length $m$ with a figure $8$ graph $E =C_k \wedge_x C_k$ we have for $k \geq 4$ and $m \geq 1$ the formula $\eta_0(L_m \times (C_k \wedge_x C_k)) =
28+(k-4) 8 - (m-1) 4 = 8k-4m$.\
[**6)**]{} So far, in all examples we have seen if $G_1$ is the Barycentric refinement, we see $|\eta_0(G_1)| \geq 2 |\eta_0(G)|$.\
[**7)**]{} For the graph $G$ obtained by filling in an equator plane, we have $\eta_0(G)=-4$ and $\eta(G)=-8$.
The functional is additive for a wedge sum and for the disjoint sum.
In both cases, the unit spheres $S(x)$ for vertices $x$ in one of the graphs $H,G$ only are not affected. For the vertex in the intersection, then $S_{G \cup H}(x)$ is the disjoint union $S_G(x) \cup S_H(x)$.
For every homotopy type of graphs, the functional $\eta_0$ is both unbounded from above and below.
Take a graph $G$ with a given homotopy type. Take a second graph $H=\eta_0(L_m \times (C_k \wedge C_k))$ with large $m$. It has $\eta_0(H)=8k-4m$. We can close one side of the graph to make it contractible. This produces a contractible graph with $\eta_0(\tilde{H}) = 8k-2m$. The graph $\tilde{H} \wedge G$ now has the same homotopy type than $G$ and has $\eta(\tilde{H} \wedge G) = 8k-2m + \eta_0(G)$. By choosing $k$ and $m$ accordingly, we can make $\eta_0$ arbitrarily large or small. The addition of the contractible graph has not changed the homotopy type of $G$.
Let $C(k)$ denote the set of connected graphs with $k$ vertices. On $C(2)$, we have $2 \leq \eta_0(G) \leq 2$, on $C(3)$, we have $3 \leq \eta_0(G) \leq 3$, on $C(4)$, we have $4 \leq \eta_0(G) \leq 8$, on $C(5)$, we have $4 \leq \eta_0(G) \leq 12$, and on $C(6)$ we have $0 \leq \eta_0(G) \leq 18$.
About the spectrum of $L$
=========================
####
We have not found any positive definite connection Laplacian $L$ yet. Since $L$ has non-negative entries, we know that $L$ has non-negative eigenvalues. By unimodularity [@Unimodularity] it therefore has some positive eigenvalue. The question about negative eigenvalues is open but the existence of some negative eigenvalues would follow from ${\rm tr}(H) \geq 0$ thanks to the following formula dealing with the column vectors $A_i=L_i-e_i$ of the adjacency matrix of $G'$.
$\eta(G) = -\sum_i (A_i, g A_i)$.
Let $A$ be the adjacency matrix of the connection graph $G'$ so that the connection Laplacian $L$ satisfies $L=1+A$. As $L$ has entries $1$ in the diagonal only, we know ${\rm tr}(L) = n$ and $g \cdot A = (1+A)^{-1} A = -(1-(1+A)^{-1})= g-1$ so that $\eta(G) = {\rm tr}(L)-{\rm tr}(g)={\rm tr}(1-g)=-{\rm tr}(g \cdot A)
= - \sum_i e_i g A e_i = - \sum_i e_i g A_i) = - \sum_i A_i g A_i$. The reason for the last step is $\sum_i (e_i+A_i) g A_i = \sum_i L_i g A_i = \sum_i e_i A_i = 0$.
This immediately implies that $g$ (and so $L=g^{-1}$) can not be positive definite if $\eta(G) \geq 0$. Indeed, if $g$ were positive definite, then $A_i g A_i>0$ for all $A_i$ and so $\sum_i A_i g A_i>0$ but $\eta(G) \geq 0$ implies $\sum_i A_i g A_i \leq 0$.
Open questions
==============
####
Since the entries of $L$ are non-negative and $L$ is invertible, the Perron-Frobenius theorem shows that $L$ has some positive eigenvalue. A similar argument to show negative eigenvalues does not work yet, at least not in such a simple way. The next result could be easy but we have no answer yet for the following question:
####
Lets call a graph a [**$d$-variety**]{}, if every unit sphere is a $(d-1)$-graph except for some isolated set of points, the singularities, where the unit sphere is allowed to be a $(d-1)$-variety. In every example of a $d$-variety with $\eta(G)<0$ seen so far, we have the singularities non-isolated
The inequality holds for $d$ graphs with boundary as for such graphs every unit sphere either has non-negative Euler characteristic $0,2$ (interior) or $1$ (at the boundary). The example shown above with $\eta(G)<0$ has some unit spheres which are not $1$-graphs (disjoint unions of circular graphs) but $1$-varieties.\
It follows that also for patched versions, graphs which are the union of two graphs such that at the intersection, the spheres add up. This happens for example, if two disks touch at a vertex.\
####
If we think of $\chi(S(x))$ as a [**curvature**]{} for the functional $\eta$, then a natural situation would be that zero total curvature implies that the curvature is zero everywhere. Here is a modification of the example with negative $\eta$ for which $\eta(G)=0$ and so $\eta(G_n)=0$ for all $n$.
####
[**Examples:**]{}\
[**1)**]{} For $1$-dimensional graphs, we have $\eta(G)=4 v_1(G) > 0$.\
[**2)**]{} For $2$-dimensional graphs, graphs with $K_3$ subgraphs and no $K_4$ subgraphs, the functional is a Dehn-Sommerville valuation $\eta(G) = 2 v_1 - 3 v_2$. It vanishes if every edge shares exactly two triangles. See [@valuation] for generalizations to multi-linear valuations.\
[**3)**]{} For $3$-dimensional graphs, graphs with $K_4$ subgraphs but no $K_5$ subgraphs, the functional is the Dehn-Sommerville valuation $2 v_1 - 3 v_2 + 4 v_3$.
####
A bit stronger but more risky is the question whether zero curvature implies that $G$ has the property that all unit spheres are unions of $d$-spheres:
####
Besides discrete manifolds, there are discrete varieties for which $\eta(G)=0$. Here is an example:
####
We could imagine for example that there are graphs which are almost $d$-graphs in the sense that the unit spheres can become discrete homology spheres, graphs with the same cohomology groups as spheres but whose geometric realiazations are not homeomorphic to spheres. An other possibility is we get graphs for which the Euler characteristic is zero but for which are also topologically different from spheres. We could imagine generalized $4$-graphs for example, where some unit spheres are $3$-graphs. All odd-dimensional graphs have then zero Euler characteristic by Dehn-Sommerville (an incarnation of Poincaré duality). But we don’t know yet of a construct of such graphs.
####
One can look at further related variational problems on graphs. One can either fix the number of elements in $G$ or the number of elements in $G_1$. In the later case, it of course does not matter whether one minimizes ${\rm tr}(L-g)$ or maximizes the trace of the Green function ${\rm tr}(g)$ as ${\rm tr}(L)=f(0)=|V(G_1)|=n$.
This is a formidable problem if one wants to explore it numerically as the number of simplicial complexes with a fixed number $n$ of faces grows very fast. A good challenge could be $n=26$ already as the $f$-vector of the octahedron $G$ is $(6,12,8)$. The simplex generating function of $G_1$ is $f(x) = 1+26x+72x^2+48 x^3$ with Euler’s Gem $\chi(G)=f(0)-f(-1)=2$. Since $f'(x)=26+144x+144 x^2$ we have $f'(-1) = 26 = {\rm tr}(g)$. It is a good guess to ask whether the trace of a Green function $g$ of a simplicial complex $G$ with $26$ simplices can get larger than $26$. In Figure (\[octahedronmatrix\]) we look at the $26 \times 26$ matrices $L$ and $g=L^{-1}$ of the Octahedron complex. Both matrices have $1$ in the diagonal. In the Green function case, we know $g_{xx} = 1-\chi(S(x)) = 1$ as all unit spheres $S(x)$ in $G_1$ are circles
####
Finally, we have seen that the simplex generating function $f(x)$ and its anti derivative $F$ can be used to compute Euler characteristic $\chi$, the Euler curvature and functional $\eta$ in a similar way $$\chi(G) = f(0)-f(-1), K = F(0)-F(-1), \eta(G) = f'(0)-f'(-1) \; .$$ This of course prompts the question whether other similar function values of $f$ are geometrically interesting. The two functionals were now linked also by the Gauss-Bonnet relation $\eta(G) = \sum_x \chi(S(x))$ as well as the old Gauss-Bonnet $\chi(G) = \sum_x K(x)$.
####
There are other algebraic-analytic relations like ${\rm tr}(g) = f'(-1)$ and as a consequence of unimodularity, ${\rm det}(g) = (-1)^{1+(f_0(-1)+f_0(1))/2}$ but where $f_0$ is the $f$-vector generating function of $G$ itself, not of the Barycentric refinement $G_1$. The reason is that ${\rm det}(L)={\rm det}(g)$ is what we called the Fermi characteristic $\phi(G) = \prod_x (-1)^{{\rm dim}(x))}$ which is $1$ if there are an even number of odd dimensional simplices present in $G$ and $-1$ if that number is odd. This number can change under Barycentric refinement unlike the Euler characteristic $\chi(G) = \sum_x (-1)^{{\rm dim}(x)}$, where $\chi(G)=f_0(0)-f_0(-1) = f(0)-f(-1)=\chi(G_1)$ is the same for the simplex generating function of $G$ and $G_1$.
Code
====
As usual, the following Mathematica procedures can be copy-pasted from the ArXiv’ed LaTeX source file to this document. Together with the text, it should be pretty clear what each procedure does.
``` {frame="single"}
UnitSphere[s_,a_]:=Module[{b=NeighborhoodGraph[s,a]},
If[Length[VertexList[b]]<2,Graph[{}],VertexDelete[b,a]]];
UnitSpheres[s_]:=Map[Function[x,UnitSphere[s,x]],VertexList[s]];
F[A_,z_]:=A-z IdentityMatrix[Length[A]]; F[A_]:=F[A,-1];
FredholmDet[s_]:=Det[F[AdjacencyMatrix[s]]];
BowenLanford[s_,z_]:=Det[F[AdjacencyMatrix[s],z]];
CliqueNumber[s_]:=Length[First[FindClique[s]]];
ListCliques[s_,k_]:=Module[{n,t,m,u,r,V,W,U,l={},L},L=Length;
VL=VertexList;EL=EdgeList;V=VL[s];W=EL[s]; m=L[W]; n=L[V];
r=Subsets[V,{k,k}];U=Table[{W[[j,1]],W[[j,2]]},{j,L[W]}];
If[k==1,l=V,If[k==2,l=U,Do[t=Subgraph[s,r[[j]]];
If[L[EL[t]]==k(k-1)/2,l=Append[l,VL[t]]],{j,L[r]}]]];l];
Whitney[s_]:=Module[{F,a,u,v,d,V,LC,L=Length},V=VertexList[s];
d=If[L[V]==0,-1,CliqueNumber[s]];LC=ListCliques;
If[d>=0,a[x_]:=Table[{x[[k]]},{k,L[x]}];
F[t_,l_]:=If[l==1,a[LC[t,1]],If[l==0,{},LC[t,l]]];
u=Delete[Union[Table[F[s,l],{l,0,d}]],1]; v={};
Do[Do[v=Append[v,u[[m,l]]],{l,L[u[[m]]]}],{m,L[u]}],v={}];v];
Barycentric[s_]:=Module[{v={},c=Whitney[s]},Do[Do[If[c[[k]]!=c[[l]]
&& (SubsetQ[c[[k]],c[[l]]] || SubsetQ[c[[l]],c[[k]]]),
v=Append[v,k->l]],{l,k+1,Length[c]}],{k,Length[c]}];
UndirectedGraph[Graph[v]]];
ConnectionGraph[s_] := Module[{c=Whitney[s],n,A},n=Length[c];
A=Table[1,{n},{n}];Do[If[DisjointQ[c[[k]],c[[l]]]||
c[[k]]==c[[l]],A[[k,l]]=0],{k,n},{l,n}];AdjacencyGraph[A]];
ConnectionLaplacian[s_]:=F[AdjacencyMatrix[ConnectionGraph[s]]];
FredholmCharacteristic[s_]:=Det[ConnectionLaplacian[s]];
GreenF[s_]:=Inverse[ConnectionLaplacian[s]];
Energy[s_]:=Total[Flatten[GreenF[s]]];
BarycentricOp[n_]:=Table[StirlingS2[j,i]i!,{i,n+1},{j,n+1}];
Fvector[s_] := Delete[BinCounts[Length /@ Whitney[s]], 1];
Fvector1[s_]:=Module[{f=Fvector[s]},BarycentricOp[Length[f]-1].f];
Fv=Fvector; Fv1=Fvector1;
GFunction[s_,x_]:=Module[{f=Fv[s]},1+Sum[f[[k]]x^k,{k,Length[f]}]];
GFunction1[s_,x_]:=Module[{f=Fv1[s]},1+Sum[f[[k]]x^k,{k,Length[f]}]];
dim[x_]:=Length[x]-1; Pro=Product; W=Whitney;
Euler[s_]:=Module[{w=W[s],n},n=Length[w];Sum[(-1)^dim[w[[k]]],{k,n}]];
Fermi[s_]:=Module[{w=W[s],n},n=Length[w];Pro[(-1)^dim[w[[k]]],{k,n}]];
Eta[s_]:=Tr[ConnectionLaplacian[s]-GreenF[s]];
Eta0[s_]:=Total[Map[Euler,UnitSpheres[s]]];
Eta1[s_]:=Total[Map[Euler,UnitSpheres[Barycentric[s]]]];
EtaG[s_]:=Module[{g=GFunction1[s,x]}, f[y_]:=g /. x->y;f'[0]-f'[-1]];
EulerG[s_]:=Module[{g=GFunction[s,x]},f[y_]:=g /. x->y;f[0] -f[-1] ];
s=RandomGraph[{23,60}]; sc = ConnectionGraph[s];
{Euler[s],Energy[s],EulerG[s]}
{Fermi[s],BowenLanford[sc,-1],FredholmCharacteristic[s]}
{Eta1[s],Eta[s],EtaG[s]}
```
Examples
========
####
Let $G$ be the house graph. It has the $f$-vector $(5,6,1)$, the generating function $f(x)=1+5x+6x^2+x^2$ and the Euler characteristic $5-6+1=0$. The unit spheres have all Euler characteristic $\chi(S(x))=2$ except the roof-top where $\chi(S(x))=1$. We therefore have $\eta_0(G)=9$. The Barycentric refinement $G_1$ of $G$ has the $f$-vector $(12,18,6)$. The unit sphere Euler characteristic spectrum is $(0, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2)$ totals to $\eta_1(G)=\eta_0(G_1)=18$. The connection graph $G'$ of $G$ already has dimension $4$. Its $f$-vector is $(12, 29, 27, 12, 2)$ and $\chi(G')=0$ still.
The connection Laplacian $L$ and its inverse $g$ are
$$\arraycolsep=2pt\def\arraystretch{1.0}
L=\left[ \begin{array}{cccccccccccc}
1&0&1&1&0&1&0&1&1&1&1&1\\
0&1&0&0&0&0&1&1&0&0&0&0\\
1&0&1&0&0&0&0&1&1&1&0&0\\
1&0&0&1&0&0&0&0&1&0&1&1\\
0&0&0&0&1&0&1&0&0&0&1&0\\
1&0&0&0&0&1&0&0&0&1&0&1\\
0&1&0&0&1&0&1&1&0&0&1&0\\
1&1&1&0&0&0&1&1&1&1&0&0\\
1&0&1&1&0&0&0&1&1&1&1&1\\
1&0&1&0&0&1&0&1&1&1&0&1\\
1&0&0&1&1&0&1&0&1&0&1&1\\
1&0&0&1&0&1&0&0&1&1&1&1\\
\end{array} \right],
g=\left[ \begin{array}{cccccccccccc}
1&0&1&1&0&1&0&0&-1&-1&0&-1\\
0&-1&-1&0&-1&0&1&1&0&0&0&0\\
1&-1&-1&0&0&0&0&1&0&0&0&-1\\
1&0&0&-1&-1&0&0&0&0&-1&1&0\\
0&-1&0&-1&-1&0&1&0&0&0&1&0\\
1&0&0&0&0&0&0&0&-1&0&0&0\\
0&1&0&0&1&0&-1&0&0&0&0&0\\
0&1&1&0&0&0&0&-1&0&0&0&0\\
-1&0&0&0&0&-1&0&0&0&1&0&1\\
-1&0&0&-1&0&0&0&0&1&0&0&1\\
0&0&0&1&1&0&0&0&0&0&-1&0\\
-1&0&-1&0&0&0&0&0&1&1&0&0\\
\end{array} \right] \; .$$
The trace of $L$ is $12$, the trace of $g$ is $-6$. The super trace of both $L$ and $g$ or the sum $\sum_{x,y} g(x,y)$ are all $\chi(G) =0$. The spectrum of $L$ is $\sigma(L)= \{$ $-1.30009$, $-0.827091$, $-0.646217$, $-0.528497$, $-0.338261$, $-0.255285$, $0.245226$, $1.20906$, $1.72111$, $2.9563$, $3.17017$, $6.59358$ $\}$. We see in most random graphs that about half of the eigenvalues are negative and that the negative spectrum has smaller amplitude.
####
The [**double pyramid**]{} $G$ is a $2$-variety with $7$ vertices. It can be obtained by making two separate pyramid construction over a wheel graph. One can also write it as the Zykov join $P_3+C_4$ of $P_3$ with $C_4$ or then $P_3+P_2+P_2=P_3 + 2 P_2$. While the octahedron $O=P_2+C_4=P_2+P_2+P_2=3P_2$ has has $\eta(O)=0$ and all unit spheres with Euler characteristic $0$, now there are $4$ vertices in $G$ where $S(x)$ has Euler characteristic $-1$. The graph $G$ has $f$-vector $(7,16,12)$ and Euler characteristic $3$ and Betti vector $(1,0,2)$. The $f$-vector of the Barycentric refinement is $(35,104,72)$. We have $\eta_0(G)=-4$. The Barycentric refinement has $8$ vertices with Euler characteristic $-1$ and $\eta_1(G)=-8$.
The connection Laplacian
$$\arraycolsep=2pt\def\arraystretch{1.0}
L=\left[ \begin{array}{ccccccccccccccccccccccccccccccccccc}
1&0&0&0&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0\\
0&1&0&0&0&0&0&1&1&0&0&1&1&1&1&0&0&0&0&1&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0\\
0&0&1&0&0&0&0&1&0&1&0&1&1&0&0&1&1&0&0&0&1&0&0&1&0&0&0&1&1&1&0&0&0&0&0\\
0&0&0&1&0&0&0&0&1&0&1&0&0&1&1&0&0&1&1&0&0&1&0&0&1&0&0&0&0&0&1&1&1&0&0\\
0&0&0&0&1&0&0&0&0&1&1&0&0&0&0&1&1&1&1&0&0&0&1&0&0&0&0&1&0&0&1&0&0&1&1\\
0&0&0&0&0&1&0&0&0&0&0&1&0&1&0&1&0&1&0&0&0&0&0&0&0&1&0&0&1&0&0&1&0&1&0\\
0&0&0&0&0&0&1&0&0&0&0&0&1&0&1&0&1&0&1&0&0&0&0&0&0&0&1&0&0&1&0&0&1&0&1\\
1&1&1&0&0&0&0&1&1&1&1&1&1&1&1&1&1&0&0&1&1&1&1&1&1&1&1&1&1&1&0&0&0&0&0\\
1&1&0&1&0&0&0&1&1&1&1&1&1&1&1&0&0&1&1&1&1&1&1&1&1&1&1&0&0&0&1&1&1&0&0\\
1&0&1&0&1&0&0&1&1&1&1&1&1&0&0&1&1&1&1&1&1&1&1&1&0&0&0&1&1&1&1&0&0&1&1\\
1&0&0&1&1&0&0&1&1&1&1&0&0&1&1&1&1&1&1&1&1&1&1&0&1&0&0&1&0&0&1&1&1&1&1\\
0&1&1&0&0&1&0&1&1&1&0&1&1&1&1&1&1&1&0&1&1&0&0&1&1&1&1&1&1&1&0&1&0&1&0\\
0&1&1&0&0&0&1&1&1&1&0&1&1&1&1&1&1&0&1&1&1&0&0&1&1&1&1&1&1&1&0&0&1&0&1\\
0&1&0&1&0&1&0&1&1&0&1&1&1&1&1&1&0&1&1&1&0&1&0&1&1&1&1&0&1&0&1&1&1&1&0\\
0&1&0&1&0&0&1&1&1&0&1&1&1&1&1&0&1&1&1&1&0&1&0&1&1&1&1&0&0&1&1&1&1&0&1\\
0&0&1&0&1&1&0&1&0&1&1&1&1&1&0&1&1&1&1&0&1&0&1&1&0&1&0&1&1&1&1&1&0&1&1\\
0&0&1&0&1&0&1&1&0&1&1&1&1&0&1&1&1&1&1&0&1&0&1&1&0&0&1&1&1&1&1&0&1&1&1\\
0&0&0&1&1&1&0&0&1&1&1&1&0&1&1&1&1&1&1&0&0&1&1&0&1&1&0&1&1&0&1&1&1&1&1\\
0&0&0&1&1&0&1&0&1&1&1&0&1&1&1&1&1&1&1&0&0&1&1&0&1&0&1&1&0&1&1&1&1&1&1\\
1&1&0&0&0&0&0&1&1&1&1&1&1&1&1&0&0&0&0&1&1&1&1&1&1&1&1&0&0&0&0&0&0&0&0\\
1&0&1&0&0&0&0&1&1&1&1&1&1&0&0&1&1&0&0&1&1&1&1&1&0&0&0&1&1&1&0&0&0&0&0\\
1&0&0&1&0&0&0&1&1&1&1&0&0&1&1&0&0&1&1&1&1&1&1&0&1&0&0&0&0&0&1&1&1&0&0\\
1&0&0&0&1&0&0&1&1&1&1&0&0&0&0&1&1&1&1&1&1&1&1&0&0&0&0&1&0&0&1&0&0&1&1\\
0&1&1&0&0&0&0&1&1&1&0&1&1&1&1&1&1&0&0&1&1&0&0&1&1&1&1&1&1&1&0&0&0&0&0\\
0&1&0&1&0&0&0&1&1&0&1&1&1&1&1&0&0&1&1&1&0&1&0&1&1&1&1&0&0&0&1&1&1&0&0\\
0&1&0&0&0&1&0&1&1&0&0&1&1&1&1&1&0&1&0&1&0&0&0&1&1&1&1&0&1&0&0&1&0&1&0\\
0&1&0&0&0&0&1&1&1&0&0&1&1&1&1&0&1&0&1&1&0&0&0&1&1&1&1&0&0&1&0&0&1&0&1\\
0&0&1&0&1&0&0&1&0&1&1&1&1&0&0&1&1&1&1&0&1&0&1&1&0&0&0&1&1&1&1&0&0&1&1\\
0&0&1&0&0&1&0&1&0&1&0&1&1&1&0&1&1&1&0&0&1&0&0&1&0&1&0&1&1&1&0&1&0&1&0\\
0&0&1&0&0&0&1&1&0&1&0&1&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&1&1&1&0&0&1&0&1\\
0&0&0&1&1&0&0&0&1&1&1&0&0&1&1&1&1&1&1&0&0&1&1&0&1&0&0&1&0&0&1&1&1&1&1\\
0&0&0&1&0&1&0&0&1&0&1&1&0&1&1&1&0&1&1&0&0&1&0&0&1&1&0&0&1&0&1&1&1&1&0\\
0&0&0&1&0&0&1&0&1&0&1&0&1&1&1&0&1&1&1&0&0&1&0&0&1&0&1&0&0&1&1&1&1&0&1\\
0&0&0&0&1&1&0&0&0&1&1&1&0&1&0&1&1&1&1&0&0&0&1&0&0&1&0&1&1&0&1&1&0&1&1\\
0&0&0&0&1&0&1&0&0&1&1&0&1&0&1&1&1&1&1&0&0&0&1&0&0&0&1&1&0&1&1&0&1&1&1\\
\end{array} \right]$$
has $659$ entries $1$ which is $|V(G')|+2|E(G')|=35+2 \cdot 312$. The Green function $g=L^{-1}$ has entries $\{-2,-1,0,1,2\}$. We have ${\rm tr}(g)=43$ and ${\rm tr}(L)=35$ and $\eta(G)=35-43=-8$. The spectrum of $L$ has the convex hull $[-3.30278,20.0327]$. There are 16 negative and 19 positive eigenvalues and 16 negative eigenvalues. The spectrum of $g$ has the convex hull $[-3.30278,16]$.\
When making a triple suspension $G=C_n + P_3$ over a larger circle we get a graph with $\eta_0(G)=-n$ and $\eta(G)=-2n$. By punching two small holes at vertices of degree 4 into $G_1$, the graph can be rendered to be contractible with $\eta(G)=8-2n$. Attaching this using using wedge sum to an other graph illustrates again that we can lower $\eta$ arbitrarily in any homotopy class of graphs.
[10]{}
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B. Bollobas. . Dover Courier Publications, 1978.
R. Forman. A discrete [M]{}orse theory for cell complexes. In [*Geometry, topology, and physics*]{}, Conf. Proc. Lecture Notes Geom. Topology, IV, pages 112–125. Int. Press, Cambridge, MA, 1995.
A.V. Ivashchenko. Graphs of spheres and tori. , 128(1-3):247–255, 1994.
O. Knill. The dimension and [Euler]{} characteristic of random graphs. http://arxiv.org/abs/1112.5749, 2011.
O. Knill. A graph theoretical [Gauss-Bonnet-Chern]{} theorem. http://arxiv.org/abs/1111.5395, 2011.
O. Knill. . http://arxiv.org/abs/1301.1408, 2012.
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O. Knill. On [F]{}redholm determinants in topology. https://arxiv.org/abs/1612.08229, 2016.
O. Knill. On primes, graphs and cohomology. https://arxiv.org/abs/1608.06877, 2016.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Along with increasingly popular virtual reality applications, the three-dimensional (3D) point cloud has become a fundamental data structure to characterize 3D objects and surroundings. To process 3D point clouds efficiently, a suitable model for the underlying structure and outlier noises is always critical. In this work, we propose a hypergraph-based new point cloud model that is amenable to efficient analysis and processing. We introduce tensor-based methods to estimate hypergraph spectrum components and frequency coefficients of point clouds in both ideal and noisy settings. We establish an analytical connection between hypergraph frequencies and structural features. We further evaluate the efficacy of hypergraph spectrum estimation in two common point cloud applications of sampling and denoising for which also we elaborate specific hypergraph filter design and spectral properties. The empirical performance demonstrates the strength of hypergraph signal processing as a tool in 3D point clouds and the underlying properties.'
author:
- 'Songyang Zhang, Shuguang Cui, , and Zhi Ding, '
title: Hypergraph Spectral Analysis and Processing in 3D Point Cloud
---
3D point clouds, hypergraph signal processing, hypergraph construction, denoising, sampling.
Introduction {#intro}
============
Recent developments in depth sensors and softwares make it easier to capture the features and create a three-dimensional (3D) model for an object and its surroundings[@c1]. In particular, with the low-cost scanners such as light detection and ranging (LIDAR) and Kinect, a new data structure known as the point cloud has achieved significant success in many areas, including virtual reality, geographic information system, reconstruction of art document and high-precision 3D maps for self-driving cars [@c2]. A point cloud consists of 3D coordinates with attributes such as color, temperature, texture, and depth [@c3]. Owing to the easy access to scanning sensors and the huge need in describing the 3D features, the use of point clouds has attracted significant attentions in areas of computer vision, virtual reality, and medical science. How to process the point clouds efficiently becomes an important topic of research in many 3D imaging and vision systems.
To analyze the features of point cloud, the first step is to construct an analytical model to represent the 3D structures. The literature provides several different models. In [@c4], the 3D space is partitioned into several boxes or voxels, and the point clouds are then discretized therein. One disadvantage of voxels is that a dense grid is required to achieve fine resolution, leading to spatial inefficiency [@c3]. A spatially efficient approach [@c5; @c6] is the octree representation of point clouds. An octree is a tree data structure in which each node has exactly eight children. It can partition a 3D space recursively, and represent the point clouds with partitioned boxes. Although efficient, octree suffers from discretization errors [@c3]. The bd-tree is another spatial decomposition technique and is robust in highly cluttered point cloud dataset. However, compared to octree structures, bd-trees are more difficult to update.
Recently, graphs and graph signal processing (GSP) have found applications in modeling point clouds. For example, the authors of [@c3] construct a graph based on pairwise point distances. Some other works, such as [@c8], construct graphs based on the $k$-nearest neighbors, where each vertex (point) has an edge connection to its $k$ nearest neighbors. There are several clear connections between graph features and point cloud characteristics. For example, the smoothness over a graph can describe the flatness of surfaces in point clouds. GSP-based tools such as filters and graph learning methods can process the point clouds and have shown great success because of the graph model’s ability to capture the underlying geometric structures. However, graph-based methods still face some challenges such as limited orders and measurement inefficiency. In a traditional graph, each edge can only connect two nodes, constraining graph-based models to describe only pairwise relationships. However, a multilateral relationship among multiple nodes is far more informative as in a point cloud model. For example, the points (nodes) on the same surface of a point cloud exhibit a strong multilateral relationship, which cannot be easily captured by an edge of a traditional graph. In fact, construction of an efficient graph for a given dataset is always an open question. Thus, studies on point clouds can benefit from more general and efficient models.
To develop an efficient model for point clouds, we explore a high-dimensional graph model, known as hypergraph [@c9]. Hypergraph can be a useful model in processing 3D point clouds. A hypergraph $\mathcal{H}=\{\mathcal{V},\mathcal{E}\}$ consists of a set of nodes $\mathcal{V}=\{\mathbf{v}_1, \dots,\mathbf{v}_K\}$ and a set of hyperedges $\mathcal{E}=\{\mathbf{e}_1, \dots,\mathbf{e}_K\}$. Each hyperedge in a hypergraph can connect more than two nodes. For example, a 3D shape together with its hypergraph model are shown as Fig. \[hyper\]. Obviously, a normal graph is a special case of hypergraph, where each hyperedge degrades to connect two nodes exactly. The hyperedge in a hypergraph can characterize the multilateral relationship among several related nodes (e.g., on a surface), thereby making hypergraph a natural and intuitive model for point clouds. Moreover, advances in hypergraph signal processing (HGSP) [@c9] are providing more hypergraph tools, such as HGSP-based filters and spectrum analysis, for effective point cloud processing.
However, processing the point clouds based on hypergraph still poses several challenges. Similar to GSP, the first problem lies in the construction of hypergraph for point clouds. The traditional hypergraph construction method for a general dataset relies on data structure. For example, in [@c11], a hypergraph model is constructed according to the sentence structure in natural language processing. The $k$-nearest neighbor model is another method to construct the hypergraph. In [@c9], a hypergraph can be formed from the feature distances for an animal dataset to achieve clustering. However, such distance-based or structure-based model may be rather lossy in information preservation. For example, the structure-based method may not preserve the correlation of some irregular structures, whereas the $k$-nearest neighbor method may narrowly emphasize the distance information. In addition to hypergraph construction, another issue in analyzing point cloud with hypergraph tools is the computation complexity of the spectrum space. In the HGSP framework, spectrum-based analysis plays an important role but needs to compute the spectrum space. Usually, the computation of hypergraph spectrum is based on orthogonal-CP decomposition, which incurs high-complexity when there are many nodes. Another challenge in point cloud processing is the effect of noise and outliers. Since a hypergraph model is constructed from observed data, noise can distort the hypergraph and degrade the performances of HGSP. Thus, mitigating noise effect and robustly estimating the hypergraph model for point clouds pose a significant challenge.
This work addresses the aforementioned problems. We propose novel spectrum-based hypergraph construction methods for both clean and noisy point clouds. For clean point clouds, we first estimate their spectrum components based on the hypergraph stationary process and optimally determine their frequency coefficients based on smoothness to recover the original hypergraph structure. For noisy point clouds, we introduce a method for joint hypergraph structure estimation and data denoising. We shall illustrate the effectiveness of the proposed hypergraph construction and spectrum estimation in two point clould applications: sampling and denoising. Our experimental results clearly establish a connection between hypergraph frequencies and point cloud features. The performance improvement in both applications demonstrates the strength and power of hypergraph in point cloud processing and the practical value of our estimation methods.
We organize the rest of the paper as follows. In Section \[pre\], we lay the foundation with respect to the preliminaries and notations of point clouds, tensor basics and hypergraph signal processing. Next, we propose means in estimating hypergraph spectrum for basic point clouds in Section \[h1\] and further develop means for hypergraph structure estimation of noisy point clouds in Section \[h2\]. With the proposed estimation methods, we study two important application scenarios and establish the effectiveness of hypergraph signal processing in Section \[appli\]. Finally, we present the conclusion and future directions in Section \[con\].
Preliminaries and Notations {#pre}
===========================
In this section, we cover basic background with respect to point cloud, tensor basics and hypergraph signal processing.
Point Clouds
------------
A point cloud is a set of 3D points obtained from sensors, where each point is attributed with coordinates and other features, like colors [@c10]. Since the coordinates are basic features of a point cloud, in this work, we mainly focus on gray-scale point clouds, where each node is characterized by its coordinates. We consider a matrix representation of the gray-scale point clouds, where a point cloud with $N$ nodes is denoted by a location matrix $$\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]=
\begin{bmatrix}
\mathbf{s}_1^T\\
\mathbf{s}_2^T\\
\ddots\\
\mathbf{s}_N^T
\end{bmatrix}\in\mathbb{R}^{N\times 3},$$ where $\mathbf{X}_i$ denotes a vector of the $i$th coordinates of all the points, and $\mathbf{s}_i$ is the three coordinates of $i$th point. With the information of coordinates, different models, such as graphs [@c3] and octrees [@c5], can be constructed to analyze the point clouds, for which we will discuss more in Section \[appli\].
Tensor Basics
-------------
Tensor is a high-dimensional generalization of matrix. A tensor can be interpreted as multi-dimensional arrays. The order of tensor is the number of indices to label the components of arrays [@c17]. For example, a scalar is a zeroth-order tensor; a vector is a first-order tensor; a matrix is a second-order tensor; and an $M$-dimensional array is an $M$th-order tensor [@c18]. In this work, an $M$th-order tensor is denoted by $\mathbf{A}\in\mathbb{R}^{I_1
\times I_2\times \cdots \times I_M}$, whose entry in position $(i_1,i_2,\cdots,i_M)$ is labeled as $a_{i_1\cdots i_M}$. Here, $I_k$ is the dimension of $k$th order.
Tensor outer product is a widely used operation to construct a higher-order tensor from lower-order tensors. The tensor outer product between an $M$th-order tensor $\mathbf{U}\in \mathbb{R}^{I_1\times I_2\times ...\times I_M }$ with entries $u_{i_1 ... i_M}$ and an $N$th-order tensor $\mathbf{V}\in \mathbb{R}^{J_1\times J_2\times ...\times J_N }$ with entries $v_{j_1 ... j_N}$ is denoted by $$\mathbf{W}=\mathbf{U} \circ \mathbf{V},$$ where the result $\mathbf{W}\in \mathbb{R}^{I_1\times I_2\times ...\times I_M \times J_1 \times J_2 \times ... \times J_N}$ is an $(M+N)$th-order tensor with entries $$w_{i_1 ... i_M j_1 ... j_N}= u_{i_1 ... i_M} \cdot v_{j_1 ... j_N}.$$
Hypergraph Signal Processing
----------------------------
Hypergraph signal processing (HGSP) is a tensor-based framework [@c9]. In the HGSP framework, a hypergraph with $N$ nodes and longest hyperedge connecting $M$ nodes, is represented by an $M$-th order $N$-dimension representing tensor $\mathbf{A}=(a_{i_1i_2\cdots i_M})\in\mathbb{R}^{N^M}$. The representing tensor can be adjacency tensor or Laplacian tensor in different purposes [@c16]. In this paper, we refer the adjacency tensor as the representing tensor, in which each entry $a_{i_1i_2\cdots i_M}$ indicates whether nodes $\{\mathbf{v}_1,\mathbf{v}_2,\cdots,\mathbf{v}_M\}$ are connected. The computation of the edge weight can be found in [@c9].
With the orthogonal-CP decomposition, the representing tensor can be decomposed via $$\label{decom}
\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{$M$ times}},$$ where $\mathbf{f}_r$’s are orthonormal basis called spectrum components and $\lambda_r$ are frequency coefficients related to the hypergraph frequency. All the spectrum components $\{\mathbf{f}_1,\cdots,\mathbf{f}_N\}$ construct the hypergraph spectral space. Each pair $(\mathbf{f}_r,\lambda_r)$ is called the spectral pair of the hypergraph.
Given an original signal $\mathbf{s}=[s_1\quad s_2\quad...\quad s_N]^{\mathrm{T}}$, the hypergraph signal is defined as the $(M-1)$ times tensor outer product of $\mathbf{s}$, i.e., $$\mathbf{s}^{[M-1]}=\underbrace{\mathbf{s\circ...\circ s}}_{\text{$M-1$ times}}.$$
The hypergraph frequency is ordered by the total variation of the spectrum component, which is defined as $$\mathbf{TV}(\mathbf{\mathbf{f}_r})=||\mathbf{f}_r-\frac{1}{\lambda_{max}}\mathbf{A}\mathbf{f}_r^{[M-1]}||_1,$$ where $\mathbf{A}\mathbf{f}_r^{[M-1]}$ is the contraction between representing tensor $\mathbf{A}$ and the hypergraph signal.
A spectrum component with larger total variation is a higher-frequency component, which indicates a faster propagation over the given hypergraph. Moreover, a supporting matrix $$\label{sup}
\mathbf{P_s}=\frac{1}{\lambda_{\max}}
\begin{bmatrix}
\mathbf{f}_1& \cdots& \mathbf{f}_N
\end{bmatrix}
\begin{bmatrix}
\lambda_1& & \\
&\ddots& \\
& &\lambda_N
\end{bmatrix}
\begin{bmatrix}
\mathbf{f}_1^{\mathrm{T}}\\
\vdots\\
\mathbf{f}_N^{\mathrm{T}}
\end{bmatrix},$$ can be defined to capture the overall spectral information of the hypergraph.
Instead of reviewing many properties of HGSP here, other aspects such as hypergraph Fourier transform, hypergraph filter design and sampling theory can be found in [@c9].
Hypergraph Spectrum Estimation for Point Clouds {#h1}
===============================================
To process the 3D point clouds, the first step is to construct an optimal hypergraph to model the point clouds. As we mentioned in the Section \[intro\], it is time-comsuming and inefficient to first construct a hypergraph structure before tensor decomposition to obtain the hypergraph spectrum. Instead, we propose to directly estimate the hypergraph spectral pairs based on the observed data, and then recover the original representing tensor with Eq. (\[decom\]). In this section, we first estimate the hypergraph spectrum components $\mathbf{f}_r$’s based on the hypergraph stationary process, and optimize the frequency coefficients $\lambda_r$’s based on the smoothness for original point clouds.
Estimation of Hypergraph Spectrum Components
--------------------------------------------
In this part, we propose a method to estimate the hypergraph spectral components based on the hypergraph stationary process.
### Hypergraph Stationary Process
Before providing details of the estimation, let us first introduce some new definitions and properties necessary for spectrum estimation.
Stationarity is a cornerstone property that facilities the analysis of random signals and observations in traditional signal processing [@c19]. It has equal importance in graph and hypergraph signal processing. Based on graph shifting introduced in [@c22], a definition of graph stationary process proposed in [@c19] can analyze the properties of the different observations of nodes, or the random signals over the graphs. Furthermore, [@c123] introduces a method to estimate the graph spectrum space and graph diffusion for multiple observations based on the graph stationary process. Similarly, the hypergraph stationary process can be defined to estimate hypergraph spectrum.
Now, let us introduce the definition of the hypergraph stationary process. In [@c9], a polynomial hypergraph filter based on supporting matrix is defined as $$\mathbf{s}'=\sum_{k=1}^{a}\alpha_k\mathbf{P}^k\mathbf{s},$$ where $\mathbf{P}=\lambda_{max}\mathbf{P_s}$.
Similarly, based on the supporting matrix, a $\tau$-step shifting operation is defined as $\mathbf{P}_{\tau}=\mathbf{P}^{\tau}$. Then, similar to the definition of the stationary process in traditional digital signal processing and graph signal processing, a strict-sense stationary process in HGSP can be defined as follows.
(Strict-Sense Stationary Process) A stochastic signal $\mathbf{x}\in\mathbb{R}^N$ is strict-sense stationary over the hypergraph with $\mathbf{P}_\tau$ if and only if $$\mathbf{x}\overset{d}{=}\mathbf{P}_{\tau}\mathbf{x}$$ holds for any $\tau$.
Since the strict-sense stationary is hard to achieve and analyze in the real datasets, we introduce the weak-sense stationary process similar to traditional digital signal processing.
(Weak-Sense Stationary Process) A stochastic signal $\mathbf{x}\in\mathbb{R}^N$ is weak-sense stationary over the hypergraph with $\mathbf{P}_\tau$ if and only if $$\label{mean}
\mathbb{E}[\mathbf{x}]=\mathbb{E}[\mathbf{P}_{\tau}\mathbf{x}]$$ and $$\label{time}
\mathbb{E}[(\mathbf{P}_{\tau_1}\mathbf{x})((\mathbf{P}^H)_{\tau_2}\mathbf{x})^H]=\mathbb{E}[(\mathbf{P}_{\tau_1+\tau}\mathbf{x})((\mathbf{P}^H)_{\tau_2-\tau}\mathbf{x})^H]$$ hold for any $\tau$, where $\mathbb{E}(\cdot)$ refers to the mean of observations and $(\cdot)^H$ is the Hermitian transpose.
From the definition of the weak-sense stationary process (WSS), Eq. (\[mean\]) implies that the mean function of the signal must be constant, which is the same condition as in traditional digital signal processing (DSP) [@c23]. From the definition of supporting matrix, the $(i,j)$-th entry of $\mathbf{P}$ is the same as the $(j,i)$-th entry of $\mathbf{P}^H$, which indicates that $\mathbf{P}^H$ is the shifting in the opposite direction of $\mathbf{P}$. Then, the condition in Eq. (\[time\]) indicates that the hypergraph covariance function $K_{\mathbf{xx}}({\tau_1},-\tau_2)=K_{\mathbf{xx}}({\tau_1}+\tau,\tau-\tau_2)=K_{\mathbf{xx}}({\tau_1}+\tau_2,0)$, which is also consistent with the definition in traditional DSP.
With the definition of the hypergraph stationary process, we have the following properties regarding the relationship between signals and hypergraph spectrum.
A stochastic signal $\mathbf{x}$ is WSS if and only if it has zero-mean and its covariance matrix has the same eigenvectors as the hypergraph spectrum basis, i.e., $$\label{s1}
\mathbb{E}[\mathbf{x}]=\mathbf{0}$$ and $$\label{s2}
\mathbb{E}[\mathbf{x}\mathbf{x}^H]=\mathbf{V}\Sigma_\mathbf{x}\mathbf{V}^{H},$$ where $\mathbf{V}=[\mathbf{f}_1,\mathbf{f}_2,\cdots,\mathbf{f}_N]\in\mathbb{R}^{N\times N}$ are the hypergraph spectrum.
Since the hypergraph spectrum basis are orthonormal, we have $\mathbf{VV}^T=\mathbf{I}$. Then, the $\tau$-step shifting based on supporting matrix can be calculated as $$\begin{aligned}
\mathbf{P}_\tau&=\underbrace{\mathbf{V}\Lambda_\mathbf{P}\mathbf{V}^T\mathbf{V}\Lambda_\mathbf{P}\mathbf{V}^T\cdots\mathbf{V}\Lambda_\mathbf{P}\mathbf{V}^T}_{\tau\quad times}\\
&=\mathbf{V}\Lambda_\mathbf{P}^\tau\mathbf{V}^T.\label{poly}
\end{aligned}$$
Now, the Eq. (\[mean\]) can be written as $$\mathbb{E}[\mathbf{x}]=\mathbf{V}\Lambda_P^\tau\mathbf{V}^T\mathbb{E}[\mathbf{x}].$$ Since $\mathbf{V}\Lambda_P^\tau\mathbf{V}^T$ does not always equal to $\mathbf{I}$, Eq. (\[mean\]) holds for arbitrary supporting matrix and $\tau$ if and only if $\mathbb{E}[\mathbf{x}]=\mathbf{0}$.
Next we show the sufficiency and necessity of the condition in Eq. (\[s2\]). The condition in Eq. (\[time\]) can be written as $$\mathbf{P}_{\tau_1}\mathbb{E}[\mathbf{xx}^H]((\mathbf{P})^H_{\tau_2})^H=\mathbf{P}_{\tau_1+\tau}\mathbb{E}[\mathbf{xx}^H]((\mathbf{P})^H_{\tau_2-\tau})^H.$$ Considering Eq. (\[poly\]) and the fact that hypergraph spectrum is real [@c9], Eq. (\[time\]) is equivalent to $$\mathbf{V}\Lambda_\mathbf{P}^{\tau_1}\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V}\Lambda_\mathbf{P}^{\tau_2}\mathbf{V}^H=\mathbf{V}\Lambda_\mathbf{P}^{\tau_1+\tau}\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V}\Lambda_\mathbf{P}^{\tau_2-\tau} \mathbf{V}^H,$$ which can be written as $$\label{eig}
(\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V})\Lambda_\mathbf{P}^{\tau}=\Lambda_\mathbf{P}^{\tau}(\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V}).$$ If Eq. (\[eig\]) holds for arbitrary $\mathbf{P}$, $(\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V})$ should be diagonal, which indicates $\mathbb{E}[\mathbf{x}\mathbf{x}^H]=\mathbf{V}\Sigma_\mathbf{x}\mathbf{V}^{H}$. Thus, the sufficiency of the condition is proved.
Similarly, we can apply Eq. (\[s2\]) on both sides of Eq. ($\ref{time}$), we can establish the necessity of the condition in Eq. (\[s2\]).
This theorem can be used to estimate the hypergraph spectrum, given multiple observations of several signal points.
### Estimation of Spectrum Components for Point Clouds
Now, we can use the property of stationary process to estimate the hypergraph spectrum of point clouds. The three coordinates of a point can be interpreted as three observations of the point from different angles, which describe the underlying multilateral relationship. Thus, we can assume that the point cloud signals follow the stationary process over the estimated underlying hypergraph structure. If the point cloud signals $\mathbf{s}$ follow the hypergraph stationarity, it should satisfy Eq. (\[s1\]) and Eq. (\[s2\]). Thus, a spectrum estimation method can be based on hypergraph staionarity. The details of the algorithm is described as follows.
: Point cloud dataset $\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]\in\mathbb{R}^{N\times 3}$. Calculate the mean of each row in $\mathbf{s}$, i.e.,\
$\mathbf{\overline s}=(\mathbf{X_1+X_2+X_3})/3$; Normalize the original point cloud data as zero-mean in each row, i.e., $\mathbf{s}'=[\mathbf{X_1-{\overline s},X_2-{\overline s},X_3-{\overline s}}]$; Calculate the eigenvectors $\{\mathbf{f}_1,\cdots,\mathbf{f}_N\}$ for $R_{\mathbf{s}'}=\mathbf{s'}(\mathbf{s'}^T)$; : Hypergraph spectrum $\mathbf{V}=[\mathbf{f}_1,\cdots,\mathbf{f}_N]$.
With *Theorem 1*, we can directly obtain an estimation of the hypergraph spectrum based on the hypergraph stationarity. Note that, here, we assume all the observations are from a clean point cloud without noise. The case of noisy point clouds will be discussed later in Section \[h2\].
Estimation of Frequency Coefficients {#es}
------------------------------------
Next, we discuss how we estimate the hypergraph frequency coefficients with the spectrum components based on the hypergraph smoothness.
In real applications, the large-scale networks are usually sparse, which makes it meaningful to infer that most entries of the hypergraph representing tensor for real datasets are zero [@c24]. In addition, the smoothness of signals is a widely-used assumption when estimating the underlying structure of graphs and hypergraphs [@c25]. Thus, the estimation of the hypergraph representing tensor with known spectrum components for a given dataset $\mathbf{s}$ can be generally formulated as $$\begin{aligned}
&\min_{\mathbf{\boldsymbol{\lambda}}}\quad \alpha \mbox{Smooth}
(\mathbf{s,\boldsymbol{\lambda}},\mathbf{f}_r)+\beta||\mathbf{A}||_T^2\label{e1}\\
s.t.\quad &\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}}. \label{dec}\\
&\quad\mathbf{A}\in \mathcal{A}.\label{cs1}\\
&||\mathbf{A}||_T=\sqrt{\sum_{i_1,i_2,\cdots, i_M=1}^N a_{i_1i_2\cdots i_M}^2}.\label{t_norm}\end{aligned}$$
The constraint set $\mathcal{A}$ in (\[cs1\]) includes the prior information of the representing tensor. For example, if the representing tensor is the adjacency tensor, its entries should be non-negative. In the constraint of $(\ref{t_norm})$, $||\mathbf{A}||_T$ is the tensor norm which controls the sparsity of the hypergraph structure. The smoothness function Smooth$(\mathbf{s,\boldsymbol{\lambda}},\mathbf{f}_r)$ can be designed for specific problems. Typical functions can be hypergraph Laplacian regularization, label ranking, and total variation [@c9]. For convenience, we use the quadratic-form total variation based on the supporting matrix to describe the hypergraph smoothness, i.e., $$\mathbf{TV}(\mathbf{s})=||\mathbf{s}-(1/\lambda_{max})\mathbf{P}\mathbf{s}||^2_2.$$ This form of smoothness function suggested in [@c9] can capture the differences between one node and its neighbors over hypergraph. Since the signals are smooth over the estimated hypergraph, observations are also smooth. Thus, the final smoothness function for point cloud $\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]$ is $$\begin{aligned}
\mbox{Smooth}(\mathbf{s,\boldsymbol{\lambda}},\mathbf{f}_r)&=\sum_{i=1}^3||\mathbf{X}_i-\mathbf{P_sX}_i||^2_2\nonumber\\
&=\sum_{i=1}^3||\mathbf{X}_i-\sum_r \sigma_r(\mathbf{f}_r^T\mathbf{X}_i)\mathbf{f}_r||^2_2\nonumber\\
&=\sum_{i=1}^3||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2,\label{sm1}\end{aligned}$$ where $\mathbf{W}_i=[(\mathbf{f}_1^T\mathbf{X}_i)\mathbf{f}_1\quad(\mathbf{f}_2^T\mathbf{X}_i)\mathbf{f}_2\quad\cdots\quad(\mathbf{f}_N^T\mathbf{X}_i)\mathbf{f}_N]$, $\sigma_r=\lambda_r/\lambda_{max}$ and $\boldsymbol{\sigma}=[\sigma_1\cdots\sigma_N]$.
Moreover, the tensor norm of a given hypergraph has the following property with the frequency coefficients.
Given a representing tensor $\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}}$, the tensor norm $||\mathbf{A}||^2_T=\sum_{i_1,i_2,\cdots,i_M=1}^{N}a_{i_1i_2\cdots i_M}^2$ can be written in the form of frequency coefficients as $$||\mathbf{A}||^2_T=\sum_{r=1}^N \lambda_r^2=\boldsymbol{\lambda}^T\boldsymbol{\lambda},$$ where $\boldsymbol\lambda=[\lambda_1 \quad\lambda_2 \quad\cdots\quad\lambda_N]$.
Since $\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}}$, we have $$a_{i_1i_2\cdots i_M}=\sum_{r=1}^N \lambda_rf_{r,i_1}f_{r,i_2}\cdots f_{r,i_M},$$ where $f_{r,i}$ is the $i$th element of $\mathbf{f}_r$. Then, the tensor norm is $$\begin{aligned}
||\mathbf{A}||^2_T
%&=\sum_{i_1,i_2,\cdots,i_M}a_{i_1i_2\cdots i_M}^2\nonumber\\
&=\sum_{i_1,i_2,\cdots,i_M}(\sum_{r=1}^N \lambda_rf_{r,i_1}f_{r,i_2}\cdots f_{r,i_M})^2\nonumber\\
&=\sum_{i_1,i_2,\cdots,i_M}(\sum_{r=1}^N \lambda_rf_{r,i_1}\cdots f_{r,i_M})(\sum_{t=1}^N \lambda_tf_{t,i_1}\cdots f_{t,i_M})\nonumber\\
&=\sum_{i_1,i_2,\cdots,i_M}\sum_{r,t}\lambda_r\lambda_tf_{r,i_1}\cdots f_{r,i_M} f_{t,i_1}\cdots f_{t,i_M}\nonumber\\
&=\sum_{r,t}\lambda_r\lambda_t\sum_{i_1,i_2,\cdots,i_M=1}^N(f_{r,i_1}f_{t,i_1})\cdots(f_{r,i_M}f_{t,i_M})\nonumber\\
% &=\sum_{r,t}\lambda_r\lambda_t(\sum_{i=1}^Nf_{r,i}f_{t,i})^M\nonumber\\
&=\sum_{r,t}\lambda_r\lambda_t(\mathbf{f}_r^T\mathbf{f}_t)^M.
\end{aligned}$$ Since $\mathbf{f}_r$ is orthogonal, $\mathbf{f}_r^T\mathbf{f}_t=1$ holds if $r=t$; otherwise, $\mathbf{f}_r^T\mathbf{f}_t=0$. Thus, we obtain $||\mathbf{A}||^2_T=\sum_{r=1}^N \lambda_r^2$.
This property can help us build a connection from the tensor norm to the frequency coefficients directly.
Now, if we consider the representing tensor as the adjacency tensor and each hyperedge consists of three nodes since at least three nodes are required to construct a surface, we optimize the normalized frequency coefficients $\boldsymbol{\sigma}=\frac{1}{\lambda_{max}}\boldsymbol{\lambda}=[\sigma_1\quad
\sigma_2 \quad \cdots\quad \sigma_N]^T$ via $$\begin{aligned}
\label{target}
& \min_{\boldsymbol{\sigma}}\quad \alpha\sum_{i=1}^3||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2+\beta{ \boldsymbol\sigma^T\boldsymbol\sigma}\\
\hspace{-3mm}s.\; t. \; &\;\;0\leq \sigma_r\leq \max_i \sigma_{i}=1,\label{non}\\
&\sum_{r=1}^N \sigma_r f_{r,i_1}f_{r,i_2}f_{r,i_3}\geq 0, \quad i_1,i_2,i_3=1,2,\cdots,N.\label{adj}\end{aligned}$$
The constraint (\[adj\]) limits the estimated representing tensor as the adjacency tensor. The constraint (\[non\]) is the nonnegative constraint on weight and the factor [@c26]. Clearly, the optimization is non-convex with the constraint $\max_i \sigma_{i}=1$. However, if the position of the maximal frequency is known, the optimization problem can be solved by tools such as cvx [@c27; @c28]. Thus, we can develop the following algorithm to estimate the frequency coefficients.
: Point cloud dataset $\mathbf{s}=[\mathbf{X_1,X_2,X_3}]\in\mathbb{R}^{N\times 3}$, hypergraph spectrum $\mathbf{V}=[\mathbf{f}_1,\cdots,\mathbf{f}_N]$. i=1,2,...,iter [**[do]{}**]{}: Set $\sigma_i=1$ as the maximal normalized eigenvalue. Solve the optimization problem in Eq. (\[target\]). Find the optimal $i$ to minimize the target function. The optimal coefficients is the solution of Eq. (\[target\]) correlated to the optimal $i$. : Frequency coefficients [$\boldsymbol\sigma$]{}.
Note that, since we consider clean point cloud without noise, we usually set parameter $\alpha\ll\beta$. Then, from the estimated spectrum pair $(\mathbf{f}_r,\sigma_r)$ under normalization, we can recover the original adjacency tensor as Eq. (\[dec\]). Hence, the hypergraph construction process for a clean point cloud can be summarized as Fig. \[fram1\]. The recovery of original adjacency tensor is not always necessary in practical applications since storing the representing tensor is less efficient than storing the spectrum pairs.
![Estimation of Hypergraph Spectral Pairs for Original Point Clouds[]{data-label="fram1"}](Framework1.jpg){width="3in"}
Joint Spectrum Estimation and Denoising {#h2}
=======================================
In practical 3D imaging, perturbations such as noises and outliers often exist when generate a point cloud of an unknown object. These noises may significantly affect the performance of point cloud processing since many existing algorithms require quality datasets [@c27]. Thus, denoising remains a vital issue in practical point cloud applications.
Usually, to denoise point sets with sharp features is difficult, especially when the noise is large, as such features are hard to distinguish from noise effect. Generally, smoothness-based methods are common. In [@c30], a method based on $L_0$ norm of differences between $k$-nearest neighbors is introduced. In [@c31], Laplacian regularization is used to describe smoothness and to denoise noisy point sets. Other works, such as [@c8; @c29], minimize the total variation over graphs to denoise the point sets. Although smoothness-based methods have achieved notable successes, how to interpret and define an effective smoothness function for a general point set remains open. Furthermore, for graph-based smoothness methods, the construction of graph model remains a critical problem, since traditional methods based on distance suffers from the imprecise location measurement. To this end, a more general definition of smoothness and a more efficient denoising method for arbitrary point clouds are highly desirable.
In this section, we introduce a joint method to simultaneously estimate the hypergraph structure and denoise noisy point clouds. In Section \[h1\], we already introduce an estimation method of spectral pair $(\mathbf{f}_r,\sigma_r)$ for clean point clouds. A similar construction process can be developed for the noisy point clouds. As the estimation of spectrum components only depends on the observed data, we need to denoise the noisy observations while optimizing the frequency coefficients. As already discussed, the problem of denoising a signal on a hypergraph can be written as a convex minimization problem with the constraints that denoised signals should be smooth over the hypergraph. Accordingly, the general process of hypergraph denoising and estimation can be summarized as the following steps:
- Step 1: Estimate the approximated hypergraph spectrum components from the observed noisy point clouds;
- Step 2: Jointly estimate frequency coefficients and denoise the noisy observations;
- Step 3: Update the noisy observations as denoised data and repeat Step 1 until enough iterations.
To estimate hypergraph spectral components of noisy data, the process is the same as Algorithm 1 based on hypergraph stationary process. To jointly estimate the frequency coefficients to recover the original underlying structure and to denoise the noisy point clouds, we propose the following objective. Given $N$ noisy points $\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]$, the joint estimation task can be formulated as
$$\begin{aligned}
\min_{\boldsymbol{\sigma},\mathbf{Y}}\quad& \sum_{i=1}^3[||\mathbf{X}_i-\mathbf{Y}_i||^2_2+\alpha ||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2]+\beta ||\mathbf{A}||_2^2\label{esfram}\\
s.t.\quad &\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}} \in \mathcal{A},\nonumber\\
&\;\;0\leq \sigma_r\leq \max_i \sigma_{i}=1, \nonumber\\
&\mathbf{W}_i=[(\mathbf{f}_1^T\mathbf{X}_i)\mathbf{f}_1\quad(\mathbf{f}_2^T\mathbf{X}_i)\mathbf{f}_2\quad\cdots\quad(\mathbf{f}_N^T\mathbf{X}_i)\mathbf{f}_N].
\nonumber\end{aligned}$$
The resulting $\mathbf{Y=[Y_1\quad Y_2 \quad Y_3]}$ is the denoised point clouds, and $(\alpha, \beta)$ are two positive regularization parameters. The first part in Eq. (\[esfram\]) lets the denoised point cloud maintain the observed structural features. The second part is the smoothness function derived from Eq. (\[sm1\]) which adjusts positions of noisy points. The third part is the tensor norm regularization to control hypergraph sparsity.
The optimization problem of Eq. (\[esfram\]) is not convex in $\mathbf{Y}$ and $\boldsymbol{\sigma}$. Therefore, similar to [@c25], we split the problem into two subproblems. For each subproblem, we fix one variable set to solve the other one. Upon convergence, the solution corresponds to a local minimum and not necessarily a global minimum.
We first initialize $\mathbf{Y}$ as the observed signals $\mathbf{X}$ and solve the following problem similar to that in Section \[h1\]. $$\label{target1}
\min_{\boldsymbol{\sigma}} \alpha\sum_{i=1}^3||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2+\beta{ \boldsymbol\sigma^T\boldsymbol\sigma}$$ $$\begin{aligned}
s.t.\quad
&\;\;0\leq \sigma_r\leq \max_i \sigma_{i}=1, \nonumber\\
&\sum_{r=1}^N \sigma_r f_{r,i_1}f_{r,i_2}f_{r,i_3}\geq 0, \quad i_1,i_2,i_3=1,2,\cdots,N.\nonumber\end{aligned}$$ This problem can be solved similarly to the solution of clean point cloud with Algorithm 2.
Once the estimated frequency coefficients are found, we solve the subproblem of point cloud denoising $$\begin{aligned}
\label{ss2}
\min_\mathbf{Y} \sum_{i=1}^3[||\mathbf{X}_i-\mathbf{Y}_i||^2_2+\alpha ||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2],\end{aligned}$$ whose close-form solution for each coordinate is $$\label{ss3}
\mathbf{Y}_i=[\mathbf{I}+\alpha(\mathbf{I-P_s})^T(\mathbf{I-P_s})]^{-1}\mathbf{X}_i.$$
Note that $\mathbf{P_s}$ is the supporting matrix. We then update the frequency components based on the denoised point clouds, and repeatedly carry out Step 1 to Step 3 until getting the final solution. In practice, we generally observe the convergence within only a few iterations. The complete algorithm is summarized in Algorithm 3 as shown in Fig. \[fram2\]. Unlike for clear point clouds, we emphasize more on the smoothness of signals over the hypergraph. The parameter $\alpha$ can be set larger than used when dealing with clean point clouds.
: Noisy observations of point clouds $\mathbf{s}=[\mathbf{X_1,X_2,X_3}]\in\mathbb{R}^{N\times 3}$. : Calculate the spectrum components $\mathbf{f}_r$’s from the observed point cloud $\mathbf{s}$ as Algorithm 1. i=1,2,...,iter [**[do]{}**]{}: Find the optimal $\boldsymbol{\sigma}$ for the first subproblem in Eq. (\[target1\]) with Algorithm 2. Solve the optimization problem in Eq. (\[ss2\]) with $\mathbf{Y}$ in Eq. (\[ss3\]). Update the observed signals as $\mathbf{Y}$ and recalculate the spectrum components $\mathbf{f}_r$’s. : Spectral pairs $(\mathbf{f}_r,\sigma_r)$’s, denoised point clouds $\mathbf{Y}$.
![Joint Hypergraph Estimation and Denoising for Noisy Point Cloud.[]{data-label="fram2"}](Framework2.jpg){width="3in"}
Application Examples {#appli}
====================
In this section, we examine two application examples to test the efficacy of the proposed method in estimating hypergraph structure for both clear and noisy point clouds.
Sampling
--------
Sampling is an important operation to facilitate analysis of very large point clouds. In this part, we consider different sampling strategies depending on different kinds of applications. Some interesting connections are found from the hypergraph frequency and point cloud features.
### Resampling using Harr-like Highpass Filtering
Filtering helps extract select features of a given dataset. In some applications such as boundary detection, accurate extraction of shape features of point clouds is important. Thus, an efficient sampling should retain the features of the original point cloud. In our estimation of hypergraph structure, smoothness is a significant feature to model point clouds. Ideally, smoothness over the original surface of a point cloud should correspond to smoothness over its hypergraph model. Therefore, we can also design a Harr-like high-pass filter to extract sharp features over the surfaces.
Let $\mathbf{I}$ be an identity matrix of appropriate size. Similar to that in GSP [@c3], a Haar-like high-pass filter is designed as $$\begin{aligned}
\mathbf{H}
&=\mathbf{I-P_s}\\
&=\mathbf{V}
\begin{bmatrix}
1-\sigma_1&0&\cdots&0\\
0&1-\sigma_2&\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
0&0&\cdots&1-\sigma_N
\end{bmatrix}\mathbf{V}^T.\end{aligned}$$ The filtered signal is $$(\mathbf{Hs})_i=\mathbf{s}_i-\sum_j {P_s}_{(ij)}\mathbf{s}_j,$$ which reflects the differences between nodes and their neighbors over the hypergraph. Note that, the frequency coefficients together with their corresponding spectral components are ordered decreasingly here, i.e., $\sigma_{i}\geq\sigma_{i+1}$. From the definition of total variation, more smoothness corresponds to larger total variation. Thus, we can extract the sharp features over the point clouds by sampling the nodes with large value of $||\mathbf{s}_i-\sum_j {P_s}_{(ij)}\mathbf{s}_j||^2_2$.
To test this application, we estimate the spectral pairs for clean point clouds and filter the signals over several synthetic datasets. We randomly generate multiple points over the surfaces of basic graphics shown as Fig. \[ori\], and sample the point clouds using the high-pass filter (HPF) given in Fig. \[sam\]. From the test results, we can see that the sampled points of the surfaces in Fig. \[sur2\] mainly congregate near the corners and edges, which are the sharp parts of the point clouds. In addition, the sampled nodes for a cube shape are also crowded near edges and corners. On the other hand, the sampled nodes of a cylinder are mostly at the boundaries of the cylinder. Our test results show that the Harr-like HPF can extract sharp features from point cloud surfaces, which correspond to the least smooth parts of the estimated hypergraph. Moreover, since the total variation measures the order of frequency, sharp features over the point cloud correspond to high frequency components. Thus, the hypergraph model and the estimated spectral pairs are efficient when extracting features of 3D point clouds.
### Down-Sampling with Hypergraph Fourier Transform
Projecting signals into a suitable orthonormal basis is a widely-used sampling method [@c32]. The work of [@c9] develops a sampling theory based on hypergraph signal processing as follows:
- Step 1: Order the spectrum components from low frequency to high frequency based on their total variations.
- Step 2: Implement hypergraph Fourier tranform as $$\mathcal{F}(\mathbf{s})=[(\mathbf{f}_1^T\mathbf{s})^{M-1}\quad (\mathbf{f}_2^T\mathbf{s})^{M-1}\quad \cdots\quad (\mathbf{f}_N^T\mathbf{s})^{M-1}]^T.$$
- Step 3: Use $C$ transformed signal components in the hypergraph frequency domain to represent $N$ signals in the original vertex domain.
More specifically, for a $K$-bandlimitted hypergraph signal, a perfect recovery is available with $K$ samples in hypergraph frequency domain. Similarly, we can sample the point clouds based on the hypergraph Fourier transform. To test the performance of the sampled signals, we implement hypergraph Fourier transform (HGFT) on each coordinates of the point clouds, i.e., $\mathcal{F}(\mathbf{X}_i)$ for all $i$. Then, we take the first $C$ transformed signals in all coordinates. Finally, we implement the inverse hypergraph Fourier transform (iHGFT) to obtain the sampled shapes of the original point clouds. Note that, perfect recovery happens with $C$ samples, if $(\mathcal{F}(\mathbf{X}_i))_{j+C}=0$ for $i,j\in {\cal Z}^+$.
We test the recovered point clouds for animal point datasets [@c33; @c34; @c35; @c36] with the GSP-based methods. For the GSP-based method, we construct the a graph adjacency matrix $\mathbf{W}$ with Guassian model, i.e., $$\label{adjj}
W_{ij}=\left\{
\begin{aligned}
\exp\left(-\frac{||\mathbf{s}_i-\mathbf{s}_j||^2_2}{\delta^2}\right),&\quad||\mathbf{s}_i-\mathbf{s}_j||^2_2\leq t;\\
0,&\quad \mbox{otherwise},
\end{aligned}
\right.$$ where $\mathbf{s}_i$ is the coordinates of the $i$th node. Then, we sample the point clouds using the signals after the graph Fourier transform (GFT).
The test point cloud is shown as Fig. \[sam2\]. We first compare the mean squared error (MSE) between the recovered point clouds and original point clouds shown as Fig. \[sam3\]. From the experimental results, we can see that the HGSP-based method has smaller error than the GSP downsampling method, clearly indicating hypergraph to be a better model. However, sometimes, MSE alone cannot tell the true story in terms of the performance for the recovered point clouds. To explore more, we compare the recovered point clouds directly in Fig. \[sam4\]. From the experimental results, we can see that HGSP-based method captures the overall structure of the point clouds with very few samples, whereas the GSP-based method requires more samples to get sufficient details. The MSE of GSP mainly stems from some outliers when taking more than 90 percent of the samples. The experiments show that HGSP-based method is a better tool for applications which need to recover an overall shape of point clouds from limited data storage. Our test shows hypergraph to be a suitable model for point clouds, and the estimated hypergraph spectral pairs capture the point cloud characteristics very well.
Denoising
---------
From estimated hypergraph spectral pairs from noisy point clouds, the performance of denoising is an intuitive metric of how good the estimates are. There are multiple methods developed to denoise noisy point clouds. The authors of [@c8] proposed a graph-based method to denoise based on total variation (GSP-TV). This method constructs a graph based on observed coordinates first before solving the denoising optimization $$\min_{\mathbf{Y}} ||\mathbf{X}-\mathbf{Y}||_2^2+\alpha \mathbf{TV}(\mathbf{Y,W}),$$ where $\mathbf{X}$ is the observed coordinates, and $\mathbf{W}$ is the adjacency matrix. Here, the graph total variation $\mathbf{TV}(\mathbf{Y,W})$ is applied in describing the smoothness over the graphs. In addition to total variation, Laplacian regularization (LR) has also been used in denoising with a basic formulation $$\min_{\mathbf{Y}} ||\mathbf{X}-\mathbf{Y}||_2^2+\alpha ||\mathbf{Y}^T\mathbf{LY}||^2_2,$$ where $\mathbf{L}$ is the Laplacian matrix. Developed from traditional Laplacian regularization methods, a mesh Laplacian smooth (MLS) method is given in [@c37].
HGSP GSP(TV) MLS LR Noisy
---------------------------- ----------- --------- -------- -------- --------
Uniform$\sim$U(-0.03,0.03) **32.60** 45.94 56.63 48.86 63.84
Uniform$\sim$U(0.08,0.16) **98.36** 160.18 205.15 168.17 220.96
Guassian$\sim$N(0,0.08) **41.10** 42.36 49.41 64.00 76.54
Guassian$\sim$N(0.02,0.08) **73.43** 76.07 83.25 123.08 142.11
Impulse (p=0.08) **34.53** 45.45 50.89 40.53 60.5
: Error in Dfferent Kinds of Noise[]{data-label="t1"}
\
To validate the performance of our denoising method, we compare with the aforementioned traditional methods using the Standford bunny dataset with 3595 points and sampled bunny with 397 points shown as Fig. \[den\]. We compare different methods in the sampled bunny dataset adding zero-mean Guassian noise with variance $\sigma^2$, and zero-mean Uniform noise with the interval $B-A$, respectively. We use the error denoted by $$Error=\sum_{i=1}^N\sum_{j=1}^3 |X_{ji}-Y_{ji}|,$$ where $X_{ij}$ and $Y_{ji}$ are the $j$th coordinates of observed and denoised point $i$, respectively, to measure the performance. We repeat the test on 1000 randomly generated noisy data. The error between the original dataset and the denoised dataset is shown in Fig. \[den1\]. The error of the noisy point clouds before denoising is also given as a reference in Fig. \[den1\]. From the test results, we can see that the HGSP-based method can achieve the lowest error, which demonstrates the effectiveness of the proposed denoising methods and estimated spectral pairs. The comparison in other types of noise is shown in Table. \[t1\]. More specifically, the methods based on total variation, i.e, HGSP and GSP-TV, have better performance than the methods based on Laplacian regularization, which indicates the total variation has a more efficient representation of the surface smoothness. The denoised bunny with 3595 samples is shown in Fig. \[den11\], using our proposed method to denoise the noisy bunny. The successful recovery of the bunny point cloud presents a strong evidence that our estimated spectral pairs and denoising method are powerful tools in processing noisy datasets.
Conclusions and Future Directions {#con}
=================================
In this work, we develop HGSP tools for effectively processing 3D point clouds. We first introduce a novel method to estimate hypergraph spectral components and presented an optimization formulation to optimally select frequency coefficients to recover the optimal hypergraph structure. We develop a HGSP algorithm to jointly estimate hypergraph spectrum pairs and denoise noisy point clouds. To test the practicality and efficacy of our proposed hypergraph tools, we study two point cloud application examples. Our results illustrate significant performance improvements for both sampling and denoising applications. Moreover, we establish a clear connection between hypergraph frequency components and features on point-cloud surface that can be exploited in future studies.
Our work establish hypergraph signal processing as an efficient tool in tackling high-dimensional interactions among multiple nodes. In addition to sampling and denoising, HGSP can find good applications in many other aspects of point clouds through estimation of spectral components and frequency coefficients. One direction is the design of filters to analyze the spectral properties and surface features of 3D point clouds. Another interesting problem is the recovery of point clouds from low dimensional samples. Beyond point clouds, HGSP can also effectively handle datasets with other complex underlying structure.
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A. M. Bronstein, M. M. Bronstein, U. Castellani, B. Falcidieno, A. Fusiello, A. Godil, L. J. Guibas, I. Kokkinos, Z. Lian, M. Ovsjanikov, G. Patané, M. Spagnuolo, and R. Toldo, “SHREC 2010: robust large-scale shape retrieval benchmark", *Proc. EUROGRAPHICS Workshop on 3D Object Retrieval (3DOR)*, 2010.
A. M. Bronstein, M. M. Bronstein, B. Bustos, U. Castellani, M. Crisani, B. Falcidieno, L. J. Guibas, I. Kokkinos, V. Murino, M. Ovsjanikov, G. Patané, I. Sipiran, M. Spagnuolo, and J. Sun, “SHREC 2010: robust feature detection and description benchmark", *Proc. EUROGRAPHICS Workshop on 3D Object Retrieval (3DOR)*, 2010.
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| {
"pile_set_name": "ArXiv"
} |
---
bibliography:
- 'main.bib'
---
[\
[Matthew W. Muterspaugh, Maciej Konacki, Benjamin F. Lane, and Eric Pfahl]{} ]{}
Why Focus Planet Searches on Binary Stars?
==========================================
Searches for planets in close binary systems explore the degree to which stellar multiplicity inhibits or promotes planet formation. There is a degeneracy between planet formation models when only systems with single stars are studied—several mechanisms appear to be able to produce such a final result. This degeneracy is lifted by searching for planets in binary systems; the resulting detections (or evidence of non-existence) of planets in binaries isolates which models may contribute to how planets form in nature. Studying relatively close pairs of stars, where dynamic perturbations are the strongest, provides the most restrictive constraints of this type [see, for example, @thebault2004; @Pfahl2005; @PfahlMute2006].
In this chapter, we consider observational efforts to detect planetary companions to binary stars in two types of hierarchical planet-binary configurations: first “S-type” planets which orbit just one of the stars, with the binary period being much longer than the planet’s; second, “P-type” or circumbinary planets, where the planet simultaneously orbits both stars, and the planetary orbital period is much longer than that of the binary [@Dvorak1982]. Simulations show each of these configurations has a large range of stable configurations .
S-Type Planets
==============
S-Type planets orbit just one of the stars in a binary, and the binary separation is much larger than that between the star and planet. Some of the binaries are so widely separated (projected semimajor axis $a_b \gtrsim 1$ arcsecond) that they can be spatially resolved by ground-based telescopes without active image correction; for these, traditional planet-finding techniques can be used. In fact, astrometric methods often perform best in this regime, as the secondary star serves as a convenient reference for the primary, and vice versa. Here, astrometric and radial velocity (RV) programs are considered as the most versatile search methods. (While transit searches might also be possible, these typically have very limited spatial resolutions, and the second star can act as a photometric “contaminant.”) When the binaries are not spatially resolved with simple imaging, modifications must be made to meet the measurement precisions required for detecting extrasolar planets.
Wide Binaries
-------------
From an observational standpoint, “wide” binaries are considered to be those that can be resolved by traditional (uncorrected) imaging techniques. Due to atmospheric seeing, this sets the projected sky separation at larger than roughly one arcsecond.
### Dualstar Astrometry
Interferometric narrow-angle astrometry [@shao92; @col94] promises astrometric performance at the 10-100 micro-arcsecond level for pairs of stars separated by 1-60 arcseconds. The lower limit of the allowable binary separation for this technique is that the binary is resolved by the individual telescopes in the interferometer; the upper limit is set by the scale over which the effects of atmospheric turbulence are correlated. This technique was first demonstrated with the Mark III interferometer for short integrations [@col94], was extended to longer integrations and shown to work at the 100 micro-arcsecond level at the Palomar Testbed Interferometer [PTI, @l00].
However, achieving such performance requires simultaneous measurement of the interferometric fringe positions of both stars, greatly complicating the instrument (two beam combiners and metrology throughout the entire array are required). In addition, the instrumental baseline vector $\overrightarrow{B}$ connecting the unit telescopes must be known to high precision ($\approx 100$ microns).
In an optical interferometer light is collected at two or more apertures and brought to a central location where the beams are combined and a fringe pattern produced on a detector. For a broadband source of central wavelength $\lambda$ and optical bandwidth $\Delta\lambda$ the fringe pattern is limited in extent and appears only when the optical paths through the arms of the interferometer are equalized to within a coherence length ($\Lambda =
\lambda^2/\Delta\lambda$). For a two-aperture interferometer, neglecting chromatic dispersion by unequal air paths, the intensity measured at one of the combined beams is given by $$\label{double_fringe}
I(x) = I_0 \left [ 1 + V \frac{\sin\left(\pi x/ \Lambda\right)}
{\pi x/ \Lambda} \sin \left(2\pi x/\lambda \right ) \right ]$$ where $V$ is the fringe contrast or “visibility”, which can be related to the morphology of the source, and $x$ is the optical path difference between arms of the interferometer; see Fig. \[fig:fringes\]. More detailed analysis of the operation of optical interferometers can be found in [*Principles of Long Baseline Stellar Interferometry*]{} [@Lawson2000].
![\[fig:fringes\] The response of an interferometer. The top two curves have been offset by 2 and 4 for clarity. The widths of the fringe packets are determined by the bandpass of the instrument, and the wavelength of fringes by an averaged wavelength of starlight. The top curve shows the intensity pattern obtained by observing two stars separated by a small angle on the sky—the observable is the distance between the fringe packets.](fringes.eps){height="3.5in"}
The location of the resulting interference fringes are related to the position of the target star and the observing geometry via $$\label{delayEquation}
d = \overrightarrow{B} \cdot \overrightarrow{S} +
\delta_a\left(\overrightarrow{S}, t\right) + c$$ where $d$ is the optical path-length one must introduce between the two arms of the interferometer to find fringes (often called the “delay”), $\overrightarrow{S}$ is the unit vector in the source direction, and $c$ is a constant additional scalar delay introduced by the instrument. The term $\delta_a\left(\overrightarrow{S}, t\right)$ is related to the differential amount of path introduced by the atmosphere over each telescope due to variations in refractive index.
If the other quantities are known or small, measurement of the instrumental path length $d$ required to observe fringes determines the position of the star $\overrightarrow{S}$. For a 100-m baseline interferometer, an astrometric precision of 10 $\mu$as corresponds to knowing $d$ to 5 nm, a difficult but not impossible proposition for all terms except that related to the atmospheric delay. Atmospheric turbulence, which changes over distances of tens of centimeters and on millisecond timescales, forces one to use very short exposures to maintain fringe contrast, and hence limits the sensitivity of the instrument. It also severely limits the astrometric accuracy of a simple interferometer, at least over large sky-angles.
However, in narrow-angle astrometry one is concerned with a close pair of stars, and the observable is a differential astrometric measurement, i.e. one is interested in knowing the angle between the two stars ($\overrightarrow{\Delta_s} = \overrightarrow{s_2} - \overrightarrow{s_1} $). The atmospheric turbulence is correlated over small angles. If the measurements of the two stars are simultaneous, or nearly so, the atmospheric term subtracts out making possible high precision “narrow-angle” astrometry.
The requirement that the target and reference stars be observed simultaneously results in a significant instrumental complexity, i.e. essentially two complete interferometers are required to share the same set of apertures (see Fig. \[fig:dsm\]). The splitting of light from the stars into two separate sets of delay lines, beam transport systems and beam combiners is done in a “dual-star module” located just after the apertures, with the split generally being accomplished using a beam-splitter. Considerable care must be taken in designing the system in order to avoid small pathlength measurement errors.
![\[fig:dsm\] Schematic of splitting the light in a dualstar interferometer. ](fig03-dualstar.eps){height="8.0cm"}
The exact level of astrometric precision that can be achieved depends on many factors, including the separation of the target/reference pair, the size of the interferometric baseline and the levels and distribution of atmospheric turbulence. For a typical Mauna Kea seeing profile the astrometric precision is $$\sigma_a \simeq 300\frac{\theta}{\sqrt{t}B^{2/3}}~{\rm arcsec}$$ where $B$ is the baseline length in meters, $\theta$ is the target/reference separation in radians, and t is the integration time in seconds. For typical baselines of $\sim 100$ m, and an angular separation of $\sim 30$ arcsecond implies an astrometric precision of 30 $\microas$ in an hour (see Fig. \[fig:naa\]).
The magnitude of the astrometric signal of the star’s motion about the center of mass (CM) between it and its planet is given by: $$\label{ast_reflex}
\Delta a_{CM} = 2 \frac{M_p}{M_s}a_p =
\frac{M_p/M_\Jupiter}{M_s/\Msun}\frac{a_p}{524}.$$ where $M_p$, $M_b$, $M_\Jupiter$, $\Msun$ are, respectively, masses of the planet,star, Jupiter, and Sun, and $a_p$ is the semimajor axis of the planet’s orbit.
Thus, the minimum mass that can be detected is roughly $$\begin{aligned}
\label{ast_ref2}
M_p/M_\Jupiter & \gtrsim & 524 \frac{\sigma_a}{a_p} \frac{M_s}{\Msun}\\
& \gtrsim & 0.1 \frac{\sigma_a/20\microas}{a_p/1\AU}
\frac{d}{10 {\rm pc}}\frac{M_s}{\Msun}\\
M_p/M_\earth & \gtrsim & 1.6 \frac{\sigma_a/1\microas}{a_p/1\AU}
\frac{d}{10 {\rm pc}}\frac{M_s}{\Msun}\end{aligned}$$ where $M_\earth$ is the mass of the Earth, and here $d$ is the distance to the target star.
### Radial Velocities {#wideRV}
When the stars in a binary can be spatially resolved without active image correction on ground based telescopes, the spectrum of each star can be recorded separately without contamination from the other, and the standard precision RV method described below can be used . Similarly, if the secondary is much fainter than the primary, precision RV might be performed on the brighter star as though it were single, though there is concern about the influence of the fainter lines. Several ($\sim 30$)exoplanet candidates in binaries have been discovered in this manner. In some cases the stars were not previously known to be binaries, and their natures were only discovered by long-term RV trends or follow-up adaptive optics imaging. Some of these efforts to detect planets in binary stellar systems include that of [@Toyota2005] for single-lined and wide binaries, that of [@Desidera2006b] targeting wide binaries, and the program targeting single-lined and wide binaries of [@Udry2004].
The highest precision RV observations are obtained either from the $I_2$ (molecular iodine) absorption cell or the use of carefully designed spectrographs with fiber scrambling. In order to achieve a RV precision of $\sim 1 m\,s^{-1}$ an iodine absorption cell is used to superimpose a reference spectrum on the stellar spectrum (by sending a starlight through the cell). The spectrum provides a fiducial wavelength scale against which radial velocity shifts are measured.
Thanks to its conceptual simplicity, the iodine technique is the most commonly adopted way to obtain precision radial velocities. Iodine absorption cells are available on many spectrographs—HIRES at the 10m Keck I (Keck Observatory), Hamilton at the 3m Shane (Lick Observatory), SARG at the 3.6m TNG (Canary Islands), UCLES at the 3.9m Anglo-Australian Telescope (Anglo-Australian Observatory), HRS at the 9m HET (McDonald Observatory), MIKE at the 6.5m Magellan (Las Campanas Observatory), UVES at the 8m Kueyen (Cerro Paranal), HDS at the 8.2m Subaru (National Astronomical Observatory of Japan) and many other—and are used for planet detections.
In the iodine absorption cell technique, the Doppler shift of a star spectrum is determined by solving the following equation [@Marcy1992] $$\label{i2::}
I_{obs}(\lambda) =
[I_{s}(\lambda+\Delta\lambda_{s})\,T_{I_{2}}(\lambda+\Delta\lambda_{I_{2}})]
\,\otimes\,PSF$$ where $\Delta\lambda_{s}$ is the shift of the star spectrum, $\Delta\lambda_{I_{2}}$ is the shift of the iodine transmission function $T_{I_{2}}$, $\otimes$ represents a convolution, and $PSF$ a spectrograph’s point-spread function. The parameters $\Delta\lambda_{s},
\Delta\lambda_{I_{2}}$ as well as parameters describing the PSF are determined by performing a least-squares fit to the observed spectrum, $I_{obs}$, as seen through the iodine cell. To this end, one also needs a high SNR star spectrum taken without the cell, $I_{s}$, which serves as a template for all the spectra observed through the cell, as well as the I$_2$ transmission function, $T_{I_{2}}$, obtained with the Fourier Transform Spectrometer at the Kitt Peak National Observatory. The Doppler shift of a star spectrum is then given by $\Delta\lambda = \Delta\lambda_{s} - \Delta\lambda_{I_{2}}$.
The velocity reflex amplitude of a star due to an unseen companion is given by $$\begin{aligned}
\label{vel_reflex}
\Delta v_b & = & 2 \frac{2 \pi a_p \sin i_p}{P_p}\frac{M_p}{M_s+M_p}
= \frac{2 \sqrt{G} M_p \sin i_p }{\sqrt{\left( M_s+M_p \right) a_p}}\nonumber\\
& = & 56.9\,{\rm m\,s^{-1}} \times \frac{ \left( M_p \sin i_p /M_\Jupiter \right)}
{\sqrt{\left( \left( M_s+M_p \right)/\Msun\right)\left(a_p/1\rm{AU} \right)}}\end{aligned}$$ where $P_p$ is the period of the planet’s orbit, $G$ is the gravitational constant, and $i_p$ is the inclination of the planet’s orbit.
For an RV precision of $\sigma_{rv}$, the minimum mass that can be detected is roughly $$\begin{aligned}
M_p \sin i_p / M_\Jupiter & \gtrsim & 0.018
\frac{\sigma_{rv}}{1{\rm m\,s^{-1}}}
{\sqrt{\left( \left( M_s+M_p \right)/\Msun\right)\left(a_p/1\rm{AU} \right)}}\\
M_p \sin i_p / M_\earth & \gtrsim & 5.6
\frac{\sigma_{rv}}{1{\rm m\,s^{-1}}}
{\sqrt{\left( \left( M_s+M_p \right)/\Msun\right)\left(a_p/1\rm{AU} \right)}}.\end{aligned}$$
### Observational Precisions
Astrometry is most sensitive to long period planets, RV to short period ones. Figure \[fig:wideS\] shows the companion masses one can detect for each method, assuming 20 $\microas$ ground-based astrometry, 1 $\microas$ space-based astrometry, and $1 {\rm m\, s^{-1}}$ RV precisions.
![\[fig:wideS\] Sensitivity to S-Type planets in wide binaries, comparing astrometric and radial velocity techniques. All calculations assume solar mass stars; astrometric sensitivity assumes a distance of 10 pc to the target system. ](wideS_Type.eps){height="3.5in"}
Close Binaries
--------------
Radial velocity surveys for extrasolar planets have been restricted largely to stars within $\simeq$100pc, and the majority of detected exoplanets are at distances of 10–50pc. Therefore, our observational definition of a “wide” binary corresponds to projected orbital separations of $\gtrsim$50AU. More compact systems form the complementary class of “close” binaries. From a theoretical standpoint, binaries with semimajor axes of $\lesssim$50AU pose important challenges to standard ideas about the formation of giant planets.
Imagine a protoplanetary disk around one star in a newly formed binary. Suppose that the orbit has semimajor axis $a$ and eccentricity $e$, and assume, for simplicity, that the stars have equal masses. The tidal gravitational field of the companion star truncates the disk at a radius of $R_t \simeq 0.26 a(1-e^2)^{1.2}$ [e.g. @Pichardo2005]. If $R_t\lesssim 3\,{\rm AU}$ (i.e., inside the so-called “ice line”), it seems unlikely that icy grains could form and grow into planetesimals, thus precluding the embyonic stage of giant planet formation in the core-accretion scenario [e.g., @Liss1993]. In more extreme cases, when $R_t \lesssim 1\,{\rm AU}$, there may be insufficient material in the disk to yield a Jovian-mass planet [@Jang-Condell2006]. Even when $R_t$ is as large as $\simeq$10AU, stirring of the disk by the tidal field and the thermal dissipation of spiral waves may inhibit planetesimal formation, as well as stabilize the disk against fragmentation [@Nelson2000; @thebault2004; @Thebault2006]. In this case, it may be that neither the core-accretion picture nor gravitational instability [e.g., @Boss2000] are accessible modes of giant planet formation. We adopt $R_t = 10\,{\rm AU}$ ($a \lesssim 40\,{\rm AU}$ for modest $e$) as a fiducial upper limit for which Jovian planet formation is significantly perturbed and perhaps strongly inhibited. This provides a simple theoretical definition of a close binary that roughly matches our observational measure.
A handful of planets in binaries with $a \lesssim 20\,{\rm AU}$ have already been discovered (see Table \[tableclose\]). The tightest of these systems [HD 188753 @Konacki2005] has a periastron separation of only $\simeq$6AU, which seems severely at odds with the conventional lore on Jovian planet formation. @Pfahl2005 and @Portegies_Zwart_2005 suggested that the planetary host star may have been acquired in a dynamical exchnage interaction in a star cluster [*after*]{} the planet formed, thus circumventing the complicating factors listed above. As most stars are born in clustered environments, one wonders how often dynamics can account for planets in close binaries. This idea was explored in @PfahlMute2006, where it was found that exchange interactions can account for only $\sim$0.1% of close binaries hosting planets. However, the (admittedly small) sample of systems in Table 1 seems to indicate that a larger fraction of $\sim$1% of close binaries harbor giant planets (see Pfahl & Muterspaugh 2006 for details), and perhaps planets do somehow form frequently in these hostile environments. It crucial that we begin to develop a census of planets in close binaries in order to test the different theories about planet formation and dynamics.
Close Binaries with Planets
Object $a_b$ (AU) $e$ $M_1/M_2$ $R_t$ (AU) Refs
----------------- ------------ ------ ----------- ------------ ---------
HD 188753 12.3 0.50 1.06/1.63 1.3 1
$\gamma$ Cephei 18.5 0.36 1.59/0.34 3.6 2, 3
GJ 86 $\sim$20 0.7/1.0 $\sim$5 4, 5, 6
HD 41004 $\sim$20 0.7/0.4 $\sim$6 7
HD 196885 $\sim$25 1.3/0.6 $\sim$7 8
: \[tableclose\] When no eccentricity is given, only the projected binary separation is known. $M_1/M_2$ is the planetary host mass divided by companion mass. In HD 188753, the secondary is a binary with semimajor axis 0.67 AU. In GJ 86, the secondary is a white dwarf; to estimate the tidal truncation radius $R_t$, an original companion mass of $1\Msun$ is assumed. (1) [@Konacki2005]; (2) [@Campbell1988] (3) [@hatzes2003]; (4) [@Queloz2000] (5) [@Mugrauer2005] (6) [@Lagrange2006] (7) [@Zuc2004] (8) [@Chauvin2006b]
### PHASES Astrometry
The dualstar astrometry method can be modified for use when the binaries are so close that the individual telescopes of an interferometer cannot resolve the pair [@LaneMute2004a]. The interferometer itself overresolves the binary, as in Fig. \[fig:fringes\]; its high spatial resolution then allows for precision astrometric measurements.
In this mode, the small separation of the binary results in both components being in the field of view of a single interferometric beam combiner. The fringe positions are measured by modulating the instrumental delay with an amplitude large enough to record both fringe packets.
However, since the fringe position measurement of the two stars is no longer truly simultaneous it is possible for the atmosphere to introduce path-length changes (and hence positional error) in the time between measurements of the separate fringes. To reduce this effect, a fraction of the incoming starlight is redirected to a separate beam-combiner. This beam-combiner is used in a “fringe-tracking” mode [@ss80; @col99] where it rapidly (10 ms) measures the phase of one of the starlight fringes, and adjusts the internal delay to keep that phase constant. The fringe tracking data is used both in real-time as a feed-back servo, after which a small residual phase error remains, and in post-processing where the measured residual error is applied to the data as a feed-forward servo. This technique—known as phase referencing—has the effect of stabilizing the fringe measured by the astrometric beam-combiner. For this observing mode, laser metrology is only required between the two beam combiners through the location of the light split (which occurs after the optical delay has been introduced), rather than throughout the entire array. Without phase referencing, the astrometric precision obtainable is a factor of a hundred worse; see Fig. \[fig:expected\].
![\[fig:expected\] The expected narrow-angle astrometric performance in mas for the phase-referenced fringe-scanning approach, for a fixed delay sweep rate, and an interferometric baseline of 110 m. Also shown is the magnitude of the temporal loss of coherence effect in the absence of phase referencing, illustrating why stabilizing the fringe via phase referencing is necessary. ](f2_expected.eps){height="2.5in"}
To analyze the data, a double fringe packet based on Eq. \[double\_fringe\] is then fit to the data, and the differential optical path between fringe packets is measured. A grid in differential right ascension and declination over which to search is constructed. For each point in the search grid the expected differential delay is calculated based on the interferometer location, baseline geometry, and time of observation for each scan. A model of a double-fringe packet is then calculated and compared to the observed scan to derive a $\chi^2$ value; this is repeated for each scan, co-adding all of the $\chi^2$ values associated with that point in the search grid. The final $\chi^2$ surface as a function of differential R.A. and declination is thus derived. The best-fit astrometric position is found at the minimum-$\chi^2$ position, with uncertainties defined by the appropriate $\chi^2$ contour—which depends on the number of degrees of freedom in the problem and the value of the $\chi^2$-minimum. The final product is a measurement of the apparent vector between the stars and associated uncertainty ellipse. Because the data were obtained with a single-baseline instrument, the resulting error contours are very elliptical, with aspect ratios that sometimes exceed 10:1.
The Palomar High-precision Astrometric Search for Exoplanet Systems (PHASES) program uses this technique to monitor $\sim 50$ binaries to search for substellar companions. $\kappa$ Pegasi is a well-known, nearby triple star system. It consists of a “wide” pair with semi-major axis 235 mas (8.14 $\AU$), one component of which is a single-line spectroscopic binary (semi-major axis 2.5 mas, physical separation 0.087 $\AU$; Fig. \[tripleOrbits\]). The perturbation due to the unseen (faint) short-period component is evident; similar sized perturbations with longer orbital periods would indicate the presence of planetary companions. Figure \[fig:207652\_phase\_space\] shows the mass-period phase space in which PHASES observations show companions do not exist in face-on, circular orbits in the 13 Pegasi system.
![ \[tripleOrbits\] The visual orbits of $\kappa$ Pegasi showing perturbations by the third component. The spiral line represents the apparent motion of the short-period pair’s center of light; the ellipses represent the $1\sigma$ uncertainties for PHASES measurements. ](206901_orbit.eps){height="2.5in"}
![ \[fig:207652\_phase\_space\] The 13 Pegasi Mass-Period companion phase space shows PHASES observations can rule out tertiary objects as small as two Jupiter masses. A few mass-period combinations introduce slight improvements over the single-Keplerian model, but none of these are more significant than $1.7\sigma$, and are probably not astrophysical in origin. There is a long period cutoff in sensitivity due to the finite span of the observations. Similar detection limits have recently been published on other PHASES targets [@Mute06Limits]. ](exclusion.eps){width="6.5cm"}
### Radial Velocities {#SRV}
In the case when a composite spectrum of a binary star is observed, the classical approach with the iodine cell (described in section \[wideRV\]) cannot be used since it is not possible to observationally obtain two separate template spectra of the binary components. To resolve this problem, one can proceed as follows. First, one always takes two sequential exposures of each (binary) target—one with and the other without the cell. This is contrary to the standard approach for single stars where an exposure without the cell (a template) is taken only once. This way one obtains an instantaneous template that is used to model only the adjacent exposure taken with the cell. Next, one performs the usual least-squares fit and obtain the parameters described in eq. \[i2::\]. Obviously, the derived Doppler shift, $\Delta\lambda_i$ (where $i$ denotes the epoch of the observation), carries no meaning since each time a different template is used. Moreover, it describes a Doppler “shift” of a composed spectrum that is typically different at each epoch. However, the parameters—in particular the wavelength solution and the parameters describing PSF—are accurately determined and can be used to extract the star spectrum, $I^{\star,i}_{obs}(\lambda)$, for each epoch $i$, by inverting the eq. \[i2::\]: $$\label{met::}
I^{\star,i}_{obs}(\lambda) = [I^{i}_{obs}(\lambda)\,\otimes^{-1}\,PSF^{i}]
/T_{I_{2}}(\lambda),$$ where $\otimes^{-1}$ denotes deconvolution, and $PSF^{i}$ represents the set of parameters describing PSF at the epoch $i$. Such a star spectrum has an accurate wavelength solution, is free of the I$_2$ lines and the influence of a varying PSF. In the final step, the velocities of both components of a binary target can be measured with the well know two-dimensional cross-correlation technique TODCOR [@Zucker1994] using as templates the synthetic spectra derived with the ATLAS 9 and ATLAS 12 programs [@Kurucz1995] and matched to the observed spectrum, $I_{s}(\lambda)$. The formal errors of the velocities can be derived from the scatter between the velocities from different echelle orders or using the formalism of TODCOR [@Zucker1994]. The technique currently produces RVs of binary stars with an average precision of 20 ${\rm m \, s^{-1}}$ [@Konacki04]. Improvements to the technique are being introduced to reach the level below 10 ${\rm m \, s^{-1}}$.
With the iodine technique [@Konacki04] has initiated the first RV survey for circumprimary or circumsecondary planets of binary or multiple stars (mainly hierarchical triples). The survey’s sample of $\sim 450$ binaries (northern and southern hemisphere) was selected based on the following criteria. (1) The apparent separation of the components had to be smaller than the width of the slit (e.g., 0.6 arcseconds for Keck-I/HIRES) to avoid possible systematic effects. Such systems will remain unresolved under most seeing conditions. (2) The brightness ratio between the components should not be too large (at the order of 10 or less) to be able to clearly identify spectra of both components. (3) The orbits of the binaries should be well known to constitute a firm ground on which one can discuss possible detection (or lack) of planets and substellar companions in the context of the binary characteristics. All these requirements could be satisfied by targeting a subset of speckle binaries that have determined orbits from the Catalog of Orbits of Visual Binary Stars [@hart01], containing 1700 binaries, of which 1300 have projected semi-major axes smaller than 1 arcsecond. The sample is sufficiently large to produce meaningful statistics. Also, such binaries have been ignored by previous RV studies.
The survey was initiated in 2003 at the Keck-I/HIRES and so far has covered about 50$\%$ of the northern hemisphere target sample, of which $\sim$150 stars have been observed at least twice. In July 2005, the first candidate planet in a triple star system HD 188753 was announced [@Konacki2005]. A few more candidates as well as HD 188753 are currently being scrutinized. In the course of the Keck survey, a dozen of new triple star systems with the third bodies in the low-mass star or even brown dwarf regime have been also identified and will be published in the near future.
### Eclipse Timing
Should a binary happen to be oriented with its orbital plane in the line of sight, it will exhibit eclipses as one star passes in front of another. For a binary with separation wide enough to allow for stable planetary systems to exist around just one component, the probability of such an alignment is extremely small, and very few targets are accessible. However, should one find such a fortunate happenstance, one can detect the planetary companions by precision timing of the eclipses. Clearly, this method has no direct analog for single systems.
This method can detect S-type planets or similarly moons of transiting planets. For this evaluation it is assumed that the depth of the planet (or moon) eclipse is sufficiently small as to be ignored (in such a detection, one could then reevaluate light curves to look for such transit signals) and the binary orbit is circular. The velocity of the binary orbit and the offset of the star-planet CM from that of the star by itself determines the timing variation observed as $$\begin{aligned}
\Delta t & = & x_{CM}/v_b \nonumber\\
& = & \left(\frac{a_p \sin \phi M_p}{M_2+M_p}\right)\left(\frac{P_b}{2\pi a_b}\right)\nonumber\\
& \approx & 57\,{\rm seconds}\times\left(P_b/{\rm month}\right)\frac{a_p}{\left( a_b/7 \right) }\frac{\left( M_p/M_\Jupiter \right) }{\left( M_2/\Msun \right) } \sin \phi\end{aligned}$$ where $M_2$ is the mass of star 2 (or the transiting planet, assumed to host the S-type companion), $M_p$ is the mass of the S-type object orbiting $M_2$, $\phi$ is angle between the planet’s orbital angular momentum vector and the direction of motion of the host star during eclipse, and $P_b$ is the period of the binary orbit. The timing delays due to orbit of $M_2$ about the $M_2$-$M_p$ CM. The factor of 7 appears in the final form is an approximate criteria for stability, that the planet semimajor axis is 7 times smaller than that of the binary (this factor varies by the system, and can be determined through detailed simulations); the above is thus an upper limit for the timing effect. Converting the se mimajor axis to orbital periods, $$\begin{aligned}
\label{STimingPert}
\Delta t & \approx & 41\,{\rm seconds}\times\left(P_b/{\rm month}\right)^{\frac{1}{3}}\left(P_p/{\rm day}\right)^{\frac{2}{3}}
\frac{M_p/M_\Jupiter}{\left(M_b^{\frac{1}{3}}M_2^{\frac{2}{3}}\right)/\Msun} \sin \phi \nonumber\\
& \approx & 65\,{\rm seconds}\times\left(P_b/{\rm month}\right)^{\frac{1}{3}}\left(P_p/{\rm day}\right)^{\frac{2}{3}}
\frac{M_p / M_\Jupiter}{M_b/\Msun} \sin \phi \nonumber\\\end{aligned}$$ where $M_b = M_1 + M_2 + M_p$. The final form assumes $M_1 \approx M_2$, in which case the maximum stable planet period is a thirteenth that of the binary period, implying days and months are the natural units for each respectively (S-type planets cannot exist in much shorter period systems, and longer period systems are even less likely to show eclipses). The equivalent relationship for a moon orbiting an eclipsing Jupiter is $$\Delta t \approx 13.3\,{\rm seconds}\times\left(P_b/{\rm month}\right)^{\frac{1}{3}}\left(P_p/{\rm day}\right)^{\frac{2}{3}}
\frac{M_p / M_\oplus}{\left( M_b / \Msun \right)^{\frac{1}{3}}\left( M_2 / M_\Jupiter \right)^{\frac{2}{3}}} \sin \phi$$ where now the $b$ subscript refers to the star-Jupiter analog system and $p$ to the Jupiter analog’s moon.
The precision with which eclipse minima can be timed is derived using standard $\chi^2$ fitting techniques. Assume a photometric data set $\{ y_i\}$ occurring at times $\{t_i\}$ with measurement precisions $\{\sigma_{i}\}$, and a model photometric light curve of flux $F(t-t_0)$. The corresponding intensity is $I(t-t_0) = f F(t-t_0) \pi D^2\Delta t / 4$, where $f$ ($0 \le f \le 1$) is the fractional efficiency and throughput of the telescope, $D$ is the telescope diameter, and $\Delta t$ is the sample integration time. ($F(t-t_0)$ might be determined to high precision by observing multiple eclipse events.) The fit parameter $t_0$ is uncertain by an amount equal to the difference between the value for which $\chi^2$ is minimized and that for which it is increased by one: $1 + \chi^2(t_0) = \chi^2(t_0 + \sigma_{t_0})$, $$\begin{aligned}
1 + \sum_{i=1}^{N}\left[\frac{y_{i} - I\left( t_i - t_0 \right)}{\sigma_{i}} \right]^2
& = & \sum_{i=1}^{N}\left[\frac{y_{i} - I\left( t_i - t_0 - \sigma_{t_0} \right)}{\sigma_{i}} \right]^2 \\
& \approx & \sum_{i=1}^{N}\left[
\frac{y_{i} - I\left( t_i - t_0\right)
- \left( \frac{\partial I\left( t \right)}{\partial t}\right)_{t_i-t_0} \sigma_{t_0}}
{\sigma_{i}}
\right]^2\nopagebreak\end{aligned}$$
$$\nopagebreak
= \sum_{i=1}^{N}
\left(
\left[ \frac{y_{i} - I\left( t_i - t_0 \right)}{\sigma_{i}} \right]^2 +
\left[ \frac{\left( \frac{\partial I\left( t \right)}{\partial t}\right)_{t=t_i-t_0} \sigma_{t_0}}{\sigma_{i}} \right]^2
- 2\frac{\left( \frac{\partial I\left( t \right)}{\partial t}\right)_{t=t_i-t_0}
\left[ y_{i} - I\left( t_i - t_0 \right) \right] \sigma_{t_0} }{\sigma_{i}^2}
\right).$$
Because $t_0$ is the minimizing point, the first derivative of $\chi^2$ at $t_0$ is zero, giving
$$\left( \frac{\partial \chi^2\left(t \right)}{\partial t}\right)_{t=t_0} =
2 \sum_{i=1}^{N}
\frac{\left( \frac{\partial I\left( t \right)}{\partial t}\right)_{t=t_i-t_0} \left( y_{i} - I\left( t_i - t_0 \right) \right) \sigma_{t_0} }{\sigma_{i}^2} = 0.$$
Rearrangement of terms leads to $$\label{lightcurve_timing_formula}
\sigma_{t_0} = \frac{1}{\sqrt{\sum_{i=1}^{N}\left( \frac{\left( \frac{\partial I\left( t \right)}{\partial t}\right)_{t=t_i-t_0}}{\sigma_{i}} \right)^2}} \nonumber \approx \frac{\sigma_I}{\sqrt{\sum_{i=1}^{N} \left( \frac{\partial I\left( t \right)}{\partial t}\right)_{t=t_i-t_0}^2}}.$$ An eclipse of length $\tau$ is approximated as a trapezoid-shape light curve (see Figure \[fig:eclipseModel\]) with maximum and minimum photon fluxes $F_0$ and $F_0(1-h/2)$ ($h$ is a dimensionless positive number producing an eclipse depth of $hF_0/2$; in the case of a faint secondary, $h$ is roughly twice the ratio of the squares of the stellar radii, $2 R_2^2/R_1^2$). The ingress and egress are each assumed to be of length $k\tau/2$ ($k\approx 2 R_2/(R_1+R_2)$ is unity in the case of an eclipsing binary with components of equal size, when the trapezoid becomes a “V”-shape). In functional form, this model is: $$F(t-t_0) = \left\{ \begin{array}{lrcl}
F_0 & & t-t_0 & \le -\tau/2 \\
F_0\left(1 - ht/\left(k \tau \right) - h/\left(2 k \right) \right) & - \tau/2 \le & t-t_0 & \le -\tau/2 + k\tau/2 \\
F_0\left(1 - h/2\right) & -\tau/2 + k\tau/2 \le & t-t_0 & \le \tau/2 - k\tau/2 \\
F_0\left(1 + ht/\left(k \tau \right) - h/\left(2 k \right) \right) & \tau/2 - k\tau/2 \le & t-t_0 & \le \tau/2 \\
F_0 & \tau/2 \le & t-t_0 &
\end{array}
\right.$$ (also, see Figure \[fig:eclipseModel\]).
![ \[fig:eclipseModel\] Eclipsing binary model light curve. ](eclipseLightcurve.eps){height="7cm"}
Only the portions of the light curve during ingress and egress have nonzero $\frac{\partial F\left( t \right)}{\partial t}$ (in more accurate models, the light curve slope will be nonzero but small in other regions, and will not contribute much to the sum); this slope is $\left| \frac{\partial F\left( t \right)}{\partial t} \right|
= h F_0/\left(k \tau \right)$.
The number of data points contributing to the sum is thus $N = g k
\tau / \Delta t$, where $0 \le g \le 1$ is the fraction of the eclipse observed (and also accounts for the fraction of time lost to, e.g., camera readout) and $\Delta t$ is the integration time for each measurement.
The measurement noise $\sigma_I$ is given by $$\label{photometryNoiseEquation}
\sigma_I =
\left(
I + \sigma^2_{sc} + I_{bg} + n_{dark} \Delta t + \sigma^2_{rn}
\right)^{\frac{1}{2}}$$ where $I_{bg} = f F_{bg} \pi D^2 \Delta t/4$ is is the sky background, $n_{dark}$ is detecter dark current, $\sigma_{rn}$ is detector read noise, and $\sigma_{sc}$ is scintillation noise given by [@Young1967] as $$\begin{aligned}
\sigma_{sc} & = & 0.09 I \left(D/1\,{\rm cm}\right)^{-2/3} X e^{-h/\left(8000\,{\rm m}\right)} / \left(2 \Delta t / 1\,{\rm second}\right)^{\frac{1}{2}} \\
& \approx & 0.003 I \left(D/1\,{\rm m}\right)^{-2/3} / \left(\Delta t /1\,{\rm second}\right)^{\frac{1}{2}}\end{aligned}$$ where $X$ is the airmass and $h$ is the altitude of the observatory. The drop in noise during eclipse is ignored (a factor less than $\approx 1.4$) and equation \[photometryNoiseEquation\] is combined with equation \[lightcurve\_timing\_formula\] to obtain an overall timing precision (in seconds) of
$$\begin{aligned}
\label{timing_prec_2}
\sigma_{t_0} & = & \sqrt{\frac{k \left( \tau / 1\,{\rm second}\right)}{gh^2}}
\left(
\frac{4}{f F_0 \pi \left(D/1\,{\rm m}\right)^2} +
\frac{9\times10^{-6}}{\left(D/1\,{\rm m}\right)^{4/3}} +
\frac{\pi D^2 f F_{bg}/4 + n_{dark} + \sigma_{rn}^2/\left( \Delta t / 1\,{\rm second}\right)}
{f^2 F_0^2 \pi^2 \left(D/1\,{\rm m}\right)^4/16}
\right)^{\frac{1}{2}}\nonumber\\
& \approx & 0.18\,{\rm seconds}\times \sqrt{\frac{k \left(\tau/1\,{\rm hr}\right)}{fgh^2}}
\left(\frac{10^{\left(V-12\right)/2.5}}{\left(D/1\,{\rm m}\right)^2} + \frac{f}{\left(D/1\,{\rm m}\right)^{4/3}}\right)^{\frac{1}{2}}.\end{aligned}$$
The final term in the first line (associated with dark current, read noise, and background) is generally smallest and will be ignored. In most cases, the second term—associated with scintillation—is dominant (though zero in the case of space-based observatories). The exponent of $(V-12)/2.5$ shows that for meter-sized telescopes, photon noise is only dominant for stars fainter than twelfth magnitude.
Systematic and astrophysical noise sources may have effects that limit the actual precisions achieved. Mass transfer between stars can cause drifts in orbital periods. Variations of this type are non-periodic, distinguishing them from companion signals. [@Applegate] has shown that gravitational coupling to the shapes of magnetically active stars can cause periodic modulations over decade timescales. This mechanism requires the star to be inherently variable; false positives can be removed using the overall calibrations of photometric data. It is possible that star spots will have large effects on timing residuals that are particularly difficult to calibrate [@Starspots]. Due to orbit-rotation tidal locking, the effect of a starspot on the light curve can be detected from the light curves of several orbits, and starspot fitting potentially can remove the timing biases introduced.
The timescale for eclipses of such long period binaries is of the order of 12 hours. Comparing eqs. \[STimingPert\] and \[timing\_prec\_2\], one finds that S-type planets with periods of a few days can be detected around either star in eclipsing binaries with month-long periods if they mass more than $$M_p/M_\earth \gtrsim 3
\frac{\left(M_b/\Msun\right)}
{\left(P_b/{\rm month}\right)^{\frac{1}{3}}\left(P_p/{\rm day}\right)^{\frac{2}{3}}}
\sqrt{\frac{k \left(\tau/12\,{\rm hr}\right)}
{fgh^2}}
\left(\frac{10^{\left(V-12\right)/2.5}}{\left(D/1\,{\rm m}\right)^2} + \frac{f}{\left(D/1\,{\rm m}\right)^{4/3}}\right)^{\frac{1}{2}}.$$
Eq. \[timing\_prec\_2\] indicates that a meter-class ground-based telescope can time a giant planet transit ($h \approx 0.02$, $k \approx 0.18$) to approximately 9.4 seconds in the regime where the photometric precision is dominated by scintillation noise, assuming a Jupiter sized planet orbiting a star of solar size and mass with period of a month (implying 6-hour duration eclipses). This precision is sufficient to find Earth mass moons. For bright stars, space-based observatories offer even better precisions. Unfortunately, no transiting exoplanets with periods this long have yet been discovered.
Space-based photometric missions such as Kepler have as their primary goal the detection of Earth-like planets via transits of the planet across the star. However, such photometric events can be explained by other astrophysical phenomena, such as a transiting Jupiter blended with a background star, so these results may be unreliable. However, Earth-like moons of transiting Jupiters might be identified through timing, and it is possible to confirm the nature of such a system. In such a scenario, a transiting Jupiter can be positively confirmed by ground-based radial velocity observations. Once this has been established, variations in the transit times would be used to detect Earth-sized moons. Because these photometric missions have limited lifetimes ($\approx 3$ years), detections of moons are only possible for short period (few months or less) Jupiters, for which many transit events can be observed (unless a follow-up ground-based campaign is pursued with large telescopes). If the planet/moon are to be in the habitable zone, one must look for such systems around late-type (cool) stars. It is possible that such systems have the greatest likelihood of being habitable; tidal-locking of the Earth-sized moon to the Jupiter-like planet would ensure that the moon has day/night cycles and stabilize its rotational axis similar to the way in which the Earth’s is stabilized by its own moon. Both of these conditions have been argued as favorable for life [see, for example, @earthAxisStability].
### Observational Precisions
Figure \[fig:narrowS\] shows the companion masses one can detect for each method, assuming 20 $\microas$ ground-based astrometry, $20 {\rm m\, s^{-1}}$ RV, and 1 m ground-based photometric telescope for eclipse timing.
![\[fig:narrowS\] Sensitivity to S-Type planets in narrow binaries, comparing astrometric, radial velocity, and eclipse timing techniques. All calculations assume solar mass stars. The PHASES sensitivity assumes 20 $\microas$ precision and a distance to the system of 20 pc. Eclipse timing assumes a 1 m photometric telescope observing a $V=10$ magnitude system with binary orbital period 2 months (longer period systems are even less likely to show eclipses); the eclipse timing sensitivity curve only extends to the region where planets are likely to have stable orbits. ](narrowS_Type.eps){height="3.5in"}
P-Type (Circumbinary) Planets
=============================
All the confirmed planets found in binary systems thus far are in S-type orbits. Discovery of circumbinary planets would consitute a new class of solar system, and would inspire new considerations to the interplay between system dynamics and planet formation.
Radial Velocities {#radial-velocities}
-----------------
A circumbinary planet will exhibit two indirect effects on the velocities of the stellar components of the system. First, the apparent system velocity will vary in a periodic manner due to the motion of the binary about the system barycenter. Second, the finite speed of light will cause apparent changes in the phase of the binary orbit. These effects may be detectable using modern observational techniques.
The first effect is that the binary will exhibit periodic changes in the apparent system velocity; this is the same effect as seen in a single star. However, it may be harder to detect for three reasons: (1) the binary system is usually more massive than a single star of the same magnitude, (2) extremely short-period planet orbits (to which system velocity measurements are most sensitive) are unstable around binaries, and (3) the presence of two sets of spectral lines may complicate the measurement, as in section \[SRV\]. Equation \[vel\_reflex\] shows that a Jupiter massed planet with the shortest period stable orbit around a 10-day period binary causes a $2 \Msun$ binary to move about its barycenter by $\approx 40\,{\rm m\,s^{-1}}$, with the amplitude decreasing as the square root of planet orbit semimajor axis. Radial velocity observations with the $20\,{\rm m\,s^{-1}}$ precision demonstrated with Konacki’s method can detect Jupiter-like planets in orbits of size $\approx 4$ or less, down to the critically stable orbit.
The second observable effect is the additional light travel time as the binary system undergoes reflex motion caused by the planet. The magnitude of this effect is given by $$\label{time_reflex}
\Delta t = 2 \frac{ a_p M_p \sin i_p}{ c M_b }\nonumber\\
= 0.95\,{\rm seconds} \times \frac{ \left(a_p/1\,\rm{AU}\right)
\left(M_p/M_\Jupiter\right) \sin i_p}{ M_b/\Msun }.$$
Following a similar derivation as that for finding the expected precision of eclipse timing, one finds the precision with which one can estimate the orbital phase of a binary based on radial velocity measurements is $$\sigma_{\phi} = \frac{\sigma_{rv}}{\sqrt{\sum_i
(\frac{\partial v_i}{\partial \phi})^2}},$$ where $\sigma_{rv}$ is the radial velocity measurement precision and $\frac{\partial v_i}{\partial \phi}$ is the derivative of the model radial velocity curve with respect to orbital phase, evaluated at times $t_i$. The timing precision corresponding to the phase precision derived is given by $\frac{\sigma_{\phi}}{2\pi} = \frac{\sigma_t}{P_b}$.
Approximating the binary orbit as circular, $v\left( t \right) \approx K\cos\left(\frac{2\pi t}{P_b}+\phi\right)$. If $N$ measurements (each with two measured velocities, one for each star) are approximately evenly distributed in phase, $$\begin{aligned}
\sigma_{\phi} & = & \frac{\sqrt{2}\sigma_{rv}}{\sqrt{\left(2N-12\right)} K}\\
\sigma_t & = & \frac{P_b\sigma_{rv}}{\sqrt{2\left(2N-12\right)}\pi K}, \end{aligned}$$ where 12 is the number of degrees of freedom for the model.
If the lines from both stars are observed, the effective $K$ is $K1+K2$ and the resulting ($1\sigma$) minimum detectable mass is thus
$$M_p = 41.4 M_\Jupiter \times \frac{\left( \sigma_{rv}/{\rm 20\,m\,s^{-1}}\right) \left(P_b/{\rm 10\,days}\right)^{4/3} \left(M_b/\Msun\right)^{2/3}}
{\sqrt{2N-12}\sin{i_b}\sin{i_p}\left( a_p/{\rm 1\,AU}\right)} ,$$
where $i_b$ and $i_p$ are the inclinations of the binary and planet orbits, respectively. Twenty-five $20\,{\rm m\,s^{-1}}$ radial velocity measurements of the “prototypical” system could detect moderate-mass brown dwarfs ($\approx 30 M_\Jupiter$) at critical orbit. Objects at the planet/brown dwarf threshold of 13 $M_\Jupiter$ are only detectable in orbits larger than 0.82 AU around a ten-day binary of sunlike stars. Alternatively, if only one set of lines are observed, the resulting expression is
$$M_p = 41.4 M_\Jupiter \times \left(1+\frac{M_1}{M_2}\right)\frac{\left( \sigma_{rv}/{\rm 20\,m\,s^{-1}}\right) \left(P_b/{\rm 10\,days}\right)^{4/3} \left(M_b/\Msun\right)^{2/3}}
{\sqrt{N-11}\sin{i_b}\sin{i_p}\left( a_p/{\rm 1\,AU}\right)},$$
where $M_1$ is the mass of the star whose lines are observed, and $M_2$ is that of the faint star.
High precision radial velocity observations are only possible on slowly rotating ($v \sin i < 10\,{\rm m\,s^{-1}}$) stars; measurements of more rapidly rotating stars are limited by line broadening to levels worse than the nominal $20\,{\rm m\,s^{-1}}$ that has been referenced by this work. This effect is particularly important for finding planets around short-period binaries, in which the stars’ rotation rates are often tidally locked to the binary orbital period; these rotation rates limit the observed precisions for systems with periods approximately five days or less.
![ \[fig:rvPhase\] Sensitivity of radial velocity measurements to circumbinary planets. The two vertical lines at the left represent the approximate critical orbits around 5-day (to the left) and 10-day period binaries; shorter period companions have unstable orbits. Stars whose rotation rates are tidally locked to orbital periods less than about 5 days show sufficient rotational line broadening to prevent $20\,{\rm m\,s^{-1}}$ radial velocity precisions. The calculations assume the binary consists of two stars each massing 1 $\Msun$. ](rv_circum_phase_space.eps){height="7cm"}
Eclipse Timing
--------------
It has long been recognized that periodic shifts in the observed times of photometric minima of eclipsing binaries can indicate the presence of an additional component to the system (see, for example, [@woltjer1922; @irwin1952; @Frieboes-Conde; @Doyle98]). The amplitude of the effect is given by eq. \[time\_reflex\]. As with RV measurements, there is a mass/inclination ambiguity; the following derivation assumes no correlation between binary and planet inclinations.
Dividing the precision of an individual measurement by $N_{obs} - 6$ (where $N_{obs}$ is the number of eclipses observed and there are 6 parameters to a timing perturbation fit, two periods and the eccentricity, angle of periastron, epoch of periastron, and mass ratio of the wide pair), converting $F_0$ to $V$ magnitude, and combining eq. \[timing\_prec\_2\] with that for the timing effect of reflex motion (eq. \[time\_reflex\]) gives a minimum detectable companion mass of $$M_p = 0.19 M_\Jupiter \times \sqrt{\frac{k \left(\tau/1\,{\rm hr}\right)}{fgh^2 \left(N_{obs}-6\right)}}
\frac{M_b/\Msun}{\left( a_p/1\,{\rm AU} \right) \sin i_p}
\left(\frac{10^{\left(V-12\right)/2.5}}{\left(D/1\,{\rm m}\right)^2} + \frac{f}{\left(D/1\,{\rm m}\right)^{4/3}}\right)^{\frac{1}{2}}.$$
![ \[fig:eclipsePhase\] Sensitivity of eclipse timing measurements to circumbinary planets. The vertical line represents the approximate critical orbit around a 10-day period binary. The calculations assume the binary consists of two stars each massing 1 $\Msun$, 6 hour eclipses, $N_{obs} = 25$ observations (150 total hours of data), $V=10$ magnitudes, and 1-$\sigma$ detections. From top to bottom, lines show sensitivity for $D=0.1$ m on the ground, $D=0.5$ m on the ground, $D=1.0$ m in space (i.e. Kepler), $D=10$ m on the ground, and $D=2.5$ m in space (HST, SOFIA). ](eclipse_circum_phase_space.eps){height="7cm"}
One might also inquire about the sensitivity of this technique to outer planets in systems comprised of a single star and a transiting “hot” Jupiter. In this case, $h \approx 2 R_p^2/R_{star}^2 \approx 0.02$ and $k \approx 2 R_p/(R_{star}+R_p) \approx 0.18$, the “binary” is half as massive, and the eclipse duration is half as long. The companion sensitivity drops by a factor of 8, and the technique is (barely) in the range of detecting additional companions of planet mass. However, for the typically $V=10$ magnitude transiting planet systems being discovered, 3 ${\rm m\,s^{-1}}$ radial velocity observations are more sensitive than half-meter telescope transit timing for companions with periods up to 60 years; even for observatories such as HST and SOFIA (for which scintillation noise is small or zero), this transition occurs at 15 year period companions.
It should be noted that the above description does not account for the possibility of resonant orbits, for which timing perturbations can be greatly enhanced by many-body dynamics—we have assumed independent Keplerian orbits for the subsystems. Resonant effects on timing perturbations have been considered by [@holman2005; @Agol2005].
Observational Precisions
------------------------
Figure \[fig:PType\] compares the sensitivity of RV and eclipse timing to circumbinary planets.
![\[fig:PType\] Sensitivity to circumbinary planets, comparing radial velocity and eclipse timing techniques. All calculations assume solar mass stars. RV assumes 20 ${\rm m\, s^{-1}}$ precision; both the system velocity and apparent period variation observables are included for the sensitivity curve, the latter assumes a 5 day binary orbital period. Eclipse timing assumes either a 1 m ground-based photometric telescope or 2.5 m space-based telescope (such as HST or SOFIA), observing a $V=10$ magnitude system, and 25 observations. ](P_Type.eps){height="3.5in"}
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'E. Congiu [^1]'
- 'M. Berton[^2]'
- 'M. Giroletti'
- 'R. Antonucci'
- 'A. Caccianiga'
- 'P. Kharb'
- 'M. L. Lister'
- |
\
L. Foschini
- 'S. Ciroi'
- 'V. Cracco'
- 'M. Frezzato'
- 'E. Järvelä'
- 'G. La Mura'
- 'J. L. Richards'
- 'P. Rafanelli'
bibliography:
- './biblio.bib'
title: |
Kiloparsec-scale emission in the\
narrow-line Seyfert 1 galaxy Mrk 783
---
Introduction
============
Narrow-line Seyfert 1 galaxies (NLS1s) are a puzzling class of active galactic nuclei (AGN), which were first classified by @Osterbrock85 according to their full width at half maximum (FWHM) of ${\rm H\beta }< 2000\,{\rm km\,s^{-1}}$. However, despite the narrowness of ${\rm H\beta }$, their ratio \[\]$\lambda5007/{\rm H\beta }< 3$ and the presence of strong multiplets in the optical and UV spectrum indicate that these objects are type 1 AGN.
Radio-quiet[^3] NLS1s (RQNLS1s) constitute $93\%$ of the total population up to redshift $0.8$ [@Komossa06] and $96.5\%$ at $\rm z<0.35$ [@Cracco16]. Radio-loud NLS1s (RLNLS1s) are relatively uncommon. They can be divided into two different classes according to their radio spectrum in the cm range. Flat-spectrum RLNLS1s (F-NLS1s) probably have a relativistic jet pointed toward Earth and can produce $\gamma$-rays [@Abdo09a; @Abdo09b], while steep-spectrum RLNLS1s (S-NLS1s) often show an extended radio morphology and are likely misaligned F-NLS1s.
One of the most interesting possibilities concerning the nature of NLS1s is that they are young and evolving objects [@Mathur00]. In particular, this appears to be true for RLNLS1s: F-NLS1s might be young flat-spectrum radio quasars (FSRQs) with a small black hole mass and S-NLS1s young radio galaxies [@Foschini15; @Berton16c]. However, a preference for low inclination might also play a role [e.g., @Shen14; @Peterson11]. Thus NLS1s are a somewhat heterogeneous group.
S-NLS1s have often been associated with compact steep-spectrum objects [CSS; @Oshlack01; @Komossa06; @Gallo06a; @Yuan08; @Caccianiga14; @Gu15; @Schulz15; @Berton16c; @Caccianiga17], which are usually believed to be young and growing radio galaxies [@Fanti95]. Only a handful of S-NLS1s were investigated in radio [@Whalen06; @Anton08; @Doi12; @Richards15; @Doi15; @Gu15; @Caccianiga17]. RLNLS1s indeed have a lower observed jet power than FSRQs [@Foschini15] because of their low black hole mass [@Heinz03; @Foschini14]. Therefore, while F-NLS1s are relatively easy to find because their luminosity is enhanced by relativistic beaming, S-NLS1s are not as easily detectable.
To study the radio properties of NLS1s, we carried out a survey with the Karl G. Jansky Very Large Array (JVLA) at $5$ GHz in A configuration. Our sample consists of 60 sources drawn from the papers by @Foschini15 and @Berton15a, and it contains radio-quiet (but not radio-silent) NLS1s, F-NLS1s, and S-NLS1s. In this paper we report the detection of extended emission in one S-NLS1s, Mrk 783. This source is one of the few NLS1 showing such an extended emission at $\rm z<0.1$. In Sect.\[sec:mrk783\] we describe the source according to results published in the literature, in Sect.\[sec:datared\] we describe the data reduction we performed, in Sect.\[sec:results\] we present our results, in Sect.\[sec:discussion\] we discuss them and, finally, in Sect.\[sec:summary\] we provide a brief summary. Throughout this work, we adopt a standard $\rm \Lambda CDM$ cosmology, with a Hubble constant $H_0 = 70\,{\rm km\,s^{-1}}Mpc^{-1}$, and $\Omega_\Lambda = 0.73$ [@Komatsu11]. Spectral indexes are specified with flux density $S_{\nu} \propto \nu^{-\alpha}$ at frequency $\nu$.
Mrk 783 {#sec:mrk783}
=======
Mrk783 (R.A. = $13$h $02$m $58.8$s Dec=$+16$d $24$m $27$s) is a NLS1 galaxy first classified by @Osterbrock85 at $z = 0.0672$ [@Hewitt91] with a bolometric luminosity of the AGN $L_{bol} = 3.3\times10^{44}\,{\rm erg\,s^{-1}}$ [@Berton15a]. Its host galaxy was classified as a lenticular galaxy [@Petrosian07], but the SDSS image clearly shows the presence of a tidal tail, or a spiral arm, extended in the east direction.
The mass of the central black hole inferred from the ${\rm H\beta }$ broad component line width is about $4.3\times10^7$ M$_{\odot}$ [@Berton15a]. ${\rm H\beta }$ shows a prominent red wing in the broad component, indicating a receding outflow with a velocity of $\sim500\,{\rm km\,s^{-1}}$. This broad component is clearly visible in all the permitted lines of the optical spectrum. Conversely, narrow lines, and particularly \[\]$\lambda5007$, do not show any outflowing component and are well reproduced by a single Gaussian profile [@Berton16b].
Mrk783 is a strong X-ray emitter that has been detected by ROSAT [@Schwope00], INTEGRAL [@Krivonos07], and Swift/XRT [@Panessa11]. @Panessa11 reported a luminosity of $9.33\times10^{43}\,{\rm erg\,s^{-1}}$ between $20$ and $100$ keV and a photon index of $1.7\pm0.2$ between $0.3$ and $100$ keV. This is consistent with nonsaturated comptonization, which occurs in the accretion disk corona and not in relativistic jets.
In the last 30 years, the galaxy was observed several times in several radio bands, for example, the WSRT at $1.4$ GHz [@Meurs81], VLA at $5$ GHz [@Ulvestad84; @Ulvestad95], and Green Bank telescope at $1.4$ GHz [@Bicay95]. However, no extended emission was found. Recently, @Doi13 observed the galaxy nucleus with the Very Long Baseline Array (VLBA) looking for extended emission near the core of the AGN. The image only shows a compact core, but the flux density recovered by the authors at $1.7$ GHz is only $4\%$ of the NRAO VLA Sky Survey (NVSS) flux density at $1.4\,$GHz [$S_{\nu}=33.2\,$mJy; @Condon98]. This discrepancy means that the vast majority of the flux emitted by the galaxy is distributed in structures with relatively low brightness temperature, which could not be seen by the instrument. Another hint of the extended emission can be found in the FIRST image of the galaxy [@Becker95]. The source is elongated along position angle (PA) $\ang{131}$ and shows a peak and a total flux density of $18.5$ mJy and $28.72$ mJy, respectively. At low frequencies, the TIFR Giant Metrewave Radio Telescope Sky Survey [TGSS; @Intema17] at $147$ MHz reports a flux density of $89.2\pm10.9$ mJy.
Mrk783 was classified as moderately radio-loud [@Berton15a] or radio-quietm [@Doi13]. The R parameter is indeed close to $10$. Therefore, a different estimate of the optical magnitude or optical variability in the source could have provided two different classifications. This is not uncommon, as has been clearly shown by @Ho01 and @Kharb14. However the radio emission does not appear to be dominant over the optical magnitude as in classical radio galaxies.
Data reduction {#sec:datared}
==============
The galaxy was observed on 2015 September 6 with the JVLA at $5$ GHz in A configuration with a bandwidth of $2$ GHz, for a total exposure time of $10$ minutes. We reduced and analyzed the data using the Common Astronomy Software Applications (CASA) version 4.5, the standard Expanded VLA (EVLA) data reduction pipeline, and the Astronomical Image Processing System (AIPS). The main calibrator was 3C 286. We split off the measurement set of the object from the main dataset, averaging over the $64$ channels of each spectral window. After that, the object was cleaned using all the spectral windows and a natural weighting to create a first image. To improve the quality of the final map, we proceeded with iterative cycles of phase only self-calibration of the visibilities. The results of the CASA self-calibration were not satisfactory because this self-calibration caused a general increase of the noise level of the maps. Therefore we tried to redo the self-calibration process using AIPS, which, indeed, significantly improved the quality of the final images. Once the presence of extended emission was confirmed, we returned to CASA and we proceeded with a second cleaning of the data to obtain the final images. In addition to the high resolution image we produced another image, using a taper of $50\,$k$\lambda$, to recover the highest possible fraction of the extended emission flux density. Fig.\[fig:map\] shows the maps of the object before and after the application of the taper. The lower panels show the radio contours superimposed on a Sloan Digital Sky Survey (SDSS) optical image.
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Results {#sec:results}
=======
The images in Fig.\[fig:map\] clearly show a compact core and extended emission at PA$=\ang{131}$, which is in agreement with the FIRST data. In the high resolution image the extended emission is observed only in the southeast region of the galaxy up to a projected distance of $\sim 8$ kpc. In the tapered image, instead, we observe it on both sides of the nucleus up to a projected distance of $14$ kpc in the southeast and $12.5$ kpc in the northwest direction.
We fitted the central component in the high resolution image with a 2D Gaussian, using the fitting algorithm in the CASA viewer. We measured a peak flux density of $3.32\pm0.04$ mJy beam$^{-1}$, an integrated flux density $S_{\nu,c}=4.03\pm0.08$ mJy, and the size of deconvolved core axes $216\pm21$ ($\sim 280$) and $173\pm24$ ($\sim225$) mas (pc) for the major and minor axis, respectively. The beam size is $0.45"\times0.40"$. With these values we calculated the core luminosity $L_{c}=2.3\times 10^{39}\,{\rm erg\,s^{-1}}$ and we used the following equation to estimate the brightness temperature [@Doi13]: $$\label{eq:Tb}
T_B=1.8\times 10^9 (1+z)\frac{S_{\nu,c}}{\nu^2\phi_{maj}\phi_{min}}\sim7600 {\rm\, K},$$ where $z$ is the redshift, $\phi_{maj}$ is the core major axis, and $\phi_{min}$ is the minor axis. This $T_B$ is only a lower limit to the peak value because our measurement includes some extended emission. The real core emission as detected with VLBI is smaller and @Doi13 quote a peak value of $T_B>7.7\times 10^7$ K at $1.7$ GHz. From the tapered image we measured a total flux density $S_{\nu,t} = 18.85 \pm 0.03$ mJy ($L_{tot}=1.03\times10^{40}\,{\rm erg\,s^{-1}}$) and we obtained the flux density of the extended emission ($S_{\nu,e} = 14.82\pm0.1$ mJy, $L_{ext}=8.5\times10^{39}\,{\rm erg\,s^{-1}}$), after subtracting the JVLA core flux density. To evaluate the relative importance of the core emission with respect to the total emission we measured the core dominance parameter (CD) as follows: $$\label{eq:cd}
{\rm CD}=S_{\nu,c}/(S_{\nu,t}-S_{\nu,c})\sim0.27.$$ The core flux is likely an upper limit, therefore the CD parameter is also an upper limit, meaning that the galaxy is not core dominated.
To better characterize the emission we measured the spectral indexes of the two components. We measured these components by dividing the spectral windows into two bins: the first centered at $4.7$ GHz and the second at $5.7$ GHz. Images were made at the two frequencies, adjusting the weighting of the visibilities using the ROBUST parameter in the AIPS task IMAGR so as to obtain similar beams; they were finally convolved with identical circular beams of size $0.45$ arcsec. The spectral index images were made using task COMB after blanking total intensity values below $1.5$ sigma. We obtained the following spectral indexes for the core component and the southern lobe, respectively: $\alpha_{c} \sim 0.67\pm0.13$, $\alpha_{ext} \sim 2.02\pm0.74$. The $\alpha_{ext}$ was estimated for the southeast lobe, avoiding the noisiest regions toward the north and east of the core.
Band Flux density (mJy) Notes Reference
------------------- -------------------- ---------- -----------
$150$ MHz $89.2$ total 1
$1.4$ GHz $33.2$ total 2
$1.4$ GHz $28.7$ total 3
$5$ GHz $4.0$ core 4
$5$ GHz $14.8$ extended 4
$5$ GHz $18.9$ total 4
$60{\,\rm \mu m}$ $310.0$ total 5
$22{\,\rm \mu m}$ $65.9$ total 6
: Fluxes of the galaxy in several radio and infrared bands.[]{data-label="tab:flux"}
Discussion {#sec:discussion}
==========
Our data revealed emission in a compact core as well as an extended component, which can be observed on both sides of the nucleus in the tapered image. The luminosity at $5$ GHz is $L_{\nu}=2.1\times10^{30}\,{\rm {\rm erg\,s^{-1}}Hz^{-1}}$, which is below the nominal $\sim 7\times10^{31}\,{\rm {\rm erg\,s^{-1}}Hz^{-1}}$, the Fanaroff-Riley (FR) cutoff luminosity. The CD ($\leq 0.27$) indicates that the emission is not core dominated and the steep spectral index of the core is different from those of highly beamed radio quasars and BL Lacs ($\alpha \sim 0$), but it is more similar to that of CSS sources ($\alpha \sim 0.5$). The extended component does not show any sign of an highly collimated jet, but it has a double-lobe shape that resembles the radio emission of some Seyfert galaxies, such as NGC6764 [@Hota06] and Mrk6 [@Kharb06].
Extended radio surveys of normal Seyfert galaxies and LINERs [e.g., @Baum93; @Gallimore06; @Singh15] have discovered that kiloparsec-scale radio emission is not uncommon in these objects. The emission usually seems to have an AGN origin [@Gallimore06; @Singh15], but in some cases star formation (SF) can contribute significantly [e.g., @Baum93]. This could be true, indeed, in the case of NLS1 galaxies where the presence of SF (both in RQ and RL objects) has often been reported in the literature [e.g., @Sani11; @Caccianiga15]. The very red mid-IR colors ($W3-W4=2.6$) measured by the Wide-field Infrared Survey Explorer (WISE) of Mrk 783 and its strong emission at $60\,{\rm\mu m}$ measured by the InfraRed Astronomical Satellite (IRAS) ($310$ mJy) seem to support this hypothesis. In order to evaluate the impact of this possible SF on the observed radio emission, we computed the parameter $q22$, defined as$$\label{eq:q22}
q22 = \log (S_{22\,{\rm \mu m}}/S_{1.4\,{\rm GHz}}),$$ where $S_{22\,{\rm \mu m}}$ and $S_{1.4\,{\rm GHz}}$ are the WISE $22\,{\rm \mu m}$ flux density and the NVSS flux density, respectively. The resulting value ($q22\sim0.3$) is significantly lower than that usually observed in SF galaxies [$q22>1$; @Caccianiga15]. This means that, even if all the observed mid-IR emissions were produced by the SF, the expected radio flux at $1.4$ GHz would be much lower than the observed flux ($\sim20$ per cent of the observed flux) . Considering that part of the IR emission is likely due to the AGN, we conclude that, even if present, the SF alone cannot explain the majority of the observed radio emission.
We compared our spectral indexes with results in the literature. Using TGSS, NVSS, and our flux densities (Tab.\[tab:flux\]) we recovered a spectral index $\alpha\sim 0.44$. This is significantly flatter with respect to our measurements. A possible explanation for this discrepancy might be the presence of a break or a cutoff at $5$ GHz. Furthermore, biases of the surveys toward extended emissions, for example, due to short integration time, might also cause an underestimate of the spectral index.
A possible explanation for the very steep spectral indexes observed in our VLA images might be that we are observing relic emission. This could be supported by the absence of collimated structure in our images and in the high resolution images from @Doi13. Relic emission has already been observed in some Seyfert galaxies [e.g., NGC 4235, @Kharb16] and they also show very steep spectra [e.g., @Jamrozy04; @Kharb16].
@Czerny09 found that in young radio sources with high accretion rates, radiation pressure instabilities of the accretion disk can result in intermittent activity of the radio jet. In their model, the activity phases last for $10^3-10^4$ yr and they are separated by periods of $10^4-10^6$ yr in which the radio jet is switched off. During the cycle, the previous activity period should manifest in the form of a relic extended emission that continues its expansion until the emitting cocoon cools down and recollapses. The maximum extension and cooling time of the cocoon depend on many factors, such as jet power and duration of the activity phase. In the case of NLS1, both quantities are considered relatively small, resulting in a very low detection rate of such emission in this class of AGN [@Czerny09; @Foschini15]. Such extended emission might appear only if the central black hole mass is in the higher part of the mass distribution [which spans between $10^6$ to $10^8$ M$_{\odot}$, e.g., @Cracco16], more precisely if M$_{BH}>10^7$ M$_{\odot}$ the BH should produce the necessary jet power to make it escape from the central regions of the AGN [@Doi12]. The mass of Mrk 783 black hole is $4.3\times10^7$ M$_{\odot}$ [@Berton15a], therefore it belongs to the objects that in principle could produce the extended emission. Some of the properties of the emission, such as its size and the absence of a collimated jet, might suggest that we are observing the galaxy in one of its quiescent phases not long after the switching off of the radio jet.
It is worth noting that the large scale structure of the radio emission in the tapered image has an S-like shape, which is typical of precessing radio jets [@Ekers78; @Parma85]. Precessing jets could arise in binary black hole systems [e.g., @Roos93; @Romero00; @Rubinur17] or result from accretion disk instabilities [@Pringle96; @Livio97]. In the latter case, it might be consistent with scenario of episodic activity described previously. The resemblance to the case of Mrk 6 is strong [@Kharb06].
Another possible explanation of why we observe such steep spectral indexes might be that there is a strong interaction between a jet and interstellar medium of the galaxy. This could cause shocks and magnetic field amplification, leading to greater radiative losses and steep spectra.
This source might represent one of the few examples of the elusive parent population of F-NLS1s, i.e., S-NLS1s, Mrk 783, which lies at the edge of the RQ/RL division. The same is true for another S-NLS1 with extended emission, Mrk 1239 [@Doi15]. This might indicate that more sources of this kind could be found among RQ objects. Very few RQNLS1s show detectable jets or diffuse radio emission [@Berton16b]. This is partly because they are by definition faint in the radio regime, but some objects that fit the RQ definition show nonthermal core emission when observed with high sensitivity [@Giroletti09].
Another hypothesis regarding the parent population of F-NLS1s is based on a different assumption about NLS1s nature. Some authors believe that the narrowness of permitted lines is not due to the low black hole mass, but instead to a flat BLR observed pole-on [e.g., @Decarli08]. Mrk 783, however, shows extended emission on both sides of the core, hence it likely has a non-negligible inclination. In this case then the low black hole mass estimate should not be significantly affected by any BLR flattening, although this might happen in other objects [@Shen14].
To better understand the nature of this object, simultaneous radio observations at different frequencies are needed to study the spectral energy distribution (SED) of the emission. In particular, it is fundamental to investigate the reason of such a difference between our spectral indexes and what we found from data in the literature. Also images with an intermediate spatial resolution between our data and the data from @Doi13 (e.g., from e-MERLIN) could be useful to investigate the presence of an intermediate scale radio jet.
Summary {#sec:summary}
=======
In this paper we present the first result of a survey of NLS1s carried out with the JVLA at $5$ GHz in A configuration. In particular we report the detection of extended emission in the S-NLS1 Mrk783. We found a compact core and extended emission that, in the tapered image, is observed on both the southeast and northwest sides of the galaxy nucleus, up to a maximum projected distance of $14$ kpc. We excluded star formation as the dominating cause of the extended emission owing to the low value of the IR-to-radio flux ratio ($q22$). The latter, together with the morphology of the emission, indeed suggests an AGN origin. At the same time, in the high resolution image (Fig.\[fig:map\]) we could not find any sign of a large scale collimated jet and @Doi13 did not find any small scale jets. These facts, and the very steep spectral indexes that we measured, led us to hypothesize that the extended emission might be a relic and that the source might be in a quiescent period of its activity cycle, which is in agreement with the theoretical model by @Czerny09 and the young age scenario for NLS1s [@Mathur00]. Further observations are needed both in radio and in other bands to confirm our results and to investigate the nature of this object in more detail.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. We thank the staff of the GMRT, that made these observations possible. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. Funding for the Sloan Digital Sky Survey has been provided by the Alfred P. Sloan Foundation and the U.S. Department of Energy Office of Science. The SDSS web site is `http://www.sdss.org`. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
[^1]: enrico.congiu@phd.unipd.it
[^2]: marco.berton@unipd.it
[^3]: The radio loudness is defined by the parameter R, the ratio between the $5$ GHz flux and the optical B-band flux [@Kellermann89]. A source is considered to be radio-loud if $\rm R>10$ and radio-quiet if $\rm R <10$.
| {
"pile_set_name": "ArXiv"
} |
The recent formation of ultracold bosonic molecules from a Fermi gas of atoms[@regal] by a Feshbach resonance allows for an experimental check of theoretical calculations for physical quantities within the BCS-BEC crossover. In particular, in Ref. the relation between the composite-boson scattering length $a_B$ and the fermionic scattering length $a_F$ was calculated in the strong-coupling limit of the BCS-BEC crossover. The summation therein of all bosonic T-matrix diagrams has led to the result $a_B=0.75 a_F$. This result corrects the value $a_B= 2 a_F$ obtained within the Born approximation for the effective residual bosonic interaction[@Haussmann; @PS-96; @epjb]. The result $a_B=0.75 a_F$ could be tested experimentally in the near future, by measuring at the same time the molecule-molecule scattering length and the fermion-fermion scattering length while scanning the magnetic field through the Feshbach resonance.
In this manuscript, we provide a condensed version of the material published in Ref. , focusing specifically on the calculation of the bosonic scattering length. We hope that this short summary of our previous work could be useful to the scientific community at the present time.
Building blocks of the diagrammatic structure for composite bosons
==================================================================
In this section, we discuss the diagrammatic structure that describes generically the composite bosons in terms of the constituent fermions. Our theory rests on a judicious choice of the fermionic interaction, which (without loss of generality) greatly reduces the number and considerably simplifies the expressions of the Feynman diagrams to be taken into account.
Regularization of the fermionic interaction
-------------------------------------------
We begin by considering the following Hamiltonian for interacting fermions (we set Planck $\hbar$ and Boltzmann $k_{B}$ constants equal to unity throughout): $$\begin{aligned}
& & H = \sum_{\sigma} \int d{\bf r} \, \psi_{\sigma}^{\dagger}({\bf r}) \left(
- \frac{\nabla^2}{2m} - \mu \right) \psi_{\sigma}({\bf r}) \nonumber \\
& & + \frac{1}{2} \sum_{\sigma, \sigma'} \int d{\bf r} \, d{\bf r'}
\psi_{\sigma}^{\dagger}({\bf r}) \psi_{\sigma'}^{\dagger}({\bf r'})
V_{{\mathrm eff}}({\bf r}-{\bf r'})
\psi_{\sigma'}({\bf r'}) \psi_{\sigma}({\bf r}) .
\label{Eq:Hamiltonian}\end{aligned}$$ Here, $\psi_{\sigma}({\bf r}$) is the fermionic field operator with spin projection $\sigma = (\uparrow, \downarrow$), $m$ the fermionic mass, $\mu$ the fermionic chemical potential, and $V_{{\mathrm eff}}({\bf r}-{\bf r'})$ the [*effective potential*]{} that provides the *attraction* between fermions. For the application to atomic gases, the two spin states correspond to two different hyperfine states of the fermionic atoms.
To simplify the ensuing many-body diagrammatic structure (yet preserving the physical effects we are after), we adopt for $V_{{\mathrm eff}}$ the form of a “contact” potential [@footnote-contact] $$V_{{\mathrm eff}}({\bf r}-{\bf r'}) = v_{0} \,\, \delta ({\bf r}-{\bf r'})
\label{Eq:deltafunc}$$ where $v_{0}$ is a negative constant. With this choice, the interaction affects only fermions with opposite spins in the Hamiltonian (\[Eq:Hamiltonian\]) owing to Pauli principle. A suitable *regularization* of the potential (\[Eq:deltafunc\]) is, however, required to get accurate control of the many-body diagrammatic structure. In particular, the equation (in the center-of-mass frame) $$\frac{m}{4 \pi a_{F}} \, = \, \frac{1}{v_{0}} \, + \, \int \! \frac{d{\bf k}}
{(2 \pi)^{3}} \frac{m}{{\bf k}^{2}} \label{ferm-scatt-ampl}$$ for the *fermionic scattering length* $a_{F}$ associated with the potential (\[Eq:deltafunc\]) is ill-defined, since the integral over the three-dimensional wave vector ${\bf k}$ is ultraviolet divergent. The delta-function potential (\[Eq:deltafunc\]) is then regularized, by introducing an ultraviolet cutoff $k_{0}$ in the integral of Eq. (\[ferm-scatt-ampl\]) and letting $v_{0} \, \rightarrow \, 0$ as $k_{0} \, \rightarrow \, \infty$, in order to keep $a_{F}$ fixed at a *finite* value. The required relation between $v_{0}$ and $k_{0}$ is obtained directly from Eq. (\[ferm-scatt-ampl\]). One finds: $$v_{0} \, = \, - \, \frac{2 \pi^{2}}{m k_{0}} \, - \, \frac{\pi^{3}}{m a_{F} k_{0}^{2}}
\label{vo}$$ when $k_{0} |a_{F}| \gg 1$.
With the regularization (\[vo\]) for the potential, the classification of the many-body diagrams gets considerably simplified, since only specific sub-structures of these diagrams survive when the limit $k_{0} \, \rightarrow \, \infty$ is eventually taken. In particular, in order to obtain a finite result for a given Feynman diagram, the vanishing strength $v_0$ of the potential should be compensated by an ultraviolet divergence in some internal wave-vector integration. For the particle-particle ladder of Fig. \[fig:pplad\], the internal wave-vector integration associated with each rung diverges in the limit $k_0\to\infty$ and compensates the vanishing of $v_0$, yielding the finite result: $$\begin{aligned}
& &\Gamma_{0}(q) = - \left\{ \frac{m}{4 \pi a_{F}} + \right.
\int \! \frac{d{\bf k}}{(2 \pi)^{3}}\nonumber\\
& &\times\left. \left[\frac{\tanh(\beta \xi({\bf k})/2)
+\tanh(\beta \xi({\bf k-q})/2)}{2(\xi({\bf k})+\xi({\bf k-q})-i
\Omega_{\nu})}
- \frac{m}{{\bf k}^{2}} \right] \right\}^{-1}\; .
\label{p-p ladder}\end{aligned}$$ Here, $\xi({\bf k}) = {\bf k}^{2} /(2m) - \mu$ and $q=({\bf q},\Omega_{\nu})$ is a four-momentum, with wave vector ${\bf q}$ and Matsubara frequency $\Omega_{\nu}$ ($\nu$ integer). In an analogous way, one can show that in the particle-particle channel the contributions of the vertex corrections and of the two-particle effective interactions other than the rung vanish for our choice of the potential.
It is thus evident from these considerations that, with our choice of the fermionic interaction, in the strong-coupling limit the *skeleton structure* of the diagrammatic theory can be constructed only with the particle-particle ladder (\[p-p ladder\]) plus an infinite number of interaction vertices. A careful diagrammatic analysis considered in detail in Ref. then shows that: (a) Bare composite bosons correspond to fermionic particle-particle ladders; (b) The interaction among bare composite bosons is described by 4-point, 6-point vertices, and so on, which correspond to the product of $4,6,\ldots$, fermionic bare Green’s functions (with one internal four-momenta integration). The correspondence rules for the bosonic Green’s function and the 4-point vertex are shown in Fig. 2.
In particular, in the strong-coupling limit (whereby $\beta |\mu| \gg 1$),[@footnote-mu] the particle-particle ladder $\Gamma_0(q)$ has the following *polar structure*:[@Haussmann] $$\Gamma_0(q)\approx-\frac{4 \pi}{m^{2} a_{F}}\frac{ 1 +
\sqrt{1 + \left(-i\Omega_{\nu} +
\frac{{\bf q}^{2}}{4 m} - \mu_{B}\right)\epsilon_{0}^{-1}}}
{i\Omega_{\nu} - \left(\frac{{\bf q}^{2}}{4m} - \mu_{B}\right)}
\label{pp-sc}$$ where we have used the definition $\mu_{B} = 2\mu + \epsilon_{0}$ for the bosonic chemical potential ($\epsilon_0=1/(m a_F^2)$ being the bound-state energy of the fermionic two-body problem). Note that (apart from the residue being different from unity) the expression (\[pp-sc\]) resembles a “free” boson propagator with mass $2 m$. The (four-point) *effective two-boson interaction* reads instead $$\begin{aligned}
& &\tilde{u}_{2}(q_{1} \dots q_{4}) \, = \, \delta_{q_1+q_2,q_3+q_4}
\left(\frac{{\cal V}}{\beta}\right)^{2} \frac{2}{\beta {\cal V}}
\label{two-body-potential}\\
& &\times \sum_k
{\cal G}^0(-k){\cal G}^0(k+q_2){\cal G}^0(-k+q_1-q_4){\cal G}^0(k+q_4)
\nonumber
\;\; .\end{aligned}$$ Here ${\cal G}^{0}(k) = [i\omega_n -{\bf k}^2/(2m)-\mu]^{-1}$ is a bare fermionic Green’s function ($\omega_n$ being a fermionic Matsubara frequency). Note that the interaction (\[two-body-potential\]) depends on wave vectors as well as on Matsubara frequencies, revealing in this way the composite nature of the bosons. The factor of $2$ in the definition (\[two-body-potential\]) corresponds to the two different sequences of spin labels that can be attached to the four fermionic Green’s functions, as shown graphically in Fig. \[fig:cobos\](b).
=3.3in
=3.3in
In the strong-coupling limit $\beta |\mu| \gg 1$, a typical value of the two-boson effective interaction is obtained by setting $q_1 = \cdots = q_4 = 0$ in Eq. (\[two-body-potential\]). One gets: $$\tilde{u}_{2}(0) \, = \, 2 \, \left( \frac{{\cal V}}{\beta} \right)^{2}
\left( \frac{m^{2} a_{F}}{8 \pi} \right)^{2} \, u_{2}(0) \label{u2-tilde-limit}$$ where [@PS-96] $$u_{2}(0) \, = \, \frac{4 \pi (2a_{F})}{2m} \, . \label{u2-limit}$$ The factor $m^{2} a_{F}/(8 \pi)$ in Eq. (\[u2-tilde-limit\]) reflects the difference between the true bosonic propagator and the particle-particle ladder in the strong-coupling limit \[cf. also Eq. (\[pp-sc\])\]. Owing to this difference, $u_{2}(0)$ given by Eq. (\[u2-limit\]) (and not $\tilde{u}_{2}(0)$ given by Eq. (\[u2-tilde-limit\])) has to be identified with the boson-boson interaction at zero four-momenta. Note that $u_{2}(0)$ is *positive* in the strong-coupling limit, thus ensuring the *stability* of the bosonic system.
Recall further that the scattering length $a^{\mathrm{Born}}_{B}$ within the Born approximation, obtained for a pair of true bosons (each of mass $2m$) mutually interacting via a two-body potential with Fourier transform $u_{2}(0)$ at zero wave vector, is given by $a^{\mathrm{Born}}_{B} = 2m u_{2}(0)/(4 \pi)$. Equation (\[u2-limit\]) then yields the following relation between the bosonic and fermionic scattering lengths within the Born approximation: $$a^{\mathrm{Born}}_{B} \, = \, 2 \, a_{F} \, .
\label{a-Born}$$ The result (\[a-Born\]) was also obtained in Ref. within the fermionic self-consistent T-matrix approximation (which corresponds to the bosonic Hartree-Fock approximation in the strong-coupling limit). In that reference, the result (\[a-Born\]) was erroneously regarded to be the full value of the scattering length $a_{B}$ for a “dilute” system of composite bosons. We will, in fact, show in the next section that the result (\[a-Born\]) differs from $a_{B}$ when *all* bosonic T-matrix diagrams are properly taken into account.
Besides the four-point vertex (\[two-body-potential\]), the composite nature of the bosons produces (an infinite set of) additional vertices. One can show that all interaction vertices can be neglected in comparison with the four-point vertex in the strong-coupling limit.[@PS00] In this limit, one can thus construct all diagrams representing the two-particle Green’s function in the particle-particle channel in terms only of the bare ladder and of the four-point interaction vertex. This is precisely what one would expect on physical grounds, since the interactions involving more than two bodies become progressively less effective as the composite bosons overlap less when approaching the strong-coupling limit.
Numerical results for the scattering length of composite bosons
===============================================================
In three dimensions the *scattering length* $a$ characterizes the low-energy collisions for the scattering from an ordinary potential. For the mutual scattering of two particles (each of mass $M$), $a$ can be expressed by the relation $t(0) = 4\pi a/M$ in terms of the ordinary T-matrix $t(0)$ in the limit of vanishing wave vector. In particular, within the Born approximation $t(0)$ is replaced by the Fourier trasform $u(0)$ of the interparticle potential for vanishing wave vector.
For composite bosons, the T-matrix $\bar{t}_{B}(q_{1},q_{2},q_{3},q_{4})$ is defined by the following integral equation (cf. Fig. \[fig:comboslo\]): $$\begin{aligned}
& &\bar{t}_{B}(q_{1},q_{2},q_{3},q_{4}) =\bar{u}_{2}(q_{1},q_{2},q_{3},q_{4})
\nonumber\\
& &-\frac{1}{\beta {\cal V}} \, \sum_{q_{5}} \,
\bar{u}_{2}(q_{1},q_{2},q_{5},q_{1}+q_{2}-q_{5})
\Gamma_{0}(q_{5}) \, \Gamma_{0}(q_{1}+q_{2}-q_{5})\nonumber\\
& &\times \bar{t}_{B}(q_{1}+q_{2}-q_{5},q_{5},q_{3},q_{4})
\label{bosonic-t-matrix}\end{aligned}$$ where $\bar{u}_{2}$ is proportional to the effective two-boson interaction of Eq. (\[two-body-potential\]): $$\begin{aligned}
& &\bar{u}_{2}(q_{1} \dots q_{4}) \, = \, \frac{1}{\beta {\cal V}} \,
\sum_{k} \nonumber\\
& &\times
{\cal G}^0(-k){\cal G}^0(k+q_2){\cal G}^0(-k+q_1-q_4){\cal G}^0(k+q_4) \; .
\label{two-body-potential-bar} \end{aligned}$$
=3.3in
Similarly to the problem of point-like bosons, we *define* the scattering length $a_{B}$ for the composite bosons (each of mass $2m$) in the strong-coupling limit and for vanishing density, by setting $t_{B}(0) = 4 \pi a_{B}/(2m)$. Here $t_{B}(0) = (8 \pi/(m^{2} a_{F}))^{2} \, \bar{t}_{B}(0)$ and $\bar{t}_{B}(0) \equiv \bar{t}_{B}(0,0,0,0)$. The rescaling between $t_{B}(0)$ and $\bar{t}_{B}(0)$ is due to the difference between the true bosonic propagator and the particle-particle ladder in the strong-coupling limit. The same rescaling is consistently used when defining the boson-boson interaction \[cf. Eqs. (\[u2-tilde-limit\]) and (\[two-body-potential\])\].
To the lowest order in the effective interaction between the composite bosons, we can replace $t_{B}(0)$ by $u_{2}(0)$ and write $u_{2}(0)= 4
\pi a^{\mathrm{Born}}_{B}/(2m)$, within the Born approximation. Comparison with Eq. (\[u2-limit\]) yields then the value $a^{\mathrm{Born}}_{B} = 2 \, a_{F}$, as anticipated by Eq. (\[a-Born\]).
In order to obtain the exact value of $\bar{t}_{B}(0)$ (and hence of the scattering length $a_B$), it is convenient to determine first the auxiliary quantity $\bar{t}_{B}(q,-q,0,0)$ by solving the following *closed-form* equation $$\begin{aligned}
& &\bar{t}_{B}(q,-q,0,0) = \bar{u}_{2}(q,-q,0,0) \label{bosonic-t-matrix-0}\\
& & - \frac{1}{\beta {\cal V}} \, \sum_{q'} \,
\bar{u}_{2}(q,-q,q',-q')\Gamma_{0}(q') \, \Gamma_{0}(-q') \,
\bar{t}_{B}(q',-q',0,0) \,\, , \nonumber \end{aligned}$$ obtained from Eq. (\[bosonic-t-matrix\]) by setting $q_{1}=-q_{2}=q$ and $q_{3}=q_{4}=0$. This integral equation can be solved by standard numerical methods, e.g., by reverting it to the solution of a system of coupled linear equations. Note that, since we are here interested in the calculation of a two-boson quantity, the zero-density limit has to be taken in Eq. (\[bosonic-t-matrix-0\]). This implies that $\mu_B=0$ (or equivalently $2\mu=\epsilon_0$) and $T=0$.[@footnote-mu] The discrete bosonic frequency $\Omega_{\nu}$ becomes a continuous variable accordingly.
Numerical calculation of Eq. (\[bosonic-t-matrix-0\]) requires us to introduce a finite-size mesh for the variables ($|{\bf q}|$, $\Omega$) and ($|{\bf q}'|$, $\Omega'$), with the angular integral over $\hat{q'}$ affecting only the function $\bar{u}_{2}(q,-q,q',-q')$. Equation (\[bosonic-t-matrix-0\]) is thus reduced to a set of coupled equations for the unknowns $\bar{t}_{B}(|{\bf q}|,\Omega;|{\bf q}|,-\Omega;0;0)$, which were solved by the Newton-Ralphson algorithm with a linear interpolation for the integrals over $|{\bf q}'|$ and $\Omega'$. In this way, we are led to the result $$a_{B} = 0.75 \, a_F$$ within an estimated $5\%$ numerical accuracy.
To summarize, we have shown that, in the strong-coupling limit of the fermionic attraction, the value $a_{B} = 0.75 \, a_{F}$ is obtained by a correct summation of the bosonic T-matrix diagrams for the composite bosons which form as bound-fermion pairs. This theoretical prediction could be tested against the experimental data with ultracold Fermi atoms in a trap, when bosonic molecules are obtained by a Feshbach resonance.
C.A. Regal, C. Ticknor, J.L. Bohn, and D. Jin, Nature [**424**]{}, 47 (2003). P. Pieri and G.C. Strinati, Phys. Rev. B [**61**]{}, 15370 (2000); see also cond-mat/9811166. R. Haussmann, Z. Phys. B [**91**]{}, 291 (1993).
F. Pistolesi and G.C. Strinati, Phys. Rev. B [**53**]{}, 15168 (1996).
N. Andrenacci, P. Pieri, and G.C. Strinati, Eur. J. Phys. B [**13**]{}, 637 (2000).
The choice of the “contact” potential (\[Eq:deltafunc\]) does not allow the fermionic attraction to extend over a *finite* range. If a finite-range fermionic attraction would instead been adopted, the effective boson-boson potential in the strong-coupling limit would acquire an *attractive* part which would dominate over the usual repulsive part due to Pauli principle, in the sense that the bosonic scattering length associated with the attractive part would not vanish in that limit. The attractive part would thus lead to an instability of the bosonic system when the finite-range fermionic attraction gets sufficiently strong. To avoid this instability, a condition of the type $\epsilon_{0} \ll k_{0}^{2}/(2m)$ has to be imposed on the BE limit, where $k_{0}$ is the wave vector specifying the finite range of the potential and $\epsilon_0$ is the bound-state energy of the fermionic two-body problem \[cf. footnote 43 of Ref. \]. In this context, it is worth mentioning the work by G. Röpke *et al.* \[Phys. Rev. Lett. [**80**]{}, 3177 (1998)\] where it has been shown that, in the strong-coupling limit, there exists a competition between pair and quartet condensation in a Fermi liquid with finite-range attraction. It is then clear that adopting a finite-range fermionic potential merely makes the BCS-BE crossover more involved, and it is thus not relevant for the present purposes.
A precise definition of what is meant by strong coupling is as follows. In this limit, the binding energy $\epsilon_{0}$ is much larger than the density $\rho$ and temperature $T$. Under this condition, the fermionic chemical potential approaches the value $-\epsilon_{0}/2$ and $\beta |\mu| \gg 1$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We measure the angular power spectrum of the WMAP first-year temperature anisotropy maps. We use SpICE (Spatially Inhomogeneous Correlation Estimator) to estimate $C_\ell$’s for multipoles $\ell=2-900$ from all possible cross-correlation channels. Except for the map-making stage, our measurements provide an independent analysis of that by [@HinshawEtal2003a]. Despite the different methods used, there is virtually no difference between the two measurements for $\ell \simlt 700$ ; the highest $\ell$’s are still compatible within $1-\sigma$ errors. We use a novel [*intra-bin variance*]{} method to constrain $C_\ell$ errors in a model independent way. Simulations show that our implementation of the technique is unbiased within 1% for $\ell \simgt 100$. When applied to WMAP data, the intra-bin variance estimator yields diagonal errors $\sim 10\%$ larger than those reported by the WMAP team for $100 < \ell < 450$. This translates into a 2.4 $\sigma$ detection of systematics since no difference is expected between the SpICE and the WMAP team estimator window functions in this multipole range. With our measurement of the $C_{\ell}$’s and errors, we get $\chi^2/d.o.f. = 1.042$ for a best-fit model, which has a 14 % probability, whereas the WMAP team [@SpergelEtal2003] obtained $\chi^2/d.o.f. = 1.066$, which has a 5 % probability. We assess the impact of our results on cosmological parameters using Markov Chain Monte Carlo simulations. From WMAP data alone, assuming spatially flat power law models, we obtain the reionization optical depth $\tau = 0.145 \pm 0.067$, spectral index $n_s = 0.99 \pm 0.04$, Hubble constant $h = 0.67 \pm 0.05$, baryon density $\Omega_b h^2 = 0.0218 \pm 0.0014$, cold dark matter density $\Omega_{cdm} h^2 = 0.122 \pm 0.018$, and $\sigma_8 = 0.92 \pm 0.12$, consistent with a reionization redshift $z_{re} = 16 \pm 5$ (68 $\%$ CL).'
author:
- 'Pablo Fosalba, István Szapudi'
title: 'The Angular Power Spectrum of the First-Year WMAP Data Reanalysed'
---
Introduction
============
The [*Wilkinson Microwave Anisotropy Probe*]{} satellite (WMAP) has provided the clearest view of the primordial universe to date. Its unprecedented high sensitivity and spatial resolution resulted in a unique set of cosmic microwave background (CMB) radiation maps with close to full sky coverage and uniformly high quality. As a result, fundamental cosmological parameters can be constrained to the highest precision ever. Thorough analysis of this dataset [@BennettEtal2003a] yielded a cosmic variance limited measurement of the angular power spectrum, $C_\ell$’s, of the CMB temperature anisotropy for multipoles $\ell \simlt 350$ ([@HinshawEtal2003a]; hereafter H03). This confirmed and improved measurements from previous experiments ([@MillerEtal1999; @deBernardisEtal2000; @HananyEtal2000; @Halverson2002; @MasonEtal2003; @ScottEtal2003; @BenoitEtal2003]). The acoustic peak structure revealed by the WMAP temperature and polarization power spectra provided strong observational support to inflation and constrained viable cosmological scenarios to the domain of flat models and its close variants.
Considering the importance of these results, our principal aim is to estimate the angular power spectrum in a completely independent way in the full range of multipoles probed by WMAP, $2 \le \ell \le 900$, and systematically compare results to H03. Our $C_{\ell}$ estimation pipeline is based on SpICE [^1] [Spatially Inhomogeneous Correlator Estimator; @SzapudiEtal2001a; @SzapudiEtal2001b], a quadratic estimator based on correlation functions. SpICE performs edge corrections and heuristic minimum variance weighting in pixel [^2] space to produce nearly optimal results. Our fast HEALPix [^3] implementation of SpICE scales as ${\rm {\cal O} (N^{3/2})}$ (${\rm N}$ is number of pixels).
Power Spectrum Estimation {#sec:ps}
=========================
Our estimation methodology closely follows that of H03, but adapted to our technique:
[*Step 1:*]{} We use the [*foreground cleaned intensity maps*]{} for the 3 highest frequency bands Q, V & W downloaded from the LAMBDA website [^4]. Strong diffuse Galactic emission and resolved point sources are masked out using the Kp0 and Kp2 masks, that leaves $76.8\%$ and $85.0\%$ of the sky useful for cosmological analyses, respectively. Monopole $\ell=0$ and dipole $\ell=1$ terms are also removed from non-masked pixels.
[*Step 2:*]{} Power-spectrum estimation is performed via SpICE: we compute the cross-correlations from 28 different pairs of channels constructed from the 8 “differencing assemblies” (DAs) Q1 through W4. Noise correlation among different channels is negligible, therefore our cross-power estimator is unbiased with respect to the noise (see H03). Like H03, we implement an heuristic $\ell$-dependent pixel noise weighting scheme that minimizes errors: we use flat weights (mask weight only) for $\ell < 200$, inverse pixel noise variance for $\ell > 450$, and a transitional inverse rms noise weight in the intermediate range $200 < \ell < 450$.
[*Step 3:*]{} A model for the power spectrum for unresolved extragalactic radio sources is subtracted from the cross-power spectrum of each channel. We implement the model given in §3.1 of H03.
[*Step 4:*]{} $C_\ell$’s from different channels are optimally combined using an inverse noise weighting, with DA sensitivities as described in the LAMBDA website. All channels are included, except for those in Q-band that are only used in the intermediate $\ell$-range. This helps minimizing galactic contamination at low $\ell$ and the window function cut-off at the highest multipoles.
[*Step 5:*]{} Our quadratic estimator is defined in pixel space, where mask effects can be easily corrected for [cf. @SzapudiEtal2001a]. The two point correlation function is then transformed into harmonic space via Gauss-Legendre quadrature to obtain the $C_{\ell}$’s deconvolved from the window function of the experiment. Symmetrized non-Gaussian beam transfer profiles [@PageEtal2003] and pixel window functions are corrected for in $\ell$-space.
Principal Results {#sec:res}
=================
Figure \[fig:cls\] shows the angular power spectrum of WMAP, $\Delta T^2_{\ell} \equiv \ell(\ell+1)C_{\ell}/2 \pi$, in $\mu$K$^2$ units, measured with SpICE. Upper panel shows the power spectrum for individual multipoles, using Kp2 sky cut. Our measurement (red line) is in excellent agreement with H03 (black line), multipole by multipole. In particular, for the quadrupole and octopole we find $\Delta T^2_{2} \sim 135 \mu$K$^2$ and $\Delta T^2_{3} \sim 591 \mu$K$^2$, respectively (H03 get $\sim 123\mu$K$^2$ and $\sim 612 \mu$K$^2$). For the highest ${\ell}$’s we find slightly different amplitudes than H03, but consistent at the 1-$\sigma$ level.
For the most part, we observe no systematic dependence of the measured $C_{\ell}$’s on the sky cut (see difference between red and blue lines in bottom panel of Figure \[fig:cls\]). However, using Kp0 instead of Kp2 yields a $15\%$ lower amplitude of the octopole $\ell=3$ and a $15-20\%$ smaller amplitudes for the 3 highest band-powers centered at $\ell_{\rm eff} \sim 660, 750, 850$. This effect might be due to imperfect foreground removal and/or the intrinsic estimator variance due to finite volume and edge effects. We estimated the dispersion in a set of WMAP simulations with Kp0 & Kp2 sky cuts to be of the same order as the measured differences in the $C_{\ell}$’s of the data. On the other hand, the cross-correlation amplitude between the clean WMAP maps and the best fit foreground templates is at the 5% and 10% level of the WMAP $C_{\ell}$’s for the lowest and highest $\ell$’s, respectively. We thus conclude that sample variance due to sky coverage can account for most of the observed difference in the $C_\ell$’s, while residual foreground contamination is always subdominant. The low level of systematics in Kp2, and the increased statistical errors due to the decreased sky fraction left by Kp0, motivate us to adopt Kp2 (as in H03) for the best estimate of the $C_{\ell}$’s.
Error Estimation {#sec:errors}
================
In order to estimate the covariance of our $C_\ell$’s, we generated MC simulations of the CMB sky and instrument noise for each of the 8 DA’s (Q1 through W4). We used the [*running index*]{} model that best fits a combination of WMAP, CBI & ACBAR data (denoted [*WMAPext*]{} in [@SpergelEtal2003]). Maps were convolved with the symmetric (non-Gaussian) beam transfer function for each DA [@PageEtal2003]. As for the noise simulations, we downloaded 100 sky maps per DA from the LAMBDA website. These simulate 1 full year of flight instrument noise and they include all known radiometric effects [@HinshawEtal2003b; @JarosikEtal2003]. Simulations were analyzed in exactly the same way as the data (see §\[sec:ps\]). All in all, we have constrained the errors from $1500$ measurements in MC simulations (combining cross spectra using 100 MC’s for each of the 6 highest frequency DA’s V1 through W4) for the multipole ranges $\ell < 200$ and $\ell > 450$, and $2800$ measurements (combining cross-spectra using 100 MC’s from each of the 8 DA’s Q1 through W4) for the intermediate $\ell$-range $200 < \ell < 450$.
For multipoles $\ell<350$, errors in the WMAP power spectrum are dominated by cosmic or sample variance (see H03) and the noise only contributes at the few percent level. Figure \[fig:lowlerr\] displays the noise contribution to the relative errors at low multipoles, $\ell \simlt 100$. Correlated noise simulation results are displayed (smooth solid line) along with results from uncorrelated noise simulations (dashed line). The latter tends to underestimate errors by $\sim 1\%$. Alternatively the noise level can be estimated from the data rms dispersion among the WMAP channels used (oscillating solid line). These results are in excellent agreement with H03 (cf. lower panel in their Figure 4).
At higher $\ell$’s pixel noise and systematic effects (beam and mode coupling, residual foregrounds) increasingly dominate the errors. MC methods assume detailed knowledge of all such effects. To provide a model independent check of the errors, we introduce a novel technique that allows estimating errors directly from the data: the [*intra-bin variance*]{} (IBV) method. IBV estimates the variance of a given $C_{\ell}$ from the rms dispersion in a bin $\rm B_{\ell}$ centered on $\ell$. The bin-width ${\Delta \ell}$ is a matter of practical consideration, balancing variance and bias. More precisely, our estimator for $\sigma(C_{\ell})$ reads $$\sigma^2(C_{\ell}) = {1 \over {\Delta {\ell} -1}}
\sum_{{\ell}^{\prime} \in {\rm B_{\ell}}}
({\Delta C_{{\ell}^{\prime}}}-{\langle{\Delta C_{\ell}\rangle}})^2
\label{eq:ibv}$$ where $\langle{\Delta C_{\ell}} \rangle = 1/\Delta {\ell} \sum_{{\ell}^{\prime} \in \rm B_{\ell}} \Delta C_{{\ell}^{\prime}},\,$ $\Delta C_{\ell} = {\bar C_{\ell}} - C^{\rm th}_{\ell}$, ${\bar{C_{\ell}}}$ is the mean of the measured $C_{\ell}$’s over channels, and $C^{\rm th}_{\ell}$ is our best guess for the data mean using a theoretical model. The latter is subtracted to decrease the bias due to the slope of the angular power spectrum. We used $C^{\rm th}_{\ell}$ from the WMAP best-fit [*running index*]{} model [@SpergelEtal2003], although this is not critical: no baseline subtraction only biases at a few percent level. By construction, IBV should not be used to obtain errors with high resolution but to assess the overall level of errors in a range of $\ell$’s, typically larger than $\Delta \ell$.
Figure \[fig:ibcal\] shows the ratio between the [*mean*]{} IBV rms dispersion to the usual MC rms dispersion, both estimated from $\sim 3000$ WMAP simulations of CMB signal and correlated noise. Narrow bins yield slightly biased (under-)estimates of the MC error at few percent level, possibly due to small mode-to-mode couplings. IBV method with $\Delta \ell = 18$ yields unbiased estimates of the error for WMAP simulations at the 1% level for $\ell \simgt 100$. Doubling $\Delta \ell$ introduces a slight high bias and significant edge effects for low $\ell$’s that could be caused by the residual slope of the $C_{\ell}$’s. The unbiased bin-width, $\Delta \ell = 18$, with $15\%$ variance is our choice for the WMAP error estimation.
Figure \[fig:ibdata\] displays the WMAP data diagonal errorbars computed with the IBV method (spiky solid line) compared to the previously published diagonal errors ([@VerdeEtal2003],[@HinshawEtal2003b],[@KogutEtal2003]). The largest IBV errors appear to correlate well with the outliers of the data $C_{\ell}$’s with respect to the best-fit model (see Figure 3 in [@SpergelEtal2003]), suggesting that our IBV estimator is closely related to a diagonal $\chi^2$ test. It is clear that the mean overall error is higher than originally estimated (otherwise the IBV curve would fluctuate around unity). The simplest and most conservative interpretation of our results yields a monotonic error increase with respect to the WMAP team diagonal errors of the form, $\sigma(C_{\ell})_{IBV}/\sigma(C_{\ell})_{WMAP} \simeq 1.08 + 8.5\cdot 10^{-5} ({\ell}-100)$ for $\ell>100$ (straight solid line in Figure \[fig:ibdata\]). This smooth prediction results from a least squares minimization to the IBV curve (large amplitude oscillating line in Figure \[fig:ibdata\]). Note that for $\ell > 450$, the error excess is consistent with the errors estimated from MC simulations with [*correlated*]{} instrument noise (see noisy line in Figure \[fig:ibdata\] growing from left to right).
In the range $100 < \ell < 450$ the mean error level is incompatible with both MC simulations that include correlated noise and the WMAP team published errors: given that there are approximately 16 independent $\Delta {\ell}$ bins, with an intrinsic 15 % error each, and that the mean error excess is 9 % in this $\ell$-range, this amounts to a 2.4 $\sigma$ detection of the error excess. We have checked that using the $C_{\ell}$’s measured by [@HinshawEtal2003a] yields comparable IBV errors in this multipole range. The excellent $\ell$-by-$\ell$ agreement between the SpICE and WMAP team’s measurement of the $C_{\ell}$’s (see Figure \[fig:cls\]) indicates that both estimators window functions are virtually identical in this regime, and thus the observed error excess points to systematics unaccounted for in the WMAP team analyses. For $\ell > 450$ the interpretation is less clear as there are hints that both window functions might be slightly different (see lower panel in Figure \[fig:cls\]). Such differences might arise in the practical implementation of the estimators. We also estimate a $\sim 5\%$ correlated noise contribution at $\ell<100$ (see Figure \[fig:lowlerr\]), that was neglected in previous likelihood analyses. A more robust assessment of errors is provided below using the full $\chi^2$ test, where off-diagonal terms are also taken into account following Eq.(15) in [@VerdeEtal2003].
Discussion: Cosmological Parameters
===================================
We investigate the implications of our measurements using a Bayesian analysis of cosmological parameter estimation. We use CosmoMC[^5], a Markov Chain Monte Carlo (MCMC) implementation [@LewisBridle2002] based on CAMB[^6] ([@LewisEtal1999]; see also CMBFAST[^7], [@SeljakZaldarriaga1996]). In order to allow direct comparison with [@SpergelEtal2003], we focus on the simplest 6-parameter cosmological model consistent with the WMAP temperature and cross-polarization data. Following [@VerdeEtal2003], we assume a set of flat models with radiation, baryons, cold dark matter and cosmological constant. Primordial fluctuations are taken to be adiabatic and Gaussian with a power-law power spectrum. We use the physical dark matter $\Omega_{cdm} h^2$ and baryon $\Omega_b h^2$ densities, the reionization optical depth $\tau$, the scalar spectral index $n_s$, the normalized Hubble constant $h$, and the dark matter power spectrum normalization $\sigma_8$ [@KosowskyEtal2002]. We estimate paramaters by combining 4 independent chains with 30000 accepted points each, and use the 6 paramater covariance matrix as proposal density from precomputed runs. This yields an excellent convergence-mixing Gelman & Rubin statistic $R-1 \simlt 0.02$ for all cases studied.
[lll]{} $\tau$ &$0.145 \pm 0.067$ & $0.151 \pm 0.069$\
$n_s$ &$0.99 \pm 0.04$ & $0.99 \pm 0.04$\
$h$ &$0.67 \pm 0.05$ & $0.70 \pm 0.05$\
$\Omega_b h^2$ &$0.0218 \pm 0.0014$ & $0.0234 \pm 0.0013$\
$\Omega_{cdm} h^2$ &$0.122 \pm 0.018$ & $0.123 \pm 0.017$\
$\sigma_8$ &$0.92 \pm 0.12$ & $0.92 \pm 0.11$\
$\chi^2_{eff}/dof$ &1398.8/1342 &1428.7/1342\
Table 1 summarizes our results. Imposing the prior $\tau < 0.3$, we find best fit values matching those of [@SpergelEtal2003]. In particular we obtain a $\chi^2/d.o.f. = 1.042$ (it has a 14 % probability) for the best-fit model (see first column in Table 1), which is a slightly better fit to the data than that of [@SpergelEtal2003], $\chi^2/d.o.f. = 1.066$ (5 % probability). Our $h$ and $\tau$ are slightly lower but still consistent at the $1-\sigma$ level. This is more significant for our estimates of the $C_{\ell}$’s and errors (see first column in Table 1). In particular, our measurement $\Omega_b h^2 = 0.0218 \pm 0.0014$ agrees with that from the latest BBN results $\Omega_b h^2 = 0.022 \pm 0.002$ [@CyburtEtal2003; @VangioniEtal2003; @CuocoEtal2003]. We have checked that relaxing the $\tau$ prior yields larger values of $\tau = 0.19 \pm 0.12$ [cf. @TegmarkEtal2004]. Our main results (see first column in Table 1) are in excellent agreement with the best-fit values from WMAP+SDSS [@TegmarkEtal2004], and suggest a redshift of (abrupt) reionization $z_{re} = 16 \pm 5$ ($68\%$ CL). Data products and additional plots from this work can be found at [http://www.ifa.hawaii.edu/cosmowave/wmap.html]{}
We thank an anonymous referee for insightful comments, Jun Pan for help and discussions, Olivier Dore, Hans K. Eriksen, Eiichiro Komatsu for useful comments, and Antony Lewis for help with CosmoMC. We acknowledge extensive use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. Some of the results in this paper have been derived using HEALPix [@GorskiEtal1998]. This research was supported by NASA through ATP NASA NAG5-12101 and AISR NAG5-11996, as well as by NSF grants AST02-06243 and ITR 1120201-128440.
ms.bbl
[^1]: http://www.ifa.hawaii.edu/cosmowave/
[^2]: The harmonic space alternative using pseudo $C_{\ell}$’s is MASTER [@HivonEtal2002].
[^3]: http://www.eso.org/science/healpix/
[^4]: http://lambda.gsfc.nasa.gov/
[^5]: http://cosmologist.info/cosmomc
[^6]: http://camb.info
[^7]: http://cmbfast.org
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose a model of soft CP violation in which the CP violating mechanism naturally lies only in the charged Higgs sector. The charged Higgs mechanism not only accounts for the measured value of the CP-violating parameter $\epsilon$ but also accommodates the current limits on $\epsilon''/\epsilon$. Our model naturally prevents tree-level Flavor-Changing Neutral Currents (FCNCs) of any kind. Unlike the Weinberg-Branco Three-Higgs Doublet Model, the deviation from the Standard Model rate for $b\to s\gamma$ is small. Furthermore, leading contributions to the electron (neutron) electric dipole moment are non-zero beginning at the three (two) loop level. Surprisingly similar to the Standard Kobayashi-Maskawa Model, our model is of milliweak character but with seemingly superweak phenomenology.'
---
6.5in
-1cm
**A Simple Charged Higgs Model of Soft CP Violation**
**without Flavor Changing Neutral Currents**
David Bowser-Chao$^{(1)}$, Darwin Chang$^{(2,3)}$, and Wai-Yee Keung$^{(1)}$
*$^{(1)}$Physics Department, University of Illinois at Chicago, IL 60607-7059, USA\
$^{(2)}$Physics Department, National Tsing-Hua University, Hsinchu 30043, Taiwan, R.O.C.\
$^{(3)}$Institute of Physics, Academia Sinica, Taipei, R.O.C.\
*
Submitted to [*Physical Review Letters*]{}
PACS numbers: 11.30.Er, 14.80.Er
Introduction {#introduction .unnumbered}
============
-1cm
Three decades after its surprising discovery in the kaon system[@ccft], CP violation has remained mysterious. A desire for deeper insight into its origin is the driving force behind many ongoing experiments and even the construction of new machines such as the two B Factories. While a profound understanding may yet be lacking, several mechanisms have been suggested to explain observed CP violation (i.e., $\epsilon \ne 0$) within a gauge field theory. Kobayashi and Maskawa(KM)[@km] proposed a third generation of fermions, so that CP violation would arise from the mixing of the three quark generations and is manifested by a single phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. Since then, many other mechanisms have been put forth, including new gauge interactions[@gauge], neutral Higgs exchange[@neutral], supersymmetric partners[@susy], and charged Higgs exchange[@weinberg; @branco]. However, the KM model has the distinguishing feature that its mechanism is of milliweak strength, though its phenomenology is manifestly superweak[@superweak], consistent with current CP related data. Such intricate character has also been the driving force behind the desire to find non-superweak CP violation in the $B$ systems.
The leading model for the charged Higgs mechanism of CP violation has long been the Weinberg Three-Doublet Model of CP violation[@weinberg], which became even more intriguing after Branco[@branco] proposed a version in which CP violation is softly or spontaneously broken. This scheme naturally avoids tree-level flavor changing neutral currents. Without hard CP violation the CKM matrix is purely real (the KM mechanism is inoperative); CP violation in the kaon system instead results from charged Higgs exchange. Many weaknesses of the Weinberg-Branco Model, however, have since been identified. Sanda and Deshpande pointed out[@sanda] that short distance contributions to $\epsilon$, if dominant, would lead to a larger $\epsilon'/\epsilon$ than experimentally allowed, although it was subsequently demonstrated that long distance contributions to $\epsilon$ could be large enough to avoid this difficulty[@chang]. More recently, however, it has become clear that this model has other problems. A charged Higgs light enough to account for the observed $\epsilon$ has already been excluded by the LEP experiments[@edm]. The large neutron electric dipole moment[@edm] (EDM) and substantial rate for $b \rightarrow s
\gamma$[@bsg] predicted are also contradicted by data, leading several authors[@edm; @bsg] to rule out this model.
As an illustrative model for charged Higgs CP violation, the Weinberg-Branco Model also has the shortcoming that its neutral Higgs sector naturally also contains CP violation, which is usually ignored in the literature to simplify analysis and highlight the charged Higgs mechanism. However, for flavor conserving CP odd observables (e.g., the neutron EDM), the neutral Higgs contribution generically can be competitive with that from charged Higgs exchange.
In this letter we propose an alternative model that may serve as a generic example in which the charged Higgs mechanism of CP violation naturally dominates completely over other mechanisms. CP is broken softly or spontaneously so that the KM mechanism is inoperative. Tree-level flavor changing neutral currents are automatically absent, and the neutral Higgs sector is CP conserving at tree level. As in the KM Model, the quark and electron EDMs are severely suppressed. The electron EDM vanishes at the two-loop level, while the first non-zero contribution to the quark EDMs is at two loops. In contrast to the Weinberg-Branco model, our model easily satisfies other experimental CP violation constraints as well as the rate for $b\to s \gamma$. Finally, the parameter $\theta_{\rm{QCD}}$ vanishes at tree-level, since we disallow hard CP breaking; we shall see that radiative corrections are mild and consistent with the limit on a non-zero $\theta_{\rm{QCD}}$.
For most of this letter, we shall assume that CP is broken softly. One can also modify our model to break CP spontaneously by introducing at least one additional CP odd scalar boson, as discussed toward the end of this work, with the bulk of the phenomenology unchanged.
General Formalism {#general-formalism .unnumbered}
=================
-1cm
The Weinberg-Branco Model augments the Standard Model (SM) with additional Higgs $SU(2)_L$ doublets, which are responsible for kaon system CP violation; in this model, then, since the charged Higgs sector must break CP, so also must the neutral Higgs sector. To mandate charged Higgs exchange as the dominant CP violation mechanism we instead introduce only additional $SU(2)_L$ singlets of quarks and scalars to the theory. The simplest model for our purposes requires two additional charged Higgs singlets, $h_\alpha (\alpha=1,2)$ and a vectorial pair of heavy quark fields, $Q_{L,R}$, of electromagnetic charge $-{4\over3}$. This vector quark charge assignment avoids fractionally charged hadrons. Relevant new terms in the Lagrangian are: $${\cal L}_{h_i} =
\left[
(g \lambda_{i\alpha} \bar Q_L d_{iR} h_\alpha
+ M_Q \bar Q_L Q_R) + \hbox{h.c.} \right]
- (m^2)_{\alpha\beta} {h_\alpha}^{\dag} h_\beta
- \kappa_{\alpha\beta}
(\phi^{\dag} \phi-|\langle\phi\rangle|^2) \,h_\alpha^{\dag}
h_\beta \;\;\label{eq:lagrangian}$$ where $\phi$ is the Standard Model Higgs doublet, and $i$ is summed over the down quark flavors ($i=d,s,b$). The vector quark has purely vectorial coupling to the photon and $Z$ boson, with respective charges $(Q_Q, -Q_Q \sin^2\theta_W)$, while the charged Higgs couples with charges $(Q_h, -Q_h \sin^2\theta_W)$ and $Q_Q = Q_d + Q_h$. The neutral Higgs sector is identical to that in the Standard Model, with neither flavor changing couplings nor CP violation. The matrices $m^2$ and $\kappa$ are hermitian. Except for the discussion at the end, we assume that CP is broken softly in this Lagrangian, implying a special basis where all the Yukawa ($\lambda, \kappa$) and the SM couplings are real. We also require (see below) that dim-3 couplings, namely $M_Q$, are also real. This leaves, as in the KM model, only a single CP violating parameter: Im$(m^2)_{12}$. We can diagonalize $(m^2)_{\alpha\beta}$ by a unitary matrix $U_{\alpha i}$ which in general is complex: $h_\alpha =
U_{\alpha i} H_i$, with $H_i$ the mass eigenstates. The quark-Higgs interaction in the mass eigenstate basis is $${\cal L}_{QqH}=g\sum_{q=d,s,b}\xi_{qj}
(\bar Q_L q_R)H^-_j \ +\ \hbox{h.c.}
\ ,
\label{eq:QqH}$$ with $\xi_{qj} \equiv \lambda_{q\alpha} U_{\alpha j}$. The CP-violating transit propagators[@weinberg] can be expressed as $
\langle h_\alpha^{\dag} h_\beta \rangle
= \sum_{i,j=1,2} U_{i \alpha }^{\dag} U_{\beta j}
\langle H_i^{\dag} H_j \rangle
= \sum_{i=1,2} U_{\beta i} U_{i \alpha }^{\dag}
\langle H_i^{\dag} H_i \rangle
.
$ With $m_1$ $(m_2)$ the mass of the lighter (heavier) charged Higgs, CP violation explicitly vanishes if $m_1=m_2$. In the limit that $m_2 \gg m_1$, these expressions reduce to $\langle h_\alpha^{\dag}
h_\beta \rangle = U_{\beta 1} U_{1 \alpha }^{\dag}
/(p^2-m_1^2+i\epsilon)\, , $ where $p$ is the momentum flowing in the propagator. The rephasing-invariant measures of CP violation are then ${\cal A}_{qq'} = \lambda_{q\alpha}\lambda_{q'\beta}U_{\beta 1}
U_{\alpha 1}^{*} = \xi_{q1}^{*} \xi_{q'1}$ with $(q,q'=d,s,b)$, and ${\cal B} = \kappa_{\alpha \beta}U_{\beta 1} U_{\alpha 1}^{*}$. For flavor changing processes, ${\cal A}_{qq'}$ plays the main role, with ${\cal B}$ its counterpart in flavor conserving processes.
Before continuing, we comment on the strong CP-violation parameter $\theta_{\rm QCD}$. With CP symmetry imposed only on the hard (dim-4) terms, the $\theta_{\rm QCD}$ parameter is naively zero at tree level, but $M_Q$ may still be complex. If so, alignment of the QCD vacuum with this complex quark mass will generate a non-zero tree level $\theta_{\rm QCD}$. To avoid this contribution, we simply impose CP symmetry on both dim-4 and dim-3 terms. $M_Q$ will then be real in the same basis that the tree-level $\theta_{\rm QCD}$ vanishes[@interplay]. A similar scheme can also be arranged if CP is broken spontaneously (see below). The first non-zero contribution to $\theta_{\rm QCD}$ (occuring at two loops) will be discussed later.
Constraint from $\epsilon$ {#constraint-from-epsilon .unnumbered}
==========================
With CP conservation modulo soft-breaking enforced, the CKM matrix is real at tree level. Leading CP violating phenomena should be due solely to the CP-violating phase in the charged Higgs sector. Making the usual “$\pi \pi (I=0)$ dominance” assumption, the CP violation parameter $\epsilon$ is approximately $$\begin{aligned}
\epsilon & \simeq &\frac{e^{i\pi/4}}{\sqrt{2}}
\left(
\frac{\hbox{Im}M_{12}}{2\hbox{Re}M_{12}}
+\frac{\hbox{Im}A_0}{\hbox{Re}A_0}
\right)\ .\end{aligned}$$ We shall postpone discussion of $A_0$, but will see later that in our model, as in the KM Model, the second term is negligible. Experimentally, $\epsilon \simeq 0.00226 \ \exp(i \pi/ 4)$. The $\Delta
S=2$ part of the effective Hamiltonian to one-loop (i.e., box diagrams) can be written as: $$\nonumber
{\cal H}^{\Delta S=2} =
\frac{G_F^2 m_W^2}{16\pi^2}
\sum_{I=R,L}C^I_{\Delta S=2}(\mu) O^I_{\Delta S=2}(\mu)
,\;\;\quad
O^{R,L}_{\Delta S=2} =\bar s \gamma_\mu (1\pm\gamma_5) d \,
\bar s \gamma^\mu (1\pm\gamma_5) d \ .$$ The $W$-boson diagrams yield a purely real Wilson coefficient $C^L_{\Delta S=2}(\mu)$; CP violation in kaon matrix elements is due solely to the operator $O^R_{\Delta S=2}$ rather than $O^L_{\Delta
S=2}$, in contrast to the KM model. The complex coefficient $C^R_{\Delta S=2}(\mu)$ is generated by the charged Higgs through a box diagram with vertices given by Eq.(\[eq:QqH\]). At the scale $\mu=
M_Q$, we have $$C^R_{\Delta S=2}(M_Q) =
2 \xi_{d1} \xi_{s1}^* \xi_{d2} \xi_{s2}^{*}
\frac{2m_W^2}{M_Q^2} \frac{f(x_2)-f(x_1)}{x_2-x_1}
+\sum_{i=1,2} (\xi_{di}\xi_{si}^{*})^2 \frac{2m_W^2}{M_Q^2}\,
{df \over dx}(x_i)
\ ,$$ with $x_{1,2}=m_{H_{1,2}}^2/M_Q^2$, $f(x)= (1-x+ x^2\log x )/(1-x)^2$, and $df/dx\-(1) = 1/3$. Clearly, $C^R_{\Delta S=2}(M_Q)$ is real when $m_2 = m_1$ as it should be. For illustration, we shall take $m_2 \gg m_1$ and $m_1 = M_Q$, in which case the first term is negligible and $C^R_{\Delta S=2}(M_Q) =
{2\over3}(\xi_{d1}\xi_{s1}^{*})^2m_W^2/M_Q^2$.
Following Ref.[@renormgroup] for the renormalization group evolution and numerical evaluation of hadronic matrix elements to leading order, we obtain $$\begin{aligned}
C^R_{\Delta S=2}(\mu \le m_c) &=&
{\left[ \frac{\alpha_s(m_c)}{\alpha_s(\mu)} \right]}^{6/27}
{\left[ \frac{\alpha_s(m_b)}{\alpha_s(m_c)} \right]}^{6/25}
{\left[ \frac{\alpha_s(m_t)}{\alpha_s(m_b)} \right]}^{6/23}
{\left[ \frac{\alpha_s(M_Q)}{\alpha_s(m_t)} \right]}^{6/21}
C^R_{\Delta S=2}(M_Q) \nonumber \\
&\approx& 0.59 \,\alpha_s^{-2/9}(\mu) \,C^R_{\Delta S=2}(M_Q) \ .\end{aligned}$$ We will assume that the $W$-boson contributions dominate the real part of all relevant matrix elements; analysis of $\epsilon^{'}/\epsilon$ below shows this to be consistent. We will thus take, [*e.g*]{}., $\hbox{Re}M_{12} = {1\over2}\Delta m_K$ from experiment, and have no need of the explicit value of $W$-boson contributions to, e.g., $C^L_{\Delta S=2}$. Let $M_{12}^R$ be the contribution of $O^R_{\Delta S=2}$ to the mass matrix. From the input parameters $ B_K=0.75, F_K=160 \,{\hbox{MeV}}, m_K= 498\,{\hbox{MeV}} ,
\Delta m_K = 3.51 \times 10^{-15} \hbox{GeV} $ and the relation $$M_{12}^R = \frac{1}{2m_K}
\langle \bar{K^0} | {\cal H}^{\Delta S=2} |K^0\rangle^{*}
= \frac{G_F^2 m_W^2}{16\pi^2} \frac{1}{2m_K} C_{\Delta S=2}^{R*}(\mu)
\langle \bar{K^0} | O^R_{\Delta S=2}(\mu) |K^0\rangle^{*} \ ,$$ $$\langle \bar{K^0} | O^R_{\Delta S=2}(\mu) |K^0\rangle =
\hbox{${8\over3}$}\alpha_s(\mu)^{2/9} B_K F_K^2 m_K^2 \ ,$$ follows the numerical prediction $ M_{12}^R / \Delta m_K = 1.2 \times 10^4 \ C_{\Delta S=2}^R(M_Q) $. Demanding that the imaginary part of $M_{12}^R$ gives enough contribution to $\epsilon$ and the corresponding real part gives just a fraction $\cal F$ of the mass difference $\Delta m_K$ ([*i.e.*]{} $ 2 \hbox{Re}( M_{12}^R ) = {\cal F} \Delta m_K$), we obtain constraints on the Wilson coefficients: $\hbox{Im}\ C_{\Delta S=2}^R(M_Q)= 2.7 \times 10^{-7} $ and $\hbox{Re}\ C_{\Delta S=2}^R(M_Q)= 4.2 \times 10^{-5}{\cal F} $. Again, with $m_2 \gg m_1, m_1=M_Q$, we then find $$\hbox{Im} \left({\cal A}_{sd} / (0.049)^2 \right)^2 R_Q^2 =1 \ ,\quad
\hbox{Re} \left({\cal A}_{sd} / (0.049)^2 \right)^2 R_Q^2 =156 {\cal F}\ ;
\label{eq:dmineq}$$ where $R_Q = 300 \hbox{ GeV}/M_Q$. The reasonable constraint $|{\cal F}|
< 1$ can be easily satisfied.
Constraints from $(\epsilon'/\epsilon)$ and $B^0$–$\bar {B^0}$ mixing {#constraints-from-epsilonepsilon-and-b0bar-b0-mixing .unnumbered}
=====================================================================
The parameter $\epsilon'$ describes direct CP violation in the kaon system. It is given in terms of the $2\pi$ decay amplitudes $A_{0,2}= {\cal A}(K \to (\pi\pi)_{0,2})$, where the subscript indicates the isospin of the outgoing state. With $\omega= |A_2/A_0|=0.045$, $\xi=\hbox{Im}A_0/\hbox{Re}A_0$, $\Phi\approx \pi/4$, and $\Omega=(1/\omega )\cdot (\hbox{Im}A_2/\hbox{Im}A_0)$, $$\epsilon' = -\frac{\omega}{\sqrt 2} \xi (1-\Omega) \exp(i\Phi) \ .$$ The dominant contributions should be the gluon and electroweak penguins mediated by $H$ and $Q$. In contrast to the KM Model, the vector coupling of the vector quark means that the $Z$ boson penguin will be suppressed by $O(m_K^2/m_Z^2)$ due to vector current conservation. The gluon penguin contributes only to $A_0$, but the isospin-breaking electromagnetic penguin (EMP) gives rise to both $A_0$ and $A_2$. Due to its suppression by $O(\alpha/\alpha_s)$, the latter affects $\epsilon'$ solely through its contribution to $\Omega$. Including the effects of evolution from the vector quark mass down to the charm mass scale, we estimate the EMP contribution to be $\Omega_{\rm EMP}
\stackrel{<}{\sim} O(1)$. There is an additional contribution to $\Omega$ from $\eta, \eta'$ isospin-breaking, with $\Omega_{\eta-\eta'}
= 0.25$. We shall ignore the electromagnetic penguin contribution here (inclusion of the electromagnetic penguin will be studied elsewhere[@UsAgain]), and set $\Omega=\Omega_{\eta-\eta'}$ to simply the analysis. The inclusion of $\Omega_{\rm EMP}$ will not change our conclusion qualitatively.
The gluon penguin diagram, which involves the virtual vector quark $Q$ and the charge Higgs boson, produces an effective Hamiltonian at the electroweak scale: $${\cal H}^{\Delta S=1}= (G_F/\sqrt{2}) \tilde{C}
( \bar s T^a \gamma_\mu(1+\gamma_5)d )
\times
\sum_q (\bar q T^a \gamma^\mu q ) \ ,$$ $$\tilde{C}=\alpha_s
\sum_i{\xi_{di}\xi^*_{si}\over 6\pi}
{m_W^2\over M_Q^2}F({m_{H_i}\over M_Q^2})
\ .$$ $$F(x)=
{x^2(2x-3)\log x\over (1-x)^4}
+ {16x^2-29x+7 \over 6(1-x)^3} ; \, F(1)={3\over4}.$$ Written in terms of the operators in Ref.[@renormgroup] (but of flipped chirality), $${\cal H}^{\Delta S=1}=
(G_F / \sqrt{2}) \sum_{i=3}^{6} \tilde{C}_i \tilde Q_i \ ,$$ with $\tilde{C}_{4,6} = \tilde{C}/4$, $\tilde{C}_{3,5} = -\tilde{C}/(4N_c) $, $Q_{3 (5)} = (\bar s_i d_i)_{V+A}
\sum_q (\bar q_j q_j)_{V+A (V-A)}$ and $Q_{4 (6)} = (\bar s_i d_j)_{V+A}
\sum_q (\bar q_j q_i)_{V+A (V-A)}$, where we have adopted the common notation $(\bar q q)_{V+A} (\bar s d)_{V-A}
= \bar q \gamma^\mu (1+\gamma_5) q \; \bar s \gamma_\mu (1 - \gamma_5) d$. Again, for simplicity, we study the scenario that $m_2 \gg m_1$ and $m_1=M_Q
\simeq 300$ GeV. Numerically, $
\tilde{C}(\mu=300 \hbox{ GeV}) = 2.8\times 10^{-4}
(\xi_{di}\xi^*_{si})
R_Q^2
\ .
$ The Wilson coefficients are then run from $M_Q$ down to the charm mass scale via the leading logarithm renormalization group equations[@renormgroup], so that $\tilde{C}_i(\mu=m_c) = r_i
\tilde{C}(\mu=300 \hbox{GeV})$, where $(r_3,\cdots, r_6) =
(-0.16,0.22,-0.036,0.51)$. We note that the two other Wilson coefficients, $\tilde{C}_1,\tilde{C}_2$, are not generated in the evolution. Terms contributing to CP violation, and thus $\epsilon^{'}$, in $A_0$ are $\langle (\pi\pi)_0 | {\cal H}^{\Delta S=1} |K \rangle =
(G_F/\sqrt{2}) \sum_{i=3}^{6} \tilde{C}_i \langle (\pi\pi)_0 |
\tilde Q_i |K \rangle$. Using the expressions for the matrix elements $\langle (\pi\pi)_0 | \tilde Q_i |K
\rangle$ found in Ref.[@renormgroup] at the scale $\mu=m_c= 1.3
\hbox{ GeV}$, we obtain $$\langle (\pi\pi)_0 | \{\tilde Q_3 \dots \tilde Q_6\} |K \rangle
(\mu=m_c) = \{0.012,0.19, -0.10, -0.30\} \hbox{ GeV}^{3} \ ,$$ $$\hbox{Im}A_0 = -
\hbox{Im}({\cal A}_{sd})
R_Q^2
\times 2.5 \times 10^{-10} \hbox{ GeV} \ .$$ For the weak couplings considered here, $\hbox{Re}A_0$ is approximated quite well by the experimental value $|A_0| = 3.33\times 10^{-7} \hbox{
GeV}$, $\omega = 0.045$, so that $$\left({\epsilon' \over \epsilon}\right)
= 1.9 \times 10^{-5} \
\hbox{Im} \left(
{{\cal A}_{sd} \over (0.049)^2}
\right)R_Q^2
= \pm 1.9 \times 10^{-5} \
\left(\sqrt{(156{\cal F})^2 +1} -156{\cal F}
\right)^{1/2} {R_Q \over \sqrt{2}} \ .$$ The second equality is derived from constraints in Eq.(\[eq:dmineq\]). For $R_Q =1$ and ${\cal F} \approx 0$, $\epsilon'/\epsilon = 1.4 \times
10^{-5}$, which is somewhat smaller than, but certainly consistent with, the results of the FNAL-E731 measurement of $(7.4 \pm 5.9) \times
10^{-4}$[@fnalE731], but further from agreement with the CERN-NA31 result of $(23\pm 7)\times 10^{-4}$[@cernNA31]. If we relax the constraint on the contribution to $\Delta m_K$ to allow ${\cal F}=-0.3$ (reflecting the uncertainty due to the large long-distance contributions), then $\epsilon^{'}/\epsilon$ rises to $1.3 \times
10^{-4}$. If the omitted electromagnetic penguin contribution $\Omega_{\rm {EMP}}$ turns out to be negative and important, it could increase the predicted value of $\epsilon^{'}$ by perhaps as much as a factor of two, still well below the experimental limit.
Another (much weaker) constraint to be considered is that from the $B^0_{s,d}$ mass splitting[@renormgroup]. Proceeding in close analogy to the calculation of the contribution to $\Delta m_K$, we obtain: $$\Delta M_{B^0} =
2 \frac{G_F^2 m_W^2 }{16\pi^2} \frac{1}{2 m_B}\eta_B
\left(\frac{2}{3}\;\hbox{Re}{\cal A}_{bd}^2\;
\frac{m_W^2}{M_Q^2}\right)
\left( \frac{8}{3} B_B F_B^2 m_B^2 \right) \, ,$$ where again $m_2 \gg m_1$, $m_1=M_Q = 300$ GeV, with the renormalization group scaling factor $\eta_B=0.55$ evaluated as for $\Delta m_K$, and $B_B=1$, $F_B=180$ MeV, $m_B= 5.28$ GeV. Given the experimental value $\Delta M_{B^0} = 3.3\times 10^{-13}$ GeV, we have $$\delta(\Delta M_{B^0}) /\Delta M_{B^0} = 1.1\times 10^{-3}\,
R_Q^2 \,
\hbox{Re}\left({\cal A}_{bd} /0.049^2\right)^2 \ .$$ Even taking ${\cal A}_{bd}= (0.13)^2$, the fractional contribution is only about $5\%$.
Other Constraints {#other-constraints .unnumbered}
=================
[*$b \to s \gamma$*]{}: Because the operator due to charged Higgs diagrams has helicity opposite to that generated in the Standard Model contribution, the two do not interfere at amplitude level. Taking $m_2 \gg m_1=M_Q\simeq 300$ GeV: $${\delta B(b\rightarrow s\gamma)\over B(b\rightarrow s\gamma)_{\rm SM}}
=3.2\times 10^{-6}
\left|0.0389 \over {V_{tb}V^*_{ts}}\right|^2
R_Q^4
\left| \frac{ {\cal A}_{bs} }{ (0.049)^2} \
\right|^2 \ .$$ Furthermore, the relevant parameter ${\cal A}_{bd}$ is not subject to constraints from $\epsilon$ or $\epsilon'$. If it is of the same size as ${\cal A}_{sd}$, the deviation from the SM would be negligible.
[*Strong CP and $\theta_{\rm QCD}$*]{}: There are no tree level complex quark masses in our model, and $\theta_{\rm QCD}$ is only induced starting at the two-loop level, via generation of complex down-flavor quark masses. A typical diagram is shown in Fig. 1; in contrast to $\epsilon$ and $\epsilon^{'}$, this effect does not require more than one flavor of down-quark. Roughly, $\theta_{\rm QCD}\sim g^2 {\cal A}_{dd}\; {\rm
Im}{\cal B}/(16\pi^2)^2$. The present constraint, $\theta_{\rm QCD} < 10^{-9}$, can easily be accommodated, assuming a moderately small $\kappa$.
[*Neutron electric dipole moment*]{}: There is no one-loop diagram to produce the electric dipole moment (EDM) of the light quarks, so our model is very weakly constrained by neutron EDM limits. A down-flavor quark EDM, however, is generated at the two-loop level, in parallel with the generation of complex down quark masses discussed above. A typical contribution is given by Fig. 1, except with an external photon is attached to internal charged lines. An estimate of the two loop contribution is consistent with the current experimental bound.
[*Electron electric dipole moment*]{}: Unlike the down-flavor quarks, the electron couples only very indirectly with the CP violating sector. The electron EDM vanishes at the two loop level. We expect the three-loop level contribution to be insignificantly small.
[*Decay of new particles*]{}: In this model, $h$ and $Q$ can be assigned a new conserved quantum number which guarantees a lightest exotic particle, either $H_1$ or $Q$. A stable charged Higgs would lead to events with possibly large missing transverse energy, while a stable vector quark might be detected through formation of its bound states[@tom]. Alternatively, one can ignore this quantum number, so that an additional interaction, $h_\alpha L_i L_j$, should be present, which can lead to $H^-$ (on-shell or off-shell) decays into $l^-\nu$. Even in this case, lepton number is still conserved, just as in the Standard Model, since the vector quark and charged Higgs will naturally carry the lepton number ($L=\pm
2)$. Another way for $H$ to decay is to introduce a second Higgs doublet and let $H$ couple to two different Higgs doublets. In that case $H$ can decay into a neutral Higgs, plus a charged Higgs which in turn decays into ordinary quarks and leptons.
Spontaneously Broken CP symmetry {#spontaneously-broken-cp-symmetry .unnumbered}
================================
We shall comment on the corresponding model in which CP is broken spontaneously. This can be implemented by adding a CP-odd scalar, $a$, which develops a non-zero vacuum expectation value (VEV) and breaks CP. However, this scalar will in general couple to $\bar{Q_L}Q_R$ and give rise a complex tree level vector quark mass and, therefore, a tree level $\theta_{\rm QCD}$. To avoid this, one can add another CP-even scalar singlet, $s$, and impose discrete symmetries which change the signs of either or both $a$ and $s$ and nothing else. As a result, a term such as $ i a \bar Q \gamma_5 Q$ is forbidden and the only additional term relevant for CP violation is $i\left[ s \,a
\,({h_1}^{\dag} h_2 - {h_2}^{\dag} h_1) \right]$. This extra term will give rise to complex $(m^2)_{12}$ after both $s$ and $a$ develop VEVs and break CP. Note that before breaking CP spontaneously, there are two possible definitions of CP symmetry, depending on which of $s$ and $a$ are defined to be CP odd; this is why both must develop develop VEVs in order to break CP. The extra neutral Higgs will of course mix with the SM Higgs, but since $a$ does not couple to fermions directly, it will have scalar-pseudoscalar coupling to fermions only at the loop level. As a result, its contribution to any CP violating phenomenology will be small.
Conclusion {#conclusion .unnumbered}
==========
We have proposed a model whose CP violation is solely mediated by charged Higgs bosons. The model is surprisingly similar to the KM model in the sense that the CP-breaking mechanism is seemingly milliweak, while its phenomenology (as studied here) is quite superweak-like. The phenomenological distinction between the two will likely be made clear in experiments planned for the B factory; although our model predicts a real CKM matrix, with corresponding collapse of the unitarity KM triangle, new CP violating contributions will be contained in all the B decay processes designed to measure this triangle. A careful and detailed analysis of such issues is clearly necessary and is in progress[@UsAgain].
D. B.-C. and W.-Y. K. are supported by a grant from the Department of Energy, and D. C. by a grant from the National Science Council of R.O.C. We thank M. Adams, W. Bardeen, G. Boyd, P. Cho, B. Grinstein, J. Hughes, T. Imbo, R. Mohapatra, and L. Wolfenstein for useful discussions. D. C. also wishes to thank the High Energy Group and Physics Group of Argonne National Laboratory its hospitality while this work was in progress.
Figures {#figures .unnumbered}
-------
- A typical digram contributing to a complex down-flavor quark mass. A chiral rotation transforms this contribution into the effective $\theta_{\rm QCD}$ term. The cross represents the CP violating insertion of $(m^2)_{12}$. The same diagram, when attached with an external photon line, produces an EDM for the the $d$-quark.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
With heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll
m_D\ll m_B$, we analyze the $B\to D^0\overline D^0$ and $B_s\to
D^0\overline D^0$ decays, which occur purely via annihilation type diagrams. As a roughly estimation, we calculate their branching ratios and CP asymmetries in Perturbative QCD approach. The branching ratio of $B\to D^0\overline D^0$ is about $3.8\times10^{-5}$ that is just below the latest experimental upper limit. The branching ratio of $B_s\to D^0\overline D^0$ is about $6.8\times10^{-4}$, which could be measured in LHC-b. From the calculation, it could be found that this branching ratio is not sensitive to the weak phase angle $\gamma$. In these two decay modes, there exist CP asymmetries because of interference between weak and strong interaction. However, these asymmetries are too small to be measured easily.
author:
- |
Ying Li[^1] and Juan Hua\
[*Physics Department, Yantai University, Yantai 264005, China*]{}
title: 'Study of Pure Annihilation Decays $B_{d,s} \to D^{0} \overline D^{0}$ '
---
Introduction {#sc:intro}
============
In the Standard Model (SM), CP-violation (CPV) arises from a complex phase in the Cabibbo-Kobayashi- Maskawa (CKM) quark mixing matrix, and the angles of unitary triangle are defined as [@Yao:2006px]: $$\begin{aligned}
\beta=\arg \Bigl[-\frac{V_{cb}^*V_{cd}}{V_{tb}^*V_{td}}\Bigl],~~~~
\alpha=\arg \Bigl[-\frac{V_{tb}^*V_{td}}{V_{ub}^*V_{ud}}\Bigl],~~~~
\gamma=\arg
\Bigl[-\frac{V_{ub}^*V_{ud}}{V_{cb}^*V_{cd}}\Bigl].\label{ckmangle}\end{aligned}$$ In order to test SM and search for new physics, many measurements of CP-violation observables can be used to constrain these above angles. It is well known that we measure $\beta$ precisely using the golden decay mode $B\to J/\psi K_s$; the angle $\alpha$ can be determined with decay $B\to\pi\pi$ and $\gamma$ could be measured precisely in Large Hadron Collider (LHC) with decay mode $B_s\to
D_sK$.
![The quark level Feynman diagrams for $B_d \to D^{0} \bar
D^{0}$ process []{data-label="fig0"}](fig1.eps)
Besides the above channels mentioned, many other channels are used to cross check the measurements. Among these decays, $B\to DD$ decay is considered to test the $\beta$ measurement. For $B\to DD$ decay, the analysis based on $SU(3)$ symmetry [@Savage:1989ub], iso-spin symmetry [@Xing:1999yx] and factorization approach [@Xing:1998ca] have been done in last several years. However, the the calculation of decay $B^0 \to D^0\overline D^0$ has difficulties. It is a pure-annihilation diagram decay, also named W-exchange diagram decay, which is power suppressed in factorization language. The quark diagrams of this decay are shown in Figure \[fig0\]. Theoretically, QCD factorization approach (QCDF) [@Beneke:1999br] and soft collinear effective theory (SCET)[@Bauer:2001yt] can not deal decays with two heavy charmed mesons effectively. In Ref.[@Keum:2003js; @Lu:2003xc], perturbative QCD (PQCD) has been exploited to $B$ meson decays with one charm meson in the final states and the results agree with experimental data well. Specially, the pure annihilation type B decays with charmed meson were studied in Ref.[@Lu:2003xc].
In the standard model picture, the $W$ boson exchange causes $\bar{b}d \to \bar{c}c $, and the $\bar{u}u$ quarks are produced from a gluon. This gluon attaches to any one of the quarks participating in the $W$ boson exchange. In decay $B \to D^{0}
\overline D^{0}$, the momentum of the final state $D$ meson is $\frac{1}{2} m_B (1-2 r^2)$, with $r=m_D/m_B$. If we consider heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll m_D\ll
m_B$, the $D$ meson momentum is nearly $m_B/2$. According to the distribution amplitude used in Ref.[@Keum:2003js], the light quark in $D$ meson carrying nearly $40\%$ of the $D$ meson momentum. So, this light quark is still a collinear quark with 1 $\mathrm{GeV}$ energy, like that in $B\to DM$ [@Keum:2003js; @Lu:2003xc], $B\to K(\pi)\pi$ [@kls; @luy] decays. The gluon could be viewed as a hard gluon approximatively, so we can treat the process perturbatively where the four-quark operator exchanges a hard gluon with $u \bar u$ quark pair. Of course, we are able to calculate the diagrams if charm quark and up quark exchange. As a roughly estimation, we give the branching ratio and CP-violation of $B_{d,s} \to D^{0} \overline D^{0}$.
In this article, the analytic formulas for the decay amplitudes will be shown in the next section. In section \[sc:result\], we give the numerical results and summarize this article in section \[sc:summary\].
Analytic formulas {#sc:analy}
=================
For simplicity, we set $B$ meson at rest in our calculation. In light-cone coordinates, the momentum of $B$, $D^0$ and$\overline
D^0$ are: $$\begin{aligned}
P_B=\frac{M_B}{\sqrt{2}}(1,1,\vec{0});
P_2=\frac{M_B}{\sqrt{2}}(1-r^2,r^2,\vec{0});
P_3=\frac{M_B}{\sqrt{2}}(r^2,1-r^2,\vec{0}).\end{aligned}$$ we define the light (anti-)quark momenta in $B$, $D^0$ and $\overline D^0$ mesons as $k_1$, $k_2$, and $k_3$ as: $$k_1 =
(x_1P_1^+,0,{\bf k}_{1T}),\ \ k_2 = (x_2 P_2^+,0,{\bf k}_{2T}),\ \
k_3 = (0, x_3 P_3^-,{\bf k}_{3T}). \label{eq:momentun2}$$
In PQCD, we factorize the decay amplitude into soft($\Phi$), hard($H$), and harder ($C$) dynamics characterized by different scales, [@kls; @luy] $$\begin{gathered}
\mathcal{A}\sim
\int\!\! d x_1 d x_2 d x_3 b_1 d b_1 b_2 d b_2 b_3
d b_3 \mathrm{Tr} \Bigl[ C(t) \Phi_B(x_1,b_1) \Phi_{D}(x_2,b_2)
\Phi_D(x_3, b_3) H(x_i, b_i,t) S_t(x_i)\, e^{-S(t)} \Bigr].
\label{eq:convolution2}\end{gathered}$$ In above equation, $b_i$ is the conjugate space coordinate of the transverse momentum ${\bf k}_{iT}$, and $t$ is the largest energy scale. $C$ is Wilson coefficient, and $\Phi$ is the wave function. The last term, $e^{-S(t)}$, contains two kinds of contributions. One is due to the resummation of the large double logarithms from renormalization of ultra-violet divergence $\ln tb$, the other is from resummation of double logarithm $\ln^2 b$ from the overlap of collinear and soft gluon corrections, which is called Sudakov form factor. The hard part $H$ can be calculated perturbatively, and it is channel dependent. More explanation of above formula and review about PQCD can be found in many reference, such as [@kls; @luy; @Ali:2007ff].
As a heavy meson, the $B$ meson wave function is not well defined, neither is $D$ meson. In heavy quark limit, we take them as: $$\Phi_{B}(x,b) = \frac{i}{\sqrt{6}}
\left[ \not \! P + M_B \right] \gamma_5\phi_B(x,b),$$ $$\Phi_{D}(x,b) = \frac{i}{\sqrt{6}}\gamma_5
\left[ \not \! P + M_D \right] \phi_D(x,b).$$ The Lorentz structure of two mesons are different because the $B$ meson is initials state and $D$ meson is final state.
The effective Hamiltonian $\bar b\to \bar q(q=d,s)$ is given by [@Buchalla:1996vs]: $$\begin{gathered}
\mathcal{H}_\mathrm{eff} =
\frac{G_F}{\sqrt{2}}\Bigl\{ V_{cq}V_{cb}^* \Bigl[
C_1(\mu) O_1^c(\mu) + C_2(\mu) O_2^c(\mu) \Bigl]+ V_{uq}V_{ub}^*
\Bigl[ C_1(\mu) O_1^u(\mu) + C_2(\mu) O_2^u(\mu) \Bigl]\\
-V_{tb}^*V_{tq}\sum_{i=3}^{10}C_i(\mu) O_i(\mu)\Bigl\},\label{hami}\end{gathered}$$ where $C_{i}(\mu)(i=1,\cdots,10)$ are Wilson coefficients at the renormalization scale $\mu$ and the four quark operators $O_{i}(i=1,\cdots,10)$ are $$\begin{array}{ll}
O_1^c = (\bar{b}_ic_j)_{V-A}(\bar{c}_jq_i)_{V-A}, &
O_2^c = (\bar{b}_ic_i)_{V-A} (\bar{c}_jq_j)_{V-A}, \\
O_1^u = (\bar{b}_iu_j)_{V-A}(\bar{u}_jq_i)_{V-A}, &
O_2^u = (\bar{b}_iu_i)_{V-A} (\bar{u}_jq_j)_{V-A}, \\
O_3 = (\bar{b}_iq_i)_{V-A}\sum_{q} (\bar{q}_jq_j)_{V-A}, &
O_4 = (\bar{b}_iq_j)_{V-A}\sum_{q} (\bar{q}_jq_i)_{V-A}, \\
O_5 = (\bar{b}_iq_i)_{V-A}\sum_{q} (\bar{q}_jq_j)_{V+A}, &
O_6 = (\bar{b}_iq_j)_{V-A} \sum_{q} (\bar{q}_jq_i)_{V+A}, \\
O_7 = \frac{3}{2}(\bar{b}_iq_i)_{V-A} \sum_{q}
e_q(\bar{q}_jq_j)_{V+A}, &
O_8 = \frac{3}{2}(\bar{b}_iq_j)_{V-A}\sum_{q} e_q
(\bar{q}_jq_i)_{V+A}, \\
O_9 = \frac{3}{2}(\bar{b}_iq_i)_{V-A}\sum_{q}
e_q(\bar{q}_jq_j)_{V-A}, &
O_{10} = \frac{3}{2}(\bar{b}_iq_j)_{V-A}\sum_{q}
e_q(\bar{q}_jq_i)_{V-A}. \label{eq:effectiv}
\end{array}$$ Here $i$ and $j$ are $SU(3)$ color indices; in $O_{3,...,10}$ the sum over $q$ runs over the quark fields that are active at the scale $\mu=O(m_{b})$, i.e., $q\in \{u,d,s,c,b\}$. For Wilson coefficients, we will also use the leading logarithm summation for QCD corrections, although the next-to -leading order calculation already exists [@Buchalla:1996vs]. This is the consistent way to cancel the explicit $\mu$ dependence in the theoretical formulae.
![The leading order Feynman diagrams for $B_d \to D^{0}
\overline D^{0}$ process in PQCD approach[]{data-label="fig1"}](fig2.eps)
According to the effective Hamiltonian in eq.(\[hami\], \[eq:effectiv\]), the lowest order diagrams of $B \to D^0\overline
D^0$ are drawn in Fig. \[fig1\]. We first calculate the usual factorizable diagrams (a), (b), (c) and (d). For the $(V-A)(V-A)$ operators, their contributions of (a) and (c) are always canceled by diagrams (b) and (d) respectively because of current conservation. For the $(V-A)(V+A)$ operators, these diagrams can not give contribution, either. That’s to say, factorizable diagrams have no contribution. For non-factorizable diagrams (e), (f), (g) and (h), we find the hard part for $(V-A)(V-A)$ operators are same to $(V-A)(V+A)$ operators. We group the contribution of diagrams (e) and (f), denoted by $M_a$, as follows: $$\begin{gathered}
M_{a}[C_i] = \frac{64 \pi C_FM_B^2}{\sqrt{2N_C}} \int_0^1 \!\!
dx_1 dx_2 dx_3
\int_0^\infty \!\! b_1 db_1\, b_2 db_2\
\phi_B(x_1,b_1) \phi_{D}(x_2,b_2)\phi_{D}(x_3,b_2) \\
\times \Bigl\{ \Bigl[x_1+x_2+(2x_3-x_2)r^2\Bigl]
C_i(t_{a}^1)E(t_{a}^1) h_a^{(1)}(x_1,
x_2,x_3,b_1,b_2) \\
+\Bigl[-x_3+(2x_1-2x_2+x_3)r^2\Bigl] C_i(t_{a}^2)E(t_{a}^2)
h_a^{(2)}(x_1, x_2,x_3,b_1,b_2) \Bigr\}, \label{eq:Ma}\end{gathered}$$ where $C_F = 4/3$ is the group factor of $\mathrm{SU}(3)_c$ gauge group, and $C_i$ is Wilson coefficient. The function $E_m$ is defined as $$E(t) = \alpha_s(t)\, e^{-S_B(t)-S_D(t)-S_D(t)},$$ and $S_B$, $S_D$ result from Sudakov factor and single logarithms due to the renormalization of ultra-violet divergence. The functions $h_a$ is the Fourier transformation of virtual quark and gluon propagators. It is defined by $$\begin{aligned}
&h^{(j)}_a(x_1,x_2,x_3,b_1,b_2) = \nonumber \\
& \biggl\{ \frac{\pi i}{2}
\mathrm{H}_0^{(1)}(M_B\sqrt{x_2x_3(1-2r^2)}\, b_1)
\mathrm{J}_0(M_B\sqrt{x_2x_3(1-2r^2)}\, b_2) \theta(b_1-b_2)
\nonumber \\
& \qquad\qquad\qquad\qquad + (b_1 \leftrightarrow b_2) \biggr\}
\times\left(
\begin{matrix}
\mathrm{K}_0(M_B F_{a(j)} b_1), & \text{for}\quad F^2_{a(j)}>0 \\
\frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{|F^2_{a(j)}|}\ b_1), &
\text{for}\quad F^2_{a(j)}<0
\end{matrix}\right),
\label{eq:propagator1}
\end{aligned}$$ with: $$\begin{aligned}
F^2_{a(1)} &=&-x_1-x_2-x_3+x_1x_3+x_2x_3+(x_2+x_3-x_1x_3-2x_2x_3)r^2;\\
F^2_{a(2)}&=&x_2x_3-x_1x_3+(x_1x_3-2x_2x_3)r^2.\end{aligned}$$ In above equation, $\mathrm{H}_0^{(1)}(z) = \mathrm{J}_0(z) + i\,
\mathrm{Y}_0(z)$. In order to reduce the large logarithmic radiative corrections, the hard scale $t$ in the amplitudes is selected as the largest energy scale in the hard part: $$\begin{aligned}
t_{a}^j &=& \mathrm{max}(M_B \sqrt{|F^2_{a(j)}|},
M_B \sqrt{(1-2r^2)x_2x_3 }, 1/b_1,1/b_2).
\end{aligned}$$ Analogically, we can get the $M_{b}$, which comes from the contribution of diagrams (g) and (h): $$\begin{gathered}
M_{b}[C_i]= \frac{64 \pi C_FM_B^2}{\sqrt{2N_C}} \int_0^1 \!\! dx_1
dx_2 dx_3
\int_0^\infty \!\! b_1 db_1\, b_2 db_2\
\phi_B(x_1,b_1) \phi_{D}(x_2,b_2)\phi_{D}(x_3,b_2) \\
\times \Bigl\{ \Bigl[1-x_3+(2+2x_1-2x_2+x_3)r^2\Bigl]C_i(t_{b}^1)
E(t_{b}^1) h_b^{(1)}(x_1,
x_2,x_3,b_1,b_2) \\
+\Bigl[x_1+x_2-1+(-2-x_2+2x_3)r^2\Bigl]C_i(t_{b}^2) E(t_{b}^2)
h_b^{(2)}(x_1, x_2,x_3,b_1,b_2) \Bigr\}, \label{eq:Mb}\end{gathered}$$ and the functions are defined as: $$\begin{aligned}
&h^{(j)}_b(x_1,x_2,x_3,b_1,b_2) = \nonumber \\
& \biggl\{ \frac{\pi i}{2}
\mathrm{H}_0^{(1)}(M_B\sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2}\,
b_1) \nonumber\\
&~~~~~~~~
\times\mathrm{J}_0(M_B\sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2}\,
b_2) \theta(b_1-b_2)
\nonumber \\
& \qquad\qquad\qquad\qquad + (b_1 \leftrightarrow b_2) \biggr\}
\times\left(
\begin{matrix}
\mathrm{K}_0(M_B F_{b(j)} b_1), & \text{for}\quad F^2_{b(j)}>0 \\
\frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{|F^2_{b(j)}|}\ b_1), &
\text{for}\quad F^2_{b(j)}<0
\end{matrix}\right);
\label{eq:propagator2}
\end{aligned}$$ $$\begin{aligned}
F^2_{b(1)}&=&-1-x_1x_3+x_2x_3+(x_1x_3-2x_2x_3)r^2,\\
F^2_{b(2)}&=&1-x_1-x_2-x_3+x_1x_3+x_2x_3+(x_2+x_3-x_1x_3-2x_2x_3)r^2,
\end{aligned}$$ $$\begin{aligned}
t_{b}^j &=& \mathrm{max}(M_B \sqrt{|F^2_{b(j)}|},
M_B \sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2 }, 1/b_1,1/b_2).
\end{aligned}$$ So, the decay amplitude of decay $B_d \to D^{0} \overline D^{0}$ can be read as: $$\begin{aligned}
\mathcal{A}_1&=&V_{cb}^*V_{cd}M_{a}[C_2]-V_{tb}^*V_{td}M_{a}[C_5+C_7]+V_{ub}^*V_{ud}M_{b}[C_2]
-V_{tb}^*V_{td}M_{b}[C_5+C_7]\nonumber\\
&=&V_{cb}^*V_{cd}T_1
-V_{tb}^*V_{td}P_1\nonumber\\
&=&V_{tb}^*V_{td}P_1(1+z_1e^{i(\beta+\delta_1)}),\label{bdd1}\end{aligned}$$ where $\beta$ is weak phase angle defined in Eq.(\[ckmangle\]), and $\delta_1$ is the strong phase, which plays an important role in studying CP-violation. In above calculation, we denote that $$\begin{aligned}
T_1&=&M_{a}[C_2]-M_{b}[C_2], \nonumber \\
P_1&=&M_{a}[C_5+C_7]+M_{b}[C_5+C_7]+M_{b}[C_2],\end{aligned}$$ and $$\begin{aligned}
z_1=\left|\frac{V_{cb}^*V_{cd}}{V_{tb}^*V_{td}}
\right|\left|\frac{T_1}{P_1}\right|,\end{aligned}$$ which describes the ratio between tree diagram and penguin diagram. The corresponding charge conjugate decay is $$\begin{aligned}
\label{bdd2}
\overline{\mathcal{A}_1}=V_{tb}V_{td}^*P_1(1+z_1e^{i(-\beta+\delta_1)}).\end{aligned}$$ Therefore, the averaged decay width $\Gamma$ for $B^0 \to D^0
\overline D^0$ decay is then given by $$\Gamma(B^0 \to D^0
\overline D^0) = \frac{G_F^2 M_B^3}{128\pi}
(1-2r^2)|V_{tb}^*V_{td}P_1|^2
\bigl|1+z_1^2+2z_1\cos \beta \cos\delta_1|. \label{eq:neut_width1}$$ >From this equation, we know that the averaged branching ratio is a function of CKM angle $\beta$, if $z_1\neq 0$. Derived from Eq.(\[bdd1\]) and Eq.(\[bdd2\]), the direct CP-violation can be formulated as: $$\begin{aligned}
\label{cp1}
A_{CP}^{dir}(B\to D^{0} \overline D^{0})=\frac{|A_{B_d\to
D^0\overline D^0}|^2-|A_{\overline B_d\to \overline
D^0D^0}|^2}{|A_{B_d\to D^0\overline D^0}|^2+|A_{\overline B_d\to
\overline
D^0D^0}|^2}=\frac{-2z_1\sin\beta\sin\delta_1}{1+z_1^2+2z_1\cos \beta
\cos\delta_1}.
\end{aligned}$$
For $ B^0_s \to D^0 \overline D^0$ and its conjugate decay, we write the decay amplitudes and rearrange them as: $$\begin{aligned}
\mathcal{A}_2&=&V_{cb}^*V_{cs}M_{a}[C_2]-V_{tb}^*V_{ts}M_{a}[C_5+C_7]+V_{ub}^*V_{us}M_{b}[C_2]
-V_{tb}^*V_{ts}M_{b}[C_5+C_7]\nonumber\\
&=&V_{ub}^*V_{us}M_{b}[C_2]-V_{tb}^*V_{ts}\Bigl\{M_{a}[C_5+C_7]
+M_{b}[C_5+C_7]-\frac{V_{cb}^*V_{cs}}{V_{tb}^*V_{ts}}M_{a}[C_2]\Bigl\}\nonumber\\
&=&V_{ub}^*V_{us}T_2-V_{tb}^*V_{ts}P_2\nonumber\\
&=&V_{ub}^*V_{us}T_2\Bigl[1+z_2e^{i(-\gamma+\delta_2)}\Bigl],\\
\overline{\mathcal{A}_2}&=&V_{ub}V_{us}^*T_2\Bigl[1+z_2e^{i(\gamma+\delta_2)}\Bigl],\label{bdd3}\end{aligned}$$ where $T_2$, $P_2$ and $z_2$ are defined as: $$\begin{aligned}
T_2&=&M_{b}[C_2],\nonumber\\
P_2&=&M_{a}[C_5+C_7]
+M_{b}[C_5+C_7]-\frac{V_{cb}^*V_{cd}}{V_{tb}^*V_{ts}}M_{a}[C_2],\nonumber\\
z_2&=&\left|\frac{V_{tb}^*V_{ts}}{V_{ub}^*V_{us}}
\right|\left|\frac{T_2}{P_2}\right|.\end{aligned}$$ So, the averaged decay width and direct CP violation can be formulated as: $$\begin{aligned}
\label{eq:neut_width2}
\Gamma(B_s \to D^0\overline D^0) &=& \frac{G_F^2 M_B^3}{128\pi} (1-2r^2) \bigl|V_{ub}V_{us}^*T_2\bigr|^2
(1+z_2^2+2z_2\cos\delta_2\cos\gamma),\\
A_{CP}^{dir}(B_s \to D^0\overline D^0)&=&\frac{|A_{B_s\to
D^0\overline D^0}|^2-|A_{\overline B_s\to \overline
D^0D^0}|^2}{|A_{B_s\to D^0\overline D^0}|^2+|A_{\overline B_s\to
\overline
D^0D^0}|^2}=\frac{2z_2\sin\gamma\sin\delta_2}{1+z_2^2+2z_2\cos
\gamma \cos\delta_2}.\end{aligned}$$ In our calculation, we set $m_c\approx m_D$, just because $m_D -m_c
\approx \Lambda_{QCD}$ and $\frac{\Lambda_{QCD}}{m_B} \rightarrow 0$ in the heavy quark limit.
Numerical Results {#sc:result}
=================
For $B$ meson, the distribution amplitude is well determined by charmless $B$ decays [@kls; @luy], which is chosen as $$\phi_B(x,b) = N_B x^2(1-x)^2 \exp \left[ -\frac{M_B^2\ x^2}{2
\omega_b^2} -\frac{1}{2} (\omega_b b)^2 \right],\label{waveb}$$ with parameters $\omega_b=0.4\mbox{ GeV}$, and $N_B=91.745\mbox{
GeV}$ which is the normalization constant using $f_B=190 \mbox{
MeV}$. For $B_s$ meson, we use the same wave function according to SU(3) symmetry, where $\omega_b=0.4\mbox{ GeV}$, $N_{B_s}=119.4
\mbox{ GeV}$ and $f_{B_s}=230 \mbox{ MeV}$.
Since the $c$ quark is much heavier than the $u$ quark, the $c$ quark shares more momentum, and this function should be asymmetric with respect to $x = 1/2$. The asymmetry is parameterized by $a_D$. Similar to the $b$-dependence on the wave function of $B$ meson, for controlling the size of charmed mesons, we also introduce the intrinsic $b$-dependence on those of charmed mesons. Hence, we use the wave function of $D$ meson as [@Chen:2003px] $$\phi_{D}(x,b) = \frac{3}{\sqrt{2 N_c}} f_{D} x(1-x)\Bigl[ 1 +
a_{D} (1 -2x)\Bigl]\exp \left[-\frac{1}{2} (\omega_D b)^2
\right].\label{waved}$$ We use $a_{D}=0.7$ and $\omega_{D}=0.4$ in above function. Other parameters, such as meson mass, decay constants, the CKM matrix elements and the lifetime of $B$ meson are list [@Yao:2006px; @Follana:2007uv]: $$\begin{gathered}
M_B = 5.28 \mbox{ GeV},\ M_{B_S} = 5.36 \mbox{ GeV},\ M_{D} = 1.87
\mbox{ GeV},\ f_{D} = 210
\mbox{ MeV}, \nonumber \\
|V_{ud}|=0.974, \ |V_{ub}|= 4.3 \times 10^{-3},\ |V_{cd}|=0.23,
\ |V_{cb}|= 41.6\times 10^{-3} \nonumber\\
|V_{td}|= 7.4\times 10^{-3}, \ |V_{tb}|=1.0,\
|V_{us}|=0.226, \ |V_{cs}|=0.957,\nonumber \\
|V_{ts}|=41.6\times 10^{-3}, \quad\tau_{B_d^0}=1.54\times 10^{-12}\mbox{ s,}\quad
\tau_{B_s^0}=1.46\times 10^{-12}\mbox{ s}.\label{parameterZ}\end{gathered}$$
$B_d\to D^0\overline D^0$ $B_s\to D^0\overline D^0$
------------- --------------------------- ---------------------------
$T(e)+T(f)$ $68+17 \, i$ $66 +27 \, i$
$P(e)+P(f)$ $ 0.80+ 0.23\, i$ $0.77 +3.68\, i$
$T(g)+T(h)$ $ 9.81- 2.99\, i$ $14.0-0.6\, i$
$P(g)+P(h)$ $ 0.08- 0.02\, i$ $-0.01 +0.01\, i$
: Amplitudes ($10^{-3}$ GeV) of $B_d\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$.[]{data-label="tb:amplitudes"}
![The branching ratio of $B_s\to D^0\overline D^0$ changes with CKM angle $\gamma$.[]{data-label="fig3"}](fig3.eps)
With these parameters fixed, we calculate the decay amplitudes of the $B^0\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$ decays in Table \[tb:amplitudes\]. From the table, we notice that the main contribution comes from the tree diagram (e) and (f). And our predictions for the branching ratio of each mode corresponding to $\beta=23^\circ$ and $\gamma=63^\circ$ are listed, $$\begin{aligned}
BR(B_d\to D^0\overline D^0) &=& 2.3\times 10^{-5}; \nonumber\\
BR(B_s\to D^0\overline D^0) &=& 6.8\times 10^{-4}.\end{aligned}$$ In Fig. \[fig3\], we plot the branching ratio of $B_s\to
D^0\overline D^0$ with different $\gamma$. In this figure, we find the branching ratio is not sensitive to CKM angle $\gamma$. For the experimental side, there are only upper limits given at $90\%$ confidence level for decay $B_d\to D^0\overline D^0$, $$\begin{aligned}
BR(B_d\to D^0\overline D^0) &< &6.0\times 10^{-5}; ~~~~~~BarBar\cite{Aubert:2006ia}\nonumber\\
BR(B_d\to D^0\overline D^0) &< &4.2\times 10^{-5}. ~~~~~~Belle\cite{:2007sk}\end{aligned}$$ Obviously, our result is consistent with the data. For $B_d\to
D^0\overline D^0$ decay mode, $z_1$ is about 6.5, and the strong phase $\delta_1$ is $34^\circ$, so $A_{CP}^{dir}$ is about $-6\%$ with the definition in Eq.(\[cp1\]). As decay mode $B_s\to
D^0\overline D^0$ is concerned, $z_2$ is about $205$ and $\delta_2=155^\circ$, and the relation between direct CP violation and $\gamma$ is shown in Fig.\[fig4\]. From the figure, we read the CP asymmetry is about $0.4\%$, which is rather tiny. It is necessary to state that the $z_1$ and $z_2$ are not the true ratio between tree contribution and penguin, because mathematical technics are used in Eq. (\[bdd1\]) and (\[bdd3\]).
![The direct CP-violation of $B_s\to D^0\overline D^0$ changes with CKM angle $\gamma$.[]{data-label="fig4"}](fig4.eps)
In addition to the perturbative annihilation contributions, there is also a hadronic picture for the $B_d\to D^0\overline D^0$, named soft final states interaction[@Cheng:2004ru]. The $B$ meson decays into $D^+$ and $D^-$, the secondary particles then exchanging a $\rho$ meson, then scatter into $D^0\overline D^0$ through final state interaction afterwards. For $B_s$ decay, the $B_s$ meson decays into $D_s^+$ and $D^+$ then scatters into $D^0\overline D^0$ by exchanging a Kaon. But this picture cannot be calculated accurately because of lack of many effective vertexes, and we will ignore this contribution here, though it may be important [@Cheng:2004ru].
---------------------------- ------------------------------- ------------------------------- ----------------------------------------- -----------------------------------------
$BR(B_d\to D^0\overline D^0)$ $BR(B_s\to D^0\overline D^0)$ $A_{CP}^{dir}(B_d\to D^0\overline D^0)$ $A_{CP}^{dir}(B_s\to D^0\overline D^0)$
$(\times 10^{-5})$ $(\times 10^{-4})$ $(\%)$ $(\%)$
$\omega_b(B\setminus B_s)$
$0.35\setminus0.45$ 4.3 7.8 -7.2 0.4
$0.40\setminus0.50$ 3.8 6.8 -5.3 0.4
$0.45\setminus0.55$ 3.2 5.9 -5.8 0.4
$\omega_D$
$0.35$ 5.0 9.7 -4.2 0.3
$0.40$ 3.8 6.8 -5.3 0.4
$0.45$ 2.2 4.2 -7.8 0.5
$a_D$
$0.6$ 3.2 5.9 -6.9 0.4
$0.7$ 3.8 6.8 -5.3 0.4
$0.8$ 4.3 7.8 -6.1 0.4
---------------------------- ------------------------------- ------------------------------- ----------------------------------------- -----------------------------------------
: The sensitivity of the decay branching ratios and CP asymmetries to change of ¥ø$\omega_b$, $\omega_D$ and $a_D$[]{data-label="table"}
There are many uncertainties in our calculation such as higher order corrections, the parameters listed in Eq.(\[parameterZ\]) and the distribution amplitudes of heavy mesons. We will not discuss uncertainty taken by high order correction as we only roughly estimate the branching ratios and CP asymmetries, though high order corrections have been done for some special channels [@Li:2005kt; @Li:2006jv] and showed $15-20\%$ uncertainty. The parameters in Eq.(\[parameterZ\]), fixed by experiments, are proportional to the amplitudes, so we will not analyze this kind uncertainty either. In our calculation, we find that the results are sensitive to the distribution amplitudes, especially to that of $D$ meson. Since the heavy $D$ wave function is less constrained, we set $a_{D}\in (0.6-0.8)\mbox{ GeV}$ and $\omega_{D}\in(0.35-0.45)\mbox{
GeV}$ to exploit the uncertainty. Table \[table\] shows the sensitivity of the branching ratios to change of $\omega_b$, $\omega_D$ and $a_{D}$. It is found that uncertainty of the predictions on PQCD is mainly due to $\omega_D$, which describes the behavior in end-point region of $D$ meson, however it is very hard to be determined. Considering the experimental upper limit, our results favor large $\omega_b$, large $\omega_D$ and small $a_D$.
At last, we give the prediction of branching ratios with err bar as follows: $$\begin{aligned}
BR(B_d\to D^0\overline D^0) &=&(3.8^{+0.5+1.2+0.5}_{-0.6-1.6-0.6})\times 10^{-5}
\left(\frac{f_B\cdot f_D\cdot f_D}{190\mathrm{ GeV}\cdot 210\mathrm{ GeV}\cdot 210\mathrm{ GeV}}\right)^2; \nonumber\\
BR(B_s\to D^0\overline D^0) &=&(6.8^{+1.0+2.9+1.0}_{-0.9-2.6-0.9})\times 10^{-4}
\left(\frac{f_{B_s}\cdot f_D\cdot f_D}{230\mathrm{ GeV}\cdot 210\mathrm{ GeV}\cdot 210\mathrm{ GeV}}\right)^2.\end{aligned}$$ We believe that the $B_d\to D^0\overline D^0$ will be measured soon because this ratio is just below the upper limit, and $B_d\to
D^0\overline D^0$ will be measured in LHC-b in next year as a channel to cross check the $\gamma$ measurements.
Summary {#sc:summary}
=======
With heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll
m_D\ll m_B$, we analyze the $B\to D^0\overline D^0$ and $B_s\to
D^0\overline D^0$ decays, which occur purely via annihilation type diagrams. As a roughly estimation, we calculate the branching ratios and CP asymmetries in PQCD approach. The branching ratios are still sizable. The branching ratio of $B\to D^0\overline D^0$ is about $3.8\times10^{-5}$, which is just below the experimental upper limited result[@Aubert:2006ia; @:2007sk], and we think that it will be measured in near future. For $B_s\to D^0\overline D^0$, the branching ratio is about $6.8\times10^{-4}$, which could be measured in LHC-b. From the calculation, it is found that this branching ratio is not sensitive to angle $\gamma$. In these two decays, there exist CP asymmetries because of interference between weak and strong interaction, though they are very small.
Acknowledgements {#acknowledgements .unnumbered}
================
Y.Li thanks Institute of High Energy Physics for their hospitality during visit where part of this work was done. This work is partly supported by Foundation of Yantai University under Grant No.WL07B19. We would like to acknowledge C.D. Lü and W. Wang for valuable discussions.
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[^1]: e-mail: liying@ytu.edu.cn
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We describe likelihood-based statistical tests for use in high energy physics for the discovery of new phenomena and for construction of confidence intervals on model parameters. We focus on the properties of the test procedures that allow one to account for systematic uncertainties. Explicit formulae for the asymptotic distributions of test statistics are derived using results of Wilks and Wald. We motivate and justify the use of a representative data set, called the “Asimov data set”, which provides a simple method to obtain the median experimental sensitivity of a search or measurement as well as fluctuations about this expectation.'
---
[Asymptotic formulae for likelihood-based tests of new physics]{}
Glen Cowan$^1$, Kyle Cranmer$^2$, Eilam Gross$^3$, Ofer Vitells$^3$
$^1$ Physics Department, Royal Holloway, University of London, Egham, TW20 0EX, U.K.\
$^2$ Physics Department, New York University, New York, NY 10003, U.S.A.\
$^3$ Weizmann Institute of Science, Rehovot 76100, Israel
Keywords: systematic uncertainties, profile likelihood, hypothesis test, confidence interval, frequentist methods, asymptotic methods
Introduction {#sec:intro}
============
In particle physics experiments one often searches for processes that have been predicted but not yet seen, such as production of a Higgs boson. The statistical significance of an observed signal can be quantified by means of a $p$-value or its equivalent Gaussian significance (discussed below). It is useful to characterize the sensitivity of an experiment by reporting the expected (e.g., mean or median) significance that one would obtain for a variety of signal hypotheses.
Finding both the significance for a specific data set and the expected significance can involve Monte Carlo calculations that are computationally expensive. In this paper we investigate approximate methods based on results due to Wilks [@Wilks] and Wald [@Wald] by which one can obtain both the significance for given data as well as the full sampling distribution of the significance under the hypothesis of different signal models, all without recourse to Monte Carlo. In this way one can find, for example, the median significance and also a measure of how much one would expect this to vary as a result of statistical fluctuations in the data.
A useful element of the method involves estimation of the median significance by replacing the ensemble of simulated data sets by a single representative one, referred to here as the “Asimov” data set.[^1] In the past, this method has been used and justified intuitively (e.g., [@quast; @CSC]). Here we provide a formal mathematical justification for the method, explore its limitations, and point out several additional aspects of its use.
The present paper extends what was shown in Ref. [@CSC] by giving more accurate formulas for exclusion significance and also by providing a quantitative measure of the statistical fluctuations in discovery significance and exclusion limits. For completeness some of the background material from [@CSC] is summarized here.
In Sec. \[sec:formalism\] the formalism of a search as a statistical test is outlined and the concepts of statistical significance and sensitivity are given precise definitions. Several test statistics based on the profile likelihood ratio are defined.
In Sec. \[sec:qdist\], we use the approximations due to Wilks and Wald to find the sampling distributions of the test statistics and from these find $p$-values and related quantities for a given data sample. In Sec. \[sec:sensitivity\] we discuss how to determine the median significance that one would obtain for an assumed signal strength. Several example applications are shown in Sec. \[sec:examples\], and numerical implementation of the methods in the RooStats package is described in Sec. \[sec:roostats\]. Conclusions are given in Sec. \[sec:conclusions\].
Formalism of a search as a statistical test {#sec:formalism}
===========================================
In this section we outline the general procedure used to search for a new phenomenon in the context of a frequentist statistical test. For purposes of discovering a new signal process, one defines the null hypothesis, $H_0$, as describing only known processes, here designated as background. This is to be tested against the alternative $H_1$, which includes both background as well as the sought after signal. When setting limits, the model with signal plus background plays the role of $H_0$, which is tested against the background-only hypothesis, $H_1$.
To summarize the outcome of such a search one quantifies the level of agreement of the observed data with a given hypothesis $H$ by computing a $p$-value, i.e., a probability, under assumption of $H$, of finding data of equal or greater incompatibility with the predictions of $H$. The measure of incompatibility can be based, for example, on the number of events found in designated regions of certain distributions or on the corresponding likelihood ratio for signal and background. One can regard the hypothesis as excluded if its $p$-value is observed below a specified threshold.
In particle physics one usually converts the $p$-value into an equivalent significance, $Z$, defined such that a Gaussian distributed variable found $Z$ standard deviations above[^2] its mean has an upper-tail probability equal to $p$. That is,
$$\label{eq:significance}
Z = \Phi^{-1}(1-p) \,,$$
where $\Phi^{-1}$ is the quantile (inverse of the cumulative distribution) of the standard Gaussian. For a signal process such as the Higgs boson, the particle physics community has tended to regard rejection of the background hypothesis with a significance of at least $Z=5$ as an appropriate level to constitute a discovery. This corresponds to $p = 2.87 \times 10^{-7}$. For purposes of excluding a signal hypothesis, a threshold $p$-value of 0.05 (i.e., 95% confidence level) is often used, which corresponds to $Z = 1.64$. It should be emphasized that in an actual scientific context, rejecting the background-only hypothesis in a statistical sense is only part of discovering a new phenomenon. One’s degree of belief that a new process is present will depend in general on other factors as well, such as the plausibility of the new signal hypothesis and the degree to which it can describe the data. Here, however, we only consider the task of determining the $p$-value of the background-only hypothesis; if it is found below a specified threshold, we regard this as “discovery”.
It is often useful to quantify the sensitivity of an experiment by reporting the expected significance one would obtain with a given measurement under the assumption of various hypotheses. For example, the sensitivity to discovery of a given signal process $H_1$ could be characterized by the expectation value, under the assumption of $H_1$, of the value of $Z$ obtained from a test of $H_0$. This would not be the same as the $Z$ obtained using Eq. (\[eq:significance\]) with the expectation of the $p$-value, however, because the relation between $Z$ and $p$ is nonlinear. The median $Z$ and $p$ will, however, satisfy Eq. (\[eq:significance\]) because this is a monotonic relation. Therefore in the following we will take the term ‘expected significance’ always to refer to the median.
A widely used procedure to establish discovery (or exclusion) in particle physics is based on a frequentist significance test using a likelihood ratio as a test statistic. In addition to parameters of interest such as the rate (cross section) of the signal process, the signal and background models will contain in general [*nuisance parameters*]{} whose values are not taken as known [*a priori*]{} but rather must be fitted from the data.
It is assumed that the parametric model is sufficiently flexible so that for some value of the parameters it can be regarded as true. The additional flexibility introduced to parametrize systematic effects results, as it should, in a loss in sensitivity. To the degree that the model is not able to reflect the truth accurately, an additional systematic uncertainty will be present that is not quantified by the statistical method presented here.
To illustrate the use of the profile likelihood ratio, consider an experiment where for each selected event one measures the values of certain kinematic variables, and thus the resulting data can be represented as one or more histograms. Using the method in an unbinned analysis is a straightforward extension.
Suppose for each event in the signal sample one measures a variable $x$ and uses these values to construct a histogram $\vec{n} = (n_1,
\ldots, n_N)$. The expectation value of $n_i$ can be written
$$\label{eq:eni}
E[n_i] = \mu s_i + b_i \;,$$
where the mean number of entries in the $i$th bin from signal and background are
$$\begin{aligned}
\label{eq:si}
s_i = s_{\rm tot} \int_{{\rm bin} \, i} f_{s}(x; \vec{\theta}_{s}) \, dx \,,
\\*[0.3 cm]
\label{eq:bi}
b_i = b_{\rm tot} \int_{{\rm bin} \, i} f_{b}(x; \vec{\theta}_{b}) \, dx \,.\end{aligned}$$
Here the parameter $\mu$ determines the strength of the signal process, with $\mu=0$ corresponding to the background-only hypothesis and $\mu=1$ being the nominal signal hypothesis. The functions $f_s(x;\vec{\theta}_s)$ and $f_b(x;\vec{\theta}_b)$ are the probability density functions (pdfs) of the variable $x$ for signal and background events, and $\vec{\theta}_s$ and $\vec{\theta}_b$ represent parameters that characterize the shapes of pdfs. The quantities $s_{\rm tot}$ and $b_{\rm tot}$ are the total mean numbers of signal and background events, and the integrals in (\[eq:si\]) and (\[eq:bi\]) represent the probabilities for an event to be found in bin $i$. Below we will use $\vec{\theta} = (\vec{\theta}_s,
\vec{\theta}_b, b_{\rm tot})$ to denote all of the nuisance parameters. The signal normalization $s_{\rm tot}$ is not, however, an adjustable parameter but rather is fixed to the value predicted by the nominal signal model.
In addition to the measured histogram $\vec{n}$ one often makes further subsidiary measurements that help constrain the nuisance parameters. For example, one may select a control sample where one expects mainly background events and from them construct a histogram of some chosen kinematic variable. This then gives a set of values $\vec{m} = (m_1, \ldots, m_M)$ for the number of entries in each of the $M$ bins. The expectation value of $m_i$ can be written
$$\label{eq:emi}
E[m_i] = u_i(\vec{\theta}) \;,$$
where the $u_i$ are calculable quantities depending on the parameters $\vec{\theta}$. One often constructs this measurement so as to provide information on the background normalization parameter $b_{\rm tot}$ and also possibly on the signal and background shape parameters.
The likelihood function is the product of Poisson probabilities for all bins:
$$\label{eq:likelihood}
L(\mu, \vec{\theta}) =
\prod_{j=1}^N \frac{ (\mu s_{j} +
b_{j} )^{n_{j}} }{ n_{j}! }
e^{- (\mu s_{j} + b_{j}) } \;\;
\prod_{k=1}^M \frac{ u_k^{m_{k}}} { m_{k}! } \,
e^{- u_k } \;.$$
To test a hypothesized value of $\mu$ we consider the profile likelihood ratio
$$\label{eq:PLR}
\lambda(\mu) = \frac{ L(\mu,
\hat{\hat{\vec{\theta}}}) } {L(\hat{\mu}, \hat{\vec{\theta}}) } \;.$$
Here $\hat{\hat{\vec{\theta}}}$ in the numerator denotes the value of $\vec{\theta}$ that maximizes $L$ for the specified $\mu$, i.e., it is the conditional maximum-likelihood (ML) estimator of $\vec{\theta}$ (and thus is a function of $\mu$). The denominator is the maximized (unconditional) likelihood function, i.e., $\hat{\mu}$ and $\hat{\vec{\theta}}$ are their ML estimators. The presence of the nuisance parameters broadens the profile likelihood as a function of $\mu$ relative to what one would have if their values were fixed. This reflects the loss of information about $\mu$ due to the systematic uncertainties.
In many analyses, the contribution of the signal process to the mean number of events is assumed to be non-negative. This condition effectively implies that any physical estimator for $\mu$ must be non-negative. Even if we regard this to be the case, however, it is convenient to define an effective estimator $\hat{\mu}$ as the value of $\mu$ that maximizes the likelihood, even this gives $\hat{\mu} <
0$ (but providing that the Poisson mean values, $\mu s_i + b_i$, remain nonnegative). This will allow us in Sec. \[sec:wald\] to model $\hat{\mu}$ as a Gaussian distributed variable, and in this way we can determine the distributions of the test statistics that we consider. Therefore in the following we will always regard $\hat{\mu}$ as an effective estimator which is allowed to take on negative values.
Test statistic $t_{\mu} = - 2 \ln \lambda(\mu)$ {#sec:tmu}
-----------------------------------------------
From the definition of $\lambda(\mu)$ in Eq. (\[eq:PLR\]), one can see that $0 \le \lambda \le 1$, with $\lambda$ near 1 implying good agreement between the data and the hypothesized value of $\mu$. Equivalently it is convenient to use the statistic
$$\label{eq:tmu}
t_{\mu} = -2 \ln \lambda(\mu)$$
as the basis of a statistical test. Higher values of $t_{\mu}$ thus correspond to increasing incompatibility between the data and $\mu$.
We may define a test of a hypothesized value of $\mu$ by using the statistic $t_{\mu}$ directly as measure of discrepancy between the data and the hypothesis, with higher values of $t_{\mu}$ correspond to increasing disagreement. To quantify the level of disagreement we compute the $p$-value,
$$\label{eq:tmupval}
p_{\mu} = \int_{t_{\mu,{\rm obs}}}^{\infty} f(t_{\mu} | \mu ) \,
d t_{\mu} \;,$$
where $t_{\mu,{\rm obs}}$ is the value of the statistic $t_{\mu}$ observed from the data and $f(t_{\mu} | \mu )$ denotes the pdf of $t_{\mu}$ under the assumption of the signal strength $\mu$. Useful approximations for this and other related pdfs are given in Sec. \[sec:tmudist\]. The relation between the $p$-value and the observed $t_{\mu}$ and also with the significance $Z$ are illustrated in Fig. \[fig:pval\].
(10.0,4.5) (1,0)
(8,0)
(0.3,3.8)[(a)]{} (14.5,3.8)[(b)]{}
When using the statistic $t_{\mu}$, a data set may result in a low $p$-value in two distinct ways: the estimated signal strength $\hat{\mu}$ may be found greater or less than the hypothesized value $\mu$. As a result, the set of $\mu$ values that are rejected because their $p$-values are found below a specified threshold $\alpha$ may lie to either side of those values not rejected, i.e., one may obtain a two-sided confidence interval for $\mu$.
Test statistic $\tilde{t}_{\mu}$ for $\mu \ge 0$ {#sec:tmutilde}
------------------------------------------------
Often one assumes that the presence of a new signal can only increase the mean event rate beyond what is expected from background alone. That is, the signal process necessarily has $\mu \ge 0$, and to take this into account we define an alternative test statistic below called $\tilde{t}_{\mu}$.
For a model where $\mu \ge 0$, if one finds data such that $\hat{\mu}
< 0$, then the best level of agreement between the data and any physical value of $\mu$ occurs for $\mu = 0$. We therefore define
$$\label{eq:lambdatilde}
\tilde{\lambda}({\mu}) =
\left\{ \! \! \begin{array}{ll}
\frac{ L(\mu,
\hat{\hat{\vec{\theta}}}(\mu)) }
{L(\hat{\mu}, \hat{\vec{\theta}}) }
& \hat{\mu} \ge 0 , \\*[0.3 cm]
\frac{ L(\mu,
\hat{\hat{\vec{\theta}}}(\mu)) }
{L(0, \hat{\hat{\vec{\theta}}}(0)) }
& \hat{\mu} < 0 \;.
\end{array}
\right.$$
Here $\hat{\hat{\vec{\theta}}}(0)$ and $\hat{\hat{\vec{\theta}}}(\mu)$ refer to the conditional ML estimators of $\vec{\theta}$ given a strength parameter of $0$ or $\mu$, respectively.
The variable $\tilde{\lambda}(\mu)$ can be used instead of $\lambda(\mu)$ in Eq. (\[eq:tmu\]) to obtain the corresponding test statistic, which we denote $\tilde{t}_{\mu}$. That is,
$$\label{eq:tmutilde}
\tilde{t}_{\mu} = - 2 \ln \tilde{\lambda}(\mu) =
\left\{ \! \! \begin{array}{ll}
- 2 \ln \frac{L(\mu, \hat{\hat{\vec{\theta}}}(\mu))}
{L(0, \hat{\hat{\theta}}(0))}
& \quad \hat{\mu} < 0 \;, \\*[0.2 cm]
-2 \ln \frac{L(\mu, \hat{\hat{\vec{\theta}}}(\mu))}
{L(\hat{\mu}, \hat{\vec{\theta}})}
& \quad \hat{\mu} \ge 0 \;.
\end{array}
\right.$$
As was done with the statistic $t_{\mu}$, one can quantify the level of disagreement between the data and the hypothesized value of $\mu$ with the $p$-value, just as in Eq. (\[eq:tmupval\]). For this one needs the distribution of $\tilde{t}_{\mu}$, an approximation of which is given in Sec. \[sec:tmutildedist\].
Also similar to the case of $t_{\mu}$, values of $\mu$ both above and below $\hat{\mu}$ may be excluded by a given data set, i.e., one may obtain either a one-sided or two-sided confidence interval for $\mu$. For the case of no nuisance parameters, the test variable $\tilde{t}_{\mu}$ is equivalent to what is used in constructing confidence intervals according to the procedure of Feldman and Cousins [@FC].
Test statistic $q_0$ for discovery of a positive signal {#sec:discovery}
-------------------------------------------------------
An important special case of the statistic $\tilde{t}_{\mu}$ described above is used to test $\mu=0$ in a class of model where we assume $\mu
\ge 0$. Rejecting the $\mu=0$ hypothesis effectively leads to the discovery of a new signal. For this important case we use the special notation $q_0 = \tilde{t}_0$. Using the definition (\[eq:tmutilde\]) with $\mu=0$ one finds
$$\label{eq:q0}
q_{0} =
\left\{ \! \! \begin{array}{ll}
- 2 \ln \lambda(0)
& \quad \hat{\mu} \ge 0 \;, \\*[0.3 cm]
0 & \quad \hat{\mu} < 0 \;,
\end{array}
\right.$$
where $\lambda(0)$ is the profile likelihood ratio for $\mu=0$ as defined in Eq. (\[eq:PLR\]).
We may contrast this to the statistic $t_{0}$, i.e., Eq. (\[eq:tmu\]), used to test $\mu = 0$. In that case one may reject the $\mu=0$ hypothesis for either an upward or downward fluctuation of the data. This is appropriate if the presence of a new phenomenon could lead to an increase or decrease in the number of events found. In an experiment looking for neutrino oscillations, for example, the signal hypothesis may predict a greater or lower event rate than the no-oscillation hypothesis.
When using $q_0$, however, we consider the data to show lack of agreement with the background-only hypothesis only if $\hat{\mu} > 0$. That is, a value of $\hat{\mu}$ much below zero may indeed constitute evidence against the background-only model, but this type of discrepancy does not show that the data contain signal events, but rather points to some other systematic error. For the present discussion, however, we assume that the systematic uncertainties are dealt with by the nuisance parameters $\vec{\theta}$.
If the data fluctuate such that one finds fewer events than even predicted by background processes alone, then $\hat{\mu} < 0$ and one has $q_0 = 0$. As the event yield increases above the expected background, i.e., for increasing $\hat{\mu}$, one finds increasingly large values of $q_0$, corresponding to an increasing level of incompatibility between the data and the $\mu=0$ hypothesis.
To quantify the level of disagreement between the data and the hypothesis of $\mu = 0$ using the observed value of $q_0$ we compute the $p$-value in the same manner as done with $t_{\mu}$, namely,
$$\label{eq:q0pval}
p_{0} = \int_{q_{0,{\rm obs}}}^{\infty} f(q_{0} | 0 ) \, d q_{0} \;.$$
Here $f(q_{0} | 0 )$ denotes the pdf of the statistic $q_0$ under assumption of the background-only ($\mu=0$) hypothesis. An approximation for this and other related pdfs are given in Sec. \[sec:q0dist\].
Test statistic $q_{\mu}$ for upper limits {#sec:qmu}
-----------------------------------------
For purposes of establishing an upper limit on the strength parameter $\mu$, we consider two closely related test statistics. First, we may define
$$\label{eq:qmu}
q_{\mu} =
\left\{ \! \! \begin{array}{ll}
- 2 \ln \lambda(\mu) & \hat{\mu} \le \mu \;, \\*[0.2 cm]
0 & \hat{\mu} > \mu \;,
\end{array}
\right.$$
where $\lambda(\mu)$ is the profile likelihood ratio as defined in Eq. (\[eq:PLR\]). The reason for setting $q_{\mu} = 0$ for $\hat{\mu} > \mu$ is that when setting an upper limit, one would not regard data with $\hat{\mu} > \mu$ as representing less compatibility with $\mu$ than the data obtained, and therefore this is not taken as part of the rejection region of the test. From the definition of the test statistic one sees that higher values of $q_{\mu}$ represent greater incompatibility between the data and the hypothesized value of $\mu$.
One should note that $q_0$ is not simply a special case of $q_{\mu}$ with $\mu=0$, but rather has a different definition (see Eqs. (\[eq:q0\]) and (\[eq:qmu\])). That is, $q_0$ is zero if the data fluctuate downward ($\hat{\mu} < 0$), but $q_{\mu}$ is zero if the data fluctuate upward ($\hat{\mu} > \mu$). With that caveat in mind, we will often refer in the following to $q_{\mu}$ with the idea that this means either $q_0$ or $q_{\mu}$ as appropriate to the context.
As with the case of discovery, one quantifies the level of agreement between the data and hypothesized $\mu$ with $p$-value. For, e.g., an observed value $q_{\mu,{\rm obs}}$, one has
$$\label{eq:qmupval}
p_{\mu} = \int_{q_{\mu,{\rm obs}}}^{\infty} f(q_{\mu} | \mu) \, d q_{\mu} \;,$$
which can be expressed as a significance using Eq. (\[eq:significance\]). Here $f(q_{\mu} | \mu)$ is the pdf of $q_{\mu}$ assuming the hypothesis $\mu$. In Sec. \[sec:qmudist\] we provide useful approximations for this and other related pdfs.
Alternative test statistic $\tilde{q}_{\mu}$ for upper limits {#sec:qtildemu}
-------------------------------------------------------------
For the case where one considers models for which $\mu \ge 0$, the variable $\tilde{\lambda}(\mu)$ can be used instead of $\lambda(\mu)$ in Eq. (\[eq:qmu\]) to obtain the corresponding test statistic, which we denote $\tilde{q}_{\mu}$. That is,
$$\label{eq:qmutilde}
\tilde{q}_{\mu} =
\left\{ \! \! \begin{array}{ll}
- 2 \ln \tilde{\lambda}(\mu) & \hat{\mu} \le \mu \\*[0.2 cm]
0 & \hat{\mu} > \mu
\end{array}
\right.
\quad = \quad \: \left\{ \! \! \begin{array}{lll}
- 2 \ln \frac{L(\mu, \hat{\hat{\vec{\theta}}}(\mu))}
{L(0, \hat{\hat{\theta}}(0))}
& \hat{\mu} < 0 \;, \\*[0.2 cm]
-2 \ln \frac{L(\mu, \hat{\hat{\vec{\theta}}}(\mu))}
{L(\hat{\mu}, \hat{\vec{\theta}})}
& 0 \le \hat{\mu} \le \mu \;, \\*[0.2 cm]
0 & \hat{\mu} > \mu \;.
\end{array}
\right.$$
We give an approximation for the pdf $f(\tilde{q}_{\mu} |
\mu^{\prime})$ in Sec. \[sec:qtildemudist\].
In numerical examples we have found that the difference between the tests based on $q_{\mu}$ (Eq. (\[eq:qmu\])) and $\tilde{q}_{\mu}$ usually to be negligible, but use of $q_{\mu}$ leads to important simplifications. Furthermore, in the context of the approximation used in Sec. \[sec:qdist\], the two statistics are equivalent. That is, assuming the approximations below, $q_{\mu}$ can be expressed as a monotonic function of $\tilde{q}_{\mu}$ and thus they lead to the same results.
Approximate sampling distributions {#sec:qdist}
==================================
In order to find the $p$-value of a hypothesis using Eqs. (\[eq:q0pval\]) or (\[eq:qmupval\]) we require the sampling distribution for the test statistic being used. In the case of discovery we are testing the background-only hypothesis ($\mu=0$) and therefore we need $f(q_0 | 0)$, where $q_0$ is defined by Eq. (\[eq:q0\]). When testing a nonzero value of $\mu$ for purposes of finding an upper limit we need the distribution $f(q_{\mu} | \mu)$ where $q_{\mu}$ is defined by Eq. (\[eq:qmu\]), or alternatively we require the pdf of the corresponding statistic $\tilde{q}_{\mu}$ as defined by Eq. (\[eq:qmutilde\]). In this notation the subscript of $q$ refers to the hypothesis being tested, and the second argument in $f(q_{\mu} | \mu)$ gives the value of $\mu$ assumed in the distribution of the data.
We also need the distribution $f(q_{\mu} | \mu^{\prime})$ with $\mu
\ne \mu^{\prime}$ to find what significance to expect and how this is distributed if the data correspond to a strength parameter different from the one being tested. For example, it is useful to characterize the sensitivity of a planned experiment by quoting the median significance, assuming data distributed according to a specified signal model, with which one would expect to exclude the background-only hypothesis. For this one would need $f(q_0 |
\mu^{\prime} )$, usually with $\mu^{\prime} = 1$. From this one can find the median $q_0$, and thus the median discovery significance. When considering upper limits, one would usually quote the value of $\mu$ for which the median $p$-value is equal to 0.05, as this gives the median upper limit on $\mu$ at 95% confidence level. In this case one would need $f(q_{\mu} | 0)$ (or alternatively $f(\tilde{q}_{\mu} | 0)$).
In Sec. \[sec:wald\] we present an approximation for the profile likelihood ratio, valid in the large sample limit. This allows one to obtain approximations for all of the required distributions, which are given in Sections \[sec:tmudist\] through \[sec:qmudist\] The approximations become exact in the large sample limit and are in fact found to provide accurate results even for fairly small sample sizes. For very small data samples one always has the possibility of using Monte Carlo methods to determine the required distributions.
Approximate distribution of the profile likelihood ratio {#sec:wald}
--------------------------------------------------------
Consider a test of the strength parameter $\mu$, which here can either be zero (for discovery) or nonzero (for an upper limit), and suppose the data are distributed according to a strength parameter $\mu^{\prime}$. The desired distribution $f(q_{\mu} | \mu^{\prime})$ can be found using a result due to Wald [@Wald], who showed that for the case of a single parameter of interest,
$$\label{eq:wald}
-2 \ln \lambda(\mu)
= \frac{(\mu - \hat{\mu})^2}{\sigma^2} + {\cal O}(1/\sqrt{N}) \;.$$
Here $\hat{\mu}$ follows a Gaussian distribution with a mean $\mu^{\prime}$ and standard deviation $\sigma$, and $N$ represents the data sample size. The standard deviation $\sigma$ of $\hat{\mu}$ is obtained from the covariance matrix of the estimators for all the parameters, $V_{ij} = \mbox{cov}[\hat{\theta}_i, \hat{\theta}_j]$, where here the $\theta_i$ represent both $\mu$ as well as the nuisance parameters (e.g., take $\theta_0 = \mu$, so $\sigma^2 = V_{00}$). In the large-sample limit, the bias of ML estimators in general tend to zero, in which case we can write the inverse of the covariance matrix as
$$\label{eq:vinvwald}
V^{-1}_{ij} = - E \left[ \frac{ \partial^2 \ln L }
{ \partial \theta_i \partial \theta_j } \right] \;,$$
where the expectation value assumes a strength parameter $\mu^{\prime}$. The approximations presented here are valid to the extent that the ${\cal O}(1/\sqrt{N})$ term can be neglected, and the value of $\sigma$ can be estimated, e.g., using Eq. (\[eq:vinvwald\]). In Sec. \[sec:asimov\] we present an alternative way to estimate $\sigma$ which lends itself more directly to determination of the median significance.
If $\hat{\mu}$ is Gaussian distributed and we neglect the ${\cal
O}(1/\sqrt{N})$ term in Eq. (\[eq:wald\]), then one can show that the statistic $t_{\mu} = -2 \ln \lambda(\mu)$ follows a [*noncentral chi-square*]{} distribution for one degree of freedom (see, e.g., [@ncc]),
$$\label{eq:ftmulambda}
f(t_{\mu};\Lambda) = \frac{1}{2 \sqrt{t_{\mu}}} \frac{1}{\sqrt{2 \pi}}
\left[ \exp \left( - \frac{1}{2}
\left( \sqrt{t_{\mu}} + \sqrt{\Lambda} \right)^2 \right) +
\exp \left( - \frac{1}{2} \left( \sqrt{t_{\mu}} - \sqrt{\Lambda} \right)^2
\right) \right] \;,$$
where the noncentrality parameter $\Lambda$ is
$$\label{eq:noncentrality}
\Lambda = \frac{(\mu - \mu^{\prime})^2}{\sigma^2} \;.$$
For the special case $\mu^{\prime} = \mu$ one has $\Lambda =
0$ and $-2 \ln\lambda(\mu)$ approaches a chi-square distribution for one degree of freedom, a result shown earlier by Wilks [@Wilks].
The results of Wilks and Wald generalize to more than one parameter of interest. If the parameters of interest can be explicitly identified with a subset of the parameters $\vec{\theta}_r = (\theta_1, \dots,
\theta_r$), then the distribution of $-2\ln\lambda(\vec{\theta}_r)$ follows a noncentral chi-square distribution for $r$-degrees of freedom with noncentrality parameter
$$\label{eq:noncentralityND}
\Lambda_r = \sum_{i,j=1}^{r} (\theta_i - \theta_i') \,\tilde{V}_{ij}^{-1}
\,(\theta_j - \theta_j') \;,$$
where $\tilde{V}_{ij}^{-1}$ is the inverse of the submatrix one obtains from restricting the full covariance matrix to the parameters of interest. The full covariance matrix is given from inverting Eq. (\[eq:vinvwald\]), and we show an efficient way to calculate it in Sec. \[sec:asimov\].
The Asimov data set and the variance of $\hat{\mu}$ {#sec:asimov}
---------------------------------------------------
Some of the formulae given require the standard deviation $\sigma$ of $\hat{\mu}$, which is assumed to follow a Gaussian distribution with a mean of $\mu^{\prime}$. Below we show two ways of estimating $\sigma$, both of which are closely related to a special, artificial data set that we call the “Asimov data set”.
We define the Asimov data set such that when one uses it to evaluate the estimators for all parameters, one obtains the true parameter values. Consider the likelihood function for the generic analysis given by Eq. (\[eq:likelihood\]). To simplify the notation in this section we define
$$\label{eq:nui}
\nu_i = \mu^{\prime} s_i + b_i \;.$$
Further let $\theta_0 = \mu$ represent the strength parameter, so that here $\theta_i$ can stand for any of the parameters. The ML estimators for the parameters can be found by setting the derivatives of $\ln L$ with respect to all of the parameters equal to zero:
$$\label{eq:dlnldtheta}
\frac{ \partial \ln L}{\partial \theta_j}
= \sum_{i=1}^N \left( \frac{n_i}{\nu_i} - 1 \right)
\frac{ \partial \nu_i}{\partial \theta_j} +
\sum_{i=1}^M \left( \frac{m_i}{u_i} - 1 \right)
\frac{ \partial u_i}{\partial \theta_j} = 0 \;.$$
This condition holds if the Asimov data, $n_{i,{\rm A}}$ and $m_{i,{\rm A}}$, are equal to their expectation values:
$$\begin{aligned}
\label{eq:asimovn}
n_{i,{\rm A}} & = & E[n_i] = \nu_i
= \mu^{\prime} s_i(\vec{\theta}) + b_i(\vec{\theta}) \;, \\*[0.2 cm]
\label{eq:asimovm}
m_{i,{\rm A}} & = & E[m_i] = u_i(\vec{\theta}) \;.\end{aligned}$$
Here the parameter values represent those implied by the assumed distribution of the data. In practice, these are the values that would be estimated from the Monte Carlo model using a very large data sample.
We can use the Asimov data set to evaluate the “Asimov likelihood” $L_{\rm A}$ and the corresponding profile likelihood ratio $\lambda_{\rm A}$. The use of non-integer values for the data is not a problem as the factorial terms in the Poisson likelihood represent constants that cancel when forming the likelihood ratio, and thus can be dropped. One finds
$$\label{eq:asimovlambdaii}
\lambda_{\rm A}(\mu)
= \frac{ L_{\rm A}( \mu, \hat{\hat{\vec{\theta}}} ) } { L_{\rm A}(\hat{\mu},
\hat{\vec{\theta}}) } = \frac{ L_{\rm A}( \mu,
\hat{\hat{\vec{\theta}}} ) } { L_{A}(\mu^{\prime}, \vec{\theta} ) } \;,$$
where the final equality above exploits the fact that the estimators for the parameters are equal to their hypothesized values when the likelihood is evaluated with the Asimov data set.
A standard way to find $\sigma$ is by estimating the matrix of second derivatives of the log-likelihood function (cf.Eq. (\[eq:vinvwald\])) to obtain the inverse covariance matrix $V^{-1}$, inverting to find $V$, and then extracting the element $V_{00}$ corresponding to the variance of $\hat{\mu}$. The second derivative of $\ln L$ is
$$\begin{aligned}
\label{eq:d2lnl}
\frac{ \partial^2 \ln L }{\partial \theta_j \partial \theta_k } & = &
\sum_{i=1}^N \left[ \left( \frac{n_i}{\nu_i} - 1 \right)
\frac{ \partial^2 \nu_i }{\partial \theta_j \partial \theta_k } -
\frac{ \partial \nu_i }{\partial \theta_j }
\frac{ \partial \nu_i }{\partial \theta_k }
\frac{n_i}{\nu_i^2} \right] \nonumber \\*[0.3 cm]
& + & \sum_{i=1}^M \left[ \left( \frac{m_i}{u_i} - 1 \right)
\frac{ \partial^2 u_i }{\partial \theta_j \partial \theta_k } -
\frac{ \partial u_i }{\partial \theta_j }
\frac{ \partial u_i }{\partial \theta_k }
\frac{m_i}{u_i^2} \right] \;.\end{aligned}$$
From (\[eq:d2lnl\]) one sees that the second derivative of $\ln L$ is linear in the data values $n_i$ and $m_i$. Thus its expectation value is found simply by evaluating with the expectation values of the data, which is the same as the Asimov data. One can therefore obtain the inverse covariance matrix from
$$\label{eq:invcov}
V^{-1}_{jk} = - E \left[ \frac{ \partial^2 \ln L }{\partial \theta_j
\partial \theta_k } \right] = - \frac{ \partial^2 \ln L_{\rm A}
}{\partial \theta_j \partial \theta_k } =
\sum_{i=1}^N \frac{\partial \nu_i}{\partial \theta_j} \frac{\partial
\nu_i}{\partial \theta_k} \frac{1}{\nu_i} + \sum_{i=1}^M
\frac{\partial u_i}{\partial \theta_j} \frac{\partial u_i}{\partial
\theta_k} \frac{1}{u_i} \;.$$
In practice one could, for example, evaluate the the derivatives of $\ln L_{\rm A}$ numerically, use this to find the inverse covariance matrix, and then invert and extract the variance of $\hat{\mu}$. One can see directly from Eq. (\[eq:invcov\]) that this variance depends on the parameter values assumed for the Asimov data set, in particular on the assumed strength parameter $\mu^{\prime}$, which enters via Eq. (\[eq:nui\]).
Another method for estimating $\sigma$ (denoted $\sigma_{\rm A}$ in this section to distinguish it from the approach above based on the second derivatives of $\ln L$) is to find find the value that is necessary to recover the known properties of $-\lambda_{\rm A}(\mu)$. Because the Asimov data set corresponding to a strength $\mu^{\prime}$ gives $\hat{\mu} = \mu^{\prime}$, from Eq. (\[eq:wald\]) one finds
$$\label{eq:2lnLambdaAsimov}
- 2 \ln \lambda_{\rm A}(\mu) \approx
\frac{ (\mu - \mu^{\prime} )^2}{\sigma^2} = \Lambda \;.$$
That is, from the Asimov data set one obtains an estimate of the noncentrality parameter $\Lambda$ that characterizes the distribution $f(q_{\mu} | \mu^{\prime})$. Equivalently, one can use Eq. (\[eq:2lnLambdaAsimov\]) to obtain the variance $\sigma^2$ which characterizes the distribution of $\hat{\mu}$, namely,
$$\label{eq:sigma2}
\sigma_{\rm A}^2 = \frac{(\mu - \mu^{\prime})^2 }{ q_{\mu,{\rm A}} } \;,$$
where $q_{\mu,{\rm A}} = - 2 \ln \lambda_{\rm A}(\mu)$. For the important case where one wants to find the median exclusion significance for the hypothesis $\mu$ assuming that there is no signal, then one has $\mu^{\prime} = 0$ and therefore
$$\label{eq:sigma2mu}
\sigma_{\rm A}^2 = \frac{\mu^2}{q_{\mu,{\rm A}}} \;,$$
and for the modified statistic $\tilde{q}_{\mu}$ the analogous relation holds. For the case of discovery where one tests $\mu=0$ one has
$$\label{eq:sigma02}
\sigma_{\rm A}^2 = \frac{\mu^{{\prime}\,2}}{q_{0,{\rm A}}} \;.$$
The two methods for obtaining $\sigma$ and $\Lambda$ — from the Fisher information matrix or from $q_{\mu,\rm A}$ — are not identical, but were found to provide similar results in examples of of practical interest. In several cases that we considered, the distribution based on $\sigma_{\rm A}$ provided a better approximation to the true sampling distribution than the standard approach based on the Fisher information matrix, leading to the conjecture that it may effectively incorporate some higher-order terms in Eq. (\[eq:wald\]).
This can be understood qualitatively by noting that under assumption of the Wald approximation, the test statistics $q_0$, $q_{\mu}$ and $\tilde{q}_{\mu}$ are monotonically related to $\hat{\mu}$, and therefore their median values can be found directly by using the median of $\hat{\mu}$, which is $\mu^{\prime}$. But monotonicity is a weaker condition than the full Wald approximation. That is, even if higher-order terms are present in Eq. (\[eq:wald\]), they will not alter the distribution’s median as long as they do not break the monotonicity of the relation between the test statistic and $\hat{\mu}$. If one uses $\sigma_{\rm A}$ one obtains distributions with medians given by the corresponding Asimov values, $q_{0,{\rm A}}$ or $q_{\mu,{\rm A}}$, and these values will be correct to the extent that monotonicity holds.
Distribution of $t_{\mu}$ {#sec:tmudist}
-------------------------
Consider first using the statistic $t_{\mu} = -2 \ln \lambda(\mu)$ of Sec. \[sec:tmu\] as the basis of the statistical test of a hypothesized value of $\mu$. This could be a test of $\mu=0$ for purposes of establishing existence of a signal process, or non-zero values of $\mu$ for purposes of obtaining a confidence interval. To find the $p$-value $p_{\mu}$, we require the pdf $f(t_{\mu} | \mu)$, and to find the median $p$-value assuming a different strength parameter we will need $f(t_{\mu} | \mu^{\prime})$.
The pdf $f(t_{\mu} | \mu^{\prime})$ is given by Eq. (\[eq:ftmulambda\]), namely,
$$\label{eq:ftmumPrime}
f(t_{\mu} | \mu^{\prime}) = \frac{1}{2 \sqrt{t_{\mu}}}
\frac{1}{\sqrt{2 \pi}} \left[ \exp \left( - \frac{1}{2} \left(
\sqrt{t_{\mu}} + \frac{\mu - \mu^{\prime}}{\sigma} \right)^2 \right) +
\exp \left( - \frac{1}{2} \left( \sqrt{t_{\mu}} - \frac{\mu -
\mu^{\prime}}{\sigma} \right)^2 \right) \right] \;.$$
The special case $\mu = \mu^{\prime}$ is simply a chi-square distribution for one degree of freedom:
$$\label{eq:ftmumu}
f(t_{\mu}|\mu) =
\frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{t_{\mu}}}
e^{- t_{\mu} / 2 } \;.$$
The cumulative distribution of $t_{\mu}$ assuming $\mu^{\prime}$ is
$$\label{eq:tmumuprimecdf}
F(t_{\mu} | \mu^{\prime}) = \Phi \left( \sqrt{t_{\mu}} + \frac{ \mu -
\mu^{\prime} }{\sigma} \right) + \Phi \left( \sqrt{t_{\mu}} - \frac{
\mu - \mu^{\prime} }{\sigma} \right) - 1 \;,$$
where $\Phi$ is the cumulative distribution of the standard (zero mean, unit variance) Gaussian. The special case $\mu =
\mu^{\prime}$ is therefore
$$\label{eq:tmumucdf}
F(t_{\mu} | \mu) =
2 \Phi \left( \sqrt{t_{\mu}} \right) - 1 \;,$$
The $p$-value of a hypothesized value of $\mu$ for an observed value $t_{\mu}$ is therefore
$$\label{eq:pmufromtmu}
p_{\mu} = 1 - F(t_{\mu} | \mu) = 2 \left( 1 -
\Phi \left( \sqrt{ t_{\mu} } \right) \right) \;,$$
and the corresponding significance is
$$\label{eq:tmusig}
Z_{\mu} = \Phi^{-1}(1 - p_{\mu}) = \Phi^{-1} \left(
2 \Phi \left( \sqrt{ t_{\mu} } \right) - 1 \right) \;.$$
If the $p$-value is found below a specified threshold $\alpha$ (often one takes $\alpha = 0.05$), then the value of $\mu$ is said to be excluded at a confidence level (CL) of $1 - \alpha$. The set of points not excluded form a confidence interval with $\mbox{CL}
= 1 - \alpha$. Here the endpoints of the interval can be obtained simply by setting $p_{\mu} = \alpha$ and solving for $\mu$. Assuming the Wald approximation (\[eq:wald\]) and using Eq. (\[eq:pmufromtmu\]) one finds
$$\label{eq:mulofromtmu}
\mu_{\rm up/lo} = \hat{\mu} \pm \sigma \Phi^{-1}(1 - \alpha/2) \;.$$
One subtlety with this formula is that $\sigma$ itself depends at some level on $\mu$. In practice to find the upper and lower limits one can simply solve numerically to find those values of $\mu$ that satisfy $p_{\mu} = \alpha$.
Distribution of $\tilde{t}_{\mu}$ {#sec:tmutildedist}
---------------------------------
Assuming the Wald approximation, the statistic $\tilde{t}_{\mu}$ as defined by Eq. (\[eq:tmutilde\]) can be written
$$\label{eq:tmutildewald}
\tilde{t}_{\mu} =
\: \left\{ \! \! \begin{array}{ll}
\frac{\mu^2}{\sigma^{2}} - \frac{2 \mu \hat{\mu}}{\sigma^{2}}
& \quad \hat{\mu} < 0 \;, \\*[0.2 cm]
\frac{(\mu - \hat{\mu})^2}{\sigma^{2}}
& \quad \hat{\mu} \ge 0 \;.
\end{array}
\right.$$
From this the pdf $f(\tilde{t}_{\mu} | \mu^{\prime})$ is found to be
$$\begin{aligned}
\label{eq:ftildetmmp}
f(\tilde{t}_{\mu}|\mu^{\prime}) & = & \frac{1}{2} \frac{1}{\sqrt{2 \pi}}
\frac{1}{\sqrt{\tilde{t}_{\mu}}} \exp \left[
-\half \left( \sqrt{\tilde{t}_{\mu}} +
\frac{\mu - \mu^{\prime}}{\sigma} \right)^2 \right] \\*[0.3 cm]
& + &
\: \left\{ \! \!
\begin{array}{ll}
\frac{1}{2} \frac{1}{\sqrt{2 \pi}}
\frac{1}{\sqrt{\tilde{t}_{\mu}}} \exp \left[
-\half \left( \sqrt{\tilde{t}_{\mu}} -
\frac{\mu - \mu^{\prime}}{\sigma} \right)^2 \right] &
\quad \tilde{t}_{\mu} \le \mu^2/\sigma^2 \;, \\*[0.5 cm]
\frac{1}{\sqrt{2 \pi}(2\mu/\sigma)} \exp \left[ - \half
\frac{ \left( \tilde{t}_{\mu} -
\frac{\mu^2 - 2 \mu \mu^{\prime}}{\sigma^2} \right)^2 }
{ (2\mu/\sigma)^2 } \right]
& \quad \tilde{t}_{\mu} > \mu^2/\sigma^2
\end{array}
\right.
\;.\end{aligned}$$
The special case $\mu = \mu^{\prime}$ is therefore
$$\label{eq:ftildetmm}
f(\tilde{t}_{\mu}|\mu^{\prime}) =
\: \left\{ \! \!
\begin{array}{ll}
\frac{1}{\sqrt{2 \pi}}
\frac{1}{\sqrt{\tilde{t}_{\mu}}} e^{- \tilde{t}_{\mu} / 2 }
& \quad \tilde{t}_{\mu} \le \mu^2/\sigma^2 \;, \\*[0.5 cm]
\frac{1}{2} \frac{1}{\sqrt{2 \pi}}
\frac{1}{\sqrt{\tilde{t}_{\mu}}} e^{- \tilde{t}_{\mu} / 2 } \: + \:
\frac{1}{\sqrt{2 \pi} (2 \mu/\sigma)} \exp \left[ - \half
\frac{ (\tilde{t}_{\mu} + \mu^2/\sigma^2)^2}{ (2 \mu/\sigma)^2 } \right]
& \quad \tilde{t}_{\mu} > \mu^2/\sigma^2 \;.
\end{array}
\right.
\;.$$
The corresponding cumulative distribution is
$$\label{eq:tildetmmpcdf}
F(\tilde{t}_{\mu}|\mu^{\prime}) = \Phi\left( \sqrt{\tilde{t}_{\mu}} +
\frac{\mu - \mu^{\prime}}{\sigma} \right) +
\: \left\{ \! \! \begin{array}{ll}
\Phi\left( \sqrt{\tilde{t}_{\mu}} -
\frac{\mu - \mu^{\prime}}{\sigma} \right) - 1
& \quad \tilde{t}_{\mu} \le \mu^2/\sigma^{2}
\;, \\*[0.5 cm]
\Phi \left( \frac{ \tilde{t}_{\mu} -
(\mu^2 - 2 \mu \mu^{\prime})/\sigma^{2}}
{2\mu/\sigma} \right) - 1
& \quad \tilde{t}_{\mu} > \mu^2/\sigma^{2} \;.
\end{array}
\right.$$
For $\mu = \mu^{\prime}$ this is
$$\label{eq:tildetmmcdf}
F(\tilde{t}_{\mu}|\mu) =
\: \left\{ \! \! \begin{array}{ll}
2 \Phi\Big( \sqrt{\tilde{t}_{\mu}} \Big) - 1
& \quad \tilde{t}_{\mu} \le \mu^2/\sigma^2
\;, \\*[0.5 cm]
\Phi\Big( \sqrt{\tilde{t}_{\mu}} \Big) +
\Phi \left( \frac{ \tilde{t}_{\mu} + \mu^2/\sigma^2}
{2\mu/\sigma} \right) - 1
& \quad \tilde{t}_{\mu} > \mu^2/\sigma^2 \;.
\end{array}
\right.$$
The $p$-value of the hypothesized $\mu$ is given by one minus the cumulative distribution, under assumption of the parameter $\mu$,
$$\label{eq:pvaltmutilde}
p_{\mu} = 1 - F(\tilde{t}_{\mu} | \mu) \;.$$
The corresponding significance is $Z_{\mu} = \Phi^{-1}(1 -
p_{\mu})$.
A confidence interval for $\mu$ at confidence level $\mbox{CL} = 1 -
\alpha$ can be constructed from the set $\mu$ values for which the $p$-value is not less than $\alpha$. To find the endpoints of this interval, one can set $p_{\mu}$ from Eq. (\[eq:pvaltmutilde\]) equal to $\alpha$ and solve for $\mu$. In general this must be done numerically. In the large sample limit, i.e., assuming the validity of the asymptotic approximations, these intervals correspond to the limits of Feldman and Cousins [@FC] for the case where physical range of the parameter $\mu$ is $\mu \ge 0$.
Distribution of $q_0$ (discovery) {#sec:q0dist}
---------------------------------
Assuming the validity of the approximation (\[eq:wald\]), one has $-2 \ln \lambda(0) = \hat{\mu}^2 / \sigma^2$. From the definition (\[eq:q0\]) of $q_0$, we therefore have
$$\label{eq:q0wald}
q_{0} =
\left\{ \! \! \begin{array}{ll}
\hat{\mu}^2 / \sigma^2
& \quad \hat{\mu} \ge 0 \;, \\*[0.3 cm]
0 & \quad \hat{\mu} < 0 \;,
\end{array}
\right.$$
where $\hat{\mu}$ follows a Gaussian distribution with mean $\mu^{\prime}$ and standard deviation $\sigma$. From this one can show that the pdf of $q_0$ has the form
$$\label{eq:fq0muprimewald}
f(q_0 | \mu^{\prime}) = \left( 1 -
\Phi \left( \frac{ \mu^{\prime}}{\sigma} \right) \right) \delta(q_0) +
\frac{1}{2}
\frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{q_0}} \exp
\left[ - \frac{1}{2} \left( \sqrt{q_0} - \frac{\mu^{\prime}}{\sigma}
\right)^2 \right]
\;.$$
For the special case of $\mu^{\prime} = 0$, this reduces to
$$\label{eq:fq00}
f(q_0 | 0) = \frac{1}{2} \delta(q_0) +
\frac{1}{2} \frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{q_0}} e^{-q_0/2} \;.$$
That is, one finds a mixture of a delta function at zero and a chi-square distribution for one degree of freedom, with each term having a weight of $1/2$. In the following we will refer to this mixture as a half chi-square distribution or $\half \chi^2_1$.
From Eq. (\[eq:fq0muprimewald\]) the corresponding cumulative distribution is found to be
$$\label{cdfq0muprimewald}
F(q_0 | \mu^{\prime}) = \Phi \left( \sqrt{q_0} - \frac{\mu^{\prime}}{\sigma}
\right) \;.$$
The important special case $\mu^{\prime} = 0$ is therefore simply
$$\label{cdfq00wald}
F(q_0 | 0) = \Phi \Big( \sqrt{q_0} \Big)
\;.$$
The $p$-value of the $\mu=0$ hypothesis (see Eq. (\[eq:q0pval\])) is
$$\label{eq:pval0}
p_0 = 1 - F(q_0 | 0) \;,$$
and therefore using Eq. (\[eq:significance\]) for the significance one obtains the simple formula
$$\label{eq:Z0}
Z_0 = \Phi^{-1}(1 - p_0) = \sqrt{q_0} \;.$$
Distribution of $q_{\mu}$ (upper limits) {#sec:qmudist}
----------------------------------------
Assuming the validity of the Wald approximation, we can write the test statistic used for upper limits, Eq. (\[eq:qmu\]) as
$$\label{eq:qmuwald}
q_{\mu} =
\quad \: \left\{ \! \! \begin{array}{lll}
\frac{(\mu - \hat{\mu})^2}{\sigma^{2}}
& \hat{\mu} < \mu \;, \\*[0.2 cm]
0 & \hat{\mu} > \mu \;,
\end{array}
\right.$$
where $\hat{\mu}$ as before follows a Gaussian centred about $\mu^{\prime}$ with a standard deviation $\sigma$.
The pdf $f(q_{\mu} | \mu^{\prime})$ is found to be
$$\label{eq:fqmmp}
f(q_{\mu}|\mu^{\prime}) =
\Phi\left( \frac{\mu^{\prime} - \mu}{\sigma} \right)
\delta(q_{\mu}) +
\frac{1}{2} \frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{q_{\mu}}}
\exp \left[ -\frac{1}{2} \left( \sqrt{q_{\mu}} -
\frac{\mu - \mu^{\prime}}{\sigma} \right)^2 \right] \;,$$
so that the special case $\mu = \mu^{\prime}$ is a half-chi-square distribution:
$$\label{eq:fqmm}
f(q_{\mu}|\mu) =
\frac{1}{2} \delta(q_{\mu}) +
\frac{1}{2} \frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{q_{\mu}}}
e^{- q_{\mu}/2} \;.$$
The cumulative distribution is
$$\label{eq:qmmpcdf}
F(q_{\mu}|\mu^{\prime}) =
\Phi\left( \sqrt{q_{\mu}} - \frac{\mu - \mu^{\prime}}{\sigma}
\right)
\;,$$
and the corresponding special case $\mu^{\prime} = \mu$ is thus the same as what was found for $q_0$, namely,
$$\label{eq:qmmcdf}
F(q_{\mu}|\mu) =
\Phi \Big( \sqrt{q_{\mu}} \Big)
\;.$$
The $p$-value of the hypothesized $\mu$ is
$$\label{eq:pvalmu}
p_{\mu} = 1 - F(q_{\mu} |\mu ) = 1 - \Phi \Big( \sqrt{q_{\mu}} \Big)$$
and therefore the corresponding significance is
$$\label{eq:Zmu}
Z_{\mu} = \Phi^{-1}(1 - p_{\mu}) = \sqrt{q_{\mu}} \;.$$
As with the statistic $t_{\mu}$ above, if the $p$-value is found below a specified threshold $\alpha$ (often one takes $\alpha = 0.05$), then the value of $\mu$ is said to be excluded at a confidence level (CL) of $1 -
\alpha$. The upper limit on $\mu$ is the largest $\mu$ with $p_{\mu}
\le \alpha$. Here this can be obtained simply by setting $p_{\mu} =
\alpha$ and solving for $\mu$. Using Eqs. (\[eq:qmuwald\]) and (\[eq:pvalmu\]) one finds
$$\label{eq:muup}
\mu_{\rm up} = \hat{\mu} + \sigma \Phi^{-1}(1 - \alpha) \;.$$
For example, $\alpha = 0.05$ gives $\Phi^{-1}(1 - \alpha) =
1.64$. Also as noted above, $\sigma$ depends in general on the hypothesized $\mu$. Thus in practice one may find the upper limit numerically as the value of $\mu$ for which $p_{\mu} = \alpha$.
Distribution of $\tilde{q}_{\mu}$ (upper limits) {#sec:qtildemudist}
------------------------------------------------
Using the alternative statistic $\tilde{q}_{\mu}$ defined by Eq. (\[eq:qmutilde\]) and assuming the Wald approximation we find
$$\label{eq:qtildemuwald}
\tilde{q}_{\mu} =
\: \left\{ \! \! \begin{array}{lll}
\frac{\mu^2}{\sigma^{2}} - \frac{2 \mu \hat{\mu}}{\sigma^{2}}
& \quad \hat{\mu} < 0 \;, \\*[0.2 cm]
\frac{(\mu - \hat{\mu})^2}{\sigma^{2}}
& \quad 0 \le \hat{\mu} \le \mu \;, \\*[0.2 cm]
0 & \quad \hat{\mu} > \mu \;.
\end{array}
\right.$$
The pdf $f(\tilde{q}_{\mu} | \mu^{\prime})$ is found to be
$$\begin{aligned}
\label{eq:ftildeqmmp}
f(\tilde{q}_{\mu}|\mu^{\prime}) & = &
\Phi \left( \frac{\mu^{\prime} - \mu}{\sigma} \right)
\delta (\tilde{q}_{\mu}) \nonumber \\*[0.3 cm]
& + &
\: \left\{ \! \! \begin{array}{lll}
\frac{1}{2} \frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{\tilde{q}_{\mu}}}
\exp \left[ -\frac{1}{2} \left( \sqrt{\tilde{q}_{\mu}} -
\frac{\mu - \mu^{\prime}}{\sigma} \right)^2 \right]
& 0 < \tilde{q}_{\mu} \le \mu^2/\sigma^{2} \;, \\*[0.5 cm]
\frac{1}{\sqrt{2 \pi} (2\mu/\sigma)} \exp \left[
-\frac{1}{2} \frac{ (\tilde{q}_{\mu} -
(\mu^2 - 2 \mu \mu^{\prime})/\sigma^{2} )^2 }
{(2 \mu/\sigma)^2} \right]
& \quad \tilde{q}_{\mu} > \mu^2/\sigma^{2} \;.
\end{array}
\right.\end{aligned}$$
The special case $\mu = \mu^{\prime}$ is therefore
$$\label{eq:ftildeqmm}
f(\tilde{q}_{\mu}|\mu) =
\frac{1}{2} \delta (\tilde{q}_{\mu}) +
\: \left\{ \! \! \begin{array}{lll}
\frac{1}{2} \frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{\tilde{q}_{\mu}}}
e^{- \tilde{q}_{\mu}/2}
& 0 < \tilde{q}_{\mu} \le \mu^2/\sigma^2 \;, \\*[0.5 cm]
\frac{1}{\sqrt{2 \pi} (2\mu/\sigma)} \exp \left[
-\frac{1}{2} \frac{ (\tilde{q}_{\mu} + \mu^2/\sigma^2 )^2 }
{(2 \mu/\sigma)^2} \right]
& \quad \tilde{q}_{\mu} > \mu^2/\sigma^2 \;.
\end{array}
\right.$$
The corresponding cumulative distribution is
$$\label{eq:tildeqmmpcdf}
F(\tilde{q}_{\mu}|\mu^{\prime}) =
\: \left\{ \! \! \begin{array}{lll}
\Phi\left( \sqrt{\tilde{q}_{\mu}} -
\frac{\mu - \mu^{\prime}}{\sigma} \right)
& \quad 0 < \tilde{q}_{\mu} \le \mu^2/\sigma^{2}
\;, \\*[0.5 cm]
\Phi \left( \frac{ \tilde{q}_{\mu} -
(\mu^2 - 2 \mu \mu^{\prime})/\sigma^{2}}
{2\mu/\sigma} \right)
& \quad \tilde{q}_{\mu} > \mu^2/\sigma^{2} \;.
\end{array}
\right.$$
The special case $\mu = \mu^{\prime}$ is
$$\label{eq:tildeqmmcdf}
F(\tilde{q}_{\mu}|\mu) =
\: \left\{ \! \! \begin{array}{lll}
\Phi\Big( \sqrt{\tilde{q}_{\mu}} \Big)
& \quad 0 < \tilde{q}_{\mu} \le \mu^2/\sigma^2
\;, \\*[0.5 cm]
\Phi \left( \frac{ \tilde{q}_{\mu} + \mu^2/\sigma^2}
{2\mu/\sigma} \right)
& \quad \tilde{q}_{\mu} > \mu^2/\sigma^2 \;.
\end{array}
\right.$$
The $p$-value of the hypothesized $\mu$ is as before given by one minus the cumulative distribution,
$$\label{eq:pvalmutilde}
p_{\mu} = 1 - F(\tilde{q}_{\mu} | \mu) \;,$$
and therefore the corresponding significance is
$$\label{eq:zmutilde}
Z_{\mu} =
\: \left\{ \! \! \begin{array}{lll}
\sqrt{\tilde{q}_{\mu}}
& \quad 0 < \tilde{q}_{\mu} \le \mu^2/\sigma^2 \;, \\*[0.5 cm]
\frac{ \tilde{q}_{\mu} + \mu^2/\sigma^2}{2\mu/\sigma}
& \quad \tilde{q}_{\mu} > \mu^2/\sigma^2 \;.
\end{array}
\right.$$
As when using $q_{\mu}$, the upper limit on $\mu$ at confidence level $1 - \alpha$ is found by setting $p_{\mu} = \alpha$ and solving for $\mu$, which reduces to the same result as found when using $q_{\mu}$, namely,
$$\label{eq:muuptilde}
\mu_{\rm up} = \hat{\mu} + \sigma \Phi^{-1}(1 - \alpha) \;.$$
That is, to the extent that the Wald approximation holds, the two statistics $q_{\mu}$ and $\tilde{q}_{\mu}$ lead to identical upper limits.
Distribution of $-2 \ln (L_{s+b}/L_b)$ {#sec:tevatron}
--------------------------------------
Many analyses carried out at the Tevatron Collider (e.g., [@TevatronSearch]) involving searches for a new signal process have been based on the statistic
$$\label{eq:qtevdef}
q = -2 \ln \frac{L_{s+b}}{L_b} \;,$$
where $L_{s+b}$ is the likelihood of the nominal signal model and $L_b$ is that of the background-only hypothesis. That is, the $s+b$ corresponds to having the strength parameter $\mu= 1$ and $L_b$ refers to $\mu = 0$. The statistic $q$ can therefore be written
$$\label{eq:qtev2}
q = -2 \ln \frac{L(\mu=1, \hat{\hat{\vec{\theta}}}(1))}
{L(\mu=0, \hat{\hat{\vec{\theta}}}(0))}
= - 2 \ln \lambda(1) + 2 \ln \lambda(0) \;.$$
Assuming the validity of the Wald approximation (\[eq:wald\]), $q$ is given by
$$\label{eq:qtevwald}
q = \frac{(\hat{\mu} - 1)^2 }{\sigma^2} - \frac{\hat{\mu}^2}{ \sigma^2 }
= \frac{ 1 - 2 \hat{\mu}}{\sigma^2} \;,$$
where as previously $\sigma^2$ is the variance of $\hat{\mu}$. As $\hat{\mu}$ follows a Gaussian distribution, the distribution of $q$ is also seen to be Gaussian, with a mean value of
$$\label{eq:meanqtev}
E[q] = \frac{ 1 - 2 \mu }{\sigma^2}$$
and a variance of
$$\label{eq:varqtev}
V[q] = \frac{4}{\sigma^2} \;.$$
That is, the standard deviation of $q$ is $\sigma_q = 2 /
\sigma$, where the standard deviation of $\hat{\mu}$, $\sigma$, can be estimated, e.g., using the second derivatives of the log-likelihood function as described in Sec. \[sec:wald\] or with the methods discussed in Sec. \[sec:asimov\]. Recall that in general $\sigma$ depends on the hypothesized value of $\mu$; here we will refer to these as $\sigma_{b}$ and $\sigma_{s+b}$ for the $\mu=0$ and $\mu=1$ hypotheses, respectively.
From Eq. (\[eq:meanqtev\]) one sees that for the $s+b$ hypothesis ($\mu=1$) the values of $q$ tend to be lower, and for the $b$ hypothesis ($\mu=0$) they are higher. Therefore we can find the $p$-values for the two hypothesis from
$$\begin{aligned}
\label{eq:psb}
p_{s+b} & = & \int_{q_{\rm obs}}^{\infty} f(q|s+b) \, dq = 1 - \Phi
\left( \frac{q_{\rm obs} + 1/\sigma_{s+b}}{2/\sigma_{s+b}} \right) \;,
\\*[0.2 cm]
\label{eq:pb}
p_{b} & = & \int_{-\infty}^{q_{\rm obs}} f(q|b) \, dq =
\Phi \left(
\frac{q_{\rm obs} - 1/\sigma_b}{2/\sigma_b} \right) \;,\end{aligned}$$
where we have used Eqs. (\[eq:meanqtev\]) and (\[eq:varqtev\]) for the mean and variance of $q$ under the $b$ and $s+b$ hypotheses.
The $p$-values from Eqs. (\[eq:psb\]) and (\[eq:pb\]) incorporate the effects of systematic uncertainties to the extent that these are connected to the nuisance parameters $\vec{\theta}$. In analyses done at the Tevatron such as in Ref. [@TevatronSearch], these effects are incorporated into the distribution of $q$ in a different but largely equivalent way. There, usually one treats the control measurements that constrain the nuisance parameters as fixed, and to determine the distribution of $q$ one only generates the main search measurement (i.e., what corresponds in our generic analysis to the histogram $\vec{n}$). The effects of the systematic uncertainties are taken into account by using the control measurements as the basis of a Bayesian prior density $\pi(\vec{\theta})$, and the distribution of $q$ is computed under assumption of the Bayesian model average
$$\label{eq:bayesmodave}
f(q) = \int f(q | \vec{\theta}) \pi(\vec{\theta}) \, d \vec{\theta}
\;.$$
The prior pdf $\pi(\vec{\theta})$ used in Eq. (\[eq:bayesmodave\]) would be obtained from some measurements characterized by a likelihood function $L_{\vec{\theta}}(\vec{\theta})$, and then used to find the prior $\pi(\vec{\theta})$ using Bayes’ theorem,
$$\label{eq:bayesthm}
\pi(\vec{\theta}) \propto L_{\vec{\theta}}(\vec{\theta})
\pi_0(\vec{\theta}) \;.$$
Here $\pi_0(\vec{\theta})$ is the initial prior for $\vec{\theta}$ that reflected one’s knowledge before carrying out the control measurements. In many cases this is simply take as a constant, in which case $\pi(\vec{\theta})$ is simply proportional to $L_{\vec{\theta}}(\vec{\theta})$.
In the approach of this paper, however, all measurements are regarded as part of the data, including control measurements that constrain nuisance parameters. That is, here to generate a data set by MC means, for a given assumed point in the model’s parameter space, one simulates both the control measurements and the main measurement. Although this is done for a specific value of $\vec{\theta}$, in the asymptotic limit the distributions required for computing the $p$-values (\[eq:psb\]) and (\[eq:pb\]) are only weakly dependent on $\vec{\theta}$ to the extent that this can affect the standard deviation $\sigma_q$. By contrast, in the Tevatron approach one generates only the main measurement with data distributed according to the averaged model (\[eq:bayesmodave\]). In the case where the nuisance parameters are constrained by Gaussian distributed estimates and the initial prior $\pi_0(\vec{\theta})$ is taken to be constant, the two methods are essentially equivalent.
Assuming the Wald approximation holds, the statistic $q$ as well as $q_0$ from Eq. (\[eq:q0\]), $q_{\mu}$ from Eq. (\[eq:qmu\]) and $\tilde{q}_{\mu}$ from Eq. (\[eq:qmutilde\]) are all monotonic functions of $\hat{\mu}$, and therefore all are equivalent to $\hat{\mu}$ in terms of yielding the same statistical test. If there are no nuisance parameters, then the Neyman–Pearson lemma (see, e.g., [@Kendall2]) states that the likelihood ratio $L_{s+b}/L_{b}$ (or equivalently $q$) is an optimal test statistic in the sense that it gives the maximum power for a test of the background-only hypothesis with respect to the alternative of signal plus background (and vice versa). But if the Wald approximation holds, then $q_0$ and $q_{\mu}$ lead to equivalent tests and are therefore also optimal in the Neyman–Pearson sense. If the nuisance parameters are well constrained by control measurements, then one expects this equivalence to remain approximately true.
Finally, note that in many analyses carried out at the Tevatron, hypothesized signal models are excluded based not on whether the $p$-value $p_{s+b}$ from Eq. (\[eq:psb\]) is less than a given threshold $\alpha$, but rather the ratio $\mbox{CL}_s = p_{s+b}/(1 - p_{b})$ is compared to $\alpha$. We do not consider this final step here; it is discussed in, e.g., Ref. [@cls].
Experimental sensitivity {#sec:sensitivity}
========================
To characterize the sensitivity of an experiment, one is interested not in the significance obtained from a single data set, but rather in the expected (more precisely, median) significance with which one would be able to reject different values of $\mu$. Specifically, for the case of discovery one would like to know the median, under the assumption of the nominal signal model ($\mu=1$), with which one would reject the background-only ($\mu=0$) hypothesis. And for the case of setting exclusion limits the sensitivity is characterized by the median significance, assuming data generated using the $\mu=0$ hypothesis, with which one rejects a nonzero value of $\mu$ (usually $\mu=1$ is of greatest interest).
The sensitivity of an experiment is illustrated in Fig. \[fig:sensitivity\], which shows the pdf for $q_{\mu}$ assuming both a strength parameter $\mu$ and also assuming a different value $\mu^{\prime}$. The distribution $f(q_{\mu}|\mu^{\prime})$ is shifted to higher value of $q_{\mu}$, corresponding on average to lower $p$-values. The sensitivity of an experiment can be characterized by giving the $p$-value corresponding to the median $q_{\mu}$ assuming the alternative value $\mu^{\prime}$. As the $p$-value is a monotonic function of $q_{\mu}$, this is equal to the median $p$-value assuming $\mu^{\prime}$.
(10.0,5) (1,-0.2)
(9.0,0.)
(5,4)\[b\]
In the rest of this section we describe the ingredients needed to determine the experimental sensitivity (median discovery or exclusion significance). In Sec. \[sec:asimov\] we introduced the Asimov data set, in which all statistical fluctuations are suppressed. This will lead directly to estimates of the experimental sensitivity (Sec. \[sec:medsig\]) as well as providing an alternative estimate of the standard deviation $\sigma$ of the estimator $\hat{\mu}$. In Sec. \[sec:multichan\] we indicate how the procedure can be extended to the case where several search channels are combined, and in Sec. \[sec:statvar\] we describe how to give statistical error bands for the sensitivity.
The median significance from Asimov values of the test statistic {#sec:medsig}
----------------------------------------------------------------
By using the Asimov data set one can easily obtain the median values of $q_0$, $q_{\mu}$ and $\tilde{q}_{\mu}$, and these lead to simple expressions for the corresponding median significance. From Eqs. (\[eq:Z0\]), (\[eq:Zmu\]) and (\[eq:zmutilde\]) one sees that the significance $Z$ is a monotonic function of $q$, and therefore the median $Z$ is simply given by the corresponding function of the median of $q$, which is approximated by its Asimov value. For discovery using $q_0$ one wants the median discovery significance assuming a strength parameter $\mu^{\prime}$ and for upper limits one is particularly interested in the median exclusion significance assuming $\mu^{\prime} = 0$, $\mbox{med}[Z_{\mu} | 0]$. For these one obtains
$$\begin{aligned}
\label{eq:medZ0}
\mbox{med}[Z_0| \mu^{\prime}] & = &
\sqrt{q_{0,{\rm A}}} \;, \\*[0.2 cm]
\label{eq:medZmu}
\mbox{med}[Z_{\mu} | 0 ] & = & \sqrt{q_{\mu,{\rm A}}} \;.\end{aligned}$$
When using $\tilde{q}_{\mu}$ for establishing upper limits, the general expression for the exclusion significance $Z_{\mu}$ is somewhat more complicated depending on $\mu^{\prime}$, but is in any case found by substituting the appropriate values of $\tilde{q}_{\mu,{\rm A}}$ and $\sigma_{\rm A}$ into Eq. (\[eq:zmutilde\]). For the usual case where one wants the median significance for $\mu$ assuming data distributed according to the background-only hypothesis ($\mu^{\prime} = 0$), Eq. (\[eq:zmutilde\]) reduces in fact to a relation of the same form as Eq. (\[eq:Zmu\]), and therefore one finds
$$\label{eq:zmutilde2}
\mbox{med}[Z_{\mu}|0] = \sqrt{\tilde{q}_{\mu,{\rm A}}} \;.$$
Combining multiple channels {#sec:multichan}
---------------------------
In many analyses, there can be several search channels which need to be combined. For each channel $i$ there is a likelihood function $L_i(\mu, \vec{\theta}_i)$, where $\vec{\theta}_i$ represents the set of nuisance parameters for the $i$th channel, some of which may be common between channels. Here the strength parameter $\mu$ is assumed to be the same for all channels. If the channels are statistically independent, as can usually be arranged, the full likelihood function is given by the product over all of the channels,
$$\label{eq:Lfull} L(\mu, \vec{\theta}) = \prod_i L_i (\mu,
\vec{\theta}_i) \;,$$
where $\vec{\theta}$ represents the complete set of all nuisance parameters. The profile likelihood ratio $\lambda(\mu)$ is therefore
$$\label{eq:lambdaFull} \lambda(\mu) =
\frac{ \prod_i L_i ( \mu, \hat{\hat{\vec{\theta}}}_i) }
{ \prod_i L_i (\hat{\mu}, \hat{\vec{\theta}}_{i} ) }
\;.$$
Because the Asimov data contain no statistical fluctuations, one has $\hat{\mu} = \mu^{\prime}$ for all channels. Furthermore any common components of $\vec{\theta}_{i}$ are the same for all channels. Therefore when using the Asimov data corresponding to a strength parameter $\mu^{\prime}$ one finds
$$\label{eq:lambdaAsimovCombo}
\lambda_{\rm A}(\mu) =
\frac{ \prod_i L_i ( \mu, \hat{\hat{\vec{\theta}}}) }
{ \prod_i L_i (\mu^{\prime}, \vec{\theta} ) }
= \prod_i \lambda_{{\rm A},i}(\mu) \;,$$
where $\lambda_{{\rm A},i}(\mu)$ is the profile likelihood ratio for the $i$th channel alone.
Because of this, it is possible to determine the values of the profile likelihood ratio entering into (\[eq:lambdaAsimovCombo\]) separately for each channel, which simplifies greatly the task of estimating the median significance that would result from the full combination. It should be emphasized, however, that to find the discovery significance or exclusion limits determined from real data, one needs to construct the full likelihood function containing a single parameter $\mu$, and this must be used in a global fit to find the profile likelihood ratio.
Expected statistical variation (error bands) {#sec:statvar}
--------------------------------------------
By using the Asimov data set we can find the median, assuming some strength parameter $\mu^{\prime}$ of the significance for rejecting a hypothesized value $\mu$. Even if the hypothesized value $\mu^{\prime}$ is correct, the actual data will contain statistical fluctuations and thus the observed significance is not in general equal to the median.
For example, if the signal is in fact absent but the number of background events fluctuates upward, then the observed upper limit on the parameter $\mu$ will be weaker than the median assuming background only. It is useful to know by how much the significance is expected to vary, given the expected fluctuations in the data. As we have formulae for all of the relevant sampling distributions, we can also predict how the significance is expected to vary under assumption of a given signal strength.
It is convenient to calculate error bands for the median significance corresponding to the $\pm N \sigma$ variation of $\hat{\mu}$. As $\hat{\mu}$ is Gaussian distributed, these error bands on the significance are simply the quantiles that map onto the variation of $\hat{\mu}$ of $\pm N \sigma$ about $\mu^{\prime}$.
For the case of discovery, i.e., a test of $\mu = 0$, one has from Eqs. (\[eq:q0wald\]) and (\[eq:Z0\]) that the significance $Z_0$ is
$$\label{eq:Z0wald}
Z_0 =
\left\{ \! \! \begin{array}{ll}
\hat{\mu} / \sigma
& \quad \hat{\mu} \ge 0 \;, \\*[0.3 cm]
0 & \quad \hat{\mu} < 0 \;.
\end{array}
\right.$$
Furthermore the median significance is found from Eq. (\[eq:medZ0\]), so the significance values corresponding to $\mu^{\prime} \pm N \sigma$ are therefore
$$\begin{aligned}
\label{eq:Z0band}
Z_0(\mu^{\prime}+N\sigma) & = &
\mbox{med}[Z | \mu^{\prime}] + N \;, \\*[0.2 cm]
Z_0(\mu^{\prime}-N\sigma) & = & \mbox{max}\left[
\mbox{med}[Z | \mu^{\prime}] - N, 0 \right] \;.\end{aligned}$$
For the case of exclusion, when using both the statistic $q_{\mu}$ as well as $\tilde{q}_{\mu}$ one found the same expression for the upper limit at a confidence level of $1 - \alpha$, namely, Eq. (\[eq:muup\]). Therefore the median upper limit assuming a strength parameter $\mu^{\prime}$ is found simply by substituting this for $\hat{\mu}$, and the $\pm N \sigma$ error bands are found similarly by substituting the corresponding values of $\mu^{\prime}
\pm N \sigma$. That is, the median upper limit is
$$\label{eq:medmuup}
\mbox{med}[\mu_{\rm up}|\mu^{\prime}] =
\mu^{\prime} + \sigma \Phi^{-1}(1 - \alpha) \;,$$
and the $\pm N\sigma$ error band is given by
$$\label{eq:muupband}
\mbox{band}_{N\sigma} =
\mu^{\prime} + \sigma ( \Phi^{-1}(1 - \alpha) \pm N ) \;.$$
The standard deviation $\sigma$ of $\hat{\mu}$ can be obtained from the Asimov value of the test statistic $q_{\mu}$ (or $\tilde{q}_{\mu}$) using Eq. (\[eq:sigma2\]).
Examples {#sec:examples}
========
In this section we describe two examples, both of which are special cases of the generic analysis described in Section \[sec:formalism\]. Here one has a histogram $\vec{n} =
(n_1, \ldots, n_N)$ for the main measurement where signal events could be present and one may have another histogram $\vec{m} = (m_1, \ldots,
m_M)$ as a control measurement, which helps constrain the nuisance parameters. In Section \[sec:counting\] we treat the simple case where each of these two measurements consists of a single Poisson distributed value, i.e., the histograms each have a single bin. We refer to this as a “counting experiment”. In Section \[sec:shape\] we consider multiple bins for the main histogram, but without a control histogram; here the measured shape of the main histogram on either side of the signal peak is sufficient to constrain the background. We refer to this as a “shape analysis”.
Counting experiment {#sec:counting}
-------------------
Consider an experiment where one observes a number of events $n$, assumed to follow a Poisson distribution with an expectation value $E[n] = \mu s + b$. Here $s$ represents the mean number of events from a signal model, which we take to be a known value; $b$ is the expected number from background processes, and as usual $\mu$ is the strength parameter.
We will treat $b$ as a nuisance parameter whose value is constrained by a control measurement. This measurement is also a single Poisson distributed value $m$ with mean value $E[m] = \tau b$. That is, $\tau
b$ plays the role of the function $u$ for the single bin of the control histogram in Eq. (\[eq:emi\]). In a real analysis, the value of the scale factor $\tau$ may have some uncertainty and could be itself treated as a nuisance parameter, but in this example we will take its value to be known. Related aspects of this type of analysis have been discussed in the literature, where it is sometimes referred to as the “on-off problem” (see, e.g., [@Cranmer03; @Cousins08]).
The data thus consist of two measured values: $n$ and $m$. We have one parameter of interest, $\mu$, and one nuisance parameter, $b$. The likelihood function for $\mu$ and $b$ is the product of two Poisson terms:
$$\label{eq:likelihoodcounting}
L(\mu, b) = \frac{(\mu s + b)^n}{n!} e^{-(\mu s + b)} \,
\frac{(\tau b)^m}{m!} e^{-\tau b} \;.$$
To find the test statistics $q_0$, $q_{\mu}$ and $\tilde{q}_{\mu}$, we require the ML estimators $\hat{\mu}$, $\hat{b}$ as well as the conditional ML estimator $\hat{\hat{b}}$ for a specified $\mu$. These are found to be
$$\begin{aligned}
\label{eq:muhatcounting}
\hat{\mu} & = & \frac{n - m/\tau}{s} \;, \\*[0.2 cm]
\label{eq:bhatcounting}
\hat{b} & = & \frac{m}{\tau} \;, \\*[0.2 cm]
\label{eq:bhathatcounting}
\hat{\hat{b}} & = & \frac{n + m - (1 + \tau) \mu s}{2 (1 + \tau)}
+ \left[ \frac{ (n + m - (1+\tau) \mu s)^2 + 4 (1 + \tau) m \mu s}{
4 (1 + \tau)^2 } \right]^{1/2} \;.\end{aligned}$$
Given measured values $n$ and $m$, the estimators from Eqs. (\[eq:muhatcounting\]), (\[eq:bhatcounting\]) and (\[eq:bhathatcounting\]) can be used in the likelihood function (\[eq:likelihoodcounting\]) to find the values of the test statistics $q_0$, $q_{\mu}$ and $\tilde{q}_{\mu}$. By generating data values $n$ and $m$ by Monte Carlo we can compare the resulting distributions with the formulae from Section \[sec:qdist\].
The pdf $f(q_0|0)$, i.e., the distribution of $q_0$ for under the assumption of $\mu=0$, is shown in Fig. \[fig:q0counting\](a). The histograms show the result from Monte Carlo simulation based on several different values of the mean background $b$. The solid curve shows the prediction of Eq. (\[eq:fq00\]), which is independent of the nuisance parameter $b$. The point at which one finds a significant departure between the histogram and the asymptotic formula occurs at increasingly large $q_0$ for increasing $b$. For $b=20$ the agreement is already quite accurate past $q_0 = 25$, corresponding to a significance of $Z =
\sqrt{q_0} = 5$. Even for $b = 2$ there is good agreement out to $q_0
\approx 10$.
(10.0,6.5) (.8,0)
(8,0)
(0.,6.)[(a)]{} (15,6.)[(b)]{}
Figure \[fig:q0counting\](b) shows distributions of $q_0$ assuming a strength parameter $\mu^{\prime}$ equal to 0 and 1. The histograms show the Monte Carlo simulation of the corresponding distributions using the parameters $s = 10$, $b = 10$, $\tau = 1$. For the distribution $f(q_0|1)$ from Eq. (\[eq:fq0muprimewald\]), one requires the value of $\sigma$, the standard deviation of $\hat{\mu}$ assuming a strength parameter $\mu^{\prime} = 1$. Here this was determined from Eq. (\[eq:sigma02\]) using the Asimov value $q_{0,{\rm A}}$, i.e., the value obtained from the Asimov data set with $n \rightarrow \mu^{\prime} s + b$ and $m \rightarrow \tau b$.
We can investigate the accuracy of the approximations used by comparing the discovery significance for a given observed value of $q_0$ from the approximate formula with the exact significance determined using a Monte Carlo calculation. Figure \[fig:q0test\](a) shows the discovery significance that one finds from $q_0 = 16$. According to Eq. (\[eq:Z0\]), this should give a nominal significance of $Z = \sqrt{q_0} = 4$, indicated in the figure by the horizontal line. The points show the exact significance for different values of the expected number of background events $b$ in the counting analysis with a scale factor $\tau = 1$. As can be seen, the approximation underestimates the significance for very low $b$, but achieves an accuracy of better than 10% for $b$ greater than around 4. It slightly overestimates for $b$ greater than around 5. This phenomenon can be seen in the tail of $f(q_0|0)$ in Fig. \[fig:q0counting\](b), which uses $b = 10$. The accuracy then rapidly improves for increasing $b$.
(10.0,6.5) (.8,0)
(8,0)
(0.,6.)[(a)]{} (15,6.)[(b)]{}
Figure \[fig:q0test\](b) shows the median value of the statistic $q_0$ assuming data distributed according to the nominal signal hypothesis from Monte Carlo (points) and the value based on the Asimov data set as a function of $b$ for different values of $s$, using a scale factor $\tau = 1$. One can see that the Asimov data set leads to an excellent approximation to the median, except at very low $s$ and $b$.
Figure \[fig:q1counting\](a) shows the distribution of the test statistic $q_1$ for $s = 6$, $b = 9$, $\tau = 1$ for data corresponding to a strength parameter $\mu^{\prime} = 1$ and also $\mu^{\prime} = 0$. The vertical lines indicate the Asimov values of $q_1$ and $\tilde{q}_1$ assuming a strength parameter $\mu^{\prime} =
0$. These lines correspond to estimates of the median values of the test statistics assuming $\mu^{\prime} = 0$. The areas under the curves $f(q_1|1)$ and $f(\tilde{q}_1|1)$ to the right of this line give the median $p$-values.
(10.0,6.5) (.8,0)
(8,0)
(0.,6.)[(a)]{} (15,6.)[(b)]{}
For the example described above we can also find the distribution of the statistic $q = - 2 \ln (L_{s+b}/L_{b})$ as defined in Sec. \[sec:tevatron\]. Figure \[fig:qtevDist\] shows the distributions of $q$ for the hypothesis of $\mu=0$ (background only) and $\mu=1$ (signal plus background) for the model described above using $b = 20$, $s = 10$ and $\tau = 1$. The histograms are from Monte Carlo, and the solid curves are the predictions of the asymptotic formulae given in Sec. \[sec:tevatron\]. Also shown are the $p$-values for the background-only and signal-plus-background hypotheses corresponding to a possible observed value of the statistic $q_{\rm obs}$.
(10.0,6.) (.5,-0.5)
(9.0,0.)
(6,4)\[b\]
### Counting experiment with known $b$ {#sec:countbknown}
An important special case of the counting experiment above is where the mean background $b$ is known with negligible uncertainty and can be treated as a constant. This would correspond to having a very large value for the scale factor $\tau$.
If we regard $b$ as known, the data consist only of $n$ and thus the likelihood function is
$$L(\mu) = \frac{(\mu s + b)^n}{n!} e^{- (\mu s + b)}
\;,$$
The test statistic for discovery $q_0$ can be written
$$\label{eq:q0count}
q_0 = \left\{ \! \! \begin{array}{ll}
- 2 \ln \frac{L(0)}{L(\hat{\mu})} & \hat{\mu} \ge 0 , \\*[0.3 cm]
0 & \hat{\mu} < 0 \;,
\end{array}
\right.$$
where $\hat{\mu} = n - b$. For sufficiently large $b$ we can use the asymptotic formula (\[eq:Z0\]) for the significance,
$$\label{eq:Z0count}
Z_0 = \sqrt{q_0} = \left\{ \! \! \begin{array}{ll}
\sqrt{ 2 \left( n \ln \frac{n}{b} + b - n \right) }
& \hat{\mu} \ge 0 , \\*[0.3 cm]
0 & \hat{\mu} < 0 .
\end{array}
\right.$$
To approximate the median significance assuming the nominal signal hypothesis ($\mu = 1$) we replace $n$ by the Asimov value $s + b$ to obtain
$$\label{eq:Zmedcount}
\mbox{med}[Z_0 | 1 ] = \sqrt{q_{0,{\rm A}}} =
\sqrt{ 2 \left( (s+b) \ln (1 + s/b) - s \right) } \;.$$
Expanding the logarithm in $s/b$ one finds
$$\label{eq:Zmedcount2}
\mbox{med}[Z_0 | 1 ] = \frac{s}{\sqrt{b}}
\left( 1 + {\cal O}(s/b) \right)\;.$$
Although $Z_0 \approx s/\sqrt{b}$ has been widely used for cases where $s + b$ is large, one sees here that this final approximation is strictly valid only for $s \ll b$. Median values, assuming $\mu=1$, of $Z_0$ for different values of $s$ and $b$ are shown in Fig. \[fig:medsig\]. The solid curve shows Eq. (\[eq:Zmedcount\]), the dashed curve gives the approximation $s/\sqrt{b}$, and the points are the exact median values from Monte Carlo. The structure seen in the points is due to the discrete nature of the data. One sees that Eq. (\[eq:Zmedcount\]) provides a much better approximation to the true median than does $s/\sqrt{b}$ in regions where $s/b$ cannot be regarded as small.
(10.0,6) (.5,-0.5)
(9.0,0.)
(6,4)\[b\]
Shape Analysis {#sec:shape}
--------------
As a second example we consider the case where one is searching for a peak in an invariant mass distribution. The main histogram $\vec{n} =
(n_1, \ldots, n_N)$ for background is shown in Fig. \[fig:pseudo\_exp\], which is here taken to be a Rayleigh distribution. The signal is modeled as a Gaussian of known width and mass (position). In this example there is no subsidiary histogram $(m_1, \ldots, m_M)$.
(10.0,5) (-.5,-4.5)
(10.0,0.)
(5,4)\[b\]
If, as is often the case, the position of the peak is not known a priori, then one will test all masses in a given range, and appearance of a signal-like peak anywhere could lead to rejection of the background-only hypothesis. In such an analysis, however, the discovery significance must take into account the fact that a fluctuation could occur at any mass within the range. This is often referred to as the “look-elsewhere effect”, and is discussed further in Ref. [@lee].
In the example presented here, however, we will test all values of the mass and $\mu$ using the statistic $q_{\mu}$ for purposes of setting an upper limit on the signal strength. Here, each hypothesis of mass and signal strength is in effect tested individually, and thus the look-elsewhere effect does not come into play.
We assume that the signal and background distributions are known up to a scale factor. For the signal, this factor corresponds to the usual strength parameter $\mu$; for the background, we introduce an analogous factor $\theta$. That is, the mean value of the number of events in the $i$th bin is $E[n_i] = \mu s_i + b_i$, where $\mu$ is the signal strength parameter and the $s_i$ are taken as known. We assume that the background terms $b_i$ can be expressed as $b_i =
\theta f_{{\rm b},i}$, where the probability to find a background event in bin $i$, $f_{{\rm b},i}$, is known, and $\theta$ is a nuisance parameter that gives the total expected number of background events. Therefore the likelihood function can be written
$$\label{eq:likelihoodshape}
L(\mu, \theta) = \prod_{i=1}^N
\frac{(\mu s_i + \theta f_{{\rm b},i})^{n_i}}{n_i!}
e^{-(\mu s_i + \theta f_{{\rm b},i})}$$
For a given data set $\vec{n} = (n_1, \ldots, n_N)$ one can evaluate the likelihood (\[eq:likelihoodshape\]) and from this determine any of the test statistics discussed previously. Here we concentrate on the statistic $q_{\mu}$ used to set an upper limit on $\mu$, and compare the distribution $f(q_{\mu} | \mu^{\prime})$ from Eq.( \[eq:fq0muprimewald\]) with histograms generated by Monte Carlo. Figure \[fig:qmu\_up\_dist\] shows $f(q_{\mu} | 0)$ (red) and $f(q_{\mu} | \mu)$ (blue).
(10.0,5.) (-1,-5)
(10.0,0.)
(5,4)\[b\]
The vertical line in Fig. \[fig:qmu\_up\_dist\] gives the median value of $q_{\mu}$ assuming a strength parameter $\mu^{\prime} = 0$. The area to the right of this line under the curve of $f(q_{\mu}|\mu)$ gives the $p$-value of the hypothesized $\mu$, as shown shaded in green. The upper limit on $\mu$ at a confidence level $\mbox{CL} = 1
- \alpha$ is the value of $\mu$ for which the $p$-value is $p_{\mu} =
\alpha$. Figure \[fig:qmu\_up\_dist\] shows the distributions for the value of $\mu$ that gave $p_{\mu} = 0.05$, corresponding to the 95% CL upper limit.
In addition to reporting the median limit, one would like to know how much it would vary for given statistical fluctuations in the data. This is illustrated in Fig. \[fig:qmu\_up+1\_dist2\], which shows the same distributions as in Figure \[fig:qmu\_up\_dist\], but here the vertical line indicates the 15.87% quantile of the distribution $f(q_{\mu} | 0)$, corresponding to having $\hat{\mu}$ fluctuate downward one standard deviation below its median.
(10.0,5.5) (-1,-5)
(10.0,0.)
(5,4)\[b\]
By simulating the experiment many times with Monte Carlo, we can obtain a histogram of the upper limits on $\mu$ at 95% CL, as shown in Fig. \[fig:mu\_up\_dist\]. The $\pm 1 \sigma$ (green) and $\pm 2
\sigma$ (yellow) error bands are obtained from the MC experiments. The vertical lines indicate the error bands as estimated directly (without Monte Carlo) using Eqs. (\[eq:medmuup\]) and (\[eq:muupband\]). As can be seen from the plot, the agreement between the formulae and MC predictions is excellent.
(10.0,5.5) (-1,-4.5)
(10.0,0.)
(5,4)\[b\]
Figures \[fig:qmu\_up\_dist\] through \[fig:mu\_up\_dist\] correspond to finding upper limit on $\mu$ for a specific value of the peak position (mass). In a search for a signal of unknown mass, the procedure would be repeated for all masses (in practice in small steps). Figure \[fig:limit\_CLsb\_modified\_1\] shows the median upper limit at 95% CL as a function of mass. The median (central blue line) and error bands ($\pm 1\sigma$ in green, $\pm 2 \sigma$ in yellow) are obtained using Eqs. (\[eq:medmuup\]) and (\[eq:muupband\]). The points and connecting curve correspond to the upper limit from a single arbitrary Monte Carlo data set, generated according to the background-only hypothesis. As can be seen, most of the plots lie as expected within the $\pm 1 \sigma$ error band.
(10.0,5.5) (-1,-4.5)
(10.0,0.)
(5,4)\[b\]
Implementation in RooStats {#sec:roostats}
==========================
Many of the results presented above are implemented or are being implemented in the framework [@Moneta:2010pm], which is a C++ class library based on the ROOT [@Brun:1997pa] and RooFit [@Verkerke:2003ir] packages. The tools in RooStats can be used to represent arbitrary probability density functions that inherit from `RooAbsPdf`, the abstract interfaces for probability density functions provided by RooFit.
The framework provides an interface with minimization packages such as `Minuit` [@James:1975dr]. This allows one to obtain the estimators required in the the profile likelihood ratio: $\hat{\mu}$, $\hat{\vec{\theta}}$, and $\hat{\hat{\vec{\theta}}}$. The Asimov dataset defined in Eq. (\[eq:asimovn\]) can be determined for a probability density function by specifying the `ExpectedData()` command argument in a call to the `generateBinned` method. The Asimov data together with the standard `HESSE` covariance matrix provided by `Minuit` makes it is possible to determine the Fisher information matrix shown in Eq. (\[eq:invcov\]), and thus obtain the related quantities such as the variance of $\hat{\mu}$ and the noncentrality parameter $\Lambda$, which enter into the formulae for a number of the distributions of the test statistics presented above.
The distributions of the various test statistics and the related formulae for $p$-values, sensitivities and confidence intervals as given in Sections \[sec:formalism\], \[sec:qdist\] and \[sec:sensitivity\] are being incorporated as well. RooStats currently includes the test statistics $t_\mu$, $\tilde{t}_\mu$, $q_0$, and $q$, $q_\mu$, and $\tilde{q}_\mu$ as concrete implementations of the `TestStatistic` interface. Together with the Asimov data, this provides the ability to calculate the alternative estimate, $\sigma_A$, for the variance of $\hat{\mu}$ shown in Eq. (\[eq:sigma2\]). The noncentral chi-square distribution is being incorporated into both RooStats and ROOT’s mathematics libraries for more general use. The various transformations of the noncentral chi-square used to obtain Eqs. (\[eq:ftmumPrime\]), (\[eq:ftildetmmp\]), (\[eq:fq0muprimewald\]), (\[eq:fqmmp\]), and (\[eq:ftildeqmmp\]) are also in development in the form of concrete implementations of the `SamplingDistribution` interface. Together, these new classes will allow one to reproduce the examples shown in Section \[sec:examples\] and to extend them to an arbitrary model within the RooStats framework.
Conclusions {#sec:conclusions}
===========
Statistical tests are described for use in planning and carrying out a search for new phenomena. The formalism allows for the treatment of systematic uncertainties through use of the profile likelihood ratio. Here a systematic uncertainty is included to the extent that the model includes a sufficient number of nuisance parameters so that for at least some point in its parameter space it can be regarded as true.
Approximate formulae are given for the distributions of test statistics used to characterize the level of agreement between the data and the hypothesis being tested, as well as the related expressions for $p$-values and significances. The statistics are based on the profile likelihood ratio and can be used for a two-sided test of a strength parameter $\mu$ ($t_{\mu}$), a one-sided test for discovery ($q_0$), and a one-sided test for finding an upper limit ($q_{\mu}$ and $\tilde{q}_{\mu}$). The statistic $\tilde{t}_{\mu}$ can be used to obtain a “unified” confidence interval, in the sense that it is one- or two-sided depending on the data outcome.
Formulae are also given that allow one to characterize the sensitivity of a planned experiment through the median significance of a given hypothesis under assumption of a different one, e.g., median significance with which one would reject the background-only hypothesis under assumption of a certain signal model. These exploit the use of an artificial data set, the “Asimov” data set, defined so as to make estimators for all parameters equal to their true values. Methods for finding the expected statistical variation in the sensitivity (error bands) are also given.
These tools free one from the need to carry out lengthy Monte Carlo calculations, which in the case of a discovery at $5 \sigma$ significance could require simulation of around $10^8$ measurements. They are are particularly useful in cases where one needs to estimate experimental sensitivities for many points in a multidimensional parameter space (e.g., for models such as supersymmetry), which would require generating a large MC sample for each point.
The approximations used are valid in the limit of a large data sample. Tests with Monte Carlo indicate, however, that the formulae are in fact reasonably accurate even for fairly small samples, and thus can have a wide range of practical applicability. For very small samples and in cases where high accuracy is crucial, one is always free to validate the approximations with Monte Carlo.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Louis Fayard, Nancy Andari, Francesco Polci and Marumi Kado for fruitful discussions. We received useful feedback at the Banff International Research Station, specifically from Richard Lockhart and Earl Lawrence. One of us (E.G.) is obliged to the Benoziyo Center for High Energy Physics, to the the Israeli Science Foundation(ISF), the Minerva Gesellschaft and the German Israeli Foundation (GIF) for supporting this work. K.C. is supported by US National Science Foundation grant PHY-0854724. G.C. thanks the U.K. Science and Technology Facilities Council as well as the Einstein Center at the Weizmann Institute of Science, where part of his work on this paper was done.
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[^1]: The name of the Asimov data set is inspired by the short story [*Franchise*]{}, by Isaac Asimov [@Asimov]. In it, elections are held by selecting the single most representative voter to replace the entire electorate.
[^2]: Some authors, e.g., Ref. [@lepcombo], have defined this relation using a two-sided fluctuation of a Gaussian variable, with a $5 \sigma$ significance corresponding to $p = 5.7 \times 10^{-7}$. We take the one-sided definition above as this gives $Z=0$ for $p=0.5$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Rank aggregation systems collect ordinal preferences from individuals to produce a global ranking that represents the social preference. Rank-breaking is a common practice to reduce the computational complexity of learning the global ranking. The individual preferences are broken into pairwise comparisons and applied to efficient algorithms tailored for independent paired comparisons. However, due to the ignored dependencies in the data, naive rank-breaking approaches can result in inconsistent estimates. The key idea to produce accurate and consistent estimates is to treat the pairwise comparisons unequally, depending on the topology of the collected data. In this paper, we provide the optimal rank-breaking estimator, which not only achieves consistency but also achieves the best error bound. This allows us to characterize the fundamental tradeoff between accuracy and complexity. Further, the analysis identifies how the accuracy depends on the spectral gap of a corresponding comparison graph.'
author:
- |
[Ashish Khetan and Sewoong Oh ]{}\
[Department of ISE, University of Illinois at Urbana-Champaign]{}\
[Email: $\{$khetan2,swoh$\}$@illinois.edu]{}
bibliography:
- '\_ranking.bib'
title: 'Data-driven Rank Breaking for Efficient Rank Aggregation'
---
Introduction {#sec:intro}
============
In several applications such as electing officials, choosing policies, or making recommendations, we are given partial preferences from individuals over a set of alternatives, with the goal of producing a global ranking that represents the collective preference of the population or the society. This process is referred to as [*rank aggregation*]{}. One popular approach is [*learning to rank*]{}. Economists have modeled each individual as a rational being maximizing his/her perceived utility. Parametric probabilistic models, known collectively as Random Utility Models (RUMs), have been proposed to model such individual choices and preferences [@McF80]. This allows one to infer the global ranking by learning the inherent utility from individuals’ revealed preferences, which are noisy manifestations of the underlying true utility of the alternatives.
Traditionally, learning to rank has been studied under the following data collection scenarios: pairwise comparisons, best-out-of-$k$ comparisons, and $k$-way comparisons. [*Pairwise comparisons*]{} are commonly studied in the classical context of sports matches as well as more recent applications in crowdsourcing, where each worker is presented with a pair of choices and asked to choose the more favorable one. [*Best-out-of-$k$ comparisons*]{} data sets are commonly available from purchase history of customers. Typically, a set of $k$ alternatives are offered among which one is chosen or purchased by each customer. This has been widely studied in operations research in the context of modeling customer choices for revenue management and assortment optimization. The [*$k$-way comparisons*]{} are assumed in traditional rank aggregation scenarios, where each person reveals his/her preference as a ranked list over a set of $k$ items. In some real-world elections, voters provide ranked preferences over the whole set of candidates [@Lun07]. We refer to these three types of ordinal data collection scenarios as ‘traditional’ throughout this paper.
For such traditional data sets, there are several computationally efficient inference algorithms for finding the Maximum Likelihood (ML) estimates that provably achieve the minimax optimal performance [@NOS12; @SBB15; @HOX14]. However, modern data sets can be unstructured. Individual’s revealed ordinal preferences can be implicit, such as movie ratings, time spent on the news articles, and whether the user finished watching the movie or not. In crowdsourcing, it has also been observed that humans are more efficient at performing batch comparisons [@NIPS2011_4187], as opposed to providing the full ranking or choosing the top item. This calls for more flexible approaches for rank aggregation that can take such diverse forms of ordinal data into account. For such non-traditional data sets, finding the ML estimate can become significantly more challenging, requiring run-time exponential in the problem parameters.
To avoid such a computational bottleneck, a common heuristic is to resort to [*rank-breaking*]{}. The collected ordinal data is first transformed into a bag of pairwise comparisons, ignoring the dependencies that were present in the original data. This is then processed via existing inference algorithms tailored for [*independent*]{} pairwise comparisons, hoping that the dependency present in the input data does not lead to inconsistency in estimation. This idea is one of the main motivations for numerous approaches specializing in learning to rank from pairwise comparisons, e.g., [@Ford57; @NOS14; @ACPX13]. However, such a heuristic of full rank-breaking defined explicitly in , where all pairwise comparisons are weighted and treated equally ignoring their dependencies, has been recently shown to introduce inconsistency [@APX14a].
The key idea to produce accurate and consistent estimates is to treat the pairwise comparisons unequally, depending on the topology of the collected data. A fundamental question of interest to practitioners is how to choose the weight of each pairwise comparison in order to achieve not only consistency but also the best accuracy, among those consistent estimators using rank-breaking. We study how the accuracy of resulting estimate depends on the topology of the data and the weights on the pairwise comparisons. This provides a guideline for the optimal choice of the weights, driven by the topology of the data, that leads to accurate estimates.
[**Problem formulation.**]{} The premise in the current race to collect more data on user activities is that, a hidden true preference manifests in the user’s activities and choices. Such data can be explicit, as in ratings, ranked lists, pairwise comparisons, and like/dislike buttons. Others are more implicit, such as purchase history and viewing times. While more data in general allows for a more accurate inference, the heterogeneity of user activities makes it difficult to infer the underlying preferences directly. Further, each user reveals her preference on only a few contents.
Traditional collaborative filtering fails to capture the diversity of modern data sets. The sparsity and heterogeneity of the data renders typical similarity measures ineffective in the nearest-neighbor methods. Consequently, simple measures of similarity prevail in practice, as in Amazon’s “people who bought ... also bought ...” scheme. Score-based methods require translating heterogeneous data into numeric scores, which is a priori a difficult task. Even if explicit ratings are observed, those are often unreliable and the scale of such ratings vary from user to user.
We propose aggregating ordinal data based on users’ revealed preferences that are expressed in the form of [*partial orderings*]{} (notice that our use of the term is slightly different from its original use in revealed preference theory). We interpret user activities as manifestation of the hidden preferences according to discrete choice models (in particular the Plackett-Luce model defined in ). This provides a more reliable, scale-free, and widely applicable representation of the heterogeneous data as partial orderings, as well as a probabilistic interpretation of how preferences manifest. In full generality, the data collected from each individual can be represented by a [*partially ordered set (poset)*]{}. Assuming consistency in a user’s revealed preferences, any ordered relations can be seamlessly translated into a poset, represented as a Hasse diagram by a directed acyclic graph (DAG). The DAG below represents ordered relations $a>\{b,d\}$, $b>c$, $\{c,d\}>e$, and $e>f$. For example, this could have been translated from two sources: a five star rating on $a$ and a three star ratings on $b,c,d$, a two star rating on $e$, and a one star rating on $f$; and the item $b$ being purchased after reviewing $c$ as well.
![ A DAG representation of consistent partial ordering of a user $j$, also called a Hasse diagram (left). A set of rank-breaking graphs extracted from the Hasse diagram for the separator item $a$ and $e$, respectively (right).[]{data-label="fig:hasse"}](hasse "fig:"){width=".2\textwidth"} (-50,60)[${\cal G}_j$]{} ![ A DAG representation of consistent partial ordering of a user $j$, also called a Hasse diagram (left). A set of rank-breaking graphs extracted from the Hasse diagram for the separator item $a$ and $e$, respectively (right).[]{data-label="fig:hasse"}](rbgraph "fig:"){width=".3\textwidth"} (-115,70)[$G_{j,1}$]{} (-40,70)[$G_{j,2}$]{}
There are $n$ users or agents, and each agent $j$ provides his/her ordinal evaluation on a subset $S_j$ of $d$ items or alternatives. We refer to $S_j\subset\{1,2,\ldots,d\}$ as [*offerings*]{} provided to $j$, and use $\kappa_j=|S_j|$ to denote the size of the offerings. We assume that the partial ordering over the offerings is a manifestation of her preferences as per a popular choice model known as Plackett-Luce (PL) model. As we explain in detail below, the PL model produces total orderings (rather than partial ones). The data collector queries each user for a partial ranking in the form of a poset over $S_j$. For example, the data collector can ask for the top item, unordered subset of three next preferred items, the fifth item, and the least preferred item. In this case, an example of such poset could be $a < \{b,c,d\} < e < f$, which could have been generated from a total ordering produced by the PL model and taking the corresponding partial ordering from the total ordering. Notice that we fix the topology of the DAG first and ask the user to fill in the node identities corresponding to her total ordering as (randomly) generated by the PL model. Hence, the structure of the poset is considered deterministic, and only the identity of the nodes in the poset is considered random. Alternatively, one could consider a different scenario where the topology of the poset is also random and depends on the outcome of the preference, which is out-side the scope of this paper and provides an interesting future research direction.
The PL model is a special case of [*random utility models*]{}, defined as follows [@WB02; @APX12]. Each item $i$ has a real-valued latent utility $\theta_i$. When presented with a set of items, a user’s reveled preference is a partial ordering according to noisy manifestation of the utilities, i.e. i.i.d. noise added to the true utility $\theta_i$’s. The PL model is a special case where the noise follows the standard Gumbel distribution, and is one of the most popular model in social choice theory [@McF73; @MT00]. PL has several important properties, making this model realistic in various domains, including marketing [@GL83], transportation [@McF80; @BL85], biology [@BIOLOGY], and natural language processing [@MCCD13]. Precisely, each user $j$, when presented with a set $S_j$ of items, draws a noisy utility of each item $i$ according to $$\begin{aligned}
u_i &=& \theta_i + Z_i\;, \label{eq:rum}\end{aligned}$$ where $Z_i$’s follow the independent standard Gumbel distribution. Then we observe the ranking resulting from sorting the items as per noisy observed utilities $u_j$’s. Alternatively, the PL model is also equivalent to the following random process. For a set of alternatives $S_j$, a ranking $\sigma_j:[|S|] \to S$ is generated in two steps: $(1)$ independently assign each item $i \in S_j$ an unobserved value $X_i$, exponentially distributed with mean $e^{-\theta_i}$; $(2)$ select a ranking $\sigma_j$ so that $X_{\sigma_j(1)} \leq X_{\sigma_j(2)} \leq \cdots \leq X_{\sigma_j(|S_j|)}$.
The PL model $(i)$ satisfies Luce’s ‘independence of irrelevant alternatives’ in social choice theory [@Ray73], and has a simple characterization as sequential (random) choices as explained below; and $(ii)$ has a maximum likelihood estimator (MLE) which is a convex program in $\theta$ in the traditional scenarios of pairwise, best-out-of-$k$ and $k$-way comparisons. Let $\prob(a>\{b,c,d\})$ denote the probability $a$ was chosen as the best alternative among the set $\{a,b,c,d\}$. Then, the probability that a user reveals a linear order $(a>b>c>d)$ is equivalent as making sequential choice from the top to bottom: $$\begin{aligned}
\prob(a>b>c>d) &=& \prob(a>\{b,c,d\})\, \,\prob(b>\{c,d\})\, \,\prob(c>d) \nonumber \\
&=& \frac{e^{\theta_a}}{(e^{\theta_a}+e^{\theta_b}+e^{\theta_c}+e^{\theta_d} ) } \,
\frac{e^{\theta_b}}{(e^{\theta_b}+e^{\theta_c}+e^{\theta_d} )} \,
\frac{e^{\theta_c}}{(e^{\theta_c}+e^{\theta_d})} \, \;.\end{aligned}$$ We use the notation $(a>b)$ to denote the event that $a$ is preferred over $b$. In general, for user $j$ presented with offerings $S_j$, the probability that the revealed preference is a total ordering $\sigma_j$ is $\prob(\sigma_j) = \prod_{i\in \{1,\ldots,\kappa_j-1\}} (e^{\theta_{\sigma^{-1}(i)}})/(\sum_{i'=i}^{\kappa_j}e^{\theta_{\sigma^{-1}(i')}}) $. We consider the true utility $\theta^* \in \Omega_b$, where we define $\Omega_b$ as $$\begin{aligned}
\Omega_b &\equiv& \Big\{ \, \theta \in \reals^d \,\big|\, \sum_{i\in[d]} \theta_i=0 \,,\, |\theta_i| \leq b \text{ for all $i\in[d]$ } \,\Big\} \;.\end{aligned}$$ Note that by definition, the PL model is invariant under shifting the utility $\theta_i$’s. Hence, the centering ensures uniqueness of the parameters for each PL model. The bound $b$ on the dynamic range is not a restriction, but is written explicitly to capture the dependence of the accuracy in our main results.
We have $n$ users each providing a partial ordering of a set of offerings $S_j$ according to the PL model. Let ${\cG}_j$ denote both the DAG representing the partial ordering from user $j$’s preferences. With a slight abuse of notations, we also let $\cG_j$ denote the set of rankings that are consistent with this DAG. For general partial orderings, the probability of observing $\cG_j$ is the sum of all total orderings that is consistent with the observation, i.e. $\prob(\cG_j)=\sum_{\sigma \in \cG_j} \prob(\sigma)$. The goal is to efficiently learn the true utility $\theta^*\in\Omega_b$, from the $n$ sampled partial orderings. One popular approach is to compute the maximum likelihood estimate (MLE) by solving the following optimization: $$\begin{aligned}
\underset{\theta \in \Omega_b }{\text{maximize}} && \sum_{j=1}^n \; \log \prob(\cG_j) \;.
\label{eq:mle}\end{aligned}$$ This optimization is a simple convex optimization, in particular a logit regression, when the structure of the data $\{\cG_j\}_{j\in[n]}$ is traditional. This is one of the reasons the PL model is attractive. However, for general posets, this can be computationally challenging. Consider an example of position-$p$ ranking, where each user provides which item is at $p$-th position in his/her ranking. Each term in the log-likelihood for this data involves summation over $O((p-1)!)$ rankings, which takes $O(n\,(p-1)!)$ operations to evaluate the objective function. Since $p$ can be as large as $d$, such a computational blow-up renders MLE approach impractical. A common remedy is to resort to rank-breaking, which might result in inconsistent estimates.
[**Rank-breaking.**]{} Rank-breaking refers to the idea of extracting a set of pairwise comparisons from the observed partial orderings and applying estimators tailored for paired comparisons treating each piece of comparisons as independent. Both the choice of which paired comparisons to extract and the choice of parameters in the estimator, which we call [*weights*]{}, turns out to be crucial as we will show. Inappropriate selection of the paired comparisons can lead to inconsistent estimators as proved in [@APX14a], and the standard choice of the parameters can lead to a significantly suboptimal performance.
A naive rank-breaking that is widely used in practice is to apply rank-breaking to all possible pairwise relations that one can read from the partial ordering and weighing them equally. We refer to this practice as [*full rank-breaking*]{}. In the example in Figure \[fig:hasse\], full rank-breaking first extracts the bag of comparisons ${\cal C}=\{(a>b),(a>c),(a>d),(a>e),(a>f),\ldots,(e>f)\}$ with 13 paired comparison outcomes, and apply the maximum likelihood estimator treating each paired outcome as independent. Precisely, the [*full rank-breaking estimator*]{} solves the convex optimization of $$\begin{aligned}
\label{eq:fullrankbreaking}
\widehat\theta &\in & \arg\max_{\theta \in \Omega_b} \;
\sum_{ (i>i') \in {\cal C} } \Big(\theta_i - \log \Big(e^{\theta_i} + e^{\theta_{\i}}\Big) \Big)\;.
\end{aligned}$$ There are several efficient implementation tailored for this problem [@Ford57; @Hun04; @NOS12; @MG15], and under the traditional scenarios, these approaches provably achieve the minimax optimal rate [@HOX14; @SBB15]. For general non-traditional data sets, there is a significant gain in computational complexity. In the case of position-$p$ ranking, where each of the $n$ users report his/her $p$-th ranking item among $\kappa$ items, the computational complexity reduces from $O(n\,(p-1)!)$ for the MLE in to $O( n\,p\,(\kappa-p))$ for the full rank-breaking estimator in . However, this gain comes at the cost of accuracy. It is known that the full-rank breaking estimator is inconsistent [@APX14a]; the error is strictly bounded away from zero even with infinite samples.
Perhaps surprisingly, Azari Soufiani et al. [@APX14a] recently characterized the entire set of consistent rank-breaking estimators. Instead of using the bag of paired comparisons, the sufficient information for consistent rank-breaking is a set of rank-breaking graphs defined as follows.
Recall that a user $j$ provides his/her preference as a poset represented by a DAG $\cG_j$. Consistent rank-breaking first identifies all [*separators*]{} in the DAG. A node in the DAG is a separator if one can partition the rest of the nodes into two parts. A partition $A_{\rm top}$ which is the set of items that are preferred over the separator item, and a partition $A_{\rm bottom}$ which is the set of items that are less preferred than the separator item. One caveat is that we allow $A_{\rm top}$ to be empty, but $A_{\rm bottom}$ must have at least one item. In the example in Figure \[fig:hasse\], there are two separators: the item $a$ and the item $e$. Using these separators, one can extract the following partial ordering from the original poset: $(a>\{b,c,d\}>e>f)$. The items $a$ and $e$ separate the set of offerings into partitions, hence the name separator. We use $\ell_j$ to denote the number of separators in the poset $\cG_j$ from user $j$. We let $p_{j,a}$ denote the ranked position of the $a$-th separator in the poset $\cG_j$, and we sort the positions such that $p_{j,1} < p_{j,2} < \ldots < p_{j,\ell_j}$. The set of separators is denoted by $\cP_j = \{p_{j,1},p_{j,2},\cdots,p_{j,\ell_j}\}$. For example, since the separator $a$ is ranked at position 1 and $e$ is at the $5$-th position, $\ell_j=2$, $p_{j,1}=1$, and $p_{j,2}=5$. Note that $f$ is not a separator (whereas $a$ is) since corresponding $A_{\rm bottom}$ is empty.
Conveniently, we represent this extracted partial ordering using a set of DAGs, which are called [*rank-breaking graphs*]{}. We generate one rank-breaking graph per separator. A rank breaking graph $G_{j,a}=(S_j,E_{j,a})$ for user $j$ and the $a$-th separator is defined as a directed graph over the set of offerings $S_j$, where we add an edge from a node that is less preferred than the $a$-th separator to the separator, i.e. $E_{j,a}=\{(i,i') \,|\, i'\text{ is the $a$-th separator, and } \sigma_j^{-1}(i) > p_{j,a} \}$. Note that by the definition of the separator, $E_{j,a}$ is a non-empty set. An example of rank-breaking graphs are shown in Figure \[fig:hasse\].
This rank-breaking graphs were introduced in [@ACPX13], where it was shown that the pairwise ordinal relations that is represented by edges in the rank-breaking graphs are sufficient information for using any estimation based on the idea of rank-breaking. Precisely, on the converse side, it was proved in [@APX14a] that any pairwise outcomes that is not present in the rank-breaking graphs $G_{j,a}$’s lead to inconsistency for a general $\theta^*$. On the achievability side, it was proved that all pairwise outcomes that are present in the rank-breaking graphs give a consistent estimator, as long as all the paired comparisons in each $G_{j,a}$ are weighted equally.
It should be noted that rank-breaking graphs are defined slightly differently in [@ACPX13]. Specifically, [@ACPX13] introduced a different notion of rank-breaking graph, where the vertices represent positions in total ordering. An edge between two vertices $i_1$ and $i_2$ denotes that the pairwise comparison between items ranked at position $i_1$ and $i_2$ is included in the estimator. Given such observation from the PL model, [@ACPX13] and [@APX14a] prove that a rank-breaking graph is consistent if and only if it satisfies the following property. If a vertex $i_1$ is connected to any vertex $i_2$, where $i_2 > i_1$, then $i_1$ must be connected to all the vertices $i_3$ such that $i_3 > i_1$. Although the specific definitions of rank-breaking graphs are different from our setting, the mathematical analysis of [@ACPX13] still holds when interpreted appropriately. Specifically, we consider only those rank-breaking that are consistent under the conditions given in [@ACPX13]. In our rank-breaking graph $G_{j,a}$, a separator node is connected to all the other item nodes that are ranked below it (numerically higher positions).
In the algorithm described in , we satisfy this sufficient condition for consistency by restricting to a class of convex optimizations that use the same weight $\lambda_{j,a}$ for all $(\kappa-p_{j,a})$ paired comparisons in the objective function, as opposed to allowing more general weights that defer from a pair to another pair in a rank-breaking graph $G_{j,a}$.
[**Algorithm.**]{} Consistent rank-breaking first identifies separators in the collected posets $\{\cG_j\}_{j\in[n]}$ and transform them into rank-breaking graphs $\{ G_{j,a}\}_{j\in[n],a\in[\ell_j]}$ as explained above. These rank-breaking graphs are input to the MLE for paired comparisons, assuming all directed edges in the rank-breaking graphs are independent outcome of pairwise comparisons. Precisely, the [*consistent rank-breaking estimator*]{} solves the convex optimization of maximizing the paired log likelihoods $$\begin{aligned}
\label{eq:likelihood_0}
\Lrb(\theta) &=&
\sum_{j=1}^n \sum_{a = 1}^{\ell_j}
\,\lambda_{j,a} \, \Big\{ \sum_{(i, \i) \in E_{j,a}}
\, \Big( \theta_{\i} - \log \Big(e^{\theta_i} + e^{\theta_{\i}}\Big) \,\Big)\, \Big\} \;,\end{aligned}$$ where $E_{j,a}$’s are defined as above via separators and different choices of the non-negative weights $\lambda_{j,a}$’s are possible and the performance depends on such choices. Each weight $\lambda_{j,a}$ determine how much we want to weigh the contribution of a corresponding rank-breaking graph $G_{j,a}$. We define the [*consistent rank-breaking estimate*]{} $\widehat\theta$ as the optimal solution of the convex program: $$\begin{aligned}
\label{eq:theta_ml}
\widehat{\theta} \;\; \in \;\; \arg\max_{\theta \in \Omega_b} \;\, \Lrb(\theta)\;. \end{aligned}$$ By changing how we weigh each rank-breaking graph (by choosing the $\lambda_{j,a}$’s), the convex program spans the entire set of consistent rank-breaking estimators, as characterized in [@APX14a]. However, only asymptotic consistency was known, which holds independent of the choice of the weights $\lambda_{j,a}$’s. Naturally, a uniform choice of $\lambda_{j,a}=\lambda$ was proposed in [@APX14a].
Note that this can be efficiently solved, since this is a simple convex optimization, in particular a logit regression, with only $O(\sum_{j=1}^n \,\ell_j\, \kappa_j)$ terms. For a special case of position-$p$ breaking, the $O(n \, (p-1)!)$ complexity of evaluating the objective function for the MLE is now significantly reduced to $O(n\,(\kappa-p))$ by rank-breaking. Given this potential exponential gain in efficiency, a natural question of interest is “what is the price we pay in the accuracy?”. We provide a sharp analysis of the performance of rank-breaking estimators in the finite sample regime, that quantifies the price of rank-breaking. Similarly, for a practitioner, a core problem of interest is how to choose the weights in the optimization in order to achieve the best accuracy. Our analysis provides a data-driven guideline for choosing the optimal weights.
[**Contributions.**]{} In this paper, we provide an upper bound on the error achieved by the rank-breaking estimator of for any choice of the weights in Theorem \[thm:main\]. This explicitly shows how the error depends on the choice of the weights, and provides a guideline for choosing the optimal weights $\lambda_{j,a}$’s in a data-driven manner. We provide the explicit formula for the optimal choice of the weights and provide the the error bound in Theorem \[thm:main2\]. The analysis shows the explicit dependence of the error in the problem dimension $d$ and the number of users $n$ that matches the numerical experiments.
If we are designing surveys and can choose which subset of items to offer to each user and also can decide which type of ordinal data we can collect, then we want to design such surveys in a way to maximize the accuracy for a given number of questions asked. Our analysis provides how the accuracy depends on the topology of the collected data, and provides a guidance when we do have some control over which questions to ask and which data to collect. One should maximize the spectral gap of corresponding comparison graph. Further, for some canonical scenarios, we quantify the price of rank-breaking by comparing the error bound of the proposed data-driven rank-breaking with the lower bound on the MLE, which can have a significantly larger computational cost (Theorem \[thm:cramer\_rao\_position\_p\]).
**Notations.** [Following is a summary of all the notations defined above. We use $d$ to denote the total number of items and index each item by $i \in \{1,2,\ldots,d\}$. $\theta \in \Omega_b$ denotes vector of utilities associated with each item. $\theta^{*}$ represents true utility and $\widehat{\theta}$ denotes the estimated utility. We use $n$ to denote the number of users/agents and index each user by $j \in \{1,2,\ldots,n\}$. $S_j \subseteq \{1,\ldots,d\}$ refer to the offerings provided to the $j$-th user and we use $\kappa_j = |S_j|$ to denote the size of the offerings. $\cG_j$ denote the DAG (Hasse diagram) representing the partial ordering from user $j$’s preferences. $\cP_j = \{p_{j,1},p_{j,2},\cdots,p_{j,\ell_j}\}$ denotes the set of separators in the DAG $\cG_j$, where $p_{j,1},\cdots,p_{j,\ell_j}$ are the positions of the separators, and $\ell_j$ is the number of separators. $G_{j,a}=(S_j,E_{j,a})$ denote the rank-breaking graph for the $a$-th separator extracted from the partial ordering $\cG_j$ of user $j$.]{}
For any positive integer $N$, let $[N] = \{1,\cdots,N\}$. For a ranking $\sigma$ over $S$, i.e., $\sigma$ is a mapping from $[|S|]$ to $S$, let $\sigma^{-1}$ denote the inverse mapping. For a vector $x$, let ${\|x\|}_2$ denote the standard $l_2$ norm. Let ${\boldsymbol{1}}$ denote the all-ones vector and ${\boldsymbol{0}}$ denote the all-zeros vector with the appropriate dimension. Let $\cS^d$ denote the set of $d \times d$ symmetric matrices with real-valued entries. For $X \in \cS^d$, let ${\lambda_1(X) \leq\lambda_2(X) \leq \cdots \leq \lambda_d(X)}$ denote its eigenvalues sorted in increasing order. Let $\Tr(X) = \sum_{i = 1}^d \lambda_i(X)$ denote its trace and ${\|X\|} = \max\{ |\lambda_1(X)|,|\lambda_d(X)| \}$ denote its spectral norm. For two matrices $X,Y \in \cS^d$, we write $X \succeq Y$ if $X-Y$ is positive semi-definite, i.e., $\lambda_1(X-Y) \geq 0$. Let $e_i$ denote a unit vector in $\reals^d$ along the $i$-th direction.
Comparisons Graph and the Graph Laplacian
=========================================
In the analysis of the convex program , we show that, with high probability, the objective function is strictly concave with $\lambda_2(H(\theta)) \leq - C_b \,\gamma\, \lambda_2(L)< 0$ (Lemma \[lem:hessian\_positionl\]) for all $\theta\in\Omega_b$ and the gradient is bounded by $\|\nabla\cL_{\rm RB}(\theta^*)\|_2 \leq C_b' \sqrt{\log d\, \sum_{j\in[n]} \ell_j}$ (Lemma \[lem:gradient\_topl\]). Shortly, we will define $\gamma$ and $\lambda_2(L)$, which captures the dependence on the topology of the data, and $C_b'$ and $C_b$ are constants that only depend on $b$. Putting these together, we will show that there exists a $\theta\in\Omega_b$ such that $$\begin{aligned}
\|\widehat\theta -\theta^* \|_2 &\leq& \frac{2\|\nabla \cL_{\rm RB}(\theta^*)\|_2 }{-\lambda_2(H(\theta))} \;\, \leq \;\, C''_b \frac{\sqrt{\log d \, \sum_{j\in[n]} \ell_j}}{\gamma \,\lambda_2(L)} \;. \label{eq:intro_bound}\end{aligned}$$ Here $\lambda_2(H(\theta))$ denotes the second largest eigenvalue of a negative semi-definite Hessian matrix $H(\theta)$ of the objective function. The reason the second largest eigenvalue shows up is because the top eigenvector is always the all-ones vector which by the definition of $\Omega_b$ is infeasible. The accuracy depends on the topology of the collected data via the comparison graph of given data.
\[def:comparison\_graph1\] (Comparison graph $\H$). We define a graph $\H([d],E)$ where each alternative corresponds to a node, and we put an edge $(i,i')$ if there exists an agent $j$ whose offerings is a set $S_j$ such that $i, \i \in S_j$. Each edge $(i,\i) \in E$ has a weight $A_{i \i}$ defined as $$\begin{aligned}
A_{i\i} &=& \sum_{j\in[n] : i,\i \in S_j} \frac{\ell_j}{\kappa_j(\kappa_j-1)}\;,
\end{aligned}$$ where $\kappa_j = |S_j|$ is the size of each sampled set and $\ell_j$ is the number of separators in $S_j$ defined by rank-breaking in Section \[sec:intro\].
Define a diagonal matrix $D = {\rm diag}(A{\boldsymbol{1}})$, and the corresponding graph Laplacian $L = D - A$, such that $$\begin{aligned}
\label{eq:comparison1_L}
L &=& \sum_{j = 1}^n \frac{\ell_j}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top.
\end{aligned}$$ Let $ 0 = \lambda_1(L) \leq \lambda_2(L) \leq \cdots \leq \lambda_d(L)$ denote the (sorted) eigenvalues of $L$. Of special interest is $\lambda_2(L)$, also called the spectral gap, which measured how well-connected the graph is. Intuitively, one can expect better accuracy when the spectral gap is larger, as evidenced in previous learning to rank results in simpler settings [@NOS14; @SBB15; @HOX14]. This is made precise in , and in the main result of Theorem \[thm:main2\], we appropriately rescale the spectral gap and use $\alpha\in[0,1]$ defined as $$\begin{aligned}
\label{eq:lambda2_L1}
\alpha &\equiv& \frac{\lambda_2(L)(d-1)}{\Tr(L)} \;\;=\;\; \frac{\lambda_2(L)(d-1)}{\sum_{j = 1}^n \ell_j } \;. \end{aligned}$$ The accuracy also depends on the topology via the maximum weighted degree defined as $D_{\max} \equiv \max_{i \in [d]} D_{ii} = \max_{i \in [d]} \{ \sum_{j: i \in S_j} \ell_j/\kappa_j\}$. Note that the average weighted degree is $\sum_i D_{ii}/ d = \Tr(L)/d$, and we rescale it by $D_{\rm max}$ such that $$\begin{aligned}
\label{eq:lambda2_L1beta}
\beta &\equiv& \frac{\Tr(L)}{d D_{\max}} \;\;=\;\; \frac{\sum_{j = 1}^n \ell_j }{d D_{\max}} \;. \end{aligned}$$ We will show that the performance of rank breaking estimator depends on the topology of the graph through these two parameters. The larger the spectral gap $\alpha$ the smaller error we get with the same effective sample size. The degree imbalance $\beta\in[0,1]$ determines how many samples are required for the analysis to hold. We need smaller number of samples if the weighted degrees are balanced, which happens if $\beta$ is large (close to one).
The following quantity also determines the convexity of the objective function. $$\begin{aligned}
\label{eq:gamma_def}
\gamma \;\equiv\; \min_{j \in [n]} \Bigg\{ \Bigg(1 - \frac{p_{j,\ell_j}}{\kappa_j} \Bigg)^{{\left \lceil{2e^{2b}} \right \rceil}-2}\Bigg\} \;.\;
\end{aligned}$$ Note that $\gamma$ is between zero and one, and a larger value is desired as the objective function becomes more concave and a better accuracy follows. When we are collecting data where the size of the offerings $\kappa_j$’s are increasing with $d$ but the position of the separators are close to the top, such that $\kappa_j = \omega(d)$ and $p_{j,\ell_j} = O(1)$, then for $b=O(1)$ the above quantity $\gamma$ can be made arbitrarily close to one, for large enough problem size $d$. On the other hand, when $p_{j,\ell_j}$ is close to $\kappa_j$, the accuracy can degrade significantly as stronger alternatives might have small chance of showing up in the rank breaking. The value of $\gamma$ is quite sensitive to $b$. The reason we have such a inferior dependence on $b$ is because we wanted to give a universal bound on the Hessian that is simple. It is not difficult to get a tighter bound with a larger value of $\gamma$, but will inevitably depend on the structure of the data in a complicated fashion. To ensure that the (second) largest eigenvalue of the Hessian is small enough, we need enough samples. This is captured by $\eta$ defined as $$\begin{aligned}
\label{eq:eta_def}
\eta \;\;\equiv\;\; \max_{j \in [n]} \{\eta_j\} \;,\; \;\;\;\;\;\text{where} \;\; \;\;\;\;\; \eta_j \;\; = \;\; \frac{\kappa_j}{\max\{\ell_j, \kappa_j - p_{j,\ell_j}\}}\,.\end{aligned}$$ Note that $1 < \eta_j \leq \kappa_j/\ell_j$. A smaller value of $\eta$ is desired as we require smaller number of samples, as shown in Theorem \[thm:main2\]. This happens, for instance, when all separators are at the top, such that $p_{j,\ell_j}=\ell_j$ and $\eta_j=\kappa_j/(\kappa_j-\ell_j)$, which is close to one for large $\kappa_j$. On the other hand, when all separators are at the bottom of the list, then $\eta$ can be as large as $\kappa_j$.
We discuss the role of the topology of data captures by these parameters in Section \[sec:role\].
Main Results {#sec:main}
============
We present the main theoretical results accompanied by corresponding numerical simulations in this section.
Upper Bound on the Achievable Error
-----------------------------------
We present the main result that provides an upper bound on the resulting error and explicitly shows the dependence on the topology of the data. As explained in Section \[sec:intro\], we assume that each user provides a partial ranking according to his/her position of the separators. Precisely, we assume the set of offerings $S_j$, the number of separators $\ell_j$, and their respective positions $\cP_j=\{p_{j,1},\ldots,p_{j,\ell_j}\}$ are predetermined. Each user draws the ranking of items from the PL model, and provides the partial ranking according to the separators of the form of $\{a>\{b,c,d\}>e>f\}$ in the example in the Figure \[fig:hasse\].
\[thm:main2\] Suppose there are $n$ users, $d$ items parametrized by $\theta^*\in\Omega_b$, each user $j$ is presented with a set of offerings $S_j\subseteq [d]$, and provides a partial ordering under the PL model. When the effective sample size $\sum_{j=1}^n \ell_j$ is large enough such that $$\begin{aligned}
\label{eq:main21}
\sum_{j=1}^n \, \ell_j \;\;\geq\;\; \frac{2^{11}e^{18b}\eta\log(\ell_{\max}+2)^2 }{\alpha^2\gamma^2\beta} d\log d\;,
\end{aligned}$$ where $b\equiv \max_{i}|\theta^*_i |$ is the dynamic range, $\ell_{\max} \equiv \max_{j\in[n]} \ell_j$, $\alpha$ is the (rescaled) spectral gap defined in , $\beta$ is the (rescaled) maximum degree defined in , $\gamma$ and $\eta$ are defined in Eqs. and , then the [*rank-breaking estimator*]{} in with the choice of $$\begin{aligned}
\lambda_{j,a} &=& \frac{1}{\kappa_j - p_{j,a}} \;,
\label{eq:deflambda}
\end{aligned}$$ for all $a\in[\ell_j]$ and $j\in[n]$ achieves $$\begin{aligned}
\label{eq:main22}
\frac{1}{\sqrt{d}}\big\|\widehat{\theta} - \theta^* \big\|_2 \;\; \leq \;\; \frac{4\sqrt{2}e^{4b}(1+ e^{2b})^2}{\alpha\gamma} \sqrt{\frac{d\, \log d}{\sum_{j=1}^n \ell_j}} \;,
\end{aligned}$$ with probability at least $ 1- 3e^3d^{-3}$.
Consider an ideal case where the spectral gap is large such that $\alpha$ is a strictly positive constant and the dynamic range $b$ is finite and $\max_{j \in[n]}p_{j,\ell_j}/\kappa_j = C$ for some constant $C <1$ such that $\gamma$ is also a constant independent of the problem size $d$. Then the upper bound in implies that we need the effective sample size to scale as $O(d\log d)$, which is only a logarithmic factor larger than the number of parameters to be estimated. Such a logarithmic gap is also unavoidable and due to the fact that we require high probability bounds, where we want the tail probability to decrease at least polynomially in $d$. We discuss the role of the topology of the data in Section \[sec:role\]. The upper bound follows from an analysis of the convex program similar to those in [@NOS12; @HOX14; @SBB15]. However, unlike the traditional data collection scenarios, the main technical challenge is in analyzing the probability that a particular pair of items appear in the rank-breaking. We provide a proof in Section \[sec:proof\_main2\].
![Simulation confirms $\|{\theta^* - \widehat{\theta}}\|_2^2 \propto 1/(\ell\,n)$, and smaller error is achieved for separators that are well spread out. []{data-label="fig:scaling_l_n"}](Plot1-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-169,50) (-115,-7)[number of separators ]{} (-52.5,87.5) (-82.5,83) (-52.5,78.5) ![Simulation confirms $\|{\theta^* - \widehat{\theta}}\|_2^2 \propto 1/(\ell\,n)$, and smaller error is achieved for separators that are well spread out. []{data-label="fig:scaling_l_n"}](Plot6-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-96,-7)[sample size ]{} ![Simulation confirms $\|{\theta^* - \widehat{\theta}}\|_2^2 \propto 1/(\ell\,n)$, and smaller error is achieved for separators that are well spread out. []{data-label="fig:scaling_l_n"}](Plot2-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-162,50) (-90,-15) (-120,-3) (-25,-3)
In Figure \[fig:scaling\_l\_n\] , we verify the scaling of the resulting error via numerical simulations. We fix $d=1024$ and $\kappa_j=\kappa = 128$, and vary the number of separators $\ell_j=\ell$ for fixed $n = 128000$ (left), and vary the number of samples $n$ for fixed $\ell_j=\ell = 16$ (middle). Each point is average over $100$ instances. The plot confirms that the mean squared error scales as $1/(\ell \, n)$. Each sample is a partial ranking from a set of $\kappa$ alternatives chosen uniformly at random, where the partial ranking is from a PL model with weights $\theta^*$ chosen i.i.d. uniformly over $[-b,b]$ with $b=2$. To investigate the role of the position of the separators, we compare three scenarios. The [*top-$\ell$-separators*]{} choose the top $\ell$ positions for separators, the [*random-$\ell$-separators among top-half*]{} choose $\ell$ positions uniformly random from the top half, and the [*random-$\ell$-separators*]{} choose the positions uniformly at random. We observe that when the positions of the separators are well spread out among the $\kappa$ offerings, which happens for [*random-$\ell$-separators*]{}, we get better accuracy. The figure on the right provides an insight into this trend for $\ell = 16$ and $n = 16000$. The absolute error $|\theta^*_i - \widehat{\theta_i}|$ is roughly same for each item $i \in [d]$ when breaking positions are chosen uniformly at random between $1$ to $\kappa-1$ whereas it is significantly higher for weak preference score items when breaking positions are restricted between $1$ to $\kappa/2$ or are top-$\ell$. This is due to the fact that the probability of each item being ranked at different positions is different, and in particular probability of the low preference score items being ranked in top-$\ell$ is very small. The third figure is averaged over $1000$ instances. Normalization constant $C$ is $n/d^2$ and $10^{3}\ell/d^2$ for the first and second figures respectively. For the first figure $n$ is chosen relatively large such that $n\ell$ is large enough even for $\ell = 1$.
The Price of Rank Breaking for the Special Case of Position-$p$ Ranking
-----------------------------------------------------------------------
Rank-breaking achieves computational efficiency at the cost of estimation accuracy. In this section, we quantify this tradeoff for a canonical example of position-$p$ ranking, where each sample provides the following information: an unordered set of $p-1$ items that are ranked high, one item that is ranked at the $p$-th position, and the rest of $\kappa_j-p$ items that are ranked on the bottom. An example of a sample with position-4 ranking six items $\{a,b,c,d,e,f\}$ might be a partial ranking of $(\{a,b,d\}>\{e\}>\{c,f\})$. Since each sample has only one separator for $2<p$, Theorem \[thm:main2\] simplifies to the following Corollary.
Under the hypotheses of Theorem \[thm:main2\], there exist positive constants $C$ and $c$ that only depend on $b$ such that if $n \geq C (\eta d \log d) /(\alpha^2\gamma^2\beta)$ then $$\begin{aligned}
\label{eq:main3}
\frac{1}{\sqrt{d}}\big\|\widehat{\theta} - \theta^* \big\|_2 \;\; \leq \;\; \frac{c}{\alpha\gamma} \sqrt{\frac{d\, \log d}{n }} \;.
\end{aligned}$$ \[coro:main2\]
Note that the error only depends on the position $p$ through $\gamma$ and $\eta$, and is not sensitive. To quantify the price of rank-breaking, we compare this result to a fundamental lower bound on the minimax rate in Theorem \[thm:cramer\_rao\_position\_p\]. We can compute a sharp lower bound on the minimax rate, using the Cramér-Rao bound, and a proof is provided in Section \[sec:proof\_cramer\_rao\_position\_p\].
\[thm:cramer\_rao\_position\_p\] Let $\mathcal{U}$ denote the set of all unbiased estimators of $\theta^*$ and suppose $b >0$, then $$\begin{aligned}
\inf_{\widehat{\theta} \in \mathcal{U}} \sup_{\theta^* \in \Omega_b} \E[{\|\widehat{\theta} - \theta^*\|}^2] &\geq&
\frac{1}{2p\log(\kappa_{\max})^2} \sum_{i = 2}^d \frac{1}{\lambda_i(L)}
\;\, \geq \;\, \frac{1}{2p\log(\kappa_{\max})^2} \frac{(d-1)^2}{n }\;,
\end{aligned}$$ where $\kappa_{\rm max} = \max_{j\in[n]} |S_j| $ and the second inequality follows from the Jensen’s inequality.
Note that the second inequality is tight up to a constant factor, when the graph is an expander with a large spectral gap. For expanders, $\alpha$ in the bound is also a strictly positive constant. This suggests that rank-breaking gains in computational efficiency by a super-exponential factor of $(p-1)!$, at the price of increased error by a factor of $p$, ignoring poly-logarithmic factors.
Tighter Analysis for the Special Case of Top-$\ell$ Separators Scenario {#sec:topl}
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The main result in Theorem \[thm:main2\] is general in the sense that it applies to any partial ranking data that is represented by positions of the separators. However, the bound can be quite loose, especially when $\gamma$ is small, i.e. $p_{j,\ell_j}$ is close to $\kappa_j$. For some special cases, we can tighten the analysis to get a sharper bound. One caveat is that we use a slightly sub-optimal choice of parameters $\lambda_{j,a} = 1/\kappa_j$ instead of $1/(\kappa_j - a)$, to simplify the analysis and still get the order optimal error bound we want. Concretely, we consider a special case of top-$\ell$ separators scenario, where each agent gives a ranked list of her most preferred $\ell_j$ alternatives among $\kappa_j$ offered set of items. Precisely, the locations of the separators are $(p_{j,1},p_{j,2},\ldots,p_{j,\ell_j})=(1,2,\ldots,\ell_j)$.
\[thm:topl\_upperbound\] Under the PL model, $n$ partial orderings are sampled over $d$ items parametrized by $\theta^* \in \Omega_b$, where the $j$-th sample is a ranked list of the top-$\ell_j$ items among the $\kappa_j$ items offered to the agent. If $$\begin{aligned}
\label{eq:topl1}
\sum_{j = 1}^n \ell_j \;\; \geq \;\; \frac{2^{12}e^{6b}}{\beta\alpha^2} d\log d\,,\end{aligned}$$ where $b \equiv \max_{i,\i} |\theta^*_i - \theta^*_{\i}|$ and $\alpha,\beta$ are defined in and , then the [*rank-breaking estimator*]{} in with the choice of $\lambda_{j,a} = 1/{\kappa_j}$ for all $a\in[\ell_j]$ and $j\in[n]$ achieves $$\begin{aligned}
\label{eq:main_topl}
\frac{1}{\sqrt{d}}\big\|\widehat{\theta} - \theta^* \big\|_2 \;\; \leq \;\; \frac{16(1+ e^{2b})^2}{\alpha} \sqrt{\frac{d\, \log d}{\sum_{j=1}^n \ell_j}} \;,
\end{aligned}$$ with probability at least $ 1- 3e^3 d^{-3}$.
A proof is provided in Section \[sec:proof\_topl\_upperbound\]. In comparison to the general bound in Theorem \[thm:main2\], this is tighter since there is no dependence in $\gamma$ or $\eta$. This gain is significant when, for example, $p_{j,\ell_j}$ is close to $\kappa_j$. As an extreme example, if all agents are offered the entire set of alternatives and are asked to rank all of them, such that $\kappa_j=d$ and $\ell_j=d-1$ for all $j\in[n]$, then the generic bound in is loose by a factor of $(e^{4b}/2\sqrt{2}) d^{\lceil2e^{2b}\rceil-2}$, compared to the above bound.
In the top-$\ell$ separators scenario, the data set consists of the ranking among top-$\ell_j$ items of the set $S_j$, i.e., ${[\sigma_j(1), \sigma_j(2),\cdots, \sigma_j(\ell_j)]}$. The corresponding log-likelihood of the PL model is $$\begin{aligned}
\label{eq:PL_likelihood}
\L(\theta) = \sum_{j = 1}^n \sum_{m = 1}^{\ell_j} \Big[ \theta_{\sigma_j(m)} - \log \Big( \exp(\theta_{\sigma_j(m)})+\exp(\theta_{\sigma_j(m+1)})+ \cdots + \exp(\theta_{\sigma_j(\kappa_j)})\Big) \Big]\;,
\end{aligned}$$ where $\sigma_j(a)$ is the alternative ranked at the $a$-th position by agent $j$. The Maximum Likelihood Estimator (MLE) for this [*traditional*]{} data set is efficient. Hence, there is no computational gain in rank-breaking. Consequently, there is no loss in accuracy either, when we use the optimal weights proposed in the above theorem. Figure \[fig:top\_l\] illustrates that the MLE and the data-driven rank-breaking estimator achieve performance that is identical, and improve over naive rank-breaking that uses uniform weights. We also compare performance of Generalized Method-of-Moments (GMM) proposed by [@ACPX13] with our algorithm. In addition, we show that performance of GMM can be improved by optimally weighing pairwise comparisons with $\lambda_{j,a}$. MSE of GMM in both the cases, uniform weights and optimal weights, is larger than our rank-breaking estimator. However, GMM is on average about four times faster than our algorithm. We choose $\lambda_{j,a} = 1/(\kappa_j-a)$ in the simulations, as opposed to the $1/\kappa_j$ assumed in the above theorem. This settles the question raised in [@HOX14] on whether it is possible to achieve optimal accuracy using rank-breaking under the top-$\ell$ separators scenario. Analytically, it was proved in [@HOX14] that under the top-$\ell$ separators scenario, naive rank-breaking with uniform weights achieves the same error bound as the MLE, up to a constant factor. However, we show that this constant factor gap is not a weakness of the analyses, but the choice of the weights. Theorem \[thm:topl\_upperbound\] provides a guideline for choosing the optimal weights, and the numerical simulation results in Figure \[fig:top\_l\] show that there is in fact no gap in practice, if we use the optimal weights. We use the same settings as that of the first figure of Figure \[fig:scaling\_l\_n\] for the figure below.
![The proposed data-driven rank-breaking achieves performance identical to the MLE, and improves over naive rank-breaking with uniform weights. []{data-label="fig:top_l"}](Plot5_new-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-171,50) (-115,-7 )[number of separators ]{} (-100,100)
To prove the order-optimality of the rank-breaking approach up to a constant factor, we can compare the upper bound to a Cramér-Rao lower bound on any unbiased estimators, in the following theorem. A proof is provided in Section \[sec:proof\_cramer\_rao\_topl\].
\[thm:cramer\_rao\_topl\] Consider ranking $\{\sigma_j(i)\}_{i \in [\ell_j]}$ revealed for the set of items $S_j$, for $j \in [n]$. Let $\mathcal{U}$ denote the set of all unbiased estimators of $\theta^*\in\Omega_b$. If $b >0$, then $$\begin{aligned}
\inf_{\widehat{\theta} \in \mathcal{U}} \sup_{\theta^* \in \Omega_b} \E[{\|\widehat{\theta} - \theta^*\|}^2]
\;\; \geq \;\; \Bigg(1 - \frac{1}{\ell_{\max}}\sum_{i= 1}^{\ell_{\max}} \frac{1}{\kappa_{\max} - i +1}\Bigg)^{-1} \sum_{i = 2}^d \frac{1}{\lambda_i(L)}
\;\; \geq \;\; \frac{(d-1)^2}{\sum_{j = 1}^n \ell_j}\;,
\label{eq:cramer_rao_topl}
\end{aligned}$$ where $\ell_{\max} = \max_{j \in [n]} \ell_j$ and $\kappa_{\max} = \max_{j \in [n]} \kappa_j$. The second inequality follows from the Jensen’s inequality.
Consider a case when the comparison graph is an expander such that $\alpha$ is a strictly positive constant, and $b=O(1)$ is also finite. Then, the Cramér-Rao lower bound show that the upper bound in is optimal up to a logarithmic factor.
Optimality of the Choice of the Weights
---------------------------------------
We propose the optimal choice of the weights $\lambda_{j,a}$’s in Theorem \[thm:main2\]. In this section, we show numerical simulations results comparing the proposed approach to other naive choices of the weights under various scenarios. We fix $d = 1024$ items and the underlying preference vector $\theta^*$ is uniformly distributed over $[-b,b]$ for $b = 2$. We generate $n$ rankings over sets $S_j$ of size $\kappa$ for $j \in [n]$ according to the PL model with parameter $\theta^*$. The comparison sets $S_j$’s are chosen independently and uniformly at random from $[d]$.
![Data-driven rank-breaking is consistent, while a random rank-breaking results in inconsistency.[]{data-label="fig:consistent"}](Plot7-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-92,-5) (-180,50)
Figure \[fig:consistent\] illustrates that a naive choice of rank-breakings can result in inconsistency. We create partial orderings data set by fixing $\kappa = 128$ and select $\ell=8$ random positions in $\{1,\ldots,127\}$. Each data set consists of partial orderings with separators at those $8$ random positions, over $128$ randomly chosen subset of items. We vary the sample size $n$ and plot the resulting mean squared error for the two approaches. The data-driven rank-breaking, which uses the optimal choice of the weights, achieves error scaling as $1/n$ as predicted by Theorem \[thm:main2\], which implies consistency. For fair comparisons, we feed the same number of pairwise orderings to a naive rank-breaking estimator. This estimator uses randomly chosen pairwise orderings with uniform weights, and is generally inconsistent. However, when sample size is small, inconsistent estimators can achieve smaller variance leading to smaller error. Normalization constant $C$ is $10^{3}\ell/d^2$, and each point is averaged over $100$ trials. We use the minorization-maximization algorithm from [@Hun04] for computing the estimates from the rank-breakings.
Even if we use the consistent rank-breakings first proposed in [@APX14a], there is ambiguity in the choice of the weights. We next study how much we gain by using the proposed optimal choice of the weights. The optimal choice, $\lambda_{j,a}=1/(\kappa_j-p_{j,a})$, depends on two parameters: the size of the offerings $\kappa_j$ and the position of the separators $p_{j,a}$. To distinguish the effect of these two parameters, we first experiment with fixed $\kappa_j=\kappa$ and illustrate the gain of the optimal choice of $\lambda_{j,a}$’s.
![There is a constant factor gain of choosing optimal $\lambda_{j,a}$’s when the size of offerings are fixed, i.e. $\kappa_j = \kappa$ (left). We choose a particular set of separators where one separators is at position one and the rest are at the bottom. An example for $\ell=3$ and $\kappa=10$ is shown, where the separators are indicated by blue (right).[]{data-label="fig:lambda_impact1"}](Plot3-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-85,102)[Top-$1$ and bottom-$(\ell-1)$ separators]{} (-180,50) (-115,-5)[number of separators ]{} ![There is a constant factor gain of choosing optimal $\lambda_{j,a}$’s when the size of offerings are fixed, i.e. $\kappa_j = \kappa$ (left). We choose a particular set of separators where one separators is at position one and the rest are at the bottom. An example for $\ell=3$ and $\kappa=10$ is shown, where the separators are indicated by blue (right).[]{data-label="fig:lambda_impact1"}](topbottom-eps-converted-to.pdf "fig:"){width=".22\textwidth"}
Figure \[fig:lambda\_impact1\] illustrates that the optimal choice of the weights improves over consistent rank-breaking with uniform weights by a constant factor. We fix $\kappa = 128$ and $n=128000$. As illustrated by a figure on the right, the position of the separators are chosen such that there is one separator at position one, and the rest of $\ell-1$ separators are at the bottom. Precisely, $(p_{j,1},p_{j,2},p_{j,3},\ldots,p_{j,\ell})=(1,128-\ell+1,128-\ell+2,\ldots,127)$. We consider this scenario to emphasize the gain of optimal weights. Observe that the MSE does not decrease at a rate of $1/\ell$ in this case. The parameter $\gamma$ which appears in the bound of Theorem \[thm:main2\] is very small when the breaking positions $p_{j,a}$ are of the order $\kappa_j$ as is the case here, when $\ell$ is small. Normalization constant $C$ is $n/d^2$.
![The gain of choosing optimal $\lambda_{j,a}$’s is significant when $\kappa_j$’s are highly heterogeneous. []{data-label="fig:lambda_impact2"}](Plot4-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-180,50) (-104,-7)
The gain of optimal weights is significant when the size of $S_j$’s are highly heterogeneous. Figure \[fig:lambda\_impact2\] compares performance of the proposed algorithm, for the optimal choice and uniform choice of weights $\lambda_{j,a}$ when the comparison sets $S_j$’s are of different sizes. We consider the case when $n_1$ agents provide their top-$\ell_1$ choices over the sets of size $\kappa_1$, and $n_2$ agents provide their top-$1$ choice over the sets of size $\kappa_2$. We take $n_1 = 1024$, $\ell_1 = 8$, and $n_2 = 10n_1\ell_1$. Figure \[fig:lambda\_impact2\] shows MSE for the two choice of weights, when we fix $\kappa_1 = 128$, and vary $\kappa_2$ from $2$ to $128$. As predicted from our bounds, when optimal choice of $\lambda_{j,a}$ is used MSE is not sensitive to sample set sizes $\kappa_2$. The error decays at the rate proportional to the inverse of the effective sample size, which is $n_1\ell_1 + n_2\ell_2 = 11n_1\ell_1$. However, with $\lambda_{j,a} = 1$ when $\kappa_2 = 2$, the MSE is roughly $10$ times worse. Which reflects that the effective sample size is approximately $n_1\ell_1$, i.e. pairwise comparisons coming from small set size do not contribute without proper normalization. This gap in MSE corroborates bounds of Theorem \[thm:main\]. Normalization constant $C$ is $10^{3}/d^2$.
The Role of the Topology of the Data {#sec:role}
====================================
We study the role of topology of the data that provides a guideline for designing the collection of data when we do have some control, as in recommendation systems, designing surveys, and crowdsourcing. The core optimization problem of interest to the designer of such a system is to achieve the best accuracy while minimizing the number of questions.
The Role of the Graph Laplacian
-------------------------------
Using the same number of samples, comparison graphs with larger spectral gap achieve better accuracy, compared to those with smaller spectral gaps. To illustrate how graph topology effects the accuracy, we reproduce known spectral properties of canonical graphs, and numerically compare the performance of data-driven rank-breaking for several graph topologies. We follow the examples and experimental setup from [@SBB15] for a similar result with pairwise comparisons. Spectral properties of graphs have been a topic of wide interest for decades. We consider a scenario where we fix the size of offerings as $\kappa_j=\kappa=O(1)$ and each agent provides partial ranking with $\ell$ separators, positions of which are chosen uniformly at random. The resulting spectral gap $\alpha$ of different choices of the set $S_j$’s are provided below. The total number edges in the comparisons graph (counting hyper-edges as multiple edges) is defined as $|E|\equiv{\kappa \choose 2}\,n$.
- Complete graph: when $|E|$ is larger than ${d \choose 2}$, we can design the comparison graph to be a complete graph over $d$ nodes. The weight $A_{ii'}$ on each edge is $n\,\ell/(d(d-1))$, which is the effective number of samples divided by twice the number of edges. Resulting spectral gap is one, which is the maximum possible value. Hence, complete graph is optimal for rank aggregation.
- Sparse random graph: when we have limited resources we might not be able to afford a dense graph. When $|E|$ is of order $o(d^2)$, we have a sparse graph. Consider a scenario where each set $S_j$ is chosen uniformly at random. To ensure connectivity, we need $n=\Omega(\log d)$. Following standard spectral analysis of random graphs, we have $\alpha=\Theta(1)$. Hence, sparse random graphs are near-optimal for rank-aggregation.
- Chain graph: we consider a chain of sets of size $\kappa$ overlapping only by one item. For example, $S_1=\{1,\ldots,\kappa\}$ and $S_2=\{\kappa,\kappa+1,\ldots,2\kappa-1\}$, etc. We choose $n$ to be a multiple of $\tau \equiv (d-1)/(\kappa-1)$ and offer each set $n/\tau$ times. The resulting graph is a chain of size $\kappa$ cliques, and standard spectral analysis shows that $\alpha=\Theta(1/d^2)$. Hence, a chain graph is strictly sub-optimal for rank aggregation.
- Star-like graph: We choose one item to be the center, and every offer set consists of this center node and a set of $\kappa-1$ other nodes chosen uniformly at random without replacement. For example, center node = $\{1\}$, $S_1=\{1,2,\ldots,\kappa\}$ and $S_2=\{1,\kappa+1,\kappa+2,\ldots,2\kappa-1\}$, etc. $n$ is chosen in the way similar to that of the Chain graph. Standard spectral analysis shows that $\alpha=\Theta(1)$ and star-like graphs are near-optimal for rank-aggregation.
- Barbell-like graph: We select an offering $S = \{S',i,j\}$, $|S'| = \kappa-2$ uniformly at random and divide rest of the items into two groups $V_1$ and $V_2$. We offer set $S$ $n\kappa/d$ times. For each offering of set $S$, we offer $d/\kappa -1$ sets chosen uniformly at random from the two groups $\{V_1,i\}$ and $\{V_2,j\}$. The resulting graph is a barbell-like graph, and standard spectral analysis shows that $\alpha=\Theta(1/d^2)$. Hence, a chain graph is strictly sub-optimal for rank aggregation.
Figure \[fig:topology\] illustrates how graph topology effects the accuracy. When $\theta^*$ is chosen uniformly at random, the accuracy does not change with $d$ (left), and the accuracy is better for those graphs with larger spectral gap. However, for a certain worst-case $\theta^*$, the error increases with $d$ for the chain graph and the barbell-like graph, as predicted by the above analysis of the spectral gap. We use $\ell = 4$, $\kappa = 17$ and vary $d$ from $129$ to $2049$. $\kappa$ is kept small to make the resulting graphs more like the above discussed graphs. Figure on left shows accuracy when $\theta^*$ is chosen i.i.d. uniformly over $[-b,b]$ with $b=2$. Error in this case is roughly same for each of the graph topologies with chain graph being the worst. However, when $\theta^*$ is chosen carefully error for chain graph and barbell-like graph increases with $d$ as shown in the figure right. We chose $\theta^*$ such that all the items of a set have same weight, either $\theta_i = 0$ or $\theta_i = b$ for chain graph and barbell-like graph. We divide all the sets equally between the two types for chain graph. For barbell-like graph, we keep the two types of sets on the two different sides of the connector set and equally divide items of the connector set into two types. Number of samples $n$ is $100(d-1)/(\kappa-1)$ and each point is averaged over $100$ instances. Normalization constant $C$ is $n\ell/d^2$.
![For randomly chosen $\theta^*$ the error does not change with $d$ (left). However, for particular worst-case $\theta^*$ the error increases with $d$ for the Chain graph and the Barbell-like graph as predicted by the analysis of the spectral gap (right).[]{data-label="fig:topology"}](Plot11_rand-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-180,50) (-95,-5) (-90,100)[Random $\theta^*$]{} ![For randomly chosen $\theta^*$ the error does not change with $d$ (left). However, for particular worst-case $\theta^*$ the error increases with $d$ for the Chain graph and the Barbell-like graph as predicted by the analysis of the spectral gap (right).[]{data-label="fig:topology"}](Plot11_0b-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-95,-5)[graph size ]{} (-95,100)[Worst-case $\theta^*$]{}
The Role of the Position of the Separators
------------------------------------------
As predicted by theorem \[thm:main2\], rank-breaking fails when $\gamma$ is small, i.e. the position of the separators are very close to the bottom. An extreme example is the bottom-$\ell$ separators scenario, where each person is offered $\kappa$ randomly chosen alternatives, and is asked to give a ranked list of bottom $\ell$ alternatives. In other words, the $\ell$ separators are placed at $(p_{j,1},\ldots,p_{j,\ell})=(\kappa_j-\ell, \ldots,\kappa-1)$. In this case, $\gamma\simeq 0$ and the error bound is large. This is not a weakness of the analysis. In fact we observe large errors under this scenario. The reason is that many alternatives that have large weights $\theta_i$’s will rarely be even compared once, making any reasonable estimation infeasible.
Figure \[fig:bottom\_l\_1\] illustrates this scenario. We choose $\ell=8$, $\kappa=128$, and $d=1024$. The other settings are same as that of the first figure of Figure \[fig:scaling\_l\_n\]. The left figure plots the magnitude of the estimation error for each item. For about 200 strong items among 1024, we do not even get a single comparison, hence we omit any estimation error. It clearly shows the trend: we get good estimates for about 400 items in the bottom, and we get large errors for the rest. Consequently, even if we only take those items that have at least one comparison into account, we still get large errors. This is shown in the figure right. The error barely decays with the sample size. However, if we focus on the error for the bottom 400 items, we get good error rate decaying inversely with the sample size. Normalization constant $C$ in the second figure is $10^2 \,x\,d/\ell$ and $10^{2}(400)d/\ell$ for the first and second lines respectively, where $x$ is the number of items that appeared in rank-breaking at least once. We solve convex program for $\theta$ restricted to the items that appear in rank-breaking at least once. The second figure of Figure \[fig:bottom\_l\_1\] is averaged over $1000$ instances.
![Under the bottom-$\ell$ separators scenario, accuracy is good only for the bottom 400 items (left). As predicted by Theorem \[thm:bottoml\_upperbound\], the mean squared error on the bottom 400 items scale as $1/n$, where as the overall mean squared error does not decay (right). []{data-label="fig:bottom_l_1"}](Plot9-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-162,50) (-20,102)[Bottom-$8$ separators]{} (-90,-15) (-127,-3) (-22,-3) ![Under the bottom-$\ell$ separators scenario, accuracy is good only for the bottom 400 items (left). As predicted by Theorem \[thm:bottoml\_upperbound\], the mean squared error on the bottom 400 items scale as $1/n$, where as the overall mean squared error does not decay (right). []{data-label="fig:bottom_l_1"}](Plot10-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-175,50) (-90,-7)[sample size ]{}
We make this observation precise in the following theorem. Applying rank-breaking to only to those weakest $\ld$ items, we prove an upper bound on the achieved error rate that depends on the choice of the $\ld$. Without loss of generality, we suppose the items are sorted such that $\theta^*_1 \leq \theta_2^* \leq \cdots \leq \theta_d^*$. For a choice of $\ld = \ell d/ (2 \kappa) $, we denote the weakest $\ld$ items by $\ltheta^* \in \reals^{\ld}$ such that $\ltheta_i^* = \theta^*_i - (1/\ld)\sum_{\i = 1}^{\ld} \theta^*_{\i}$, for $i \in [\ld]$. Since $\theta^* \in \Omega_b$, $\ltheta^* \in [-2b,2b]^{\ld}$. The space of all possible preference vectors for $[\ld]$ items is given by $\lOmega = \{ \ltheta \in \reals^{\ld} : \sum_{i =1}^{\ld} \ltheta_i = 0\}$ and $\lOmega_{2b} = \lOmega \cap [-2b,2b]^{\ld}$.
Although the analysis can be easily generalized, to simplify notations, we fix $\kappa_j = \kappa$ and $\ell_j = \ell$ and assume that the comparison sets $S_j$, $|S_j| = \kappa$, are chosen uniformly at random from the set of $d$ items for all $j \in [n]$. The rank-breaking log likelihood function $\Lrb(\ltheta)$ for the set of items $[\ld]$ is given by $$\begin{aligned}
\label{eq:likelihood_bl_0}
\Lrb(\ltheta) &=&
\sum_{j=1}^n \sum_{a = 1}^{\ell_j}
\,\lambda_{j,a} \, \Big\{ \sum_{(i, \i) \in E_{j,a}}
\, \I_{\big\{i, \i \in [\ld] \big\}} \Big( \theta_{\i} - \log \Big(e^{\theta_i} + e^{\theta_{\i}}\Big) \,\Big)\, \Big\} \;.\end{aligned}$$ We analyze the rank-breaking estimator $$\begin{aligned}
\label{eq:theta_ml_bl}
\widehat{\ltheta} \;\; \equiv \;\; \max_{\ltheta \in \lOmega_{2b}} \Lrb(\ltheta)\;.\end{aligned}$$ We further simplify notations by fixing $\lambda_{j,a} = 1$, since from Equation , we know that the error increases by at most a factor of $4$ due to this sub-optimal choice of the weights, under the special scenario studied in this theorem.
\[thm:bottoml\_upperbound\] Under the bottom-$\ell$ separators scenario and the PL model, $S_j$’s are chosen uniformly at random of size $\kappa$ and $n$ partial orderings are sampled over $d$ items parametrized by $\theta^* \in \Omega_b$. For $\ld=\ell d / (2 \kappa)$ and any $\ell\geq 4$, if the effective sample size is large enough such that $$\begin{aligned}
\label{eq:bottoml_1}
n\ell \;\; \geq \;\; \bigg(\frac{2^{14}e^{8b}}{\chi^2 }\frac{\kappa^3}{\ell^3}\bigg) d\log d\;,
\end{aligned}$$ where $$\begin{aligned}
\chi & \equiv & \frac14 \Bigg(1 - \exp\bigg(-\frac{ 2}{9(\kappa-2)} \,\bigg)\,\Bigg),
\end{aligned}$$ then the [*rank-breaking*]{} estimator in achieves $$\begin{aligned}
\label{eq:bottoml_3}
\frac{1}{\sqrt{\ld}}\big\|\widehat{\ltheta} - \ltheta^*\big\|_2 \; \leq \; \frac{128(1+ e^{4b})^2}{\chi}\frac{\kappa^{3/2}}{{\ell}^{3/2}}\sqrt{\frac{d\log d}{n\ell} }\;,
\end{aligned}$$ with probability at least $1 - 3e^3 d^{-3}$.
Consider a scenario where $\kappa=O(1)$ and $\ell=\Theta(\kappa)$. Then, $\chi$ is a strictly positive constant, and also $\kappa/\ell$ is s finite constant. It follows that rank-breaking requires the effective sample size $n\ell=O(d\log d / \varepsilon^2 )$ in order to achieve arbitrarily small error of $\varepsilon>0$, on the weakest $\ld=\ell\,d/(2 \kappa)$ items.
Real-World Data Sets {#sec:real}
====================
On real-world data sets on sushi preferences [@Kam03], we show that the data-driven rank-breaking improves over Generalized Method-of-Moments (GMM) proposed by [@ACPX13]. This is a widely used data set for rank aggregation, for instance in [@ACPX13; @APX12; @MG15a; @LLN15; @LB11; @LB11b]. The data set consists of complete rankings over $10$ types of sushi from $n=5000$ individuals. Below, we follow the experimental scenarios of the GMM approach in [@ACPX13] for fair comparisons.
To validate our approach, we first take the estimated PL weights of the 10 types of sushi, using [@Hun04] implementation of the ML estimator, over the entire input data of $5000$ complete rankings. We take thus created output as the ground truth $\theta^*$. To create partial rankings and compare the performance of the data-driven rank-breaking to the state-of-the-art GMM approach in Figure \[fig:sushi\_10\_mse\], we first fix $\ell=6$ and vary $n$ to simulate top-$\ell$-separators scenario by removing the known ordering among bottom $10-\ell$ alternatives for each sample in the data set (left). We next fix $n=1000$ and vary $\ell$ and simulate top-$\ell$-separators scenarios (right). Each point is averaged over $1000$ instances. The mean squared error is plotted for both algorithms.
![The data-driven rank-breaking achieves smaller error compared to the state-of-the-art GMM approach. []{data-label="fig:sushi_10_mse"}](sushi10_n_mse-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-170,50) (-90,-7)[sample size ]{} (-100,100) ![The data-driven rank-breaking achieves smaller error compared to the state-of-the-art GMM approach. []{data-label="fig:sushi_10_mse"}](sushi10_l_mse-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-100,100) (-115,-7 )[number of separators ]{}
Figure \[fig:sushi\_10\_ken\] illustrates the Kendall rank correlation of the rankings estimated by the two algorithms and the ground truth. Larger value indicates that the estimate is closer to the ground truth, and the data-driven rank-breaking outperforms the state-of-the-art GMM approach.
![The data-driven rank-breaking achieves larger Kendall rank correlation compared to the state-of-the-art GMM approach. []{data-label="fig:sushi_10_ken"}](sushi10_n_ken-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-100,100) (-220,50) (-90,-7)[sample size ]{} ![The data-driven rank-breaking achieves larger Kendall rank correlation compared to the state-of-the-art GMM approach. []{data-label="fig:sushi_10_ken"}](sushi10_l_ken-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-100,100) (-115,-7 )[number of separators ]{}
To validate whether PL model is the right model to explain the sushi data set, we compare the data-driven rank-breaking, MLE for the PL model, GMM for the PL model, Borda count and Spearman’s footrule optimal aggregation. We measure the Kendall rank correlation between the estimates and the samples and show the result in Table \[tab:sushi\_10\_all\]. In particular, if $\sigma_1,\sigma_2,\cdots, \sigma_n$ denote sample rankings and $\widehat{\sigma}$ denote the aggregated ranking then the correlation value is $(1/n)\sum_{i = 1}^n \big(1-\frac{4\mathcal{K}(\widehat{\sigma},\sigma_i)}{\kappa(\kappa-1)}\big)$, where $\mathcal{K}(\sigma_1,\sigma_2) = \sum_{i < j \in [\kappa]} \mathbb{I}_{\{(\sigma_1^{-1}(i) - \sigma_1^{-1}(j))(\sigma_2^{-1}(i) - \sigma_2^{-1}(j)) < 0 \}}$. The results are reported for different number of samples $n$ and different values of $\ell$ under the top-$\ell$ separators scenarios. When $\ell=9$, we are using all the complete rankings, and all algorithms are efficient. When $\ell < 9$, we have partial orderings, and Spearman’s footrule optimal aggregation is NP-hard. We instead use scaled footrule aggregation (SFO) given in [@DKNS01]. Most approaches achieve similar performance, except for the Spearman’s footrule. The proposed data-driven rank-breaking achieves a slightly worse correlation compared to other approaches. However, note that none of the algorithms are necessarily maximizing the Kendall correlation, and are not expected to be particularly good in this metric.
[ C[2.5cm]{} | C[1.5cm]{} C[2.cm]{} C[1.5cm]{} C[1.5cm]{} C[2cm]{} ]{} & MLE under PL & data-driven RB & GMM & Borda count & Spearman’s footrule\
$n = 500$, $\ell = 9$ & 0.306 & 0.291 & 0.315 & 0.315 & 0.159\
$n = 5000$, $\ell = 9$ & 0.309 & 0.309& 0.315 &0.315 & 0.079\
$n = 5000$, $\ell = 2$ & 0.199 &0.199 & 0.201&0.200 & 0.113\
$n = 5000$, $\ell = 5$ & 0.217& 0.200& 0.217& 0.295& 0.152\
We compare our algorithm with the GMM algorithm on two other real-world data-sets as well. We use jester data set [@GRG01] that consists of over $4.1$ million continuous ratings between $-10$ to $+10$ of $100$ jokes from $48,483$ users. The average number of jokes rated by an user is $72.6$ with minimum and maximum being $36$ and $100$ respectively. We convert continuous ratings into ordinal rankings. This data-set has been used by [@MP00; @PD05; @CMR07; @LM07] for rank aggregation and collaborative filtering.
Similar to the settings of sushi data experiments, we take the estimated PL weights of the 100 jokes over all the rankings as ground truth. Figure \[fig:jest\] shows comparative performance of the data-driven rank-breaking and the GMM for the two scenarios. We first fix $\ell = 10$ and vary $n$ to simulate random-$10$ separators scenario (left). We next take all the rankings $n = 73421$ and vary $\ell$ to simulate random-$\ell$ separators scenario (rights). Since sets have different sizes, while varying $\ell$ we use full breaking if the setsize is smaller than $\ell$. Each point is averaged over $100$ instances. The mean squared error is plotted for both algorithms.
![jester data set: The data-driven rank-breaking achieves smaller error compared to the state-of-the-art GMM approach. []{data-label="fig:jest"}](jest_n-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-175,50) (-100,-7)[sample size ]{} (-110,100) ![jester data set: The data-driven rank-breaking achieves smaller error compared to the state-of-the-art GMM approach. []{data-label="fig:jest"}](jest_l-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-110,100) (-115,-7 )[number of separators ]{}
We perform similar experiments on American Psychological Association (APA) data-set [@Dia89]. The APA elects a president each year by asking each member to rank order a slate of five candidates. The data-set represents full rankings given by 5738 members of the association in 1980’s election. The mean squared error is plotted for both algorithms under the settings similar to that of jester data-set.
![APA data set: The data-driven rank-breaking achieves smaller error compared to the state-of-the-art GMM approach. []{data-label="fig:apa"}](apa_n-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-175,50) (-90,-7)[sample size ]{} (-110,100) ![APA data set: The data-driven rank-breaking achieves smaller error compared to the state-of-the-art GMM approach. []{data-label="fig:apa"}](apa_l-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-110,100) (-115,-7 )[number of separators ]{}
Related Work
============
Initially motivated by elections and voting, rank aggregation has been a topic of mathematical interest dating back to Condorcet and Borda [@Con1785; @Borda1781]. Using probabilistic models to infer preferences has been popularized in operations research community for applications such as assortment optimization and revenue management. The PL model studied in this paper is a special case of MultiNomial Logit (MNL) models commonly used in discrete choice modeling, which has a long history in operations research [@McF80]. Efficient inference algorithms has been proposed to either find the MLE efficiently or approximately, such as the iterative approaches in [@Ford57; @Dyk60], minorization-maximization approach in [@Hun04], and Markov chain approaches in [@NOS12; @MG15]. These approaches are shown to achieve minimax optimal error rate in the traditional comparisons scenarios. Under the pairwise comparisons scenario, Negahban et al. [@NOS12] provided Rank Centrality that provably achieves minimax optimal error rate for randomly chosen pairs, which was later generalized to arbitrary pairwise comparisons in [@NOS14]. The analysis shows the explicit dependence on the topology of data shows that the spectral gap of comparisons graph similar to the one presented in this paper. This analysis was generalized to $k$-way comparisons in [@HOX14] and generalized to best-out-of-$k$ comparisons with sharper bounds in [@SBB15]. In an effort to give a guarantee for exact recovery of the top-$\ell$ items in the ranking, Chen et al. in [@CS15] proposed a new algorithm based on Rank Centrality that provides a tighter error bound for $L_\infty$ norm, as opposed to the existing $L_2$ error bounds. Another interesting direction in learning to rank is non-parametric learning from paired comparisons, initiated in several recent papers such as [@DMJ10; @RA14; @SBGW15; @SW15].
More recently, a more general problem of learning [*personal*]{} preferences from ordinal data has been studied [@YJJ13; @LB11b; @DIS15]. The MNL model provides a natural generalization of the PL model to this problem. When users are classified into a small number of groups with same preferences, mixed MNL model can be learned from data as studied in [@AOSV14; @OS14; @WXSMLH15]. A more general scenario is when each user has his/her individual preferences, but inherently represented by a lower dimensional feature. This problem was first posed as an inference problem in [@LN14] where convex relaxation of nuclear norm minimization was proposed with provably optimal guarantees. This was later generalized to $k$-way comparisons in [@OTX15]. A similar approach was studied with a different guarantees and assumptions in [@PNZSD15]. Our algorithm and ideas of rank-breaking can be directly applied to this collaborative ranking under MNL, with the same guarantees for consistency in the asymptotic regime where sample size grows to infinity. However, the analysis techniques for MNL rely on stronger assumptions on how the data is collected, and especially on the independence of the samples. It is not immediate how the analysis techniques developed in this paper can be applied to learn MNL.
In an orthogonal direction, new discrete choice models with sparse structures has been proposed recently in [@FJS09] and optimization algorithms for revenue management has been proposed [@FJS13]. In a similar direction, new discrete choice models based on Markov chains has been introduced in [@BGG13], and corresponding revenue management algorithms has been studied in [@FT14]. However, typically these models are analyzed in the asymptotic regime with infinite samples, with the exception of [@AS11]. A non-parametric choice models for pairwise comparisons also have been studied in [@RA14; @SBGW15]. This provides an interesting opportunities to studying learning to rank for these new choice models.
We consider a fixed design setting, where inference is separate from data collection. There is a parallel line of research which focuses on adaptive ranking, mainly based on pairwise comparisons. When performing sorting from noisy pairwise comparisons, Braverman et al. in [@BM09] proposed efficient approaches and provided performance guarantees. Following this work, there has been recent advances in adaptive ranking [@Ail11; @JN11; @MG15a].
Discussion
==========
We study the problem of learning the PL model from ordinal data. Under the traditional data collection scenarios, several efficient algorithms find the maximum likelihood estimates and at the same time provably achieve minimax optimal performance. However, for some non-traditional scenarios, computational complexity of finding the maximum likelihood estimate can scale super-exponentially in the problem size. We provide the first finite-sample analysis of computationally efficient estimators known as rank-breaking estimators. This provides guidelines for choosing the weights in the estimator to achieve optimal performance, and also explicitly shows how the accuracy depends on the topology of the data.
This paper provides the first analytical result in the sample complexity of rank-breaking estimators, and quantifies the price we pay in accuracy for the computational gain. In general, more complex higher-order rank-breaking can also be considered, where instead of breaking a partial ordering into a collection of paired comparisons, we break it into a collection of higher-order comparisons. The resulting higher-order rank-breakings will enable us to traverse the whole spectrum of computational complexity between the pairwise rank-breaking and the MLE. We believe this paper opens an interesting new direction towards understanding the whole spectrum of such approaches. However, analyzing the Hessian of the corresponding objective function is significantly more involved and requires new technical innovations.
Proofs
======
Proof of Theorem \[thm:main2\] {#sec:proof_main2}
------------------------------
We prove a more general result for an arbitrary choice of the parameter $\lambda_{j,a}>0$ for all $j\in[n]$ and $a\in[\ell_j]$. The following theorem proves the (near)-optimality of the choice of $\lambda_{j,a}$’s proposed in , and implies the corresponding error bound as a corollary.
\[thm:main\] Under the hypotheses of Theorem \[thm:main2\] and any $\lambda_{j,a}$’s, the rank-breaking estimator achieves $$\begin{aligned}
\label{eq:main1}
\frac{1}{\sqrt{d}} \big\|\,\widehat{\theta} - \theta^* \,\big\|_2 \; \,\leq\, \; \frac{4\sqrt{2}e^{4b}(1+ e^{2b})^2 \sqrt{d \log d} }{\alpha\, \gamma} \frac{\sqrt{\sum_{j=1}^n \sum_{a=1}^{\ell_j} \big(\lambda_{j,a}\big)^2 \big(\kappa_j - p_{j,a}\big)\big(\kappa_j- p_{j,a}+1\big)}}{ \sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a})}\;,
\end{aligned}$$ with probability at least $ 1- 3e^{3}d^{-3}$, if $$\begin{aligned}
\label{eq:main2}
\sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a}) \;\;\geq\;\; 2^{6}e^{18b} \frac{\eta\delta}{\alpha^2 \beta \gamma^2\tau} d\log d \;,
\end{aligned}$$ where $\gamma$, $\eta$, $\tau$, $\delta$, $\alpha$, $\beta$, are now functions of $\lambda_{j,a}$’s and defined in , , , and .
We first claim that $\lambda_{j,a} = 1/(\kappa_j-p_{j,a}+1)$ is the optimal choice for minimizing the above upper bound on the error. From Cauchy-Schwartz inequality and the fact that all terms are non-negative, we have that $$\begin{aligned}
\label{eq:cauchy-schwartz}
\frac{\sqrt{\sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \big(\lambda_{j,a}\big)^2(\kappa_j - p_{j,a})(\kappa_j - p_{j,a}+1)}}
{\sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a})}
\;\;\geq\;\; \frac{1}{\sqrt{ \sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \frac{(\kappa_j - p_{j,a})}{(\kappa_j - p_{j,a}+1)}}}\,,\end{aligned}$$ where $\lambda_{j,a} = 1/(\kappa_j-p_{j,a}+1)$ achieves the universal lower bound on the right-hand side with an equality. Since $\sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \frac{(\kappa_j - p_{j,a})}{(\kappa_j - p_{j,a}+1)} \geq \sum_{j=1}^n \ell_j$, substituting this into gives the desired error bound in . Although we have identified the optimal choice of $\lambda_{j,a}$’s, we choose a slightly different value of $\lambda=1/(\kappa_j-p_{j,a})$ for the analysis. This achieves the same desired error bound in , and significantly simplifies the notations of the sufficient conditions.
We first define all the parameters in the above theorem for general $\lambda_{j,a}$. With a slight abuse of notations, we use the same notations for $\H$, $L$, $\alpha$ and $\beta$ for both the general $\lambda_{j,a}$’s and also the specific choice of $\lambda_{j,a}=1/(\kappa_j-p_{j,a})$. It should be clear from the context what we mean in each case. Define $$\begin{aligned}
\tau &\equiv & \min_{j \in [n]} \tau_j \;, \; \;\;\;\;\text{where}\;\; \tau_{j} \equiv \frac{\sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a})}{\ell_j}\label{eq:tau_def}\\
\delta_{j,1} & \equiv & \bigg\{ \max_{a \in [\ell_j]} \Big\{\lambda_{j,a}(\kappa_j - p_{j,a})\Big\} + \sum_{a = 1}^{\ell_j} \lambda_{j,a} \bigg\} \;\;, \;\text{and}\;\;\;\;\;\; \delta_{j,2} \equiv \sum_{a = 1}^{\ell_j} \lambda_{j,a} \label{eq:delta12_def} \\
\delta & \equiv & \max_{j \in [n]} \bigg\{ 4 \delta_{j,1}^2 + \frac{2\big(\delta_{j,1}\delta_{j,2} + \delta_{j,2}^2\big)\kappa_j}{\eta_{j}\ell_j} \bigg\} \;\;\,. \label{eq:delta_def} \end{aligned}$$ Note that $\delta \geq \delta_{j,1}^2 \geq \max_a \lambda_{j,a}^2 (\kappa_j-p_{j,a})^2 \geq \tau^2$, and for the choice of $\lambda_{j,a}=1/(\kappa_j-p_{j,a})$ it simplifies as $\tau=\tau_j=1$. We next define a comparison graph $\H$ for general $\lambda_{j,a}$, which recovers the proposed comparison graph for the optimal choice of $\lambda_{j,a}$’s
\[def:comparison\_graph2\] (Comparison graph $\H$). Each item $i \in [d]$ corresponds to a vertex $i$. For any pair of vertices $i,\i$, there is a weighted edge between them if there exists a set $S_j$ such that $i, \i \in S_j$; the weight equals $\sum_{j: i,\i \in S_j} \frac{\tau_{j}\ell_j}{\kappa_j(\kappa_j-1)}$.
Let $A$ denote the weighted adjacency matrix, and let $D = {\rm diag}(A {\boldsymbol{1}})$. Define, $$\begin{aligned}
\label{eq:posl_Dlmax}
D_{\max} \;\;\equiv \;\; \max_{i \in [d]} D_{ii} \;= \;\max_{i \in [d]} \bigg\{ \sum_{j: i \in S_j} \frac{\tau_{j}\ell_j}{\kappa_j} \bigg\} \;\;\geq\;\; \tau_{\rm min} \max_{i \in [d]} \bigg\{ \sum_{j: i \in S_j} \frac{\ell_j}{\kappa_j} \bigg\} \,. \end{aligned}$$ Define graph Laplacian $L $ as $L = D - A$, i.e., $$\begin{aligned}
\label{eq:comparison2_L}
L \; =\; \sum_{j = 1}^n \frac{\tau_{j}\ell_j}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top.\end{aligned}$$ Let $ 0 = \lambda_1(L) \leq \lambda_2(L) \leq \cdots \leq \lambda_d(L)$ denote the sorted eigenvalues of $L$. Note that $\Tr(L ) = \sum_{i =1}^d \sum_{j: i \in S_j}\tau_{j}\ell_j/\kappa_j = \sum_{j = 1}^n \tau_{j}\ell_j$. Define $\alpha $ and $\beta $ such that $$\begin{aligned}
\label{eq:lambda2_L2}
\alpha \equiv \frac{\lambda_2(L )(d-1)}{\Tr(L)} = \frac{\lambda_2(L)(d-1)}{ \sum_{j = 1}^n \tau_{j}\ell_j} \;\; \text{and} \;\; \beta \equiv \frac{\Tr(L)}{d D_{\max}} = \frac{ \sum_{j = 1}^n \tau_{j}\ell_j}{d D_{\max}} \;.
\end{aligned}$$
For the proposed choice of $\lambda_{j,a} = 1/(\kappa_j-p_{j,a})$, we have $\tau_j = 1$ and the definitions of $\H$, $L$, $\alpha$, and $\beta$ reduce to those defined in Definition \[def:comparison\_graph1\]. We are left to prove an upper bound, $\delta\leq 32 (\log(\ell_{\max}+2))^2$, which implies the sufficient condition in and finishes the proof of Theorem \[thm:main2\]. We have, $$\begin{aligned}
\label{eq:main4}
\delta_{j,1} = \max_{a \in [\ell_j]} \Big\{\lambda_{j,a}(\kappa_j - p_{j,a})\Big\} + \sum_{a = 1}^{\ell_j} \lambda_{j,a}
&=& 1 + \sum_{a = 1}^{\ell_j} \frac{1}{\kappa_j - p_{j,a}} \nonumber\\
&\leq& 1 + \sum_{a=1}^{\ell_j} \frac{1}{a}\nonumber\\
&\leq& 2\log(\ell_j+2) \,, \end{aligned}$$ where in the first inequality follows from taking the worst case for the positions, i.e. $p_{j,a}= \kappa_j-\ell_j+a-1$ Using the fact that for any integer $x$, $\sum_{a=0}^{\ell-1} 1/(x+a) \leq \log((x +\ell -1)/(x -1))$, we also have $$\begin{aligned}
\label{eq:main5}
\frac{\delta_{j,2}\kappa_j}{\eta_j\ell_j} &\leq& \sum_{a = 1}^{\ell_j} \frac{1}{\kappa_j - p_{j,a}} \frac{\max{\{\ell_j,\kappa_j - p_{j,\ell_j}\}}}{\ell_j}\nonumber\\
&\leq& \min\Big\{\,\log(\ell_j+2) \,,\, \log\Big(\frac{\kappa_j-p_{j,\ell_j}+\ell_j -1}{\kappa_j-p_{j,\ell_j} -1 }\Big)\,\Big\} \frac{\max{\{\ell_j,\kappa_j - p_{j,\ell_j}\}}}{\ell_j}
\nonumber\\
&\leq& \frac{\log(\ell_j+2)\ell_j}{\max{\{\ell_j,\kappa_j - p_{j,\ell_j} -1}\}} \frac{\max{\{\ell_j,\kappa_j - p_{j,\ell_j}\}}}{\ell_j}\nonumber\\
&\leq& 2\log(\ell_j+2)\,,\end{aligned}$$ where the first inequality follows from the definition of $\eta_j$, Equation . From , , and the fact that $\delta_{j,2}\leq\log(\ell_j+2)$, we have $$\begin{aligned}
\label{eq:main6}
\delta = \max_{j \in [n]} \bigg\{ 4 \delta_{j,1}^2 + \frac{2\big(\delta_{j,1}\delta_{j,2} +\delta_{j,2}^2\big)\kappa_j}{\eta_{j}\ell_j} \bigg\} \;\;\leq\;\; 28 (\log(\ell_{\max} +2))^2\,.\end{aligned}$$
Proof of Theorem \[thm:main\]
-----------------------------
We first introduce two key technical lemmas. In the following lemma we show that $\E_{\theta^*}[\nabla \Lrb(\theta^*)] = 0$ and provide a bound on the deviation of $\nabla \Lrb(\theta^*)$ from its mean. The expectation $\E_{\theta^*}[\cdot]$ is with respect to the randomness in the samples drawn according to $\theta^*$. The log likelihood Equation can be rewritten as $$\begin{aligned}
\label{eq:likelihood}
\Lrb(\theta) =
\sum_{j=1}^n \sum_{a = 1}^{\ell_j}\sum_{i < \i \in S_j}\I_{\big\{(i,\i) \in \Gja\big\}}
\lambda_{j,a} \Big(\theta_i\I_{\big\{\sigma_j^{-1}(i) < \sigma_j^{-1}(\i)\big\}} + \theta_{\i}\I_{\big\{\sigma_j^{-1}(i) > \sigma_j^{-1}(\i)\big\}} - \log \Big(e^{\theta_i}
+ e^{\theta_{\i}}\Big) \Big)\;.\end{aligned}$$ We use $(i,\i) \in G_{j,a}$ to mean either $(i,\i)$ or $(\i,i)$ belong to $E_{j,a}$. Taking the first-order partial derivative of $\Lrb(\theta)$, we get $$\begin{aligned}
\label{eq:liklihood_grad}
\nabla_i\Lrb(\theta^*) \;\, =\;\, \sum_{j:i\in S_j} \sum_{a=1}^{\ell_j} \sum_{\substack{\i \in S_j \\ \i \neq i}} \,\lambda_{j,a}\,\I_ {\big\{(i,\i) \in G_{j,a}\big\}} \,\Bigg(\I_{\big\{\sigma_j^{-1}(i) < \sigma_j^{-1}(\i)\big\}} - \frac{\exp(\theta_i^*)}{\exp(\theta_i^*) + \exp(\theta_{\i}^*)} \Bigg)\;.\end{aligned}$$
\[lem:gradient\_topl\] Under the hypotheses of Theorem \[thm:main2\], with probability at least $1 - 2e^{3}d^{-3}$, $$\begin{aligned}
\big\|\nabla\Lrb(\theta^*)\big\|_2 \;\;\leq\;\; \sqrt{ 6\log d \, \sum_{j=1}^n \sum_{a=1}^{\ell_j} \big(\lambda_{j,a}\big)^2 \big(\kappa_j - p_{j,a}\big)\big(\kappa_j- p_{j,a}+1\big)} \,.
\end{aligned}$$
The Hessian matrix $H(\theta) \in \cS^d$ with $H_{i\i}(\theta) = \frac{\partial^2\Lrb(\theta)}{\partial\theta_i \partial\theta_{\i}}$ is given by $$\begin{aligned}
\label{eq:hessian}
H(\theta) = -\sum_{j=1}^n \sum_{a=1}^{\ell_j} \sum_{i<\i \in S_j} \I_{\big\{(i,\i) \in G_{j,a}\big\}} \lambda_{j,a} \Bigg( (e_i - e_{\i})(e_i - e_{\i})^\top \frac{\exp(\theta_i + \theta_{\i})}{[\exp(\theta_i) + \exp(\theta_{\i})]^2}\Bigg).\end{aligned}$$ It follows from the definition that $-H(\theta)$ is positive semi-definite for any $\theta \in \reals^d$. The smallest eigenvalue of $-H(\theta)$ is equal to zero and the corresponding eigenvector is all-ones vector. The following lemma lower bounds its second smallest eigenvalue $\lambda_2(-H(\theta))$.
\[lem:hessian\_positionl\] Under the hypotheses of Theorem \[thm:main2\], if $$\begin{aligned}
\label{eq:posl_lam_cond}
\sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a}) \geq 2^{6}e^{18b} \frac{\eta\delta}{\alpha^2\beta\gamma^2\tau} d\log d
\end{aligned}$$ then with probability at least $ 1- d^{-3}$, the following holds for any $\theta\in\Omega_b$: $$\begin{aligned}
\label{eq:lambda2_bound_positionl}
\lambda_2(-H(\theta)) \;\geq\; \frac{e^{-4b}}{(1+e^{2b})^2}\frac{\alpha \gamma}{d-1} \sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a})\,.
\end{aligned}$$
Define $\Delta = \widehat{\theta} - \theta^*$. It follows from the definition that $\Delta$ is orthogonal to the all-ones vector. By the definition of $\hat{\theta}$ as the optimal solution of the optimization , we know that $\Lrb(\widehat{\theta}) \geq \Lrb(\theta^*)$ and thus $$\begin{aligned}
\Lrb(\widehat{\theta}) - \Lrb(\theta^*) - \langle\nabla\Lrb(\theta^*),\Delta\rangle \;\geq\; -\langle\nabla\Lrb(\theta^*),\Delta\rangle \;\geq\; -{\|\nabla\Lrb(\theta^*)\|}_2{\|\Delta\|}_2, \label{eq:thm_ml_1}\end{aligned}$$ where the last inequality follows from the Cauchy-Schwartz inequality. By the mean value theorem, there exists a $\theta = a\widehat{\theta} + (1-a)\theta^*$ for some $a \in [0,1]$ such that $\theta \in \Omega_b$ and $$\begin{aligned}
\label{eq:thm_ml_2}
\Lrb(\widehat{\theta}) - \Lrb(\theta^*) - \langle\nabla\Lrb(\theta^*),\Delta\rangle \; =\; \frac{1}{2}\Delta^\top H(\theta)\Delta \leq -\frac{1}{2}\lambda_2(-H(\theta)){\|\Delta\|}_2^2,\end{aligned}$$ where the last inequality holds because the Hessian matrix $-H(\theta)$ is positive semi-definite with $H(\theta){\boldsymbol{1}} = {\boldsymbol{0}}$ and $\Delta^\top{\boldsymbol{1}} = 0$. Combining and , $$\begin{aligned}
\label{eq:thm_ml_3}
{\|\Delta\|}_2 \;\;\leq\;\; \frac{2{\|\nabla\Lrb(\theta^*)\|}_2}{\lambda_2(-H(\theta))}.\end{aligned}$$ Note that $\theta \in \Omega_b$ by definition. Theorem \[thm:main\] follows by combining Equation with Lemma \[lem:gradient\_topl\] and Lemma \[lem:hessian\_positionl\].
### Proof of Lemma \[lem:gradient\_topl\]
The idea of the proof is to view $\nabla\Lrb(\theta^*)$ as the final value of a discrete time vector-valued martingale with values in $\reals^d$. Define $\nabla\L_{G_{j,a}}(\theta^*)$ as the gradient vector arising out of each rank-breaking graph $\{G_{j,a}\}_{j\in [n], a \in [\ell_j]}$ that is $$\begin{aligned}
\label{eq:rev1}
\nabla_i\L_{G_{j,a}}(\theta^*) \equiv \sum_{\substack{\i \in S_j \\ \i \neq i}} \,\lambda_{j,a}\,\I_ {\big\{(i,\i) \in G_{j,a}\big\}} \,\Bigg(\I_{\big\{\sigma_j^{-1}(i) < \sigma_j^{-1}(\i)\big\}} - \frac{\exp(\theta_i^*)}{\exp(\theta_i^*) + \exp(\theta_{\i}^*)} \Bigg)\;.\end{aligned}$$ Consider $\nabla\L_{G_{j,a}}(\theta^*)$ as the incremental random vector in a martingale of $\sum_{j=1} \ell_j$ time steps. Lemma \[lem:consistency\] shows that the expectation of each incremental vector is zero. Observe that the conditioning event $\{i'' \in S \,:\,\sigma^{-1}(i'')<p_{j,a} \}$ given in is equivalent to conditioning on the history $\{G_{j,a'}\}_{a'<a}$. Therefore, using the assumption that the rankings $\{\sigma_j\}_{j \in [n]}$ are mutually independent, we have that the conditional expectation of $\nabla\L_{G_{j,a}}(\theta^*)$ conditioned on $\{G_{j',a''}\}_{j' < j, a'' \in [\ell_{j'}]}$ is zero. Further, the conditional expectation of $\nabla\L_{G_{j,a}}(\theta^*)$ is zero even when conditioned on the rank breaking due to previous separators $\{G_{j,a'}\}_{a'<a}$ that are ranked higher (i.e. $a'<a$), which follows from the next lemma.
\[lem:consistency\] For a position-$p$ rank breaking graph $G_p$, defined over a set of items $S$, where $p \in [|S|-1]$, $$\begin{aligned}
\label{eq:grad_eq6}
\P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \;\Big|\; \big(i,\i\big) \in G_p \Big] \;=\; \frac{\exp(\theta^*_{i})}{\exp(\theta^*_{i})+\exp(\theta^*_{i'})} \;,\end{aligned}$$ for all $i,i'\in S$ and also $$\begin{aligned}
\label{eq:grad_eq8}
\P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \;\Big|\; \big(i,\i\big) \in G_p \text{ and } \{i'' \in S \,:\,\sigma^{-1}(i'')<p \} \Big] \;=\; \frac{\exp(\theta^*_{i})}{\exp(\theta^*_{i})+\exp(\theta^*_{i'})} \;.\end{aligned}$$
This is one of the key technical lemmas since it implies that the proposed rank-breaking is consistent, i.e. $\E_{\theta^*}[\nabla \Lrb(\theta^*)] = 0$. Throughout the proof of Theorem \[thm:main2\], this is the only place where the assumption on the proposed (consistent) rank-breaking is used. According to a companion theorem in [@APX14a Theorem 2], it also follows that any rank-breaking that is not union of position-$p$ rank-breakings results in inconsistency, i.e. $\E_{\theta^*}[\nabla \Lrb(\theta^*)] \neq 0$. We claim that for each rank-breaking graph $G_{j,a}$, ${\|\nabla\L_{G_{j,a}}(\theta^*)\|}_2^2 \leq (\lambda_{j,a})^2 (\kappa_j - p_{j,a})(\kappa_j- p_{j,a}+1)$. By Lemma \[lem:az\_gen\] which is a generalization of the vector version of the Azuma-Hoeffding inequality found in [@hayes2005large Theorem 1.8], we have $$\begin{aligned}
\P\big[\big\|\nabla\Lrb(\theta^*)\big\|_2 \geq \delta \big] \;\;\leq\;\; 2e^{3}\exp\Bigg(\frac{-\delta^2}{2\sum_{j=1}^n \sum_{a=1}^{\ell_j} \big(\lambda_{j,a}\big)^2 \big(\kappa_j - p_{j,a}\big)\big(\kappa_j- p_{j,a}+1\big)}\Bigg)\,,\end{aligned}$$ which implies the result.
\[lem:az\_gen\] Let $(X_1,X_2,\cdots, X_n)$ be real-valued martingale taking values in $\reals^d$ such that $X_0 = 0$ and for every $1 \leq i \leq n$, ${\|X_i-X_{i-1}\|}_2 \leq c_i$, for some non-negative constant $c_i$. Then for every $\delta > 0$, $$\begin{aligned}
\P[{\|X_n\|}_2 \geq \delta] & \leq & 2e^{3}e^{-\frac{\delta^2}{2\sum_{i=1}^n c_i^2}}\,.\end{aligned}$$
It follows from the upper bound on ${\|\nabla\L_{G_{j,a}}(\theta^*)\|}_2^2 \leq c_i^2$ with $c_i^2=\lambda^2\big( (k_j-p_{j,a})^2 + (k_j-p_{j,a}) \big)$. In the expression , $\nabla\L_{G_{j,a}}(\theta^*)$ has one entry at $p_{j,a}$-th position that is compared to $(k_j-p_{j,a})$ other items and $(k_j-p_{j,a})$ entries that is compared only once, giving the bound $$\begin{aligned}
{\|\nabla\L_{G_{j,a}}(\theta^*)\|}_2^2 &\leq&
\lambda_{j,a}^2(k_j-p_{j,a})^2 + \lambda_{j,a}^2 (k_j-p_{j,a}) \;.\end{aligned}$$
### Proof of Lemma \[lem:consistency\]
Define event $E \equiv \{(i,\i) \in G_p \}$. Observe that $$\begin{aligned}
E = \Big\{\Big(\I_ {\{(\sigma^{-1}(i) = p\}} + \I_{\{\sigma^{-1}(\i)) = p\}} = 1\Big) \wedge \Big(\sigma^{-1}(i),\sigma^{-1}(\i) \geq p \Big)\Big\} \;.\end{aligned}$$ Consider any set $\Omega \subset S\setminus\{i,\i\}$ such that $|\Omega| = p-1$. Let $M$ denote an event that items of the set $\Omega$ are ranked in top-$(p-1)$ positions in a particular order. It is easy to verify the following: $$\begin{aligned}
\P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \Big| E, M\Big] &=& \frac{\P\Big[\big(\sigma^{-1}(i) < \sigma^{-1}(\i)\big), E, M\Big]}{\P\Big[E, M\Big]}\\
&=& \frac{\P\Big[\big(\sigma^{-1}(i)= p\big), M\Big]}{\P\Big[\big(\sigma^{-1}(i)= p\big), M\Big] + \P\Big[\big(\sigma^{-1}(\i) =p \big), M\Big]} \\
&=& \frac{\exp(\theta^*_i)}{\exp(\theta^*_i) + \exp(\theta^*_{\i})} = \P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \Big]\;.\end{aligned}$$ Since $M$ is any particular ordering of the set $\Omega$ and $\Omega$ is any subset of $S\setminus\{i,\i\}$ such that $|\Omega| = p -1$, conditioned on event $E$ probabilities of all the possible events $M$ over all the possible choices of set $\Omega$ sum to $1$.
### Proof of Lemma \[lem:az\_gen\]
It follows exactly along the lines of proof of Theorem 1.8 in [@hayes2005large].
### Proof of Lemma \[lem:hessian\_positionl\]
The Hessian $H(\theta)$ is given in . For all $j\in [n]$, define $M^{(j)} \in \cS^d$ as $$\begin{aligned}
\label{eq:posl_M_j_def}
M^{(j)} &\equiv& \sum_{a=1}^{\ell_j} \lambda_{j,a} \sum_{i<\i \in S_j} \I_{\big\{(i,\i)\; \in \; G_{j,a}\big\}} (e_i - e_{\i})(e_i - e_{\i})^\top,\end{aligned}$$ and let $M \equiv \sum_{j=1}^n M^{(j)}$. Observe that $M$ is positive semi-definite and the smallest eigenvalue of $M$ is zero with the corresponding eigenvector given by the all-ones vector. If $|\theta_i| \leq b$, for all $i \in [d]$, $\frac{\exp(\theta_i + \theta_{\i})}{[\exp(\theta_i) + \exp(\theta_{\i})]^2} \geq \frac{e^{2b}}{(1+ e^{2b})^2}$. Recall the definition of $H(\theta)$ from Equation . It follows that $-H(\theta) \succeq \frac{e^{2b}}{(1+ e^{2b})^2} M$ for $\theta \in \Omega_b$. Since, $-H(\theta)$ and $M$ are symmetric matrices, from Weyl’s inequality we have, $\lambda_2(-H(\theta)) \geq \frac{e^{2b}}{(1+ e^{2b})^2} \lambda_2(M)$. Again from Weyl’s inequality, it follows that $$\begin{aligned}
\lambda_2(M) &\geq& \lambda_2(\E[M]) - {\|M-\E[M]\|} \;, \end{aligned}$$ where $\|\cdot\|$ denotes the spectral norm. We will show in that $\lambda_2(\E[M]) \geq 2 \gamma e^{-6b} (\alpha/(d-1)) \sum_{j = 1}^n \tau_j\ell_j$, and in that ${\|M-\E[M]\|}\leq 8e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} $.
$$\begin{aligned}
\label{eq:lambda2_M}
\lambda_2(M) &\geq& \frac{2e^{-6b} \alpha \gamma}{d-1} \sum_{j = 1}^n \tau_j\ell_j - 8e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} \;\geq\; \frac{e^{-6b} \alpha \gamma}{d-1} \sum_{j = 1}^n \tau_j\ell_j \;, \end{aligned}$$
where the last inequality follows from the assumption that $\sum_{j = 1}^n \tau_j \ell_j \geq 2^{6}e^{18b} \frac{\eta\delta}{\alpha^2\beta \gamma^2\tau} d\log d$. This proves the desired claim. To prove the lower bound on $\lambda_2(\E[M])$, notice that $$\begin{aligned}
\label{eq:posl_expec1a}
\E[M] &=& \sum_{j = 1}^n \sum_{a =1}^{\ell_j} \lambda_{j,a} \sum_{i<\i \in S_j} \P\Big[(i,\i) \in G_{j,a} \Big| (i,\i \in S_j) \Big] (e_i - e_{\i})(e_i - e_{\i})^\top\;.\end{aligned}$$ The following lemma provides a lower bound on $\P[(i,\i) \in G_{j,a} | (i,\i \in S_j)]$.
\[lem:posl\_lowerbound\] Consider a ranking $\sigma$ over a set $S \subseteq [d]$ such that $|S| = \kappa$. For any two items $i,\i \in S$, $\theta\in\Omega_b$, and $1 \leq \ell \leq \kappa-1$, $$\begin{aligned}
\label{eq:posl_lowerbound_eq}
\P_{\theta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \;\;\geq\;\; \frac{e^{-6b}(\kappa-\ell)}{\kappa(\kappa-1)} \bigg(1 - \frac{\ell}{\kappa}\bigg)^{\alpha_{i,i',\ell,\theta} -2} \;,\end{aligned}$$ where the probability $\prob_\theta$ is with respect to the sampled ranking resulting from PL weights $\theta\in\Omega_b$, and $\alpha_{i,i',\ell,\theta}$ is defined as $1 \leq \alpha_{i,i',\ell,\theta} = {\left \lceil{\widetilde{\alpha}_{i,i',\ell,\theta}} \right \rceil}$, and $\widetilde{\alpha}_{i,i',\ell,\theta}$ is, $$\begin{aligned}
\label{eq:posl_alpha}
\widetilde{\alpha}_{i,i',\ell,\theta} \;\; \equiv \;\;
\max_{\ell'\in[\ell]} \max_{\substack{\Omega \subseteq S\setminus\{i,\i\} \\ : |\Omega| = \kappa-\ell'}} \Bigg\{\frac{\exp(\theta_i)+\exp(\theta_{\i})}{\big(\sum_{j\in \Omega} \exp(\theta_j)\big)/|\Omega|} \Bigg \}\;.
\end{aligned}$$
Note that we do not need $\max_{\ell' \in[\ell]}$ in the above equation as the expression achieves its maxima at $\ell' = \ell$, but we keep the definition to avoid any confusion. In the worst case, $2e^{-2b} \leq \widetilde{\alpha}_{i,i',\ell,\theta} \leq 2e^{2b}$. Therefore, using definition of rank breaking graph $G_{j,a}$, and Equations and we have, $$\begin{aligned}
\E[M] &\succeq& \gamma e^{-6b} \sum_{j = 1}^n \sum_{a =1}^{\ell_j} \lambda_{j,a} \frac{2(\kappa_j-p_{j,a})}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \nonumber\\
&\succeq& 2\gamma e^{-6b} \sum_{j = 1}^n \frac{1}{\kappa_j(\kappa_j-1)} \sum_{a =1}^{\ell_j} \lambda_{j,a}(\kappa_j-p_{j,a}) \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \nonumber\\
&=& 2\gamma e^{-6b} L, \label{eq:positionl_expec}\end{aligned}$$ where we used $\gamma\leq (1-p_{j,\ell_j}/\kappa_j )^{\alpha_1-2}$ which follows for the definition in . follows from the definition of Laplacian $L $, defined for the comparison graph $\H $ in Definition \[def:comparison\_graph2\]. Using $\lambda_2(L) = (\alpha /(d-1)) \sum_{j = 1}^n \tau_j\ell_j$ from , we get the desired bound $\lambda_2(\E[M]) \geq 2 \gamma e^{-6b} (\alpha /(d-1)) \sum_{j = 1}^n \tau_j\ell_j$.
Next we need to upper bound $\|\sum_{j =1}^n\E[(M^{j})^2]\|$ to bound the deviation of $M$ from its expectation. To this end, we prove an upper bound on $\P[\sigma_j^{-1}(i) = p_{j,a} \; | \;i \in S_j ]$ in the following lemma.
\[lem:posl\_upperbound\] Under the hypotheses of Lemma \[lem:posl\_lowerbound\], $$\begin{aligned}
\label{eq:posl_upperbound_eq}
\P_{\theta}\Big[ \sigma^{-1}(i) = \ell \Big] \;\;\leq\;\; \frac{e^{6b}}{\kappa} \bigg(1 - \frac{\ell}{\kappa+\alpha_{i,\ell,\theta}} \bigg)^{\alpha_{i,\ell,\theta} -1}
\;\; \leq \;\; \frac{e^{6b}}{\kappa-\ell} \;,
\end{aligned}$$ where $0 \leq \alpha_{i,\ell,\theta} = {\left \lfloor{\widetilde{\alpha}_{i,\ell,\theta}} \right \rfloor}$, and $\widetilde{\alpha}_{i,\ell,\theta}$ is, $$\begin{aligned}
\label{eq:posl_upper1}
\widetilde{\alpha}_{i,\ell,\theta} \;\; \equiv \;\; \min_{\ell' \in [\ell]} \min_{\substack{\Omega \in S\setminus\{i\} \\ : |\Omega| = \kappa-\ell'+1}} \Bigg\{\frac{\exp(\theta_i)}{\big(\sum_{j\in \Omega} \exp(\theta_j)\big)/|\Omega|} \Bigg \}\;.\end{aligned}$$
In the worst case, $e^{-2b} \leq \widetilde{\alpha}_{i,\ell,\theta} \leq e^{2b}$. Note that $\alpha_{i,\ell,\theta} =0$ gives the worst upper bound.
Therefore using Equation , for all $i \in [d]$, we have, $$\begin{aligned}
\label{eq:hess_posl_16}
\P\Big[\sigma_j^{-1}(i) \in \cP_j \Big] \leq \min \Bigg\{1, \frac{e^{6b}\ell_j}{\kappa_j - p_{j,\ell_j}} \Bigg\} \;\leq\; \frac{e^{6b}\ell_j}{\max\{ \ell_j, \kappa_j - p_{j,\ell_j}\}} \leq \frac{e^{6b}\eta \ell_j}{ \kappa_j}\,,\end{aligned}$$ where we used $\eta$ defined in Equation . Define a diagonal matrix $D^{(j)} \in \cS^{d}$ and a matrix $A^{(j)} \in \cS^d$, $$\begin{aligned}
A^{(j)}_{i\i} &\equiv & \I_{\big\{i,\i \in S_j \big\}} \,\sum_{a=1}^{\ell_j} \lambda_{j,a} \I_{\big\{(i,\i) \in G_{j,a}\big\}}\;,\; \text{for all} \;\; i,\i \in [d] \,, \label{eq:hess_posl_8}\end{aligned}$$ and $D^{(j)}_{ii} = \sum_{i'\neq i} A^{(j)}_{ii'}$. Observe that $M^{(j)} = D^{(j)} - A^{(j)}$. For all $i \in [d]$, we have, $$\begin{aligned}
D^{(j)}_{ii} &=& \I_{\big\{i \in S_j \big\}} \sum_{\i = 1}^{\kappa_j} \I_{\big\{\sigma_j^{-1}(i) = \i \big\}} \sum_{a=1}^{\ell_j} \lambda_{j,a} \deg_{G_{j,a}}(\sigma_j^{-1}(\i)) \nonumber\\
&\leq& \I_{\big\{i \in S_j \big\}}\Bigg\{ \I_{\big\{\sigma^{-1}_j(i) \in \cP_j\big\}}\Bigg( \max_{a \in [\ell_j]} \Big\{\lambda_{j,a}(\kappa_j - p_{j,a})\Big\} + \sum_{a = 1}^{\ell_j} \lambda_{j,a} \Bigg) + \I_{\big\{\sigma^{-1}_j(i) \notin \cP_j\big\}} \Bigg( \sum_{a = 1}^{\ell_j} \lambda_{j,a} \Bigg)\Bigg\} \nonumber\\
&=& \I_{\big\{i \in S_j \big\}}\bigg\{ \I_{\big\{\sigma^{-1}_j(i) \in \cP_j\big\}}\delta_{j,1} \; + \; \I_{\big\{\sigma^{-1}_j(i) \notin \cP_j\big\}} \delta_{j,2}\bigg\}, \label{eq:hess_posl_9}\end{aligned}$$ where the last equality follows from the definition of $\delta_{j,1}$ and $\delta_{j,2}$ in Equation . Note that $\max_{i \in [d]} \{D_{ii}\} = \delta_{j,1}$. Using and , we have, $$\begin{aligned}
\label{eq:hess_posl_14}
\E\Big[D^{(j)}_{ii}\Big] &\leq & \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{e^{6b}\eta\ell_j}{\kappa_j} \bigg(\delta_{j,1} + \frac{\delta_{j,2}\kappa_j}{\eta\ell_j} \bigg) \Bigg\} \,.\end{aligned}$$ Similarly we have, $$\begin{aligned}
\label{eq:hess_posl_10}
\E\Big[\big(D^{(j)}_{ii}\big)^2\Big] &\leq & \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{ e^{6b}\eta\ell_j}{\kappa_j} \bigg( \delta_{j,1}^2 + \frac{\delta_{j,2}^2\kappa_j}{\eta\ell_j} \bigg) \Bigg\} \end{aligned}$$ For all $i \in [d]$, we have, $$\begin{aligned}
\label{eq:hess_posl_12}
\E\Bigg[\sum_{\i = 1}^d \big(\big(A^{(j)}\big)^2\big)_{i\i} \Bigg] & \leq & \E\Bigg[ \bigg(\sum_{\i =1 }^d A^{(j)}_{i\i} \bigg) \max_{i \in [d]} \bigg\{ \sum_{\i =1 }^d A^{(j)}_{i\i}\bigg\} \Bigg] \nonumber\\
&\leq & \E\bigg[ D^{(j)}_{ii} \delta_{j,1} \bigg] \nonumber\\
&\leq& \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{e^{6b}\eta\ell_j}{\kappa_j} \bigg(\delta_{j,1}^2 + \frac{\delta_{j,1}\delta_{j,2}\kappa_j}{\eta\ell_j} \bigg)\Bigg\} \,.
\end{aligned}$$ Using and , we have, for all $i \in [d]$, $$\begin{aligned}
&&\sum_{\i = 1}^d \Big|\E\Big[\big(\big(M^{(j)}\big)^2\big)_{i\i}\Big]\Big| \nonumber\\
& = & \sum_{\i = 1}^d \Bigg|\E\Big[\big(\big(D^{(j)}\big)^2\big)_{i\i}\Big] - \E\Big[\big(D^{(j)} A^{(j)}\big)_{i\i}\Big]
- \E\Big[\big( A^{(j)} D^{(j)} \big)_{i\i}\Big]
+ \E\Big[\big(\big(A^{(j)}\big)^2\big)_{i\i}\Big] \Bigg| \nonumber\\
&\leq& 2\E\Big[\big(D^{(j)}_{ii}\big)^2\Big] + \sum_{\i = 1}^d \bigg( \E\Big[\delta_{j,1}\big(A^{(j)}\big)_{i\i}\Big] + \E\Big[\big(\big(A^{(j)}\big)^2\big)_{i\i}\Big] \bigg) \nonumber\\
&\leq& \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{e^{6b}\eta\ell_j}{\kappa_j}\bigg( 4 \delta_{j,1}^2 + \frac{2\big(\delta_{j,1}\delta_{j,2} +\delta_{j,2}^2\big)\kappa_j}{\eta\ell_j} \bigg) \Bigg\}\nonumber\\
&=& \I_{\big\{i \in S_j \big\}} \bigg\{ \frac{e^{6b}\delta\eta\ell_j}{\kappa_j} \bigg\}\,, \label{eq:hess_posl_15}\end{aligned}$$ where the last equality follows from the definition of $\delta$, Equation .
To bound $\|\sum_{j =1}^n \E[(M^{(j)})^2]\|$, we use the fact that for $J \in \reals^{d\times d}, {\|J\|} \leq \max_{i \in [d]}\sum_{\i = 1}^d|J_{i\i}|$. Therefore, we have $$\begin{aligned}
\Bigg\|\sum_{j =1}^n \E\Big[(M^{(j)})^2\Big]\Bigg\| & \leq & e^{6b}\delta\eta \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j}{\kappa_j} \Bigg\} \nonumber\\
& = & \frac{e^{6b}\eta\delta}{\tau} D_{\max} \label{eq:hess_posl_4}\\
& = & \frac{e^{6b}\eta\delta}{\beta \tau d} \sum_{j = 1}^n \tau_{j}\ell_j\;, \label{eq:hess_posl_5}\end{aligned}$$ where follows from the definition of $D _{\max}$ in Equation and follows from the definition of $\beta$ in . Observe that from Equation , ${\|M^{(j)}\|} \leq 2\delta_{j,1} \leq 2\sqrt{\delta}$. Applying matrix Bernstein inequality, we have, $$\begin{aligned}
\mathbb{P}\Big[\big\|M - \E[M]\big\| \geq t\Big] \leq d \,\exp\Bigg(\frac{-t^2/2}{\frac{e^{6b}\eta\delta}{\beta \tau d} \sum_{j = 1}^n \tau_{j}\ell_j + 4\sqrt{\delta}t/3}\Bigg). \end{aligned}$$ Therefore, with probability at least $1 - d^{-3}$, we have, $$\begin{aligned}
\label{eq:hess_posl_6}
\big\|M - \E[M]\big\| \leq 4e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} +\frac{64 \sqrt{\delta}\log d}{3} \leq 8e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} \;,\end{aligned}$$ where the second inequality uses $\sum_{j = 1}^n \tau_j \ell_j \geq 2^{6} (\beta \tau /\eta)d\log d$ which follows from the assumption that $\sum_{j = 1}^n \tau_j \ell_j \geq 2^{6}e^{18b} \frac{\eta\delta}{\tau\gamma^2\alpha^2 \beta } d\log d$ and the fact that $\alpha, \beta \leq 1$, $\gamma \leq 1$, $\eta\geq 1$, and $\delta> \tau^2$.
### Proof of Lemma \[lem:posl\_lowerbound\]
Since providing a lower bound on $\P_{\theta}\big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \big] $ for arbitrary $\theta$ is challenging, we construct a new set of parameters $\{\ltheta_j\}_{j\in[d]}$ from the original $\theta$. These new parameters are constructed such that it is both easy to compute the probability and also provides a lower bound on the original distribution. We denote the sum of the weights by $W \equiv \sum_{j \in S} \exp(\theta_j)$. We define a new set of parameters $\{\ltheta_j\}_{j \in S}$: $$\begin{aligned}
\ltheta_j &=& \left\{ \begin{array}{rl}
\log(\widetilde{\alpha}_{i,i',\ell,\theta}/2) &\; \text{for} \; j = i \text{ or }\i\;, \\
0&\;\text{otherwise}\;. \end{array}\right. \end{aligned}$$ Similarly define $\widetilde{W} \equiv \sum_{j \in S} \exp(\ltheta_j) = \kappa-2+\widetilde{\alpha}_{i,i',\ell,\theta}$. We have, $$\begin{aligned}
\label{eq:posl_3}
&& \P_{\theta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \nonumber\\
&=& \sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg(\frac{\exp(\theta_{j_1})}{W} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg(\frac{\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})}\cdots \nonumber \\
&& \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})}\frac{\exp(\theta_i)}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})}\Bigg) \cdots\Bigg)\Bigg) \nonumber\\
&=&\frac{\exp(\theta_i)}{W} \,\sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg( \frac{\exp(\theta_{j_1})}{W-\exp(\theta_{j_1})} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg( \frac{\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})-\exp(\theta_{j_2})}\cdots \nonumber\\
&& \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\ j_1,\cdots,j_{\ell-2}}} \Bigg( \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})} \Bigg)\cdots \Bigg)\Bigg) \nonumber\\ \end{aligned}$$ Consider the last summation term in the above equation and let $\Omega_\ell = S\setminus\{i,i',j_1,\ldots,j_{\ell-2}\}$. Observe that, $|\Omega_\ell| = \kappa-\ell$ and from equation , $\frac{\exp(\theta_{i})+\exp(\theta_{\i})}{\sum_{j \in \Omega_\ell} \exp(\theta_j)} \leq \frac{\widetilde{\alpha}_{i,i',\ell,\theta}}{\kappa-\ell}$. We have, $$\begin{aligned}
&&\sum_{j_{\ell-1} \in \Omega_\ell} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})} \nonumber\\
&=& \sum_{j_{\ell-1} \in \Omega_\ell } \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k}) - \exp(\theta_{j_{\ell-1}})} \nonumber\\
&\geq& \frac{\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})-\big(\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})\big)/|\Omega_\ell|} \label{eq:posl_jensen_ineq}\\
&=& \frac{\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})}{\exp(\theta_i)+\exp(\theta_{\i}) +\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})-\big(\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})\big)/|\Omega_\ell|} \nonumber\\
&=&\Bigg({\frac{\exp(\theta_{i})+\exp(\theta_{\i})}{\sum_{j_{\ell-1} \in \Omega_\ell} \exp(\theta_{j_{\ell-1}})} + 1 - \frac{1}{\kappa-\ell}}\Bigg)^{-1} \nonumber\\
&\geq& \Bigg(\frac{\widetilde{\alpha}_1}{\kappa-\ell} + 1 - \frac{1}{\kappa-\ell}\Bigg)^{-1} \label{eq:posl_1}\\
&=& \frac{\kappa-\ell}{\widetilde{\alpha}_1 + \kappa-\ell-1} \nonumber\\
&=& \sum_{j_{\ell-1} \in \Omega_\ell } \frac{\exp(\ltheta_{j_{\ell-1}})}{\widetilde{W}-\sum_{k=j_1}^{j_{\ell-2}}\exp(\ltheta_{k}) - \exp(\ltheta_{j_{\ell-1}})} \label{eq:posl_2}\;,\end{aligned}$$ where follows from the Jensen’s inequality and the fact that for any $c >0$, $0 < x < c$, $\frac{x}{c-x}$ is convex in $x$. Equation follows from the definition of $\widetilde{\alpha}_{i,i',\ell,\theta}$, , and the fact that $|\Omega_\ell| = \kappa-\ell$. Equation uses the definition of $\{\ltheta_j\}_{j \in S}$.
Consider $\{\Omega_{\widetilde{\ell}}\}_{2 \leq \widetilde{\ell} \leq \ell - 1}$, $|\Omega_{\widetilde{\ell}}| = \kappa - \widetilde{\ell}$, corresponding to the subsequent summation terms in . Observe that $\frac{\exp(\theta_i)+\exp(\theta_{\i})}{\sum_{j \in \Omega_{\widetilde{\ell}}} \exp(\theta_j)} \leq \widetilde{\alpha}_{i,i',\ell,\theta}/|\Omega_{\widetilde{\ell}}|$. Therefore, each summation term in equation can be lower bounded by the corresponding term where $\{\theta_j\}_{j \in S}$ is replaced by $\{\ltheta_j\}_{j \in S}$. Hence, we have $$\begin{aligned}
\label{eq:posl_4}
&&\P_{\theta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \nonumber\\
&\geq& \frac{\exp(\theta_i)}{W} \sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg( \frac{\exp(\ltheta_{j_1})}{\widetilde{W}-\exp(\ltheta_{j_1})} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg( \frac{\exp(\ltheta_{j_2})}{\widetilde{W}-\exp(\ltheta_{j_1})-\exp(\ltheta_{j_2})}\cdots \nonumber\\
&& \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\ j_1,\cdots,j_{\ell-2}}} \Bigg( \frac{\exp(\ltheta_{j_{\ell-1}})}{\widetilde{W}-\sum_{k=j_1}^{j_{\ell-1}}\exp(\ltheta_{k})} \Bigg)\Bigg)\Bigg) \nonumber\\
&\geq& \frac{e^{-4b} \exp(\ltheta_i)}{\widetilde{W}} \sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg( \frac{\exp(\ltheta_{j_1})}{\widetilde{W}-\exp(\ltheta_{j_1})} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg( \frac{\exp(\ltheta_{j_2})}{\widetilde{W}-\exp(\ltheta_{j_1})-\exp(\ltheta_{j_2})}\cdots \nonumber\\
&& \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\j_1,\cdots,j_{\ell-2}}} \Bigg( \frac{\exp(\ltheta_{j_{\ell-1}})}{\widetilde{W}-\sum_{k=j_1}^{j_{\ell-1}}\exp(\ltheta_{k})} \Bigg)\Bigg)\Bigg)\nonumber\\
&=& \big(e^{-4b}\big) \P_{\ltheta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \;.\end{aligned}$$ The second inequality uses $\frac{\exp(\theta_i)}{W} \geq e^{-2b}/\kappa$ and $\frac{\exp(\ltheta_i)}{\widetilde{W}} \leq e^{2b}/\kappa$. Observe that $\exp(\ltheta_j) = 1$ for all $j \neq i,\i$ and $\exp(\ltheta_i) + \exp(\ltheta_{\i}) = \widetilde{\alpha}_{i,i',\ell,\theta} \leq {\left \lceil{\widetilde{\alpha}_{i,i',\ell,\theta}} \right \rceil} = \alpha_{i,i',\ell,\theta} \geq 1$. Therefore, we have $$\begin{aligned}
&&\P_{\ltheta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \nonumber\\
&=& {\kappa-2 \choose \ell-1} \frac{(\widetilde{\alpha}_{i,i',\ell,\theta}/2)(\ell-1)!}{(\kappa-2+\widetilde{\alpha}_{i,i',\ell,\theta})(\kappa-2+\widetilde{\alpha}_{i,i',\ell,\theta} - 1)\cdots(\kappa-2+\widetilde{\alpha}_{i,i',\ell,\theta} - (\ell-1))} \nonumber\\
&\geq& \frac{(\kappa-2)!}{(\kappa-\ell-1)!} \frac{e^{-2b}}{(\kappa + \alpha_{i,i',\ell,\theta}-2)(\kappa + \alpha_{i,i',\ell,\theta} - 3)\cdots(\kappa + \alpha_{i,i',\ell,\theta} - (\ell+1))} \label{eq:posl_5} \\
& =& \frac{e^{-2b}(\kappa-\ell + \alpha_{i,i',\ell,\theta}-2)(\kappa -\ell +\alpha_{i,i',\ell,\theta}-3)\cdots (\kappa -\ell)}{(\kappa+\alpha_{i,i',\ell,\theta}-2)(\kappa+\alpha_{i,i',\ell,\theta}-3)\cdots(\kappa-1)} \nonumber\\
&=& \frac{e^{-2b}}{(\kappa-1)} \frac{(\kappa-\ell + \alpha_{i,i',\ell,\theta}-2)(\kappa -\ell +\alpha_{i,i',\ell,\theta}-3)\cdots (\kappa -\ell)}{(\kappa+\alpha_{i,i',\ell,\theta}-2)(\kappa+\alpha_{i,i',\ell,\theta}-3)\cdots(\kappa)}\nonumber\\
&\geq& \frac{e^{-2b}}{(\kappa-1)} \bigg( 1- \frac{\ell}{\kappa}\bigg)^{\alpha_{i,i',\ell,\theta}-1}\nonumber\\
&=& \frac{e^{-2b}(\kappa-\ell)}{\kappa(\kappa-1)} \bigg( 1- \frac{\ell}{\kappa}\bigg)^{\alpha_{i,i',\ell,\theta}-2}, \label{eq:posl_6}\end{aligned}$$ where follows from the fact that $\widetilde{\alpha}_{i,i',\ell,\theta} \geq 2e^{-2b}$. Claim follows by combining Equations and .
### Proof of Lemma \[lem:posl\_upperbound\]
Analogous to the proof of Lemma \[lem:posl\_lowerbound\], we construct a new set of parameters $\{\ltheta_j\}_{j\in[d]}$ from the original $\theta$. We denote the sum of the weights by $W \equiv \sum_{j \in S} \exp(\theta_j)$. We define a new set of parameters $\{\ltheta_j\}_{j \in S}$: $$\begin{aligned}
\ltheta_j &=& \left\{ \begin{array}{rl}
\log(\widetilde{\alpha}_{i,\ell,\theta}) &\; \text{for} \; j = i \;, \\
0&\;\text{otherwise}\;. \end{array}\right. \end{aligned}$$ Similarly define $\widetilde{W} \equiv \sum_{j \in S} \exp(\ltheta_j) = \kappa-1+\widetilde{\alpha}_{i,\ell,\theta}$. We have, $$\begin{aligned}
& \P_{\theta}\Big[\sigma^{-1}(i) = \ell \Big] \nonumber\\
&= \sum_{\substack{j_1 \in S \\ j_1 \neq i}} \Bigg(\frac{\exp(\theta_{j_1})}{W} \sum_{\substack{j_2 \in S \\ j_2 \neq i,j_1}} \Bigg(\frac{\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})}\cdots \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})}\frac{\exp(\theta_i)}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})}\Bigg)\Bigg)\Bigg) \nonumber\\
&\leq \sum_{\substack{j_1 \in S \\ j_1 \neq i}} \Bigg(\frac{\exp(\theta_{j_1})}{W} \sum_{\substack{j_2 \in S \\ j_2 \neq i,j_1}} \Bigg(\frac{\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})}\cdots \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})}\Bigg)\Bigg)\Bigg) \frac{e^{2b}}{\kappa-\ell+1} \label{eq:posl_upper2}\end{aligned}$$ Consider the last summation term in the equation , and let $\Omega_\ell = S\setminus\{i,j_1,\ldots,j_{\ell-2}\}$, such that $|\Omega_\ell| = \kappa-\ell+1$. Observe that from equation , $\frac{\exp(\theta_i)}{\sum_{j \in \Omega_\ell} \exp(\theta_j)} \geq \frac{\lalpha_{i,\ell,\theta}}{\kappa-\ell+1}$. We have, $$\begin{aligned}
\sum_{j_{\ell-1} \in \Omega_\ell} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})} &=& \frac{\sum_{j_{\ell-1} \in \Omega_\ell} \exp(\theta_{j_{\ell-1}}) }{ \exp(\theta_i)+ \sum_{j_{\ell-1} \in \Omega_\ell} \exp(\theta_{j_{\ell-1}})} \nonumber\\
&\leq& \bigg( \frac{\lalpha_{i,\ell,\theta}}{\kappa-\ell+1} + 1\bigg)^{-1} \nonumber\\
&=& \frac{\kappa-\ell+1}{\lalpha_{i,\ell,\theta} + \kappa - \ell +1} \nonumber\\
&=& \sum_{j_{\ell-1} \in \Omega_\ell} \frac{\exp(\ltheta_{j_{\ell-1}})}{\lW-\sum_{k=j_1}^{j_{\ell-2}}\exp(\ltheta_{k})}, \label{eq:posl_upper3}\end{aligned}$$ where follows from the definition of $\{\ltheta\}_{j \in S}$.
Consider $\{\Omega_{\widetilde{\ell}}\}_{2 \leq \widetilde{\ell} \leq \ell - 1}$, $|\Omega_{\widetilde{\ell}}| = \kappa - \widetilde{\ell} +1$, corresponding to the subsequent summation terms in . Observe that $\frac{\exp(\theta_i)}{\sum_{j \in \Omega_{\widetilde{\ell}}} \exp(\theta_j)} \geq \lalpha_{i,\ell,\theta}/|\Omega_{\widetilde{\ell}}|$. Therefore, each summation term in equation can be lower bounded by the corresponding term where $\{\theta_j\}_{j \in S}$ is replaced by $\{\ltheta_j\}_{j \in S}$. Hence, we have $$\begin{aligned}
\label{eq:posl_upper4}
&&\P_{\theta}\Big[\sigma^{-1}(i) = \ell\Big] \nonumber\\
&\leq & \sum_{\substack{j_1 \in S \\ j_1 \neq i}} \Bigg(\frac{\exp(\ltheta_{j_1})}{\lW} \sum_{\substack{j_2 \in S \\ j_2 \neq i,j_1}} \Bigg(\frac{\exp(\ltheta_{j_2})}{\lW-\exp(\ltheta_{j_1})}\cdots \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\ltheta_{j_{\ell-1}})}{\lW-\sum_{k=j_1}^{j_{\ell-2}}\exp(\ltheta_{k})}\Bigg)\Bigg)\Bigg) \frac{e^{2b}}{\kappa-\ell+1} \nonumber\\
&\leq & e^{4b} \sum_{\substack{j_1 \in S \\ j_1 \neq i}} \Bigg(\frac{\exp(\ltheta_{j_1})}{\lW} \sum_{\substack{j_2 \in S \\ j_2 \neq i,j_1}} \Bigg(\frac{\exp(\ltheta_{j_2})}{\lW-\exp(\ltheta_{j_1})}\cdots \nonumber\\
&& \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\ltheta_{j_{\ell-1}})}{\lW-\sum_{k=j_1}^{j_{\ell-2}}\exp(\ltheta_{k})} \frac{\exp(\ltheta_i)}{\lW - \sum_{k = j_1}^{j_{\ell-1}} \exp(\ltheta_k)} \Bigg)\Bigg)\Bigg) \nonumber\\
&\leq & e^{4b} \P_{\ltheta}\Big[\sigma^{-1}(i) = \ell\Big] \label{eq:posl_upper5}\end{aligned}$$ The second inequality uses $\lalpha_2/(\kappa- \ell+\lalpha_{i,\ell,\theta}) \geq e^{-2b}/(\kappa - \ell +1)$. Observe that $\exp(\ltheta_j) = 1$ for all $j \neq i$ and $\exp(\ltheta_i) = \widetilde{\alpha}_{i,\ell,\theta} \geq {\left \lfloor{\widetilde{\alpha}_{i,\ell,\theta}} \right \rfloor} = \alpha_{i,\ell,\theta} \geq 0$. Therefore, we have $$\begin{aligned}
\P_{\ltheta}\Big[\sigma^{-1}(i) = \ell \Big]
&=& {\kappa-1 \choose \ell-1} \frac{\lalpha_{i,\ell,\theta}(\ell-1)!}{(\kappa-1+\widetilde{\alpha}_{i,\ell,\theta})(\kappa-2+\widetilde{\alpha}_{i,\ell,\theta})\cdots(\kappa-\ell+\widetilde{\alpha}_{i,\ell,\theta}) } \nonumber\\
&\leq& \frac{(\kappa-1)!}{(\kappa-\ell)!} \frac{e^{2b}}{(\kappa -1 + \alpha_{i,\ell,\theta})(\kappa -2+ \alpha_{i,\ell,\theta} )\cdots(\kappa -\ell + \alpha_{i,\ell,\theta} )} \nonumber\\
&\leq& \frac{e^{2b}}{\kappa} \bigg( 1- \frac{\ell}{\kappa+\alpha_{i,\ell,\theta}}\bigg)^{\alpha_{i,\ell,\theta}-1}, \label{eq:posl_upper6}\end{aligned}$$ Note that equation holds for all values of $\alpha_{i,\ell,\theta} \geq 0$. Claim \[eq:posl\_upperbound\_eq\] follows by combining Equations and .
Proof of Theorem \[thm:cramer\_rao\_position\_p\] {#sec:proof_cramer_rao_position_p}
-------------------------------------------------
Let $H(\theta) \in \mathcal{S}^d$ be Hessian matrix such that $H_{i\i}(\theta) = \frac{\partial^2\L(\theta)}{\partial\theta_i \partial \theta_{\i}}$. The Fisher information matrix is defined as $I(\theta) = -\E_\theta[H(\theta)]$. Fix any unbiased estimator $\widehat{\theta}$ of $\theta \in \Omega_b$. Since, $\widehat{\theta} \in \mathcal{U}$, $\widehat{\theta} - \theta$ is orthogonal to ${\boldsymbol{1}}$. The Cramér-Rao lower bound then implies that ${\E[{\|\widehat{\theta} - \theta^*\|}^2] \geq \sum_{i = 2}^d \frac{1}{\lambda_i(I(\theta))}}$. Taking the supremum over both sides gives $$\begin{aligned}
\sup_{\theta}\E[{\|\widehat{\theta} - \theta\|}^2] \;\; \geq \;\; \sup_{\theta} \sum_{i=2}^d \frac{1}{\lambda_i(I(\theta))} \geq \sum_{i = 2}^d \frac{1}
{\lambda_i(I({\boldsymbol{0}}))}\;.\end{aligned}$$ The following lemma provides a lower bound on $\E_\theta[H({\boldsymbol{0}})]$, where ${\boldsymbol{0}}$ indicates the all-zeros vector.
\[lem:cr\_lem\] Under the hypotheses of Theorem \[thm:cramer\_rao\_position\_p\], $$\begin{aligned}
\label{eq:cr0}
\E_\theta[H({\boldsymbol{0}})] \;\;\succeq\;\; - \sum_{j=1}^n \frac{2p\log(\kappa_j)^2}{\kappa_j(\kappa_j-1)} \sum_{\i<i \in S_j}(e_i - e_{\i})(e_i - e_{\i})^{\top} \,.\end{aligned}$$
Observe that $I({\boldsymbol{0}})$ is positive semi-definite. Moreover, $\lambda_1(I({\boldsymbol{0}}))$ is zero and the corresponding eigenvector is the all-ones vector. It follows that $$\begin{aligned}
I(0) &\preceq & \sum_{j=1}^n \frac{2p\log(\kappa_j)^2}{\kappa_j(\kappa_j-1)} \sum_{\i<i \in S_j}(e_i - e_{\i})(e_i - e_{\i})^{\top} \\
& \preceq & 2p\log(\kappa_{\max})^2 \underbrace{\sum_{j=1}^n \frac{1}{\kappa_j(\kappa_j-1)} \sum_{\i<i \in S_j}(e_i - e_{\i})(e_i - e_{\i})^{\top}}_{=L}\;,\end{aligned}$$ where $L$ is the Laplacian defined for the comparison graph $\H$, Definition \[def:comparison\_graph1\], as $\ell_j = 1$ for all $j \in [n]$ in this setting. By Jensen’s inequality, we have $$\begin{aligned}
\sum_{i = 2}^d \frac{1}{\lambda_i(L)} \geq \frac{(d-1)^2}{\sum_{i = 2}^d \lambda_i(L)} = \frac{(d-1)^2}{\Tr(L)} = \frac{(d-1)^2}{n}.\end{aligned}$$
### Proof of Lemma \[lem:cr\_lem\]
Define $\L_j(\theta)$ for $j \in [n]$ such that $\L(\theta) = \sum_{j = 1}^n \L_j(\theta)$. Let $H^{(j)}(\theta) \in \mathcal{S}^d$ be the Hessian matrix such that $H^{(j)}_{i\i}(\theta) = \frac{\partial^2\L_j(\theta)}{\partial\theta_i \partial \theta_{\i}}$ for $i,\i \in S_j$. We prove that for all $j \in [n]$, $$\begin{aligned}
\label{eq:cr01}
\E_\theta[H^{(j)}({\boldsymbol{0}})] \;\; \succeq\;\; - \frac{2p\log(\kappa_j)^2}{\kappa_j(\kappa_j-1)} \sum_{\i<i \in S_j}(e_i - e_{\i})(e_i - e_{\i})^{\top} \,.\end{aligned}$$ In the following, we omit superscript/subscript $j$ for brevity. With a slight abuse of notation, we use $\I_{\{\Omega^{-1}(i) = a\}} = 1$ if item $i$ is ranked at the $a$-th position in all the orderings $\sigma \in \Omega$. Let $\P[\theta]$ be the likelihood of observing $\Omega^{-1}(p) = i^{(p)}$ and the set $\Lambda$ (the set of the items that are ranked before the $p$-th position). We have, $$\begin{aligned}
\label{eq:cr1}
\P(\theta) = \sum_{\sigma \in \Omega} \Bigg(\frac{\exp\big(\sum_{m = 1}^{p} \theta_{\sigma(m)} \big)}{\prod_{a=1}^{p} \Big(\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big) \Big)}\Bigg)\,.\end{aligned}$$ For $i,\i \in S_j $, we have $$\begin{aligned}
\label{eq:cr7}
H_{i\i}(\theta) = \frac{1}{\P(\theta)} \frac{\partial^2\P(\theta)}{\partial\theta_i \partial \theta_{\i}} - \frac{\nabla_i \P(\theta) \nabla_{\i} \P(\theta)}{\big(\P(\theta)\big)^2} \end{aligned}$$ We claim that at $\theta = {\boldsymbol{0}}$,
$$\begin{aligned}
-H_{i\i}({\boldsymbol{0}}) = \left\{ \begin{array}{rl}
C_1 & \;\; \text{if} \; i = \i, \; \big\{\Omega^{-1}(i) \geq p \big\} \label{eq:cr81} \\
C_2 + A_3^2 - C_3 & \;\; \text{if} \; i = \i,\; \big\{\Omega^{-1}(i) < p \big\} \label{eq:cr82}\\
-B_1 & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) \geq p, \;\Omega^{-1}(\i) \geq p \big\} \label{eq:cr83} \\
-B_2 & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) \geq p, \; \Omega^{-1}(\i) < p \big\} \label{eq:cr84} \\
-B_2 & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) < p, \; \Omega^{-1}(\i) \geq p\big\} \label{eq:cr85} \\
-(B_3 + B_4 - A_3^2) & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) < p,\; \Omega^{-1}(\i) < p\big\} \;. \label{eq:cr86}
\end{array}
\right.\end{aligned}$$
where constants $A_3, B_1, B_2, B_3, B_4, C_1, C_2$ and $C_3$ are defined in Equations , , , , , , and respectively. From this computation of the Hessian, note that we have $$\begin{aligned}
\label{eq:cr11}
H({\boldsymbol{0}}) = \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \Big(H_{i\i}({\boldsymbol{0}}) \Big) \;. \end{aligned}$$ which follows directly from the fact that the diagonal entries are summations of the off-diagonals, i.e. $C_1 = B_1(\kappa-p) + B_2(p-1)$ and $C_2 + A_3^2 - C_3 = B_2(\kappa-p+1) + (B_3 + B_4 - A_3^2)(p-2)$. The second equality follows from the fact that $C_2 = B_2(\kappa-p+1) + B_3(p-2)$ and $A_3^2(p-1) = B_4(p-2) + C_3$. Note that since $\theta = {\boldsymbol{0}}$, all items are exchangeable. Hence, $\E[H_{i\i}({\boldsymbol{0}})] = \E[H_{ii}({\boldsymbol{0}})]/(\kappa-1)$, and substituting this into and using Equations , we get $$\begin{aligned}
&& \E\Big[ H({\boldsymbol{0}})\Big] \nonumber\\
&=& -\frac{1}{\kappa-1}\bigg(\P\big[\Omega^{-1}(i) \geq p \big]C_1 + \P\big[\Omega^{-1}(i) < p \big](C_2 + A_3^2 - C_3)\bigg) \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \nonumber\\
&\succeq & - \frac{1}{\kappa(\kappa-1)} \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \nonumber\\
&& \Bigg((\kappa-p+1)\log\bigg(\frac{\kappa}{\kappa-p}\bigg) + (p-1)\bigg(\log\bigg(\frac{\kappa}{\kappa-p+1}\bigg) + \log\bigg(\frac{\kappa}{\kappa-p+1}\bigg)^2 \bigg)\Bigg) \nonumber\\\label{eq:cr12}\\
&\succeq & -\frac{2p\log(\kappa)^2}{\kappa(\kappa-1)} \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \;, \label{eq:cr13} \end{aligned}$$ where uses $\sum_{a = 1}^p \frac{1}{\kappa -a+1} \leq \log\big(\frac{\kappa}{\kappa-p}\big)$ and $C_3 \geq 0$. Equation follows from the fact that for any $x>0$, $\log(1+x) \leq x$. To prove , we have the first order partial derivative of $\P(\theta)$ given by $$\begin{aligned}
\label{eq:cr2}
\nabla_i \P(\theta) &=& \I_{\{\Omega^{-1}(i) \leq p \}}\P(\theta) - \sum_{\sigma \in \Omega} \Bigg(\frac{\exp\big(\sum_{m = 1}^{p} \theta_{\sigma(m)} \big)}{\prod_{a=1}^{p} \Big(\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big) \Big)} \Bigg( \sum_{a = 1}^p \frac{\I_{\{\sigma^{-1}(i) \geq a \}}\exp(\theta_i)}{\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big)} \Bigg) \Bigg) \,.\end{aligned}$$ Define constants $A_1$, $A_2$ and $A_3$ such that $$\begin{aligned}
A_1 & \equiv & \P(\theta) \big|_{\{\theta = {\boldsymbol{0}}\}} = \frac{(p-1)!}{\kappa(\kappa-1)\cdots(\kappa-p+1)}, \label{eq:crA1}\\
A_2 & \equiv & \Bigg( \sum_{a = 1}^p \frac{\exp(\theta_i)}{\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big)} \Bigg)\Bigg|_{\{\theta = {\boldsymbol{0}}\}} = \Bigg(\frac{1}{\kappa} + \frac{1}{\kappa-1}+\cdots + \frac{1}{\kappa-p+1}\Bigg), \label{eq:crA2} \\
A_3 & \equiv & \Bigg(\frac{(p-1)(p-2)!}{(p-1)!(\kappa)} + \frac{(p-2)(p-2)!}{(p-1)!(\kappa-1)} + \cdots + \frac{(p-2)!}{(p-1)!(\kappa-p+2)} \Bigg) \,. \label{eq:crA3}\end{aligned}$$ Observe that, for all $i \in [d]$, $$\begin{aligned}
\label{eq:cr4}
\nabla_i \P(\theta) \big|_{\{\theta = {\boldsymbol{0}}\}} = A_1 \Big( \I_{\{\Omega_j^{-1}(i) = p\}}(1 - A_2) + \I_{\{\Omega_j^{-1}(i) < p\}}(1 - A_3) - \I_{\{\Omega_j^{-1}(i) > p\}}A_2 \Big) \;\; \,.\end{aligned}$$ Further define constants $B_1$, $B_2$, $B_3$ and $B_4$ such that $$\begin{aligned}
B_1 &\equiv & \Bigg(\frac{1}{\kappa^2} + \frac{1}{(\kappa-1)^2} + \cdots + \frac{1}{(\kappa - p+1)^2}\Bigg), \label{eq:crB1}\\
B_2 & \equiv & \Bigg(\frac{p-1}{(p-1)\kappa^2} + \frac{p-2}{(p-1)(\kappa-1)^2} + \cdots + \frac{1}{(p-1)(\kappa-p+2)^2} \Bigg), \label{eq:crB2} \\
B_3 & \equiv & \Bigg( \frac{(p-1)(p-2)(p-3)!}{(p-1)!\kappa^2} + \frac{(p-2)(p-3)(p-3)!}{(p-1)!(\kappa-1)^2} + \cdots + \frac{2(p-3)!}{(p-1)!(\kappa-p+3)^2} \Bigg), \label{eq:crB3} \\
B_4 & \equiv & \frac{(p-3)!}{(p-1)!} \Bigg(\sum_{a,b \in [p-1], b \neq a} \bigg(\frac{1}{\kappa} + \frac{1}{\kappa -1} + \cdots + \frac{1}{\kappa - a+1} \bigg) \bigg(\frac{1}{\kappa} + \frac{1}{\kappa -1} + \cdots + \frac{1}{\kappa - b+1} \bigg) \Bigg) \,. \label{eq:crB4}\end{aligned}$$ Observe that, $$\begin{aligned}
\label{eq:cr6}
&&\frac{\partial^2\P(\theta)}{\partial\theta_i \partial \theta_{\i}}\bigg|_{\theta = {\boldsymbol{0}}} \nonumber\\
&=& \I_{\big\{\Omega^{-1}(i),\Omega^{-1}(\i) > p\big\}} A_1 \Big((-A_2)(-A_2) + B_1 \Big) \nonumber\\
&&+ \;\Big(\I_{\big\{\Omega^{-1}(i) > p, \Omega^{-1}(\i) = p\big\}} + \I_{\big\{\Omega^{-1}(i) = p, \Omega^{-1}(\i) > p\big\}} \Big) A_1 \Big((-A_2)(1-A_2) + B_1 \Big) \nonumber\\
&&+ \; \Big( \I_{\big\{\Omega^{-1}(i) = p, \Omega^{-1}(\i) < p\big\}} + \I_{\big\{\Omega^{-1}(i) < p, \Omega^{-1}(\i) = p\big\}}\Big) A_1 \Big((1-A_3) + (-A_2)(1-A_3) + B_2 \Big) \nonumber\\
&& + \; \Big(\I_{\big\{\Omega^{-1}(i) > p, \Omega^{-1}(\i) < p\big\}} + \I_{\big\{\Omega^{-1}(i) < p, \Omega^{-1}(\i) > p\big\}} \Big) A_1 \Big((-A_2)(1-A_3) + B_2 \Big) \nonumber\\
&& + \;\I_{\big\{\Omega^{-1}(i) < p, \Omega^{-1}(\i) < p\big\}} A_1 \Big((1-A_3) + (-A_3) + B_4 + B_3 \Big)\,.\end{aligned}$$ The claims in are easy to verify by combining Equations and with . Also, define constants $C_1$, $C_2$ and $C_3$ such that, $$\begin{aligned}
C_1 &\equiv &\Bigg( \frac{\kappa-1}{(\kappa)^2} + \frac{\kappa - 2}{(\kappa-1)^2} + \cdots + \frac{\kappa - p}{(\kappa-p+1)^2} \Bigg)\,,\label{eq:crC1}\\
C_2 & \equiv & \Bigg(\frac{(p-1)(p-2)!(\kappa-1)}{(p-1)!(\kappa)^2} + \frac{(p-2)(p-2)!(\kappa-2)}{(p-1)!(\kappa-1)^2} + \cdots + \frac{(p-2)!(\kappa-p+1)}{(p-1)!(\kappa-p+2)^2} \Bigg) \,, \label{eq:crC2}\\
C_3 &\equiv & \frac{(p-2)!}{(p-1)!} \Bigg(\sum_{a,b \in [p-1], b=a} \bigg(\frac{1}{\kappa} + \frac{1}{\kappa-1} + \cdots + \frac{1}{\kappa-a+1}\bigg) \bigg(\frac{1}{\kappa} + \frac{1}{\kappa-1} + \cdots + \frac{1}{\kappa-b+1}\bigg) \Bigg) \,, \label{eq:crC3}\end{aligned}$$ such that, $$\begin{aligned}
\label{eq:cr10}
\frac{\partial^2\P(\theta)}{\partial\theta_i^2}\bigg|_{\theta = {\boldsymbol{0}}} &=& \I_{\{ \Omega^{-1}(i) > p \}}A_1\Big((-A_2)(-A_2) - C_1 \Big) + \I_{\{ \Omega^{-1}(i) = p \}}A_1\Big((1-A_2) - A_2(1-A_2) - C_1 \Big) \nonumber\\
&& + \, \I_{\{ \Omega^{-1}(i) < p \}}A_1 \Big((1-A_3) - A_3 - C_2 + C_3 \Big)\,.\end{aligned}$$ The claims is easy to verify by combining Equations and with .
Proof of Theorem \[thm:topl\_upperbound\] {#sec:proof_topl_upperbound}
-----------------------------------------
The proof is analogous to the proof of Theorem \[thm:main\]. It differs primarily in the lower bound that is achieved for the second smallest eigenvalue of the Hessian matrix $H(\theta)$, .
\[lem:hessian\_topl\] Under the hypotheses of Theorem \[thm:topl\_upperbound\], if $\sum_{j = 1}^n \ell_j \geq (2^{12}e^{6b}/\beta\alpha^2) d\log d$ then with probability at least $ 1- d^{-3}$, $$\begin{aligned}
\label{eq:lambda2_bound_topl}
\lambda_2(-H(\theta)) \;\geq\; \frac{\alpha}{2(1+ e^{2b})^2} \frac{1}{d-1} \sum_{j = 1}^n \ell_j\,. \end{aligned}$$
Using Lemma \[lem:gradient\_topl\] that is derived for the general value of $\lambda_{j,a}$ and $p_{j,a}$, and by substituting $\lambda_{j,a} = 1/(\kappa_j-1)$ and $p_{j,a} = a$ for each $j \in [n]$, we get that with probability at least $1 - 2e^3d^{-3}$, $$\begin{aligned}
\label{eq:gradient_bound_topl}
\|\nabla\Lrb(\theta^*)\|_2 \;\leq\; \sqrt{16\log d\sum_{j=1}^n \ell_j} \;. \end{aligned}$$ Theorem \[thm:topl\_upperbound\] follows from Equations , and .
### Proof of Lemma \[lem:hessian\_topl\]
Define $M^{(j)} \in \cS^d$ as $$\begin{aligned}
\label{eq:M_j_def_topl}
M^{(j)} &=& \frac{1}{\kappa_j -1} \sum_{i<\i \in S_j} \sum_{a = 1}^{\ell_j} \I_{\{(i,\i)\; \in \; G_{j,a}\}} (e_i - e_{\i})(e_i - e_{\i})^\top,\end{aligned}$$ and let $M = \sum_{j=1}^n M^{(j)}$. Similar to the analysis carried out in the proof of Lemma \[lem:hessian\_positionl\], we have $\lambda_2(-H(\theta)) \geq \frac{e^{2b}}{(1 + e^{2b})^2} \lambda_2(M)$, when $\lambda_{j,a} = 1/(\kappa_j-1)$ is substituted in the Hessian matrix $H(\theta)$, Equation . From Weyl’s inequality we have that $$\begin{aligned}
\label{eq:topl3}
\lambda_2(M) \;\; \geq \lambda_2(\E[M]) - {\|M - \E[M]\|}\,.\end{aligned}$$ We will show in that $\lambda_2(\E[M]) \geq e^{-2b}(\alpha/(d-1))\sum_{j = 1}^n \ell_j$ and in that ${\|M - \E[M]\|} \leq 32e^{b}\sqrt{\frac{\log d}{\beta d}\sum_{j=1}^n \ell_j}$. $$\begin{aligned}
\label{eq:topl4}
\lambda_2(M) \; \geq \; \frac{\alpha e^{-2b}}{d-1} \sum_{j=1}^n \ell_j \;-\; 32e^{b} \sqrt{\frac{\log d}{\beta d}\sum_{j=1}^n \ell_j} \;\geq \; \frac{\alpha e^{-2b}}{2(d-1)} \sum_{j=1}^n \ell_j\;, \end{aligned}$$ where the last inequality follows from the assumption that $\sum_{j=1}^n \ell_j \geq (2^{12}e^{6b}/\beta\alpha^2) d\log d$. This proves the desired claim.
To prove the lower bound on $\lambda_2(\E[M])$, notice that $$\begin{aligned}
\label{eq:topl5}
\E[M] &=& \sum_{j = 1}^n \frac{1}{\kappa_j -1} \sum_{i<\i \in S_j} \E\Bigg[ \sum_{a = 1}^{\ell_j} \I_{\{(i,\i) \in G_{j,a}\}} \Big| (i,\i \in S_j) \Bigg] (e_i - e_{\i})(e_i - e_{\i})^\top \;.\end{aligned}$$ Using the fact that $p_{j,a} = a$ for each $j \in [n]$, and the definition of rank-breaking graph $G_{j,a}$, we have that $$\begin{aligned}
\label{eq:topl6}
\E\Bigg[ \sum_{a = 1}^{\ell_j} \I_{\{(i,\i) \in G_{j,a}\}} \Big| (i,\i \in S_j) \Bigg] &=& \P\Big[\I_{\{\sigma_j^{-1}(i) \leq \ell_j\}} + \I_{\{\sigma_j^{-1}(\i) \leq \ell_j\}} \geq 1 \Big| (i,\i \in S_j) \Big] \nonumber\\
& \geq & \P\Big[(\sigma^{-1}(i) \leq \ell_j \Big| (i,\i \in S_j)\Big]\,.\end{aligned}$$ The following lemma provides a lower bound on $\P[(\sigma^{-1}(i) \leq \ell_j | (i,\i \in S_j)]$.
\[lem:prob\_toplbound\] Consider a ranking $\sigma$ over a set of items $S$ of size $\kappa$. For any item $i \in S$, $$\begin{aligned}
\label{eq:prob_toplbound_eq}
\P[(\sigma^{-1}(i) \leq \ell] \geq e^{-2b}\frac{\ell}{\kappa}\;.\end{aligned}$$
Therefore, using the fact that $(e_i - e_{\i})(e_i - e_{\i})^\top$ is positive semi-definite, and Equations , and we have $$\begin{aligned}
\label{eq:topl_expec}
\E[M] &\succeq& e^{-2b} \sum_{j = 1}^n \frac{\ell_j}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top = e^{-2b} L,\end{aligned}$$ where $L$ is the Laplacian defined for the comparison graph $\H$, Definition \[def:comparison\_graph1\]. Using $\lambda_2(L) = (\alpha/(d-1))\sum_{j = 1}^n \ell_j$ from , we get the desired bound $\lambda_2(\E[M]) \geq e^{-2b}(\alpha/(d-1))\sum_{j = 1}^n \ell_j$.
For top-$\ell_j$ rank breaking, $M^{(j)}$ is also given by $$\begin{aligned}
\label{eq:topl7}
M^{(j)} = \frac{1}{\kappa_j -1}\Big((\kappa_j - \ell_j)\diag(e_{\{I_j\}}) +\ell_j \diag(e_{\{S_j\}}) - e_{\{I_j\}}e_{\{S_j\}}^\top - e_{\{S_j\}}e_{\{I_j\}}^\top + e_{\{I_j\}}e_{\{I_j\}}^\top \Big),\end{aligned}$$ where $e_{\{S_j\}},e_{\{I_j\}} \in \reals^d$ are zero-one vectors, $e_{\{S_j\}}$ has support corresponding to the set of items $S_j$ and $e_{\{I_j\}}$ has support corresponding to the random top-$\ell_j$ items in the ranking $\sigma_j$. $I_j = \{\sigma_j(1), \sigma_j(2),\cdots, \sigma_j(\ell_j)\}$ for $j \in [n]$. $(M^{(j)})^2$ is given by $$\begin{aligned}
(M^{(j)})^2 &=& \frac{1}{(\kappa_j -1)^2}\Big((\kappa_j^2 - \ell_j^2)\diag(e_{\{I_j\}}) + {\ell_j}^2\diag(e_{\{S_j\}}) - \nonumber\\
&& \hspace{5em}(\kappa_j +\ell_j)(e_{\{I_j\}}e_{\{S_j\}}^\top + e_{\{S_j\}}e_{\{I_j\}}^\top -e_{\{I_j\}}e_{\{I_j\}}^\top ) + \ell_j e_{\{S_j\}}e_{\{S_j\}}^\top \Big).\end{aligned}$$ Note that $\P[i \in I_j| i \in S_j] \leq \ell_j e^{2b}/\kappa_j$ for all $i \in S_j$. Its proof is similar to the proof of Lemma \[lem:prob\_toplbound\]. Therefore, we have $\E[\diag(e_{\{I_j\}})] \preceq \ell_j e^{2b}/\kappa_j \diag(e_{\{{\boldsymbol{1}}\}})$. To bound $\|\sum_{j =1}^n\E[(M^{(j)})^2]\|$, we use the fact that for $J \in \reals^{d\times d}, {\|J\|} \leq \max_{i \in [d]}\sum_{\i = 1}^d|J_{i\i}|$. Maximum of row sums of $\E[e_{\{I_j\}}e_{\{I_j\}}^\top]$ is upper bounded by $\max_{i \in [d]}\big\{\ell_j\P[i \in I_j| i \in S_j]\big\} \leq {\ell_j}^2 e^{2b}/\kappa_j$. Therefore using triangle inequality, we have, $$\begin{aligned}
&&\Bigg\|\sum_{j =1}^n\E\big[(M^{(j)})^2\big]\Bigg\| \nonumber\\
& \leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{1}{(\kappa_j-1)^2} \Bigg(\frac{(\kappa_j^2 - {\ell_j}^2)\ell_j e^{2b}}{\kappa_j} + {\ell_j}^2 + e^{2b}(\kappa_j+\ell_j)(2\ell_j + {\ell_j}^2/\kappa_j) + \ell_j\kappa_j \Bigg)\Bigg\} \nonumber\\
&\leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j e^{2b}}{\kappa_j}\Bigg(\frac{(\kappa_j^2 -{\ell_j}^2)}{(\kappa_j-1)^2} + \frac{\ell_j\kappa_j}{{(\kappa_j-1)^2}} + \frac{2(\kappa_j+\ell_j)\kappa_j}{(\kappa_j-1)^2} + \frac{(\kappa_j+\ell_j)\ell_j}{(\kappa_j-1)^2} + \frac{\kappa_j^2}{(\kappa_j-1)^2} \Bigg)\Bigg\} \nonumber\\
& \leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j e^{2b}}{\kappa_j}\Bigg(\frac{(\kappa_j^2 -1)}{(\kappa_j-1)^2} + \frac{\kappa_j(\kappa_j-1)}{{(\kappa_j-1)^2}} + \frac{4\kappa_j^2}{(\kappa_j-1)^2} + \frac{2\kappa_j(\kappa_j-1)}{(\kappa_j-1)^2} + \frac{\kappa_j^2}{(\kappa_j-1)^2} \Bigg)\Bigg\} \nonumber\\
& \leq & \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j e^{2b}}{\kappa_j}\Bigg(3 + 2 + 16 + 4 + 4 \Bigg)\Bigg\} \label{eq:topl_grad1}\\
& \leq & 29 e^{2b} \max_{i \in [d]}\bigg \{\sum_{j:i \in S_j}\frac{\ell_j}{\kappa_j}\bigg\} \nonumber\\
& = & 29 e^{2b}D_{\max} \label{eq:topl_grad2}\\
& = & \frac{29 e^{2b}}{\beta d} \sum_{j = 1}^n \ell_j\;, \label{eq:topl_grad3}\end{aligned}$$ where uses the fact that $\kappa_j \geq 2$ and $1 \leq \ell_j \leq \kappa_j -1$ for all $j \in [n]$. follows from the definition of $D_{\max}$, Definition \[def:comparison\_graph1\] and follows from the Equation . Also, note that ${\|M_j\|} \leq 2$ for all $j \in [n]$. Applying matrix Bernstien inequality, we have, $$\begin{aligned}
\mathbb{P}\Big[{\|M - \E[M]\|} \geq t\Big] \leq d \,\exp\Bigg(\frac{-t^2/2}{\frac{29e^{2b}}{\beta d}\sum_{j=1}^n\ell_j + 4t/3}\Bigg). \end{aligned}$$ Therefore, with probability at least $1 - d^{-3}$, we have, $$\begin{aligned}
\label{eq:topl_error}
{\|M - \E[M]\|} \leq 22e^{b}\sqrt{\frac{\log d}{\beta d} \sum_{j=1}^n \ell_j} +\frac{64 \log d}{3} \leq 32e^{b}\sqrt{\frac{\log d}{\beta d}\sum_{j=1}^n \ell_j} \;,\end{aligned}$$ where the second inequality follows from the assumption that $\sum_{j = 1}^n \ell_j \geq 2^{12} d\log d$ and $\beta \leq 1$.
### Proof of Lemma \[lem:prob\_toplbound\]
Define $i_{\min} \equiv \arg \min_{i \in S} \theta_i$. We claim the following. For all $i \in S$ and any $1 \leq \ell \leq |S|-1$, $$\begin{aligned}
\label{eq:prob_topl_eq}
\mathbb{P}[\sigma^{-1}(i) > \ell] \;\leq\; \mathbb{P}[\sigma^{-1}(i_{\min}) > \ell] \;\; \text{and} \;\; \mathbb{P}[\sigma^{-1}(i_{\min}) = \ell] \; \geq \; \mathbb{P}[\sigma^{-1}(i_{\min}) = 1]\,.\end{aligned}$$ Therefore $\mathbb{P}[\sigma^{-1}(i) \leq \ell] \;\geq\; \mathbb{P}[\sigma^{-1}(i_{\min}) \leq \ell]$. Using $\mathbb{P}[\sigma^{-1}(i_{\min}) = 1] > e^{-2b}/\kappa$, we get the desired bound $\mathbb{P}[\sigma^{-1}(i) \leq \ell] > e^{-2b} \ell/\kappa$.
To prove the claim , let $\widehat{\sigma}_1^\ell$ denote a ranking of top-$\ell$ items of the set $S$ and $\P[\widehat{\sigma}_1^\ell]$ be the probability of observing $\widehat{\sigma}_1^\ell$. Let ${i \in (\widehat{\sigma}_1^\ell)^{-1}}$ denote that $i = (\widehat{\sigma}_1^\ell)^{-1}(j)$ for some $1 \leq j \leq \ell$. Let $$\begin{aligned}
\Omega_1 = \Big\{ \widehat{\sigma}_1^\ell : {i \notin (\widehat{\sigma}_1^\ell)^{-1}}, {i_{\min} \in (\widehat{\sigma}_1^\ell)^{-1}} \Big\} \;\; \text{and} \;\; \Omega_2 = \Big\{ \widehat{\sigma}_1^\ell : {i \in (\widehat{\sigma}_1^\ell)^{-1}}, {i_{\min} \notin (\widehat{\sigma}_1^\ell)^{-1}} \Big\}.\end{aligned}$$ We have $\mathbb{P}[\sigma^{-1}(i) > \ell] - \mathbb{P}[\sigma^{-1}(i_{\min}) > \ell] = \sum_{\widehat{\sigma}_1^\ell \in \Omega_1}\P[\widehat{\sigma}_1^\ell] - \sum_{\widehat{\sigma}_1^\ell \in \Omega_2} \P[\widehat{\sigma}_1^\ell].$ Now, take any ranking $\widehat{\sigma}_1^\ell \in \Omega_1$ and construct another ranking $\widetilde{\sigma}_1^\ell$ from $\widehat{\sigma}_1^\ell$ by replacing $i_{\min}$ with $i$-th item. Observe that $ \P[\widehat{\sigma}_1^\ell] \leq \P[\widetilde{\sigma}_1^\ell]$ and $\widetilde{\sigma}_1^\ell \in \Omega_2$. Moreover, such a construction gives a bijective mapping between $\Omega_1$ and $\Omega_2$. Hence, the first claim is proved. For the second claim, let $$\begin{aligned}
\widehat{\Omega}_1 = \Big\{ \widehat{\sigma}_1^\ell : {(\widehat{\sigma}_1^\ell)^{-1}(i_{\min}) = 1} \Big\} \;\; \text{and} \;\; \widehat{\Omega}_2 = \Big\{ \widehat{\sigma}_1^\ell : {(\widehat{\sigma}_1^\ell)^{-1}(i_{\min}) = \ell} \Big\}.\end{aligned}$$ We have $\mathbb{P}[\sigma^{-1}(i_{\min}) =1] - \mathbb{P}[\sigma^{-1}(i_{\min}) = \ell] =\sum_{\widehat{\sigma}_1^\ell \in \widehat{\Omega}_1}\P[\widehat{\sigma}_1^\ell]- \sum_{\widehat{\sigma}_1^\ell \in \widehat{\Omega}_2} \P[\widehat{\sigma}_1^\ell].$ Now, take any ranking $\widehat{\sigma}_1^\ell \in \widehat{\Omega}_1$ and construct another ranking $\widetilde{\sigma}_1^\ell$ from $\widehat{\sigma}_1^\ell$ by swapping items at $1$st position and $\ell$-th position. Observe that $ \P[\widehat{\sigma}_1^\ell] \leq \P[\widetilde{\sigma}_1^\ell]$ and $\widetilde{\sigma}_1^\ell \in \widehat{\Omega}_2$. Moreover, such a construction gives a bijective mapping between $\widehat{\Omega}_1$ and $\widehat{\Omega}_2$. Hence, the claim is proved.
Proof of Theorem \[thm:cramer\_rao\_topl\] {#sec:proof_cramer_rao_topl}
------------------------------------------
The first order partial derivative of $\L(\theta)$, Equation , is given by $$\begin{aligned}
&&\nabla_i\L(\theta) \nonumber\\
&=&\sum_{j:i\in S_j} \sum_{m =1}^{\ell_j} \I_{\{\sigma_j^{-1}(i) \geq m \}} \Big[ \I_{\{\sigma_j(m) = i\}} - \frac{\exp(\theta_i)}{\exp(\theta_{\sigma_j(m)})+\exp(\theta_{\sigma_j(m+1)})+ \cdots + \exp(\theta_{\sigma_j(\kappa_j)})} \Big], \; \forall i \in [d]\end{aligned}$$ and the Hessian matrix $H(\theta) \in \mathcal{S}^d$ with $H_{i\i}(\theta) = \frac{\partial^2\L(\theta)}{\partial\theta_i \partial \theta_{\i}}$ is given by $$\begin{aligned}
H(\theta) = - \sum_{j = 1}^n \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \sum_{m = 1}^{\ell_j} \frac{\exp(\theta_i+ \theta_{\i})\I_{\{\sigma_j^{-1}(i),\sigma_j^{-1}(\i) \geq m\}}}{[\exp(\theta_{\sigma_j(m)})+\exp(\theta_{\sigma_j(m+1)})+ \cdots + \exp(\theta_{\sigma_j(\kappa_j)})]^2}.\end{aligned}$$ It follows from the definition that $-H(\theta)$ is positive semi-definite for any $\theta \in \reals^n$.
The Fisher information matrix is defined as $I(\theta) = -\E_\theta[H(\theta)]$ and given by $$\begin{aligned}
I(\theta) = \sum_{j = 1}^n \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \sum_{m = 1}^{\ell_j} \E\Bigg[ \frac{\I_{\{\sigma_j^{-1}(i),\sigma_j^{-1}(\i) \geq m\}}}{[\exp(\theta_{\sigma_j(m)})+ \cdots + \exp(\theta_{\sigma_j(\kappa_j)})]^2}\Bigg]\exp(\theta_i+ \theta_{\i}).\end{aligned}$$ Since $-H(\theta)$ is positive semi-definite, it follows that $I(\theta)$ is positive semi-definite. Moreover, $\lambda_1(I(\theta))$ is zero and the corresponding eigenvector is the all-ones vector. Fix any unbiased estimator $\widehat{\theta}$ of $\theta \in \Omega_b$. Since, $\widehat{\theta} \in \mathcal{U}$, $\widehat{\theta} - \theta$ is orthogonal to ${\boldsymbol{1}}$. The Cramér-Rao lower bound then implies that ${\E[{\|\widehat{\theta} - \theta^*\|}^2] \geq \sum_{i = 2}^d \frac{1}{\lambda_i(I(\theta))}}$. Taking the supremum over both sides gives $$\begin{aligned}
\sup_{\theta}\E[{\|\widehat{\theta} - \theta\|}^2] \geq \sup_{\theta} \sum_{i=2}^d \frac{1}{\lambda_i(I(\theta))} \geq \sum_{i = 2}^d \frac{1}{\lambda_i(I({\boldsymbol{0}}))}\;.\end{aligned}$$ If $\theta$ equals the all-zero vector, then $$\begin{aligned}
\P_\theta[\sigma_j^{-1}(i),\sigma_j^{-1}(\i) \geq m] = \frac{{\kappa_j-m+1 \choose 2}}{{\kappa_j \choose 2}} = \frac{(\kappa_j - m +1)(\kappa_j - m)}{\kappa_j(\kappa_j - 1)}.\end{aligned}$$ It follows from the definition that $$\begin{aligned}
I(0) &=& \sum_{j = 1}^n \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \sum_{m = 1}^{\ell_j} \frac{(\kappa_j - m)}{\kappa_j(\kappa_j - 1)(\kappa_j - m +1)} \\
&\preceq& \ell\Big(1 - \frac{1}{\ell_j}\sum_{m= 1}^{\ell_j} \frac{1}{\kappa_{\max} - m +1}\Big) \underbrace{\sum_{j = 1}^n \frac{1}{\kappa_j(\kappa_j - 1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top}_{=L}\;,\end{aligned}$$ where $L$ is the Laplacian defined for the comparison graph $\H$, Definition \[def:comparison\_graph1\]. By Jensen’s inequality, we have $$\begin{aligned}
\sum_{i = 2}^d \frac{1}{\lambda_i(L)} \geq \frac{(d-1)^2}{\sum_{i = 2}^d \lambda_i(L)} = \frac{(d-1)^2}{\Tr(L)} = \frac{(d-1)^2}{n}.\end{aligned}$$
Proof of Theorem \[thm:bottoml\_upperbound\]
--------------------------------------------
We prove a slightly more general result that implies the desired theorem. For $\ell\geq 4$, we can choose $\beta_1=1/2$. Then, the condition that $\gamma_{\beta_1}\leq1$ implies $\ld\leq (\ell/2+1)(d-2)/(\kappa-2)$, which implies $\ld \leq \ell d / (2 \kappa)$. With the choice of $\ld = \ell d / (2 \kappa) $, this implies Theorem \[thm:bottoml\_upperbound\].
\[thm:bottoml\_upperbound\_general\] Under the bottom-$\ell$ separators scenario and the PL model, $n$ partial orderings are sampled over $d$ items parametrized by $\theta^* \in \Omega_b$. For any $\beta_1$ with $ 0 \leq \beta_1 \leq \frac{\ell-2}{\ell}$, define $$\begin{aligned}
\label{eq:bottoml_2_genreal}
\gamma_{\beta_1} \;\; \equiv \;\; \frac{\ld(\kappa-2)}{({\left \lfloor{\ell\beta_1} \right \rfloor}+1)(d-2)}, \;\;
\end{aligned}$$ and for $\gamma_{\beta_1}\leq1$, $$\begin{aligned}
\chi_{\beta_1} & \equiv& \big(1-{\left \lfloor{\ell\beta_1} \right \rfloor}/\ell\big)^2\Bigg(1 - \exp\bigg(-\frac{({\left \lfloor{\ell\beta_1} \right \rfloor}+1)^2(1-\gamma_{\beta_1})^2}{2(\kappa-2)}\bigg)\Bigg) \;.
\end{aligned}$$ If $$\begin{aligned}
\label{eq:bottoml_1_general}
n\ell \;\; \geq \;\; \bigg(\frac{2^{12}e^{8b}}{\chi_{\beta_1}^2}\frac{d^2}{{\ld}^2}\frac{\kappa}{\ell}\bigg) d\log d\;, \;\;
\end{aligned}$$ then the [*rank-breaking*]{} estimator in achieves $$\begin{aligned}
\label{eq:bottoml_3}
\frac{1}{\sqrt{\ld}}\big\|\widehat{\ltheta} - \ltheta^*\big\|_2 \; \leq \; \frac{32\sqrt{2}(1+ e^{4b})^2}{\chi_{\beta_1}}\frac{d^{3/2}}{{\ld}^{3/2}}\sqrt{\frac{d\log d}{n\ell} }\;,
\end{aligned}$$ with probability at least $1 - 3e^3d^{-3}$.
Proof is very similar to the proof of Theorem \[thm:main\]. It mainly differs in the lower bound that is achieved for the second smallest eigenvalue of the Hessian matrix $H(\ltheta)$ of $\Lrb(\ltheta)$, Equation . Equation can be rewritten as $$\begin{aligned}
\label{eq:likelihhod_bl}
\Lrb(\ltheta) = \sum_{j=1}^n \sum_{a = 1}^{\ell} \sum_{\substack{i <\i \in S_j \\ : i,\i \in [\ld]}} \I_ {\big\{(i,\i) \in G_{j,a}\big\}} \lambda_{j,a} \Big(\ltheta_i\I_{\big\{\sigma_j^{-1}(i) < \sigma_j^{-1}(\i)\big\}} + \ltheta_{\i}\I_{\big\{\sigma_j^{-1}(i) > \sigma_j^{-1}(\i)\big\}} - \log \Big(e^{\ltheta_i} + e^{\ltheta_{\i}}\Big) \Big)\;,\end{aligned}$$ where $(i,\i) \in G_{j,a}$ implies either $(i,\i)$ or $(\i,i)$ belong to $E_{j,a}$. The Hessian matrix $H(\ltheta) \in \cS^{\ld}$ with $H_{i\i}(\ltheta) = \frac{\partial^2 \Lrb(\ltheta)}{\partial\ltheta_i \partial\ltheta_{\i}}$ is given by $$\begin{aligned}
\label{eq:limited_hessian}
H(\ltheta) = -\sum_{j=1}^n \sum_{a = 1}^{\ell} \sum_{\substack{i<\i \in S_j :\\ i,\i \in [\ld]}} \I_{\big\{(i,\i) \in G_{j,a}\big\}}\Bigg( (\le_i - \le_{\i})(\le_i - \le_{\i})^\top \frac{\exp(\ltheta_i + \ltheta_{\i})}{[\exp(\ltheta_i) + \exp(\ltheta_{\i})]^2}\Bigg).\end{aligned}$$
The following lemma gives a lower bound for $\lambda_2(-H(\ltheta))$.
\[lem:hessian\_bottoml\] Under the hypothesis of Theorem \[thm:bottoml\_upperbound\_general\], with probability at least $1 - d^{-3}$, $$\begin{aligned}
\label{eq:lambda2_bound_bottoml}
\lambda_2(-H(\ltheta)) \geq \frac{\chi_{\beta_1}}{8(1+ e^{4b})^2} \frac{n\ld\ell^2}{d^2}\;.\end{aligned}$$
Observe that although $\ltheta^* \in \reals^{\ld}$, Lemma \[lem:gradient\_topl\] can be directly applied to upper bound ${\|\nabla\Lrb(\ltheta^*)\|}_2$. It might be possible to tighten the upper bound, given that $\ld \leq d$. However, for $\ell \ll \kappa$, for the smallest preference score item, $i_{\min} \equiv \arg \min_{i \in [d]} \ltheta^*_i$, the upper bound $\P[\sigma^{-1}(i_{\min}) > \kappa-\ell] \leq 1$ is tight upto constant factor (Lemma \[lem:posl\_upperbound\]). Substituting $\lambda_{j,a} = 1$ and $p_{j,a} = \kappa - \ell +a$ for each $j \in [n]$, $a \in [\ell]$, in Lemma \[lem:gradient\_topl\], we have that with probability at least $1 - 2e^3 d^{-3}$, $$\begin{aligned}
\label{eq:gradient_bound_bottoml}
{\|\nabla\Lrb(\ltheta^*)\|}_2 \;\; \leq \;\; (\ell-1)\sqrt{8n\ell\log d}.\end{aligned}$$ Theorem \[thm:bottoml\_upperbound\_general\] follows from Equations , and .
### Proof of Lemma \[lem:hessian\_bottoml\]
Define $\lM^{(j)} \in \cS^{\ld}$, $$\begin{aligned}
\label{eq:limited_M_j_def}
\lM^{(j)} = \sum_{\substack{i<\i \in S_j : i,\i \in [\ld]}} \sum_{a = 1}^\ell \I_{\{(i,\i) \in G_{j,a}\}} (\le_i - \le_{\i})(\le_i - \le_{\i})^\top,\end{aligned}$$ and let $\lM = \sum_{j = 1}^n \lM^{(j)}$. Similar to the analysis in Lemma \[lem:hessian\_positionl\], we have $\lambda_2(-H(\ltheta)) \geq \frac{e^{4b}}{(1+ e^{4b})^2} \lambda_2(\lM)$. Note that we have $e^{4b}$ instead of $e^{2b}$ as $\ltheta \in \lOmega_{2b}$. We will show a lower bound on $\lambda_2(\E[\lM])$ in and an upper bound on ${\|\lM - \E[\lM]\|}$ in . Therefore using $\lambda_2(\lM) \geq \lambda_2(\E[\lM]) - {\|\lM - \E[\lM]\|}$, $$\begin{aligned}
\label{eq:bottoml_lambda2_M}
\lambda_2(\lM) \; \geq\; \frac{e^{-4b}}{4}\underbrace{(1-\beta_1)^2\Bigg(1 - \exp\bigg(-\frac{({\left \lfloor{\ell\beta_1} \right \rfloor}+1)^2(1-\gamma_{\beta_1})^2}{2(\kappa-2)}\bigg)\Bigg)}_{ \equiv \chi_{\beta_1}}\frac{n\ld\ell^2}{d^2} - 8\ell\sqrt{\frac{n\kappa\log d}{d}} \;.\end{aligned}$$ The desired claim follows from the assumption that $n\ell \geq \big( \frac{2^{12}e^{8b}}{\chi_{\beta_1}^2}\frac{d^2}{{\ld}^2}\frac{\kappa}{\ell} \big) d\log d$, where $\chi_{\beta_1}$ is defined in . To prove the lower bound on $\lambda_2(\E[\lM])$, notice that $$\begin{aligned}
\label{eq:bottoml_hess2}
\E\big[\lM\big] &=& \sum_{j = 1}^n \sum_{i<\i \in [\ld]} \E\Bigg[ \sum_{a = 1}^{\ell} \I_{\big\{(i,\i) \in G_{j,a}\big\}} \Big| (i,\i \in S_j) \Bigg] \P\Big[i,\i \in S_j\Big] (\le_i - \le_{\i})(\le_i - \le_{\i})^\top \;. \end{aligned}$$ Since the sets $S_j$ are chosen uniformly at random, $\P[i,\i \in S_j] = \kappa(\kappa -1)/d(d-1)$. Using the fact that $p_{j,a} = \kappa - \ell +a$ for each $j \in [n]$, and the definition of rank breaking graph $G_{j,a}$, we have that $$\begin{aligned}
\label{eq:bottoml_hess3}
\E\Bigg[ \sum_{a = 1}^{\ell} \I_{\big\{(i,\i) \in G_{j,a}\big\}} \Big| (i,\i \in S_j) \Bigg] = \P\Big[\big(\sigma_j^{-1}(i), \sigma_j^{-1}(\i) > \kappa - \ell\big) \Big| (i,\i \in S_j) \Big]\;.\end{aligned}$$ The following lemma provides a lower bound on $\P[(\sigma_j^{-1}(i),\sigma_j^{-1}(\i)) > \kappa - \ell | (i,\i \in S_j)]$.
\[lem:prob\_bottomlbound\] Under the hypotheses of Theorem \[thm:bottoml\_upperbound\_general\], for any two items $i,\i \in [\ld]$, the following holds: $$\begin{aligned}
\label{eq:prob_bottomlbound_eq}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \;\Big|\; i,\i \in S\Big] \;\geq\;
\frac{e^{-4b}(1-\beta_1)^2(1 - \exp({-\eta_{\beta_1}(1-\gamma_{\beta_1})^2}))}{2}\frac{\ell^2}{\kappa^2}\;,\end{aligned}$$ where $\gamma_{\beta_1} \equiv \ld (\kappa-2)/(\lfloor \ell\beta_1 \rfloor+1 )(d-2) $ and $\eta_{\beta_1} \equiv (\lfloor \ell \beta_1 \rfloor +1)^2/2(\kappa-2)$.
Therefore, using Equations , and we have, $$\begin{aligned}
\label{eq:bottoml_expec}
\E\big[\lM\big] &\succeq& \frac{e^{-4b}(1-\beta_1)^2(1 - \exp({-\eta_{\beta_1}(1-\gamma_{\beta_1})^2}))}{2} \frac{\ell^2}{\kappa^2} \frac{\kappa(\kappa-1)}{d(d-1)} \sum_{j = 1}^n \sum_{i<\i \in [\ld]} (\le_i - \le_{\i})(\le_i - \le_{\i})^\top\;.\end{aligned}$$ Define $\lL = \sum_{j = 1}^n \sum_{i<\i \in [\ld]} (\le_i - \le_{\i})(\le_i - \le_{\i})^\top$. We have, $\lambda_1(\lL) = 0$ and $\lambda_2(\lL)=\lambda_3(\lL)=\cdots=\lambda_{\ld}(\lL)$. Therefore, using $\lambda_2(\lL) = \Tr(\lL)/(\ld-1) = n\ld$. Using the fact that $\E[\lM]$ and $\lL$ are symmetric matrices, we have, $$\begin{aligned}
\label{eq:lambda2_bottoml_expec}
\lambda_2(\E\big[\lM\big]) \geq \frac{e^{-4b}(1-\beta_1)^2(1 - \exp({-\eta_{\beta_1}(1-\gamma_{\beta_1})^2}))}{4} \frac{n\ld\ell^2}{d^2}. \end{aligned}$$
To get an upper bound on ${\|\lM - \E[\lM]\|}$, notice that $\lM^{(j)}$ is also given by, $$\begin{aligned}
\label{eq:bottoml_hess4}
\lM^{(j)} \;\; =\;\; \ell\, \diag(\le_{\{I_j\}}) - \le_{\{I_j\}}\le_{\{I_j\}}^\top\;,\end{aligned}$$ where $\le_{\{I_j\}} \in \reals^{\ld}$ is a zero-one vector, with support corresponding to the bottom-$\ell$ subset of items in the ranking $\sigma_j$. $I_j = \{\sigma_j(\kappa-\ell+1),\cdots, \sigma_j(\kappa)\}$ for $j \in [n]$. $(\lM^{(j)})^2$ is given by $$\begin{aligned}
\label{eq:bottoml_hess5}
(\lM^{(j)})^2 \;\; =\;\; \ell^2 \,\diag(\le_{\{I_j\}}) - \ell\, \le_{\{I_j\}}\le_{\{I_j\}}^\top\;.\end{aligned}$$ Using the fact that sets $\{S_j\}_{j \in [n]}$ are chosen uniformly at random and $\P[i \in \I_j | i \in S_j] \leq 1$, we have $\E[\diag(\le_{\{I_j\}})] \preceq (\kappa/d) \diag(\le_{\{{\boldsymbol{1}}\}})$. Maximum of row sums of $\E\big[\le_{\{I_j\}}\le_{\{I_j\}}^\top\big]$ is upper bounded by $\ell\kappa/d$. Therefore, from triangle inequality we have ${\|\sum_{j=1}^n \E[(\lM^{(j)})^2]\|} \leq 2n\ell^2\kappa/d$. Also, note that ${\|\lM^{(j)}\|} \leq 2\ell$ for all $j \in [n]$. Applying matrix Bernstien inequality, we have that $$\begin{aligned}
\label{eq:bottoml_hess6}
\mathbb{P}\Big[{\|\lM - \E[\lM]\|} \geq t\Big] \leq d\,\exp\Big(\frac{-t^2/2}{2n\ell^2\kappa/d + 4\ell t/3}\Big). \end{aligned}$$ Therefore, with probability at least $1 - d^{-3}$, we have, $$\begin{aligned}
\label{eq:bottoml_error}
{\|\lM - \E[\lM]\|} \leq 4\ell\sqrt{\frac{2n\kappa\log d}{d}} + \frac{64\ell\log d}{3} \leq 8\ell\sqrt{\frac{n\kappa\log d}{d}}\;,\end{aligned}$$ where the second inequality follows from the assumption that $n\ell \geq 2^{12}d\log d$.
### Proof of Lemma \[lem:prob\_bottomlbound\]
Without loss of generality, assume that $\i < i$, i.e., $\ltheta^*_{\i} \leq \ltheta^*_i$. Define $\Omega$ such that $\Omega = \{j: j\in S, j \neq i,\i\}$. For any $\beta_1 \in [0,(\ell-2)/\ell]$, define event $E_{\beta_1}$ that occurs if in the randomly chosen set $S$ there are at most ${\left \lfloor{\ell\beta_1} \right \rfloor}$ items that have preference scores less than $\ltheta^*_i$, i.e., $$\begin{aligned}
\label{eq:bl_prob_7}
E_{\beta_1} \; \equiv \; \Big\{\textstyle\sum_{j \in \Omega} \I_{\{\ltheta^*_i > \ltheta^*_j\}} \leq {\left \lfloor{\ell\beta_1} \right \rfloor} \Big\} \;.\end{aligned}$$ We have, $$\begin{aligned}
\label{eq:bl_prob_1}
&&\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) \;>\; \kappa - \ell \;\Big|\; i,\i \in S\Big] \nonumber\\
&>& \P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \;\Big|\; i,\i \in S; E_{\beta_1} \Big] \P\Big[E_{\beta_1} \;\Big|\; i,\i \in S\Big]\end{aligned}$$ The following lemma provides a lower bound on $\P[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \;|\; i,\i \in S; E_{\beta_1}]$.
\[lem:bl\_prob1\] Under the hypotheses of Lemma \[lem:prob\_bottomlbound\], $$\begin{aligned}
\label{eq:bl_prob2}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) \; > \; \kappa - \ell \;\Big|\; i,\i \in S; E_{\beta_1} \Big] \;\geq\; \frac{e^{-4b}(1-{\left \lfloor{\ell\beta_1} \right \rfloor}/\ell)^2}{2}\frac{\ell^2}{\kappa^2}\;.\end{aligned}$$
Next, we provide a lower bound on $\P[E_{\beta_1} \;|\;i,\i \in S]$. Fix $i,\i$ such that $i,\i \in S$. Selecting a set uniformly at random is probabilistically equivalent to selecting items one at a time uniformly at random without replacement. Without loss of generality, assume that $i,\i$ are the $1$st and $2$nd pick. Define Bernoulli random variables $Y_{\j}$ for $ 3 \leq \j \leq \kappa$ corresponding to the outcome of the $\j$-th random pick from the set of $(d-\j-1)$ items to generate the set $\Omega$ such that $Y_{\j} = 1$ if and only if $\ltheta^*_{i} > \ltheta^*_{\j}$.
Recall that $\gamma_{\beta_1} \equiv \ld (\kappa-2)/(\lfloor \ell\beta_1 \rfloor+1 )(d-2) $ and $\eta_{\beta_1} \equiv (\lfloor \ell \beta_1 \rfloor +1)^2/2(\kappa-2)$. Construct Doob’s martingale $(Z_2,\cdots,Z_{\kappa})$ from $\{Y_{\k}\}_{ 3 \leq \k \leq \kappa}$ such that $Z_{\j} = \E[\sum_{\k=3}^{\kappa} Y_{\k}\;|\;Y_3,\cdots,Y_{\j}]$, for $2 \leq \j\leq \kappa$. Observe that, $Z_2 = \E[\sum_{\k=3}^{\kappa} Y_{\k}] \leq \frac{(i-2)(\kappa-2)}{d-2} \leq \gamma_{\beta_1}({\left \lfloor{\ell\beta_1} \right \rfloor}+1)$, where the last inequality follows from the assumption that $i \leq \ld $. Also, $|Z_{\j} - Z_{\j-1}| \leq 1$ for each $\j$. Therefore, we have $$\begin{aligned}
\label{eq:bl_prob_3}
\P\Big[\textstyle\sum_{j \in \Omega} \I_{\{\ltheta^*_{i} > \ltheta^*_j\}} \leq {\left \lfloor{\ell\beta_1} \right \rfloor} \Big] & = &\P\Big[ \textstyle\sum_{\j=3}^{\kappa} Y_{\j} \leq {\left \lfloor{\ell\beta_1} \right \rfloor} \Big]\nonumber\\
& = & 1 - \P\Big[ \textstyle\sum_{\j=3}^{\kappa} Y_{\j} \geq {\left \lfloor{\ell\beta_1} \right \rfloor} +1 \Big] \nonumber\\
& \geq & 1 - \P\Big[Z_{\kappa-2} - Z_2 \geq (\ell\beta_1 +1) - \gamma({\left \lfloor{\ell\beta_1} \right \rfloor}+1) \Big] \nonumber\\
& \geq & 1 - \exp\Big(-\frac{({\left \lfloor{\ell\beta_1} \right \rfloor}+1)^2(1-\gamma_1)^2}{2(\kappa-2)} \Big) \nonumber\\
& = & 1 - \exp\Big(-\eta_{\beta_1}(1-\gamma_{\beta_1})^2\Big),\end{aligned}$$ where the inequality follows from the Azuma-Hoeffding bound. Since, the above inequality is true for any fixed $i,\i \in S$, for random indices $i,\i$ we have $ \P[E_{\beta_1} \;|\; i,\i \in S] \geq 1 - \exp(-\eta_{\beta_1}(1-\gamma_{\beta_1})^2)$. Claim follows by combining Equations , and .
### Proof of Lemma \[lem:bl\_prob1\]
Without loss of generality, assume that $\i < i$, i.e., $\ltheta^*_{\i} \leq \ltheta^*_i$. Define $\Omega = \{j: j\in S, j \neq i,\i\}$, and event $E_{\beta_1} = \{ i,\i \in S; \textstyle\sum_{j \in \Omega} \I_{\{\ltheta^*_i > \ltheta^*_j\}} \leq {\left \lfloor{\ell\beta_1} \right \rfloor} \}$. Since set $S$ is chosen randomly, $i,\i$ and $j \in \Omega$ are random. Throughout this section, we condition on the random indices $i,\i$ and the set $\Omega$ such that event $E_{\beta_1}$ holds. To get a lower bound on $\P[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell ]$, define independent exponential random variables $X_j \sim \exp(e^{\ltheta^*_j})$ for $j \in S$. Observe that given event $E_{\beta_1}$ holds, there exists a set $\Omega_1 \subseteq \Omega$ such that $$\begin{aligned}
\label{bl_prob_8}
\Omega_1 = \Big\{j\in S :\ltheta^*_i \leq \ltheta^*_j \Big\}\;,\end{aligned}$$ and $|\Omega_1| = \kappa-{\left \lfloor{\ell\beta_1} \right \rfloor} -2$. In fact there can be many such sets, and for the purpose of the proof we can choose one such set arbitrarily. Note that ${\left \lfloor{\ell\beta_1} \right \rfloor} +2 \leq \ell$ by assumption on $\beta_1$, so $|\Omega_1| \geq \kappa-\ell$. From the Random Utility Model (RUM) interpretation of the PL model, we know that the PL model is equivalent to ordering the items as per [*random cost*]{} of each item drawn from exponential random variable with mean $e^{\tilde\theta^*_i}$. That is, we rank items according to $X_j$’s such that the lower cost items are ranked higher. From this interpretation, we have that $$\begin{aligned}
\label{eq:bl_prob_4}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \Big] &=& \P\Big[ \sum_{j \in \Omega} \I_{\big\{\min\{X_i, X_{\i}\}\; > \;X_j\big\}} \geq \kappa-\ell \Big] \nonumber\\
&>& \P\Big[ \sum_{\j \in \Omega_1} \I_{\big\{\min\{X_i, X_{\i}\} \;>\; X_{\j}\big\}} \geq \kappa-\ell \Big] \end{aligned}$$ The above inequality follows from the fact that $\Omega_1 \subseteq \Omega$ and $|\Omega_1| \geq \kappa -\ell$. It excludes some of the rankings over the items of the set $S$ that constitute the event $\{\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \}$. Define $\Omega_2 = \{\Omega_1,i,\i\}$. Observe that items $i,\i$ have the least preference scores among all the items in the set $\Omega_2$. Therefore, the term in Equation is the probability of the least two preference score items in the set $\Omega_2$, that is of size $(\kappa-{\left \lfloor{\ell\beta_1} \right \rfloor})$, being ranked in bottom $(\ell-{\left \lfloor{\ell\beta_1} \right \rfloor})$ positions.
The following lemma shows that the probability of the least two preference score items in a set being ranked at any two positions is lower bounded by their probability of being ranked at $1$st and $2$nd position.
\[lem:bl\_prob2\] Consider a set of items $S$ and a ranking $\sigma$ over it. Define $i_{\min_1} \equiv \arg \min_{i \in S} \theta_i$, $i_{\min_2} \equiv \arg \min_{i \in S\setminus i_{min_1}} \theta_i$. For all $ 1 \leq i_1, i_2 \leq |S|$, $i_1 \neq i_2$, following holds: $$\begin{aligned}
\P\Big[ \sigma^{-1}(i_{\min_1}) = i_1, \sigma^{-1}(i_{\min_2}) = i_2 \Big] \geq \P\Big[ \sigma^{-1}(i_{\min_1}) = 1, \sigma^{-1}(i_{\min_2}) = 2 \Big]. \end{aligned}$$
Using the fact that $\i = \arg \min_{j \in \Omega_2} \ltheta^*_j $, $i = \arg \min_{j \in \Omega_2 \setminus \i} \ltheta^*_j$, for all $1 \leq i_1, i_2 \leq \kappa-{\left \lfloor{\ell\beta_1} \right \rfloor}$, $i_1 \neq i_2$, we have that $$\begin{aligned}
\label{eq:bl_prob_5}
\P\Big[ \sigma^{-1}(\i) = i_1, \sigma^{-1}(i) = i_2 \Big] \geq \P\Big[ \sigma^{-1}(\i) = 1, \sigma^{-1}(i) = 2 \Big] \geq e^{-4b}\frac{1}{\kappa^2}\;, \end{aligned}$$ where the second inequality follows from the definition of the PL model and the fact that $\ltheta^* \in \lOmega_{2b}$. Together with Equation and the fact that there are a total of $(\ell-{\left \lfloor{\ell\beta} \right \rfloor})(\ell-{\left \lfloor{\ell\beta} \right \rfloor}-1) \geq (\ell-{\left \lfloor{\ell\beta} \right \rfloor})^2/2$ pair of positions that $i,\i$ can occupy in order to being ranked in bottom $(\ell-{\left \lfloor{\ell\beta} \right \rfloor})$, we have, $$\begin{aligned}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \Big] \geq \frac{e^{-4b}(1-{\left \lfloor{\ell\beta_1} \right \rfloor}/\ell)^2}{2}\frac{\ell^2}{\kappa^2}.\end{aligned}$$ Since, the above inequality is true for any fixed $i,\i$ and $j \in \Omega$ such that event $E$ holds, it is true for random indices $i,\i$ and $j \in \Omega$ such that event $E$ holds, hence the claim is proved.
### Proof of Lemma \[lem:bl\_prob2\]
Let $\widehat{\sigma}$ denote a ranking over the items of the set $S$ and $\P[\widehat{\sigma}]$ be the probability of observing $\widehat{\sigma}$. Let $$\begin{aligned}
\widehat{\Omega}_1 = \Big\{ \widehat{\sigma}: \widehat{\sigma}^{-1}(i_{\min_1}) = i_1, \widehat{\sigma}^{-1}(i_{\min_2}) = i_2 \Big\} \;\; \text{and} \;\; \widehat{\Omega}_2 = \Big\{ \widehat{\sigma} : \sigma^{-1}(i_{\min_1}) = 1, \sigma^{-1}(i_{\min_2}) = 2 \Big\}.\end{aligned}$$ Now, take any ranking $\widehat{\sigma} \in \widehat{\Omega}_1$ and construct another ranking $\widetilde{\sigma}$ from $\widehat{\sigma}$ as following. If $i_1 =2, i_2 = 1$, then swap the items at $i_1$-th and $i_2$-th position in ranking $\widehat{\sigma}$ to get $\widetilde{\sigma}$. Else, if $i_1 < i_2$, then first: swap items at $i_1$-th position and $1$st position, and second: swap items at $i_2$-th position and $2$nd position, to get $\widetilde{\sigma}$; if $i_2 < i_1$, then first: swap items at $i_2$-th position and $2$nd position, and second: swap items at $i_1$-th position and $1$st position, to get $\widetilde{\sigma}$.
Observe that $\P[\widetilde{\sigma}] \leq \P[\widehat{\sigma}]$ and $\widetilde{\sigma}_1^\ell \in \widehat{\Omega}_2$. Moreover, such a construction gives a bijective mapping between $\widehat{\Omega}_1$ and $\widehat{\Omega}_2$. Hence, the claim is proved.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank the anonymous reviewers for their constructive feedback. This work was partially supported by National Science Foundation Grants MES-1450848, CNS-1527754, and CCF-1553452.
| {
"pile_set_name": "ArXiv"
} |
[**About neutral mesons and particle oscillations in the light of field-theoretical prescriptions of Weinberg**]{}
L.M. Slad[^1]\
[*Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia*]{}
The postulated universality of the Weinberg’s prescriptions on the diagonalization of the mass term of the Lagrangian without increasing the total number of entities leads to the following conclusions: the set of neutral $K$-mesons consists of two elements, $K_{S}^{0}$ and $K_{L}^{0}$; the states $K^{0}$ and $\bar{K}^{0}$ do not exist as physical objects (in the form of particles or “particle mixtures”); the absence of the states $K^{0}$ and $\bar{K}^{0}$ destroys grounds for introducing the notion of their oscillations. The conclusions concerning the neutral $K$-mesons are also applicable to the neutral $D$-, $B$- and $B_{s}$-mesons.A theoretical and experimental vulnerability of the neutrino oscillation concept is noted.
The initial judgments about the family of four neutral $K$-mesons still remain almost unchanged and, furthermore, extend on the families of neutral $D$- and $B$-mesons. The concept of mutual transition of $K^{0}$- and $\bar{K}^{0}$-mesons in vacuum, originated long ago and retained up to present day, has served initially [@1] and continues to serve now [@2] as the only theoretical argument in favor of the hypothesis of neutrino oscillations by analogy with the letter.
In the present paper, we propose to put the status of neutral $K$-mesons in full accordance with field-theoretical prescriptions of Weinberg [@3] that have led to the prodigious gauge theory of electroweak interactions by making use of the diagonalization of the mass term in the Lagrangian without increasing the total number of entities. The specified prescriptions are an exact realization of general scientific principle existing for hundreds of years that “entities must not be multiplied beyond necessity”, which, having been accepted as a universal rule of field theory and particle physics, inevitably eliminates any reason for the meson oscillation.
For further comparisons, we focus on particular steps of Weinberg’s procedure [@3] on eliminating the item $cA_{\mu}^{3}(x)B^{\mu}(x)$ ($c$ is a constant) in the Lagrangian mass term, which is nondiagonal on the initial gauge fields and appears due to Higgs mechanism of spontaneous breaking of the original symmetry. This item could serve as the reason for the judgment on the possibility of a transition of one field to another in vacuum. First step: on the basis of two suitable superpositions of classical fields $A_{\mu}^{3}(x)$ and $B_{\mu}(x)$, Weinberg introduces orthonormal classical fields with definite masses $Z_{\mu}(x)$ and $A_{\mu}(x)$. Second step: Weinberg expresses the fields $A_{\mu}^{3}(x)$ and $B_{\mu}(x)$, and then all terms of the gauge Lagrangian, through the fields $Z_{\mu}(x)$ and $A_{\mu}(x)$. Third step: Weinberg gives status of quantized fields to the fields $Z_{\mu}(x)$ and $A_{\mu}(x)$ and identifies them with such particles as the intermediate meson $Z$ and the photon. First note: at any stage of constructing the gauge model, Weinberg does not connect the fields $A_{\mu}^{3}(x)$ and $B_{\mu}(x)$ with any quanta. Second note: the original gauge field $A_{\mu}^{3}(x)$ and $B_{\mu}(x)$ that serve as the cornerstone in the Weinberg’s construction disappear in the final model of electroweak interactions. Third note: the initial fields possess well-defined values of weak isospin and its third projection, but the final fields $Z_{\mu}(x)$ and $A_{\mu}(x)$ do not have such definite values, and, therefore, the corresponding terms of the electroweak interaction Lagrangian violate the isospin symmetry.
We now turn to a number of current opinions about the neutral $K$-mesons. They mainly reproduce part of the judgments stated in the works by Gell-Mann [@4] and Gell-Mann and Pais [@5] with adding some corrections for the results of the subsequent experiments concerning the violation of $CP$-symmetry.
Starting from the assumption of strict conservation of the isotopic spin in strong interactions, Gell-Mann [@4] concludes that, among the two long-lived neutral particles produced in the collision of the $\pi^{-}$meson with the proton, one particle ($K^{0}$) is a boson with the isospin 1/2 and its projection -1/2 and that there exists an antiparticle ($\bar{K}^{0}$) with the isospin projection +1/2 which corresponds to the boson $K^{0}$ and is different from it. The mentioned assumption of Gell-Mann is the key element in the subsequently formed structure of the family of neutral $K$-mesons.
Gell-Mann and Pais [@5] consider that, if there exists the decay $K^{0} \rightarrow \pi ^{+}+\pi^{-}$, then there should exist the charge-conjugate process $\bar{K}^{0} \rightarrow \pi ^{+}+\pi^{-}$, and thus, the weak interaction causes the virtual transition $K^{0} \rightleftharpoons \pi^{+}+\pi^{-} \rightleftharpoons \bar{K}^{0}$. The last judgment and the aspiration for providing the $C$-parity conservation in weak decays lead to the introduction of the quanta $K_{1}^{0}$ and $K_{2}^{0}$, which fields are expressed in the form of normalized sum and difference of the fields $K^{0}$ and $\bar{K}^{0}$, respectively. According to Gell-Mann and Pais, $K^{0}$ and $\bar{K}^{0}$ are the primary objects in production phenomena, whereas the decay process is best described in terms of $K_{1}^{0}$ and $K_{2}^{0}$. Each of the latter can be assigned a definite lifetime, that is not true to the $K^{0}$ and $\bar{K}^{0}$.
Note that a long-lived neutral boson produced, for example, in the collision of a $\pi^{-}$-meson with a proton, cannot make any experimental manifestation between the procuction moment and the decay moment and, consequently, it does not allow experimental identification in this time interval. The opinion stated in [@4] and [@5] that such a boson must have a well-defined value of the isospin and its third projection remains nothing more but an assumption. The fact that the introduced hadrons $K_{1}^{0}$ and $K_{2}^{0}$ do not possess definite values of the third projection of the isospin, does not induce Gell-Mann and Pais to reconsider their assumption of strict isotopic spin conservation in strong interactions, and this essentially prohibits the participation of these hadrons in strong interactions, causing at least a surprise.
Essential is the position of the authors of work [@5] about reserving the word “particle” for an object with a definite lifetime and recognizing the quanta $K_{1}^{0}$ and $K_{2}^{0}$ as true “particles”, and about that the $K^{0}$ and $\bar{K}^{0}$ must, strictly speaking, be considered as “particle mixtures” is presented essential. The mathematical definition of “particle mixtures” has resulted by Pais and Piccioni [@6] in introducing and describing the concept of oscillations, mutual transitions of the $K^{0}$ and $\bar{K}^{0}$-mesons in vacuum.
Let us now present a view of the neutral $K$-mesons obtained on the basis of exactly applying to them the field-theoretical prescriptions of Weinberg on the diagonalization of the mass term in the Lagrangian without increasing the total number of entities.
Following Weinberg, we shall deal initially not with particles but with suitable classical fields and assume that the initial Lagrangian of strong interactions is invariant under transformations of the isospin group $SU(2)$ (or the internal symmetry group $SU(3)$).
We introduce two neutral fields $\Phi_{+1 \frac{1}{2} -\frac{1}{2}}(x)$ and $\Phi_{-1 \frac{1}{2} +\frac{1}{2}}(x)$ that are pseudoscalar under the orthochronous Lorentz group and possess strangenesses $\pm 1$, isospin $1/2$ and its projections $\mp 1/2$ (these fields can also be considered as the components of vectors in the octet representation space of the group $SU(3)$).
We assume at the stage of preliminary analysis that, in the absence of weak interactions, these fields could describe the lower bound states of quark-antiquark systems, respectively, $d\bar{s}$ and $s\bar{d}$. In the presence of weak interactions, these two systems would virtually pass into one another due to double exchange of $W$-bosons with changing the third projection of isospin and strangeness. (Feynman diagrams corresponding to such an exchange can be found in the monograph [@7]). This means that the mass term in the Lagrangian of fields $\Phi_{\pm 1 \frac{1}{2} \mp \frac{1}{2}}(x)$, in view of the influence of weak interactions, should be presented in the following form $${\cal L} = -m_{+}^{2}\Phi_{+1\frac{1}{2} -\frac{1}{2}}^{*}(x)
\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x)
-m_{-}^{2}\Phi_{-1\frac{1}{2} +\frac{1}{2}}^{*}(x)
\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)$$ $$-a\Phi_{+1\frac{1}{2} -\frac{1}{2}}^{*}(x)\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)
-a^{*}\Phi_{-1\frac{1}{2} +\frac{1}{2}}^{*}(x)
\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x),
\label{1}$$ where $m_{+}$ and $m_{-}$ are real constants.
If the mass term of Lagrangian (\[1\]) does not change under the transformation $\Phi_{\pm 1 \frac{1}{2} \mp \frac{1}{2}}(x)\rightarrow \Phi_{\mp 1 \frac{1}{2} \pm \frac{1}{2}}(x)$ (it is possible to consider this term as possessing $SU(3)$-symmetry) then the constant $a$ is real, and the quantities $m_{+}^{2}$ and $m_{-}^{2}$ are equal. The diagonalization of the mass term leads to the fields $\Phi_{1}(x)$ and $\Phi_{2}(x)$ with certain masses given by the expressions $$\Phi_{1}(x) = \frac{1}{\sqrt{2}}[\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x)+
\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)], \quad
\Phi_{2}(x) = \frac{1}{\sqrt{2}}[\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x)-
\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)],
\label{2}$$ whose form is the same for all nonzero values of the constants $a$ and $m_{\pm}^{2}$. The definite values of the $CP$-parity of fields $\Phi_{1}(x)$ and $\Phi_{2}(x)$ are the result and not the ground for the formation of expressions (\[2\]).
Since the experiments indicate absence of definite values of $CP$-parity of the decaying neutral $K$-mesons, one should consider the case when the mass term in Lagrangian (\[1\]) does not possess the mentioned $SU(3)$-invariance. Then the first step of the appropriate procedure of Weinberg gives orthonormal fields $\Phi_{S}(x)$ and $\Phi_{L}(x)$ with definite masses $m_{S}$ and $m_{L}$ specified by the formulas $$\Phi_{S}(x) = \frac{(1-\varepsilon^{*})\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x)+
(1+\varepsilon^{*})\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)}
{\sqrt{2(1+|\varepsilon|^{2})}} =
\frac{\Phi_{1}(x)-\varepsilon^{*}\Phi_{2}(x)}{\sqrt{1+|\varepsilon|^{2}}},
\label{3}$$ $$\Phi_{L}(x) = \frac{(1+\varepsilon)\Phi_{+1\frac{1}{2} -\frac{1}{2}}(x)-
(1-\varepsilon)\Phi_{-1\frac{1}{2} +\frac{1}{2}}(x)}
{\sqrt{2(1+|\varepsilon|^{2})}} =
\frac{\varepsilon \Phi_{1}(x)+\Phi_{2}(x)}{\sqrt{1+|\varepsilon|^{2}}},
\label{4}$$ with $$|m_{L}^{2}-m_{S}^{2}| = \sqrt{(m_{+}^{2}-m_{-}^{2})^{2}+4|a|^{2}},
\label{5}$$ $$\varepsilon = \frac{m_{+}^{2}-m_{-}^{2}+2i{\rm Im} a}
{m_{L}^{2}-m_{S}^{2}-2{\rm Re} a}.
\label{6}$$
The next step of Weinberg’s procedure consists in finding expressions for the fields $\Phi_{\pm 1 \frac{1}{2} \mp \frac{1}{2}}(x)$ through the fields $\Phi_{S}(x)$ and $\Phi_{L}(x)$ on the basis of the relations(\[3\]) and (\[4\]) and in substituting these expressions into all terms in the initial Lagrangian describing both strong and weak interactions.
Then, in completing the Weinberg’s procedure, we should obtain Euler equations for each of the fields $\Phi_{S}(x)$ and $\Phi_{L}(x)$ from the transformed Lagrangian and perform the second quantization of the solutions of these equations, that would consist in identifying these solutions as vectors in the spaces of suitable irreducible representations of the Poincare group with neutral mesons $K_{S}^{0}$ and $K_{L}^{0}$. As a result, the processes which were thought to be accompanied by the production or are caused by an interaction of one of the hypothetical $K^{0}$- and $\bar{K}^{0}$-bosons, with never being accompanied with the production or caused by an interaction of other one, can now proceed within the considered concept with involving both $K_{S}^{0}$- and $K_{L}^{0}$-bosons with almost equal probabilities (though, slightly different probabilities, that is important for a number of experiments). So, the production of $\Lambda$-hyperon in a $\pi^{-}p$-collision is accompanied with either $K_{S}^{0}$- or $K_{L}^{0}$-meson, with both of the latter being able to produce $K^{-}$-meson (when interacting with neutrons) and $K^{+}$-meson (when interacting with protons).
Fields $\Phi_{\pm 1 \frac{1}{2} \mp \frac{1}{2}}(x)$ and $\Phi_{1,2}(x)$, which are involved in mathematical operations to form the fields $\Phi_{S}(x)$ and $\Phi_{L}(x)$ and, that is especially important, to form the $K_{S}^{0}$- and $K_{L}^{0}$-mesons interaction constants, are not identified with any quanta at any step of such a formation.
Thus, the postulated universality of the Weinberg’s prescriptions on the diagonalization of the mass term in the Lagrangian without increasing the total number of entities leads to the following conclusions:
\(1) The states $K^{0}$ and $\bar{K}^{0}$, as physical objects (in the form of particles or “particle mixtures”), do not exist. Accordingly, the quark-antiquark bound states $d\bar{s}$ and $s\bar{d}$ are not realized;
\(2) The set of neutral $K$-mesons consists of two elements, $K_{S}^{0}$ and $K_{L}^{0}$, which do not possess definite values of the third projection of the isospin. Accordingly, the bound states are formed only by the superpositions of quark-antiquark pairs $d\bar{s}$ and $s\bar{d}$ which are given by the expressions, obtainable in an obvious way from the formulas (\[3\]) and (\[4\]);
\(3) The antiparticles of the mesons $K_{S}^{0}$ and $K_{L}^{0}$ coincide with themselves;
\(4) In strong interactions involving neutral $K$-mesons, the isospin is not conserved;
\(5) The absence of the states $K^{0}$ and $\bar{K}^{0}$ destroys the ground for introducing the notion of their oscillations made by Pais and Piccioni [@6].
All the listed conclusions about the family of neutral $K$-mesons should, in our view, also be extended to the sets of neutral $D$-, $B$- and $B_{s}$-mesons. In particular, each of these sets consists of two elements, namely, $D^{0}_{1}$ and $D^{0}_{2}$, $B^{0}_{H}$ and $B^{0}_{L}$, $B^{0}_{sH}$ and $B^{0}_{sL}$, and the states $D^{0}$ and $\bar{D}^{0}$, $B^{0}$ and $\bar{B}^{0}$, $B^{0}_{s}$ and $\bar{B}^{0}_{s}$ do not exist.
The outlined concept of neutral mesons, precisely following the Weinberg’s prescriptions on the transition from the initial gauge fields to the observed particles, is simple, elegant and consistent both in the theoretical and experimental aspects. None of its parts can serve as an analogy or basis for the hypothesis of neutrino oscillations. The radical difference between the judgements on the family of neutral $K$-mesons and the present day dominanting judgements on the family of neutrinos start with the origin of nondiagonal mass terms in the Lagrangian of the initial fields and are strengthened by the total number of entities. Namely, such terms in the Lagrangian of neutral meson fields with isospin 1/2 are necessitated, as it was indicated above, by weak interactions, and the number of the initial and final meson fields remains the same. At the same time, the nondiagonal mass terms in the Lagrangian of the known neutrino fields are only arbitrarily introduced (see, e.g., [@8]) to justify adding new entities, the massive neutrinos $\nu_{j}$, $j=1,2,3$, as “true particles”, to justify declaring the known neutrinos $\nu_{\alpha}$, $\alpha = e, \mu, \tau$, as “particle mixtures” and to realize the oscillation scenario a la Pais and Piccioni. It is worth noting that the “true particles” $\nu_{j}$ do not directly participate in any production, scattering or absorption processes.
The redundancy of neutrino family inevitably leads to logical contradictions and uncertainties. Attempts to avoid some of the contradictions in the description of neutrino oscillations through the revision of quantum mechanics can be found, e.g., in the works [@9] and [@10].
We would now like to draw attention to only one essential logical aspect related to the concept of neutrino oscillations. Note in the beginning, that the realization of the notion of “particle mixtures”, since the times of Pais and Piccioni [@6] up to present day (see, e.g., [@2]), contains the elements both quantum and classical mechanics. With respect to massive neutrino states whose superpositions correspond to the states describing the production of any of the known neutrinos $\nu_{\alpha}$, $\alpha = e, \mu, \tau$, ones adopt the assumption (see, e.g.,[@2]), that each of them is described by its well-defined value of 4-momentum. Let us briefly elucidate the possible consequences of non-fulfilment of the specified assumptions.
It is accepted that the state vector $|\nu_{\alpha}(t_{0},{\bf r})\rangle$ of the neutrino $\nu_{\alpha}$, if exists, does not change in time, and that at the time $t_{0}$ of the neutrino production it is linearly expressed through the state vectors of all neutrinos $\nu_{j}$ $$|\nu_{\alpha}(t_{0},{\bf r})\rangle = \sum_{j=1,2,3} u_{\alpha j}
|\nu_{j}(t_{0},{\bf r})\rangle, \qquad \alpha = e, \mu, \tau,
\label{7}$$ where the coefficients $u_{\alpha j}$ do not depend on the production conditions of the neutrino $\nu_{\alpha}$. As the state vector $|\nu_{j}(t_{0},{\bf r})\rangle$ belongs to the space of irreducible representations of the Poincare group characterized by the mass $m_{j}$, its dependence on the spatial coordinates determines the distribution of the state over 3-momentum and thus specifies the evolution in time $$|\nu_{j}(t,{\bf r})\rangle = \int |\nu_{j}({\bf p})\rangle
\exp[-i\sqrt{{\bf p}^{2}+m_{j}^{2}}(t-t_{0})+i{\bf p}{\bf r}] d^{3}{\bf p}.
\label{8}$$
Let the state of an evolving in time quantum-mechanical object identified at its production moment $t_{0}$ with neutrinos $\nu_{\alpha}$ be denoted as $|Z_{\alpha}(t,{\bf r})\rangle$: $|Z_{\alpha}(t_{0},{\bf r})\rangle = |\nu_{\alpha}(t_{0},{\bf r})\rangle$. On one hand, for any time $t > t_{0}$ it is natural to describe this state by a time-dependent superposition of states presented in the right part of relation (\[7\]) $$|Z_{\alpha}(t,{\bf r})\rangle = \sum_{j=1,2,3} u_{\alpha j}
|\nu_{j}(t,{\bf r})\rangle.
\label{9}$$ On the other hand, the essence of the concept of neutrino oscillations is that the state $|Z_{\alpha}(t,{\bf r})\rangle$, $\alpha = e, \mu, \tau$, is expressed as superposition of the three states $|\nu_{\beta}(t_{0},{\bf r})\rangle$ $$|Z_{\alpha}(t,{\bf r})\rangle = \sum_{\beta = e, \mu, \tau} v_{\alpha \beta}(t)
|\nu_{\beta}(t_{0},{\bf r})\rangle,
\label{10}$$ where $v_{\alpha \beta}(t)$ are some $c$-number valued functions of time.
From relations (\[9\]), (\[8\]) and (\[7\]), we conclude that the equality (\[10\]) is feasible if and only if each of the three states $|\nu_{j}(t,{\bf r})\rangle$ (\[8\]) is characterized by its definite the 3-momentum absolute value $|{\rm p}_{j}|$, $j=1,2,3$. If such a condition concerning the distributions of the massive neutrino states over 3-momentum is not fulfilled, that is likely enough from the field-theoretical point of view, then the realization of the concept of neutrino oscillations in such a situation becomes impossible, namely, the states of $|Z_{\alpha}(t,{\bf r})\rangle$ for $t > t_{0}$ are not representable as superpositions of the neutrino states $\nu_{e}$, $\nu_{\mu}$ and $\nu_{\tau}$. Because of that, we lose the possibility to calculate the rate of any expected process involving the object $|Z_{\alpha}(t,{\bf r})\rangle$.
Undoubtedly, an indicator of acceptability of some physical hypothesis is its capacity to explain some class of experiments. If an alternative description of this class of experiments appears, then the number of new entities and arbitrary parameters of both hypotheses lies on a bowl of scales.
The most meaningful experimental application of the neutrino oscillation hypothesis is aimed at solving the solar neutrino problem. It has been initiated by the work [@8] in connection with negative results of Davis and others [@11] on detecting the transitions ${}^{37}{\rm Cl} \rightarrow {}^{37}{\rm Ar}$. Now we have results of five types of experiments with solar neutrinos: the transitions ${}^{37}{\rm Cl} \rightarrow {}^{37}{\rm Ar}$, the transitions ${}^{71}{\rm Ga} \rightarrow {}^{71}{\rm Ge}$, the elastic scattering on electrons, the disintegration of the deuteron by charged current and the disintegration of the deuteron by neutral currents (see review [@2]).
We consider it necessary to note the following. First, as it is not possible to achieve a satisfactory solution to the solar neutrino problem by only using the hypothesis of the neutrino oscillation in vacuum, the mechanism of Wolfenstein, Mikheev and Smirnov comes into play, that involves significant influence of the Sun environment on the transitions of neutrinos from one type to another and has its own arbitrariness. Secondly, it is difficult, if at all possible, to find in the literature a summary of theoretical results for the rates of the processes of the types listed above that would be calculated on the basis of formulas of the neutrino oscillation model under some, recognized as optimal, values of its parameters.
45 years after its first emergence, the solar neutrino problem has received an alternative solution based on the hypothesis of the existence of semiweak interactions between electron neutrinos and nucleons mediated by (almost) massless pseudoscalar isoscalar boson, and the description of such interactions contains only one free parameter [@12]. The relevant theoretical results are in good agreement with the results of four of the five listed above types of experiments with solar neutrinos. The rate of the deuteron disintegration by solar neutrinos induced by the exchange of a massless pseudoscalar boson where the mass difference between the neutron and proton would have been taken into account remains yet not calculated.
Over a long time, many neutrino experiments are being in operation searching for a manifestation of neutrino oscillations. Among them, the KamLAND experiment with reactor antineutrinos is. There, the observed results differ significantly from the expected ones, that one explains by neutrino oscillations (see review [@2]). In our work [@13], an alternative explanation is given for this discrepancy, which is based on significant role of light attenuation in the KamLAND liquid scintillator, that has not been taken into account in theoretical calculations of the observability of expected events.
So, the present work together with the works [@12] and [@13] significantly narrows the theoretical and experimental base for the hypothesis of particle oscillations. It would be very desirable if it could stimulate a new comprehensive analysis of all calculations and conditions of the accomplished and running neutrino experiments.
I am sincerely grateful to S.P. Baranov, M.Z. Iofa, A.E. Lobanov, A.M. Snigirev and S.P. Volobuev for the numerous discussions of problems connected with the present work.
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[^1]: slad@theory.sinp.msu.ru
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The effect of narrow dibaryon resonances to nuclear matter and structure of neutron stars is investigated in the mean-field theory (MFT) and in the relativistic Hartree approximation (RHA). The existence of massive neutron stars imposes constraints to the coupling constants of the $\omega $- and $%
\sigma $-mesons with dibaryons. We conclude that the experimental candidates to dibaryons d$_1$(1920) and d’(2060) if exist form in nuclear matter a Bose condensate stable against compression. This proves stability of the ground state for nuclear matter with a Bose condensate of the light dibaryons.
author:
- |
Amand Faessler$^1$, A. J. Buchmann$^1$, M. I. Krivoruchenko$^{1,2}$\
[$^1$[*Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14* ]{}]{}\
\
[$^2$[*Institute for Theoretical and Experimental Physics, B.Cheremushkinskaya 25*]{}]{}\
title: 'Constraints to Coupling Constants of the $\omega $- and $\sigma $-Mesons with Dibaryons'
---
=-1.3truecm 16 truecm
The prospect to observe the long-lived H-particle predicted in 1977 by R. Jaffe [@Jaf] stimulated considerable activity in the experimental searches of dibaryons. It was proposed to examine the H-particle production in different reactions [@San]. The experiments [@Aok] did not give a positive sign for the H-particle, however, the existence of the H-particle remains an open question which must eventually be settled by experiment. The non-strange dibaryons with exotic quantum numbers, which have a small width due to zero coupling to the $NN$-channel, are promising candidates for experimental searches [@Mul]. The data on pion double charge exchange (DCE) reactions on nuclei [@Bil] exhibit a peculiar energy dependence, which can be interpreted [@Mar] as evidence for the existence of a narrow d’ dibaryon with quantum numbers $T=0$, $J^p=0^{-}$ and the total resonance energy of $2063$ MeV. Recent experiments at TRIUMPF (Vancouver) and CELSIUS (Uppsala) seem to support the existence of the d’ dibaryon [@Mey]. A method for searching narrow, exotic dibaryon resonances in the double proton-proton bremsstrahlung reaction is discussed in Ref. [@Ger]. Recently, some indications for a d$_1$(1920) dibaryon in this reaction have been found [@Khr].
When density of nuclear matter is increased beyond a critical value, production of dibaryons becomes energetically favorable. Dibaryons are Bose particles, so they condense in the ground state and form a Bose condensate [@Bal; @Kri]. An exactly solvable model for a one-dimensional Fermi-system of fermions interacting through a potential leading to a resonance in the two-fermion channel is analyzed in Ref. [@Buc]. The behavior of the system with increasing the density can be interpreted in terms of a Bose condensation of two-fermion resonances. The effect of narrow dibaryon resonances on nuclear matter in the mean field theory (MFT) is analyzed in Refs. [@Fae; @Faes]. In the limit of vanishing decay width, a dibaryon can be approximately described as an elementary field.
Despite the dibaryon Bose condensate does not exist in ordinary nuclei, dibaryons affect properties of nuclear matter and the ordinary nuclei through a Casimir effect. Presence of the background $\sigma $-meson mean field inside of nuclei modifies the nucleon and dibaryon masses and in turn modifies the zero-point vacuum fluctuations of the nucleon and dibaryon fields. This effect contributes to the energy density and pressure. It can be evaluated within the relativistic Hartree approximation (RHA). For nucleons, this effect is well known [@Wal]. In the loop expansion of quantum hadrodynamics (QHD), MFT corresponds to the lowest approximation (no loops), while RHA corresponds to the one-loop approximation in a calculation of the equation of state for nuclear matter.
At zero temperature, a uniformly distributed system of bosons with attractive potential is energetically unstable against compression and collapses [@Abr]. In such a case, the long wave excitations (sound in the medium) have imaginary dispersion law: The square of the sound velocity is negative $a_s^2<0$. The amplitude of these excitations increases with the time, providing instability of the system. It is necessary to analyze dispersion laws of other elementary excitations also. We shall see, however, that in MFT and RHA only sound waves can generate an instability. The ground state of nuclear matter with a Bose condensate of dibaryons is stable or unstable against small perturbations according as the repulsive $\omega $-meson exchange or the attractive $\sigma $-meson exchange is dominant between dibaryons.
In this paper, we investigate the hypothesis that the dibaryon matter is unstable against compression. In such a case, formation of dibaryons in nuclear matter can be treated as a possible mechanism for a phase transition into the quark matter. If central density of a massive neutron stars exceeds a critical value for formation of dibaryons, the neutron star should convert into a quark star, a strange star, or a black hole. Some of the observed pulsars are identified quite reliably with ordinary neutron stars [@Sha]. From the requirement that the dibaryon formation is not energetically favorable at densities lower than the central density of neutron stars with a mass $1.3M_{\odot }$, we derive constraints to the coupling constants of the mesons and dibaryons d$_1$(1920) and d$^{\prime }$(2060) and conclude that narrow dibaryons in this mass range can form a Bose condensate stable against perturbations only. The effect of the dibaryons to stability and structure of neutron stars in different phenomenological models is analyzed in Refs. [@Kri; @Tam]. Constraints to the binding energy of strange matter [@Wit] from the existence of massive neutron stars are discussed in Ref. [@KrMa].
The dibaryonic extension of the Walecka model [@Wal] is obtained by including dibaryons to the Lagrangian density [@Fae; @Faes] $$\label{I}
\begin{array}{c}
{\cal L}=\bar \Psi (i\partial _\mu \gamma _\mu -m_N-g_\sigma \sigma
-g_\omega \omega _\mu \gamma _\mu )\Psi +\frac 12(\partial _\mu \sigma
)^2-\frac 12m_\sigma ^2\sigma ^2 \\ -\frac 14F_{\mu \nu }^2+\frac 12m_\omega
^2\omega _\mu ^2+(\partial _\mu -ih_\omega \omega _\mu )\varphi
^{*}(\partial _\mu +ih_\omega \omega _\mu )\varphi -(m_D+h_\sigma \sigma
)^2\varphi ^{*}\varphi .
\end{array}$$ Here, $\Psi $ is the nucleon field, $\omega _\mu $ and $\sigma $ are fields of the $\omega $- and $\sigma $-mesons, $F_{\mu \nu }=\partial _\nu \omega
_\mu -\partial _\mu \omega _\nu $, $\varphi $ is the dibaryon isoscalar-scalar (or isoscalar-pseudoscalar) field. The values $m_\omega \ $ and $m_\sigma $ are the $\omega $- and $\sigma $-meson masses and the values $g_\omega $, $g_\sigma $, $h_\omega $, $h_\sigma $ are coupling constants of the $\omega $- and $\sigma $-mesons with nucleons ($g$) and dibaryons ($h$).
The $\sigma $-meson mean field $\sigma _c$ determines the effective nucleon and dibaryon masses in the medium $$\label{II}m_N^{*}=m_N+g_\sigma \sigma _c,$$ $$\label{III}m_D^{*}=m_D+h_\sigma \sigma _c.$$
The nucleon scalar density in the RHA is defined by expression [@Wal]
$$\label{IV}\rho _{NS}=<\bar \Psi (0)\Psi (0)>=\gamma \int \frac{d{\bf p}}{%
(2\pi )^3}\frac{m_N^{*}}{E^{*}({\bf p})}\theta (p_F-|{\bf p}|)-4m_N^3\zeta
(m_N^{*}/m_N)$$
where
$$4\pi ^2\zeta (x)=x^3lnx+1-x-\frac 52(1-x)^2+\frac{11}2(1-x)^3.$$ The last term in Eq.(4) occurs after the renormalization of the scalar density. Here, $\gamma =2$ for neutron matter and $\gamma =4$ for nuclear matter.
We investigate here properties of the nuclear matter below the critical density for formation of dibaryons, so the dibaryon condensate is zero $%
<|\varphi (0)|>=0$. The vacuum contribution to the scalar density of dibaryons can be found to be $$\label{V}2m_D^{*}\rho _{DS}=2m_D^{*}<\varphi (0)^{*}\varphi (0)>=m_D^3\zeta
(m_D^{*}/m_D).$$ It differs from the nucleon term in the sign and in the statistical factor (one should replace $4$ ($=2_s\times 2_I)\rightarrow 1$). The self-consistency condition for the nucleon effective mass has the form $$\label{VI}m_N^{*}=m_N-\frac{g_\sigma }{m_\sigma ^2}(g_\sigma \rho
_{NS}+h_s2m_D^{*}\rho _{DS}).$$
The renormalized vacuum contribution to the nucleon energy-momentum tensor has the form [@Wal] $$\label{VII}<T_{\mu \nu }^N(0)>_{vac}=-4g_{\mu \nu }m_N^4\eta (m_N^{*}/m_N)$$ where$$16\pi ^2\eta (x)=x^4lnx+1-x-\frac 72(1-x)^2+\frac{13}3(1-x)^3-\frac{25}{12}%
(1-x)^4.$$
For dibaryons, we get expression $$\label{VIII}<T_{\mu \nu }^D(0)>_{vac}=g_{\mu \nu }m_D^4\eta (m_D^{*}/m_D).$$
The elementary excitations in nuclear matter with a Bose condensate of dibaryons correspond to nucleons and antinucleons, $\omega $-mesons, $\sigma
$-mesons, and dibaryons and antidibaryons. The dispersion laws for these quasiparticles are found in Ref. [@Faes]. The nucleon and antinucleon dispersion laws have the same form as in the vacuum with a replacement $%
m_N\rightarrow m_N^{*}$ and therefore cannot generate an instability of the system. The dispersion laws of $\omega $-mesons, $\sigma $-mesons, and antidibaryons turn out to be real also. The possible source of the instability are the dibaryon quasiparticle excitations only, which are responsible for a long wave perturbations of the system and connected with existence of the sound in the medium.
The square of the sound velocity has the form [@Faes] $$\label{IX}a_s^2=\frac \alpha {1+\alpha }$$ where $$\label{X}\alpha =2\rho _{DS}\frac{m_\sigma ^2}{\tilde m_\sigma ^2}(\frac{%
h_\omega ^2}{m_\omega ^2}-\frac{h_\sigma ^2}{m_\sigma ^2})$$ and $\tilde m_\sigma ^2=m_\sigma ^2+2h_\sigma ^2\rho _{DS}$. We see that $%
a_s^2>0$ for $$\label{XI}\frac{h_\omega ^2}{m_\omega ^2}>\frac{h_\sigma ^2}{m_\sigma ^2}.$$ The validity of inequality (11) is sufficient condition for stability of the ground state of nuclear matter with a Bose condensate of dibaryons.
The physical meaning of the inequality (11) can be clarified by considering the interaction energy of a uniformly distributed dibaryon matter $\rho
_{DV}({\bf x}_1)=\rho _{DV}=$ constant: $$\label{XII}W=\frac 12\int d{\bf x}_1d{\bf x}_2\rho _{DV}({\bf x}_1)\rho
_{DV}({\bf x}_2)V(|{\bf x}_1-{\bf x}_2|).$$ The Yukawa potential $V(r)$ for two dibaryons has the form $$\label{XIII}V(r)=\frac{h_\omega ^2}{4\pi }\frac{e^{-m_\omega r}}r-\frac{%
h_\sigma ^2}{4\pi }\frac{e^{-m_\sigma r}}r.$$ The integration gives $$\label{XIV}W=\frac 12N_D\rho _{DV}(\frac{h_\omega ^2}{m_{_\omega }^2}-\frac{%
h_\sigma ^2}{m_{_\sigma }^2})$$ where $N_D$ is the number of dibaryons. When the dibaryon density $\rho
_{DV} $ increases, the energy for $a_s^2>0$ increases also, the pressure is positive, and so the system is stable.
At present, the coupling constants of the mesons with dibaryons are not known with precision good enough to draw a definite conclusion concerning the stability of the dibaryon matter.
Here, we show that violation of the inequality (11) for light dibaryons is in contradiction with the existence of massive neutron stars.
Given that the ratio $h_\sigma /(2g_\sigma )$ between the $\sigma $-meson couplings with dibaryons and nucleons is fixed, one can find the $\omega $- and $\sigma $-meson couplings with nucleons by fitting the nuclear matter binding energy $E/A-m_N=-15.75$ MeV at the empirical equilibrium density $%
\rho _0=0.148$ fm$^{-3}$ determined from the density in the interior of $%
^{208}Pb$ [@Wal]. It corresponds to the equilibrium Fermi wavenumber $%
k_F=1.3$ fm$^{-1}$. The dependence of the effective nucleon mass at the equilibrium density on the ratio $h_\sigma /(2g_\sigma )$ is shown on Fig.1 (a). In Figs.1 (b) and (c) we give dependence of the incompressibility $%
K=9\rho _0(\partial ^2\varepsilon /\partial \rho ^2)|_{\rho =\rho _0}$ and the asymmetry coefficient $a_4$ on the ratio $h_\sigma /(2g_\sigma )$. In Fig.1 (d), the values $C_s^2=g_\sigma ^2(m_N/m_\sigma )^2$ and $C_\omega
^2=g_\omega ^2(m_N/m_\omega )^2$ are plotted. Notice that $h_\sigma
/(2g_\sigma )=0$ is equivalent to RHA with no dibaryons. For comparison, we give the results of MFT where the effect of dibaryons to the nuclear matter below the critical density for occurrence of a Bose condensate of dibaryons is absent. The results RHA for different dibaryons are not much different. In this work, we study the physical implications of the effective Lagrangian (1) describing nucleon and dibaryon degrees of freedom. Other baryons can be included in the QHD framework in a similar way. Their effect is quite small. We have checked that the Casimir effect caused by the inclusion of other octet baryons ($2 \times 8 = 16$ degrees of freedom) shift the critical value $h_{\sigma }$ only by about 25$\%$ if one assumes that the ${\sigma }$-meson is an SU(3)$_{f}$ singlet. The inclusion of octet and decuplet baryons ($2 \times 8 + 4 \times 10 = 56$ degrees of freedom) with a universal sigma-meson coupling constant increases the critical value of $h_{\sigma }/(2g_{\sigma })$ to 1.2.
When the ratio $h_\sigma /(2g_\sigma )$ approaches a value $0.8$, the system of equations blows up and the empirical equilibrium properties of the nuclear matter can no longer be reproduced. When $x\rightarrow \ 0$ $\zeta
(x)=O(x^4)$, so the zero-point contributions to the scalar density of nucleons and dibaryons, which have the opposite signs, are comparable for $%
4g_\sigma ^4/m_N\approx h_\sigma ^4/m_D$. The dibaryon effects become large for $h_\sigma /(2g_\sigma )\approx 0.5(4m_D/m_N)^{1/4}\approx 0.84$. The greater dibaryon mass, the greater the upper limit to the ratio $h_\sigma
/(2g_\sigma )$. This effect is seen in Fig.1.
The saturation curve is shown in Fig.2 and the effective nucleon mass dependence on the Fermi wavenumber $k_F$ is shown in Fig.3 for $h_\sigma
/(2g_\sigma )=0.6$ in case of the H-particle. EOS in RHA is softer than in MFT. The contributions of the vacuum zero-point fluctuations of nucleons and dibaryons partially cancel each other, so the inclusion of the dibaryons makes the EOS stiffer. In Fig.2, we see that the dashed curves corresponding to RHA with dibaryons lie above the solid lined corresponding to the RHA with no dibaryons. The same effect is seen on Fig.3. In MFT, the nucleon effective mass decreases with the density faster then in RHA. Due to the partial compensation of the nucleon and dibaryon contributions to the vacuum scalar density, the dashed lines lie below the solid lines.
In Fig.4 we show the critical densities for occurrence of dibaryons H(2220), d’(2060), and d$_1$(1920) in MFT and RHA in nuclear and neutron matter versus the ratio $h_\sigma /(2g_\sigma )$ for $h_\omega /h_\omega ^{max}=1$, $0.8$, and $0.6$ where $h_\omega ^{max}=h_\sigma m_\omega /m_\sigma $ is the maximum value for the $\omega $-meson coupling constant with dibaryons at which the inequality (11) is violated ($h_\omega /h_\omega ^{max}=1$ corresponds to $a_s^2=0$ and $h_\omega /h_\omega ^{max}=0.8$ and $0.6$ correspond to $a_s^2<0$). In RHA, dibaryons occur at higher densities. The coupling constant $h_\omega $ determines the energy of dibaryons in the positive $\omega $-meson mean field. The greater the $h_\omega $, the greater the density is required to make production of dibaryons energetically favorable. This effect is seen in Fig.4: The solid lines $%
h_\omega /h_\omega ^{max}=1$ lie above the long-dashed and dashed lines $%
h_\omega /h_\omega ^{max}=0.8$ and $0.6$, respectively.
When the dibaryon matter is unstable against compression, production of dibaryons with increasing the density results to instability of neutron stars with subsequent phase transition into the quark matter and conversion of neutron stars into quark stars, strange stars, or black holes. In such a case, the maximum masses of neutron stars are determined by the mass and the coupling constants of the mesons with the lightest dibaryon. In Fig.5 we show the minimal neutron star masses in which dibaryons can occur.
The MFT and RHA EOS for neutron matter at supranuclear densities are matched smoothly with the BBP EOS [@BBP] at densities $\rho _{drip}<\rho
<0.8\rho _0$ where $\rho _{drip}=4.3$ $10^{11}$g/cm$^3$ and then with the BPS EOS [@BPS] at densities $\rho <\rho _{drip}$. The maximum neutron star masses are sensitive to the value of the equilibrium Fermi wavenumber. If we chose $k_F=1.42$ fm$^{-1}$ instead of $k_F=1.3$ fm$^{-1}$, the maximum masses in MFT (with no dibaryons) are reduced from $3M_{\odot }$ down to $%
2.6M_{\odot }$ [@Wal]. The choice $k_F=1.3$ fm$^{-1}$ provides less stringent (therefore more conservative) constraints to the meson-dibaryon coupling constants. We do not show the results for the d$_1$(1920) dibaryon, since its condensation starts at a density $\rho \approx $ $\rho _0$ providing conversion to quark stars of neutron stars with very low masses $%
M<0.2M_{\odot }$.
In Fig. 6 we show the parameter space for the coupling constants of dibaryons with the mesons. Our discussion and the validity of our arguments are restricted to the region $a_s^2<0$ in which the dibaryon matter is unstable against compression. The requirement of stability of the normal nuclear matter at the saturation density allows to get constraints to the coupling constants. The corresponding curves (straight lines in the MFT case) marked by arrows with the white ends restrict from below the parameter space of the coupling constants. The dotted line in the MFT case, which refers to the d$_1$(1920) dibaryon, is very close to the dashed-dotted line $%
a_s^2=0$. For low values $h_\sigma $ the dotted line lies above the line $%
a_s^2=0$. In such a case, the dibaryon matter unstable against compression cannot exist. The window in the parameter space for the unstable dibaryon matter for d$_1$(1920) is, however, much greater for RHA.
If a conservative assumption is used, namely, that pulsars with a mass $%
1.3M_{\odot }$ are ordinary neutron stars, the constraints to the meson coupling constants with dibaryons can further be improved. The corresponding curves (straight lines in the MFT case) are shown on Fig.6. We see that the dotted and dashed lines lie above or very close to the line $a_s^2=0$. It means that for d$_1$(1920) and d’(2060) dibaryons, the dibaryon matter unstable against compression cannot exist (owing to a very small window for d’(2060) at higher values of the $h_\sigma $). For the H-particle, there is a window in the parameter space between the line $a_s^2=0$ and the solid curves marked by arrows with the black ends, which corresponds to the dibaryon matter unstable against compression.
The H-particle interaction were studied in the non-relativistic quark cluster model [@Str; @Oka] successful in describing the $NN$-phase shifts. The coupling constants of the mesons with the H-particle can be fixed by fitting the depth and the position of the minimum of the HH-adiabatic potential [@Sak] to give $h_\omega /(2g_\omega )=0.89$ and $h_\sigma
/(2g_\sigma )=0.80$. These values are marked on Fig.6 with a cross. These estimates are used in the MFT-calculations [@Fae; @Faes]. They correspond to the unstable dibaryon matter and are in the allowed region of the parameter space for the H-particle. The energetically favorable compression of the H-matter can lead to the formation of the absolutely stable strange mater [@Wit], producing conversion of neutron stars to strange stars [@Oli].
MFT and RHA EOS both are very stiff. In the soft models like the Reid one (for a review of the nuclear matter models see [@Sha]), the central density of neutron stars is much greater than in stiff models. Respectively, conditions for occurrence of new forms of nuclear matter are more favorable. From Figs. 5 and 6, we see that the softer RHA EOS produces lower upper limits to the neutron star masses and, respectively, more stringent constraints to the meson-dibaryon coupling constants as compared to the stiffer MFT EOS, despite in RHA dibaryons occur at higher densities (see Fig.4). One can assume that this effect is of the general validity and that softer EOS like the Reid one give more stringent constraints to the meson-dibaryon coupling constants. We consider therefore the constraints given in Fig.6 as the conservative ones.
In the conclusion, we showed that the hypothesis of instability of the dibaryon matter against compression is in contradiction with the hypothesis that pulsars of a mass $1.3M_{\odot }$ are ordinary neutron stars for dibaryons d$_1$(1920) and d’(2060). This conclusion is valid for all narrow dibaryons with the same quantum numbers in the same mass range. The H-particle is sufficiently heavy, its condensation starts at higher densities, respectively, constraints to the meson-dibaryon coupling constants are not much stringent, allowing a possibility for the H-particle to form a condensate unstable against compression with subsequent formation of the strange matter. The meson coupling constants with dibaryons d$_1$(1920) and d’(2060) should obey the inequality (11). The meson coupling constants with the H-particle should lie above the solid curves on Fig.6.
The authors are grateful to B. V. Martemyanov for discussions of the results. One of the authors (M. I. K.) acknowledges hospitality of Institute for Theoretical Physics of University of Tuebingen, Alexander von Humboldt Stiftung for support with a Forschungsstipendium and DFG-RFBR for Grant No. Fa-67/20-1.
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[**Figure captions**]{}
[**Fig.1**]{} The effective nucleon mass (a), the asymmetry coefficient (b), the incompressibility of nuclear matter (c) at the saturation density, and the coupling constants $C_s^2=g_s^2(m_N/m_s)^2$ and $C_\omega ^2=g_\omega
^2(m_N/m_\omega )^2$ in the RHA versus the ratio $h_\sigma /(2g_\sigma )$ between the $\sigma $-meson coupling constants with dibaryons and nucleons. The MFT results are shown for comparison. The coupling constants are fixed by fitting the minimum and depth of the energy per baryon number at the saturation density of nuclear matter. The solid, dashed, and dotted curves correspond to the dibaryons H(2220), d$^{\prime }$(2060), and d$_1$(1920). The reasonable description of the properties of the nuclear matter at the saturation density is possible for $h_\sigma /(2g_\sigma )<0.8$.
[**Fig.2**]{} Saturation curve for nuclear matter in RHA with no dibaryons (solid line) and in RHA with inclusion of the H-dibaryon (dashed line) for $%
h_\sigma /(2g_\sigma )=0.6$.
[**Fig.3**]{} The effective nucleon mass versus the fermi momentum of nucleons in the nuclear matter (upper curves) and in the neutron matter (lower curves). The solid lines correspond to RHA with no dibaryons, the dashed lines correspond to RHA with inclusion of the H-dibaryon.
[**Fig.4**]{} Critical densities for occurrence of the dibaryons H(2220), d’(2060), and d$_1$(1920) in nuclear and neutron matter in MFT and RHA versus the $\sigma $-meson coupling constant with dibaryons for $h_\omega
/h_\omega ^{max}=1$, $0.8$, and $0.6$ (the solid, longed-dashed, and dashed lines, respectively), where $h_\omega ^{max}=h_\sigma m_\omega /m_\sigma $ is the maximum value for the $\omega $-meson coupling constant with dibaryons at which the dibaryon matter is unstable against compression (see the text).
[**Fig.5**]{} The minimum neutron star masses, in which the dibaryon formation becomes energetically favorable, versus the $\sigma $-meson coupling constant with dibaryons for $h_\omega /h_\omega ^{max}=1$, $0.8$, and $0.6$ (the solid, longed-dashed, and dashed lines, respectively) where $h_\omega
^{max}$ is the maximum value for the $\omega $-meson coupling constant at which the dibaryon matter is unstable against compression ($a_s^2<0$). The results are given in MFT and RHA and for two dibaryons H(2220) and d’(2060). The minimum neutron star masses for the d$_1$(1920) are very small ($%
<0.2M_{\odot }$).
[**Fig.6**]{} Parameter space for the coupling constants of the $\sigma $- and $\omega $-mesons with dibaryons in MFT and RHA. The dashed-dotted line $%
a_s^2=0$ divides the parameter space into two parts. The upper left part of the parameter space corresponds to the dibaryon matter stable against compression (square of the sound velocity is positive $a_s^2>0$), the lower right part corresponds to the nuclear matter unstable against compression ($%
a_s^2>0$). The solid, dashed, and doted curves constrain from below the regions in which the dibaryon formation is energetically not favorable in ordinary nuclei (curves marked by arrows with white ends) and at the density equal to the central density of a $1.3M_{\odot }$ mass neutron star (curves marked by arrows with black ends). The cross refers to the H-particle coupling constants with the mesons, determined from the adiabatic potential [@Sak].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: '[Despite the great progress of current cosmological measurements, the nature of the dominant component of the universe, coined [*dark energy*]{}, is still an open question. [*Early Dark Energy*]{} is a possible candidate which may also alleviate some fine tuning issues of the standard paradigm. Using the latest available cosmological data, we find that the 95% CL upper bound on the early dark energy density parameter is ${\Omega_{\textrm{eDE}}}\,<\,0.009$. On the other hand, the dark energy component may be a stressed and inhomogeneous fluid. If this is the case, the effective sound speed and the viscosity parameters are unconstrained by current data. Future omniscope-like $21$ cm surveys, combined with present CMB data, could be able to distinguish between standard quintessence scenarios from other possible models with $2\sigma$ significance, assuming a non-negligible early dark energy contribution. The precision achieved on the ${\Omega_{\textrm{eDE}}}$ parameter from these $21$ cm probes could be below $\mathcal{O} (10\%)$.]{}'
author:
- Maria Archidiacono
- 'Laura Lopez-Honorez'
- Olga Mena
title: |
Current constraints on early and stressed dark energy models\
and future 21 cm perspectives
---
Introduction
============
The nature of the mysterious dark energy component that currently dominates the energy content of the universe reveals new physics missing from our universe’s picture, and constitutes the fundamental key to understand the fate of the universe. The most economical explanation of the dark energy component attributes this energy density to the one of the vacuum, i.e., a cosmological constant scenario. Together with cold dark matter (CDM), the so-called $\Lambda$CDM scenario can account for present data with a flat universe made up of roughly $30\%$ dark matter and $70\%$ dark energy. In this minimal model, the dark energy equation of state, $w$, which corresponds to the ratio of the dark energy pressure to the dark energy density, is constant and equal to $-1$. However, this simple picture suffers from severe fine tuning theoretical issues (see Ref. [@Copeland:2006wr] and references therein) as well as from problems with observations related to the matter power spectrum on scales of a few Mpc and below [@Moore:1999gc; @Bode:2000gq; @Penarrubia:2012bb; @BoylanKolchin:2011dk; @Ferrero:2011au; @Weinberg:2013aya]. Possible alternatives to alleviate them have been extensively explored. Perfect dark energy fluids, characterised either by a constant ($w\neq-1$) or by a time varying dark energy equation of state $w(a(t))$, or scalar field models, are the most popular options considered in the cosmological data analyses, as their parameterizations require few extra parameters (two at most) to be added to the usual $\Lambda$CDM scenario.
There exists also alternative scenarios, in which the gravitational sector is modified, leading to a modification of Einstein’s equations of gravity on large scales. Modifications of gravity (see e.g. [@DeFelice:2010aj] and references therein) incorporate models with extra spatial dimensions or an action which is non-linear in the Ricci scalar. There are also non-perfect fluid models, as Chaplygin gas cosmologies [@Bento:2002ps], which involve more parameters than just one equation of state $w$. Of particular interest here is the [*Early Dark Energy*]{} (hereafter EDE) case, as it arises as a natural hypothesis of dark energy [@Wetterich:2004pv; @Doran:2006kp; @odea; @Calabrese:2010uf]. EDE differs from the cosmological constant because it is not negligible in the early universe and the contribution depends on the initial density parameter ${\Omega_{\textrm{eDE}}}$. Furthermore, the EDE model considered here is based on a generic dark energy fluid which is inhomogeneous. Density and pressure are time varying, therefore the equation of state is not constant in time. The phenomenological analyses of these inhomogeneous dark energy models usually require additional dark energy clustering parameters, i.e. the dark energy effective sound speed and the dark energy anisotropic stress. The sound speed ${c^2_{\textrm{eff}}}$ [@Hu:1998kj; @Hu:1998tk; @ceff] is defined as the ratio between the dark energy pressure perturbation and the dark energy density contrast in the rest frame of the fluid, ${c^2_{\textrm{eff}}}\equiv(\delta P/\delta \rho)_{\rm rest}$. In the simplest quintessence models, ${c^2_{\textrm{eff}}}=1$, while the anisotropic stress is zero. The effective sound speed determines the clustering properties of dark energy and consequently it affects the growth of matter density fluctuations. Therefore, in principle, its presence could be revealed in large scale structure observations. The growth of perturbations can also be affected by the anisotropic stress contributions [@Hu:1998kj; @Hu:1998tk; @cvis] which lead to a damping in the velocity perturbations. In the parametrization used here, the damping effect is driven by the viscosity parameter ${c^2_{\textrm{vis}}}$ which links the anisotropic stress to the velocity perturbation and the metric shear.
Despite the precision achieved by the combination of Cosmic Microwave Background (CMB) measurements from the Planck satellite [@planck], Baryon Acoustic Oscillation (BAO) data from a number of galaxy surveys [@Anderson:2013zyy; @Beutler:2011hx; @Busca:2012bu; @Kirkby:2013fh; @Slosar:2013fi] and Supernovae Ia luminosity distance measurements [@Suzuki:2011hu] in the extraction of the dark energy equation of state parameter, $w=-1.06\pm 0.06$ at $68\%$ CL [@Anderson:2013zyy], the nature of the dark energy component remains unknown. Therefore, it is mandatory to carefully study other possibilities including the one of an EDE component, as well as the clustering properties of the dark energy fluid. In this paper we shall address both issues, relaxing the perfect fluid assumption and considering current cosmological data, in addition to the recent BICEP2 measurements of the B-modes power spectrum [@Ade:2014xna] .
We also explore the possibility of constraining an EDE component and/or a stressed dark energy fluid with future $21$ cm surveys. The next generation of radio experiments, which will image the neutral intergalactic medium (IGM) in $21$ cm emission/absorption, will provide a unique probe of the universe at higher redshifts ($z>6$) which lie out of the reach of galaxy surveys and CMB experiments. The $21$ cm line signal presents several advantages compared to traditional cosmic and astrophysical probes, see e.g. [@Loeb:2003ya], and it could be used to test the nature of dark energy [@Wyithe:2007rq]. The future generation of radio interferometers testing the $21$ cm signal, including the Squared Kilometer Array (SKA) [@Mellema:2012ht] and omniscopes [@Tegmark:2008au; @Zheng:2013tpz], may provide extra constraints on the cosmological parameters probing the Epoch of Reionisation (EoR) or the high redshift window, see e.g. [@Mao:2008ug; @Clesse:2012th]. In addition, the $21$ cm signal can also be used at low redshifts ($z<5$), offering a competitive cosmological probe for unraveling the nature of the component responsible for the present universe’s accelerated expansion [@Chang:2007xk; @Hall:2012wd].
The structure of the paper is as follows. Sections \[sec:ede\] and \[sec:stress\] describe the early and stressed dark energy models evolution in terms of the background and perturbation variables. In Sec. \[sec:methodanddata\] we present the method and data followed in the numerical analyses presented in Sec. \[sec:current\]. Section \[sec:future\] addresses the future perspective and constraints from $21$ cm surveys by means of a Fisher matrix forecast analysis. Finally, we draw our conclusions in Sec. \[sec:conclusions\].
Early Dark energy models {#sec:ede}
========================
The concept of EDE cosmology was introduced in [@Wetterich:2004pv] and studied in several subsequent works following different possible effective parametrizations of the evolution of the dark energy fluid, see e.g. [@Doran:2006kp; @Calabrese:2010uf; @Pettorino:2013ia; @planck]. Here we follow Ref. [@Doran:2006kp] to describe the evolution of the background dark energy density from the high redshift, constant value ${\Omega_{\textrm{eDE}}}$ until its present-day value $\Omega_{\rm DE}^0$ (assuming a flat universe with $\Omega_{\rm DE}^0+\Omega_{\rm m}^0=1$): $$\Omega_{\rm DE}(a) =\frac{\Omega_{\rm DE}^0 - {\Omega_{\textrm{eDE}}}\left(1- a^{-3 w_0}\right) }{\Omega_{\rm DE}^0 + \Omega_{m}^{0} a^{3w_0}} + {\Omega_{\textrm{eDE}}}\left (1- a^{-3 w_0}\right).
\label{eq:odea}$$ The evolution of $w(a)$ in this EDE parametrization reads $$w(a) = -\frac{1}{3[1-\Omega_{\rm DE}(a)]} \frac{d\ln \Omega_{\rm DE}(a)}{d\ln a} + \frac{a_{eq}}{3(a + a_{eq})},
\label{eq:wa}$$ where $a_{eq}$ is the scale factor at matter-radiation equality era. The time dependent equation of state $w(a)$ typically traces the dominant component of the universe at each epoch: first $w\simeq1/3$ during the radiation dominated period, then $w\simeq 0$ during the matter dominated era and finally $w\rightarrow w_0$ in the present epoch. The current value of the equation of state parameter $w_0$ might be different[^1] from $-1$.
Stressed Dark energy models {#sec:stress}
===========================
Using the notation of Ref. [@Ma] and assuming the synchronous gauge, we follow [@Calabrese:2010uf] to describe the dark energy scalar perturbation evolution equations in Fourier space for the density contrast ($\delta$), the velocity divergence ($\theta$) and the anisotropic stress perturbation ($\sigma$): $$\begin{aligned}
\frac{\dot{\delta}}{1+w}&=&
-\left[k^{2}+9\left(\frac{\dot{a}}{a}\right)^{2}\left({c^2_{\textrm{eff}}}-w+\frac{\dot{w}}{3(1+w)(\dot{a}/a)}\right)\right]\frac{\theta}{k^{2}} \nonumber\\
&&-\frac{\dot{h}}{2}-3\frac{\dot{a}}{a}({c^2_{\textrm{eff}}}-w)\frac{\delta}{1+w}; \\
\label{eq:pertdelta}
\dot{\theta}&=&-\frac{\dot{a}}{a}(1-3{c^2_{\textrm{eff}}})\,\theta+
\frac{\delta}{1+w}{c^2_{\textrm{eff}}}k^{2}-k^{2}\sigma;\\
\label{eq:perttheta}
\dot{\sigma}&=&-3\frac{\dot{a}}{a}\left[1-\frac{\dot{w}}{3w(1+w)(\dot a/a)}\right]\,\sigma\nonumber\\
& & +\frac{8{c^2_{\textrm{vis}}}}{3(1+w)}\left[\theta+\frac{\dot{h}}{2}+3\dot{\eta}\right]~,
\label{eq:pertsigma}\end{aligned}$$ where ${c^2_{\textrm{eff}}}$ denotes the effective sound speed. In the last equation, the velocity and the metric shear (sometimes referred to as $H_T= -(h/2 + 3 \eta)$) are related to the dark energy shear stress through the viscosity parameter ${c^2_{\textrm{vis}}}$. The latter relation was first introduced in Ref. [@Hu:1998kj] [^2] and relates directly the anisotropic stress with the damping of velocity fluctuations on shear-free frames ($H_T$ = 0), if ${c^2_{\textrm{vis}}}> 0$. We have also addressed the contribution of the dark energy shear stress to the evolution equations for the tensor perturbations.
The differential equations above govern the clustering properties of the dark energy fluid, and we shall solve them and compare the results to current and future observations using the methods detailed in the following sections.
Method and data for current constraints {#sec:methodanddata}
=======================================
We have modified the latest version of the Boltzmann equations solver [@camb] in order to account for Eqs. (\[eq:odea\])-(\[eq:pertsigma\]).
The parameter space contains the six standard parameters of the $\Lambda$CDM model $$\{\Omega_{\rm b}h^2,\,\Omega_{\rm c}h^2,\,\theta,\,\tau,\,n_{\rm s},\,\ln{(10^{10} A_{\rm s})}\},$$ where $\Omega_{\rm b}h^2=\omega_{\rm b}$ is the present physical energy density in baryons, $\Omega_{\rm c}h^2=\omega_{\rm c}$ is the present physical cold dark matter energy density, $\theta$ is the angular scale of the sound horizon, $\tau$ is optical depth to reionisation and $n_{\rm s}$ and $A_{\rm s}$ are the spectral index and the amplitude of primordial scalar perturbations at a pivot scale $k=0.05\,{\rm Mpc}^{-1}$, respectively.
Since we include tensor perturbations, we have also considered the tensor-to-scalar ratio $r$ parameter, defined relatively to the same pivot scale of the scalar perturbations, $k=0.05\,{\rm
Mpc}^{-1}$. Finally, we include all the parameters describing the EDE model evolution (see Secs. \[sec:ede\] and \[sec:stress\]): $$\{{\Omega_{\textrm{eDE}}},\,w_0,\,{c^2_{\textrm{vis}}},\,{c^2_{\textrm{eff}}}\}.$$ We assume flat priors on the parameters as listed in Tab. \[tab:priors\]. The sampling of the parameter space is performed through the Monte Carlo Markov Chain (MCMC) public package [@cosmomc].
Parameter Prior
--------------------------- -----------------
$\Omega_{\rm b}h^2$ $0.005 \to 0.1$
$\Omega_{\rm c}h^2$ $0.01 \to 0.99$
$\theta$ $0.5 \to 10$
$\tau$ $0.01 \to 0.8$
$n_{\rm s}$ $0.5 \to 1.5$
$\ln{(10^{10} A_{s})}$ $2.7 \to 4$
$r$ $0\to1$
${\Omega_{\textrm{eDE}}}$ $0 \to 0.1$
$w_0$ $-1 \to 0$
${c^2_{\textrm{eff}}}$ $0\to1$
${c^2_{\textrm{vis}}}$ $0\to1$
: Range of the flat priors for the cosmological parameters considered here.[]{data-label="tab:priors"}
The Bayesian inference is based on the CMB temperature anisotropy power spectrum of the Planck experiment, implemented following the prescriptions of Ref. [@Ade:2013kta]. We have also considered the CMB polarization measurements from the nine-year data release of the WMAP satellite [@Bennett:2012zja]. In the following, we shall refer to the former data as WP. The maximum multipole number of the Planck temperature power spectra is $\ell_{\rm max}=2500$. The WP measurements reach a maximum multipole $\ell=23$, see Ref. [@Bennett:2012zja]. In order to directly constrain the tensor-to-scalar ratio $r$, the nine-bins measurements of the B-modes polarization power spectrum from the BICEP-2 collaboration[@Ade:2014xna] are included.
Current cosmological constraints {#sec:current}
================================
In this section we apply the data sets described above, using the MCMC method, to four possible scenarios:
- [**Case 1**]{}: In this scenario, both the early dark energy component ${\Omega_{\textrm{eDE}}}$ and the dark energy perturbation parameters ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$ are free parameters, with the priors specified in Tab. \[tab:priors\]. We also consider in this case the current value of the dark energy equation of state, $w_0$, see Eq. (\[eq:odea\]), as a free parameter. \[case1\]
- [**Case 2**]{}: The early dark energy component ${\Omega_{\textrm{eDE}}}$ and $w_0$ are free parameters, but the dark energy perturbations are fixed to their standard values: ${c^2_{\textrm{eff}}}=1$ and ${c^2_{\textrm{vis}}}=0$ (i.e. no anisotropic stress contribution is considered in this case). \[case2\]
- [**Case 3**]{}: We consider no early dark energy component (${\Omega_{\textrm{eDE}}}=0$) but the dark energy perturbations ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$ are both free parameters, varying with a flat prior in the range $[0,1]$, as well as a constant dark energy equation of state $w$, which varies with a prior in the range $[-1,0]$. \[case3\]
- [**Case 4**]{}: We consider a simple $w$CDM cosmology, i.e., a cosmological scenario with a constant dark energy equation of state, which is allowed to freely vary in the range $[-1,0]$. \[case4\]
Planck+ WP Planck +WP + BICEP-2
----------------------------------------------------------------------------- ----------------- ----------------------
Case 1
${{\Omega_{\textrm{eDE}}}}$ $<0.015$ $<0.010$
$w_0$ $<-0.658$ $<-0.722$
$r$ $<0.09$ $0.15\pm0.04$
$n_{\rm s}$ $0.960\pm0.008$ $0.963\pm0.007$
Case 2
${{\Omega_{\textrm{eDE}}}}({c^2_{\textrm{eff}}}=1, {c^2_{\textrm{vis}}}=0)$ $<0.012$ $<0.009$
$w_0({c^2_{\textrm{eff}}}=1, {c^2_{\textrm{vis}}}=0)$ $<-0.659$ $<-0.722$
$r$ $<0.10$ $0.16\pm0.04$
$n_{\rm s}$ $0.960\pm0.007$ $0.963\pm0.008$
Case 3
${{\Omega_{\textrm{eDE}}}}$ $0$ $0$
$w$ $<-0.647$ $<-0.709$
$r$ $<0.11$ $0.16\pm0.04$
$n_{\rm s}$ $0.960\pm0.007$ $0.964\pm0.007$
Case 4
${{\Omega_{\textrm{eDE}}}}({c^2_{\textrm{eff}}}=1, {c^2_{\textrm{vis}}}=0)$ $0$ $0$
$w({c^2_{\textrm{eff}}}=1, {c^2_{\textrm{vis}}}=0)$ $<-0.655$ $<-0.705$
$r$ $<0.11$ $0.16\pm0.04$
$n_{\rm s}$ $0.960\pm0.008$ $0.964\pm0.007$
Table \[tab:ede\_finaltest\] shows the mean values with $1\sigma$ errors and the $2\sigma$ upper bounds for the EDE parameters following the case order listed above. Notice first that we do not show the values for the dark energy perturbation parameters (${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$), since current CMB measurements are unable to constrain them. Secondly, when setting ${c^2_{\textrm{eff}}}=1$ and ${c^2_{\textrm{vis}}}=0$ (see Case 2 above), we find an upper limit on the early dark energy parameter ${\Omega_{\textrm{eDE}}}< 0.012$ at $95\%$ CL. The former bound is looser than the one reported by the Planck collaboration, ${\Omega_{\textrm{eDE}}}<
0.010$ at $95\%$ CL with the same data sets (Planck temperature and WP data). The larger value that we get on ${\Omega_{\textrm{eDE}}}$ is related to the degeneracy between this parameter and the tensor-to-scalar ratio $r$, as we shall explain below. The addition of the BICEP2 data makes our $95\%$ CL upper limit on ${\Omega_{\textrm{eDE}}}$ tighter (${\Omega_{\textrm{eDE}}}<
0.009$ at $95\%$ CL). When allowing the dark energy perturbations ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$ to be free parameters (Case 1 above), the $95\%$ CL upper bound ${\Omega_{\textrm{eDE}}}$ degrades but not significantly: we find ${\Omega_{\textrm{eDE}}}< 0.015$ (${\Omega_{\textrm{eDE}}}< 0.010$) at $95\%$ CL before (after) combining Planck and WP measurements with BICEP2 data.
In general, the results for the standard $\Lambda$CDM cosmological parameters do not deviate significantly from their expected mean values and errors. This can be noticed by comparing the first three cases depicted in Tab. \[tab:ede\_finaltest\] with the last rows, which show the expectations within the $w$CDM cosmological scenario. Indeed, the current value of the dark energy equation of state $w_0$ does not show a very strong dependence on the dark energy perturbation parameters, as its $95\%$ CL upper bound remains unaffected when ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$ are both freely varying. Concerning the value of $n_s$, its mean value is strongly affected when including BICEP2 data in our numerical analyses, regardless of the dark energy scenario.
Figure \[ede\_finaltest\] shows the marginalised 2D plots and the posteriors involving the most relevant cosmological parameters here in the case in which both the early dark energy component ${\Omega_{\textrm{eDE}}}$ and the perturbation parameters ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$ are allowed to vary freely (see Case 1 above). The red contours refer to the results arising from the analysis of Planck + WP data, while the blue contours include BICEP2 as well. The marginalised 2D plot in the bottom left corner, in the (${\Omega_{\textrm{eDE}}}$, $r$) plane, shows the degeneracy between the EDE component and the tensor-to-scalar ratio $r$. There exists a mild anti-correlation between these two parameters, which can be easily understood: both parameters show an effect at very large scales, increasing the power at very low multipoles. As the BICEP2 data constrain $r$ to be different from zero, the $2\sigma$ upper bound on ${\Omega_{\textrm{eDE}}}$ is tighter, in order to compensate the contribution from the tensor modes at large scales. A similar effect can also be noticed in the 2D marginalised plot in the ($w_0$, $r$) plane: given the anti-correlation between $w_0$ and $r$, the BICEP2 measurements of $r$ reduce the upper bound on $w_0$. There also exists a degeneracy between the ${\Omega_{\textrm{eDE}}}$ and $w_0$ parameters, as can be noticed from the right lower panel of Fig. \[fig:ede\_finaltest\]: larger (smaller) values of the present dark energy equation of state, $w_0$, allow for smaller (larger) values of the EDE parameter, ${\Omega_{\textrm{eDE}}}$. Therefore, these two parameters are anti-correlated, as can be learnt from Eq. (\[eq:odea\]): for a given value of the ${\Omega_{\textrm{eDE}}}$ parameter and the scale factor $a$, the quantity $\Omega_{\rm DE}$ grows as the value of $w_0$ does.
{width="18cm"}
$21$ cm Forecasts {#sec:future}
=================
In this section, we follow the description of Ref. [@Clesse:2012th] for the 21 cm brightness background temperature $T_b(z)$, for the evolution equations of the linear perturbation $\delta T_b(z)$ as well as for the reionisation model implementation.
The study of the 21 cm signal requires to deal with the angular location on the sky plane ${\bf \theta}$, and with the frequency difference $\Delta f$ of the signal to a central 21 cm line of redshift $z$. The dual coordinates of this system are denoted by ${\bf {u_\perp}}$ and ${u_\parallel}$, and they are related to the standard comoving wavevector ${\bf k}$ components as follows: $${ {{\bf u}_\perp}= {D_{\mathrm{A}}}(z) {\bf {k}_\perp},} \qquad {u_\parallel}= y(z) {k_\parallel},$$ where ${D_{\mathrm{A}}}(z)$ is the angular comoving distance and $$y(z) = \frac{{\lambda_{21}}(1+z)^2 }{H(z)}\,,$$ where ${\lambda_{21}}$ is the 21 cm wavelength (in the rest frame) and $H(z)$ is the Hubble rate. The 21 cm brightness temperature power spectrum relevant for our analyses, ${P_{{\delta T_b}}}({\bf u})$, is related to ${P_{{\delta T_b}}}({\bf
k})$ as follows: $${P_{{\delta T_b}}}({\bf u}) = \frac{{P_{{\delta T_b}}}({\bf k})}{{D_{\mathrm{A}}}(z)^2 y(z)}.$$
For the Fisher matrix analysis, we have adopted the formalism of Refs. [@Mao:2008ug; @Clesse:2012th]. Assuming that ${P_{{\delta T_b}}}({\bf u})$ is gaussian-distributed, we can approximate the Fisher matrix by $$F_{ab} = \dfrac{1}{2} \sum_{{u_\parallel},{u_\perp}} \frac{N_c}{\left[{P_{{\delta T_b}}}({\bf u}) +
P_{noise}\right]^2}
\frac{\partial {P_{{\delta T_b}}}({\bf u})}{\partial\lambda_a} \dfrac{\partial
{P_{{\delta T_b}}}({\bf u})}{\partial \lambda_b}\,,
\label{eq:fish}$$ where $\lambda_{a,b}$ are the cosmological parameters involved in the Fisher forecast analysis, and $$N_c = \frac{4 \pi f_{sky}}{\Theta^2} 2 \pi {k_\perp}\delta {k_\perp}\delta {k_\parallel}\dfrac{V}{(2\pi)^3}\,,$$ is the number of independent cells probed for a given value of ${\bf
u}$ (or ${\bf k}$), $V$ is the comoving volume covered and $\Theta$ is the angular patch in the sky [^3]. In Eq. (\[eq:fish\]), $P_{noise}$ is given by [@Zaldarriaga:2003du; @Clesse:2012th]: $$\label{eq:Pnoiseu}
P_{noise}({\bf u}) \simeq \frac{4 \pi f_{sky}}{\Omega_{\rm fov}} \frac{\lambda^2}{D^2 f_{cover}^2}
\frac{T_{sys}^2}{B_W t_{obs}} \,,$$ with $f_{sky}$ the fraction of the sky covered by the survey, $\Omega_{\rm fov}$ the field of view, $\lambda$ the redshifted wavelength of the signal, $T_{sys}$ the system temperature, $D$ the size of the array, $B_W$ the experiment’s bandwidth and $f_{cover}$ the covering factor of the array. Beam effects at small scales can be incorporated by multiplying Eq. (\[eq:Pnoiseu\]) by the factor $\exp{[
{{\bf u}_\perp}^2/(4\sqrt{\ln 2}/\theta_{fw})^2]}$, with $\theta_{fw}=0.89\lambda/D$, see Ref. [@Clesse:2012th].
In what follows, we consider two possible $21$ cm experiment configurations. The first one is a CHIME-like [@Newburgh:2014toa] experiment, covering a low redshift range $0.8<z<2.5$. In our analyses we use a setup similar to the one considered in [@Hall:2012wd]. The second one is an omniscope-like instrument sensitive to the EoR. In the latter case, we follow the setup of Ref. [@Clesse:2012th]. In our treatment of the Fisher matrix, we use a convolution of the signal with the frequency window function associated with the mean redshift of observation. This method helps in reducing the degeneracy between the cosmological parameters $\tau$ and $\ln (A_S)$ when considering one single redshift slice for an omniscope-like experiment [@Clesse:2012th]. Notice that, in what follows, we shall assume that most of the foregrounds can be eliminated, assumption which is still under active research (see e.g. [@forgnd]). We also neglect the fact that ionising sources could affect the 21cm perturbations, providing extra contributions to the power spectrum [@Mao:2008ug; @Clesse:2012th]. Therefore the analysis presented here should be regarded as an optimistic appraisal of the 21 cm signal potential to constrain both an EDE component and its clustering properties.
We present results for two fiducial cosmological models: the fiducial model 1 (2) with ${\Omega_{\textrm{eDE}}}=0.01$ ($0.03$), ${c^2_{\textrm{vis}}}=0$ ($0.33$) and ${c^2_{\textrm{eff}}}=1$ ($0.33$), both of them assuming the same value for the dark energy equation of state at present, $w_0=-0.9$. Figure \[fig:fid\] shows the evolution of the background quantities $\Omega_{\rm DE}(z)$ and $w(z)$, see Eqs. (\[eq:odea\]) and (\[eq:wa\]), as a function of the redshift, for these two possible fiducial cosmologies. The redshift ranges tested by the two possible $21$ cm future experiments considered here are depicted by the grey rectangular zones. Notice that both experiments are located where the difference among the expansion histories for these two fiducial models is non-negligible. Therefore, one would expect to have sensitivity to distinguish between different cosmological backgrounds when exploring the $21$ cm power spectrum in the two redshift ranges depicted in Fig. \[fig:fid\].
![Evolution of the background quantities for the fiducial models of Tabs. \[tab:chime12\] and \[tab:omn12\]. The redshift ranges tested by the 21 cm experiments considered here are shown by the grey rectangular areas.[]{data-label="fig:fid"}](fid.png){width="9cm"}
CHIME $0.8<z<2.5$
-----------------
In Tab. \[tab:chime\], we provide the value of the parameters specifying the CHIME experiment considered in our analyses, which are similar to those considered in Ref. [@Hall:2012wd]. For the system temperature, we have taken $T_{sys}= \left[ 40 + 5 \left(\nu /710 {\rm
Mhz}\right)^{-2.6}\right]$ K, where $\nu$ is the redshifted frequency of the 21 cm signal. We have also considered a comoving number density of sources of $0.03h^3$Mpc$^{-3}$, contributing to shot-noise.
\[h!\]
redshift slices $B_W$ $D$ $f_{cover}$ $t_{obs}$ $f_{sky}$
----------------- ------- ------- ------------- ----------- -----------
0.8/1/2/2.5 2 Mhz 100 m 1 1 yr 0.5
: Specifications of the CHIME-like experiment, see also Ref. [@Hall:2012wd].[]{data-label="tab:chime"}
The results for the two possible fiducial models described in the previous section are presented in Tab. \[tab:chime12\], using ${{u_\perp}^{\min}}=
2\pi/\theta_{res}(z)$ with $\theta_{res}={\lambda_{21}}(1+z)/D$. Notice that the CHIME configuration can provide a high precision measurement of $w_0$. However, the precision in the extraction of the EDE background parameter ${\Omega_{\textrm{eDE}}}$, as well as in the measurements of the dark energy clustering parameters ${c^2_{\textrm{vis}}}$ and ${c^2_{\textrm{eff}}}$, is quite poor. Concerning the standard cosmological parameters, the constraints on both $\ln A_s$ and $\Omega_{\rm b} h^2$ are worse than those obtained with current CMB data. Indeed, these two parameters affect the overall amplitude of the 21 cm signal, while the CMB amplitude signal is mainly driven by the $\ln A_s$ parameter, with $\Omega_{\rm b} h^2$ controlling the CMB even-odd peak ratio. However, the constraints on both $\tau$ and $n_s$ are tighter for the 21 cm experiment. Let us emphasise that the addition of BICEP2 data does not change the results presented here.
--------------------------- ----------------------------- ----------------------------------- --------------------------------------- --
fiducial$_1$ (fiducial$_2$) CHIME CHIME
+ Planck & WP
$\Omega_{\rm b} h^2 $ $ 0.02258 $ $ 2.07\,(2.15) \cdot 10^{-3} $ $ 2.55 \,(2.22)\cdot 10^{-4} $
$h $ $ 0.71 $ $ 1.4\, (2.21)\cdot 10^{-2} $ $ 0.88\,(1.11)\cdot 10^{-2} $
$\Omega_{\rm c} h^2 $ $ 0.1109 $ $ 7.07\, (9.57)\cdot 10^{-3} $ $ 1.54\,(1.66)\cdot 10^{-3} $
${\Omega_{\textrm{eDE}}}$ $ 0.01\, (0.03) $ $ 1.92\, (2.97) \cdot 10^{-2} $ $ 3.31\, (3.8)\cdot 10^{-3} $
${c^2_{\textrm{vis}}}$ $ 0.\, (0.33) $ $ 13.6\, (2.24) $ $ 2.82\, (2.63)\cdot 10^{-1} $
$w_0 $ $ -0.9 $ $ 5.35\, (7.78)\cdot 10^{-2} $ $ 2.65\, (3.23)\cdot 10^{-2} $
${c^2_{\textrm{eff}}}$ $ 1.\, (0.33) $ $ 0.214 \, (1.41) $ $ 2.89 (2.72)\cdot 10^{-1} $
$n_s $ $ 0.963 $ $ 1.9\, (3.63)\cdot 10^{-2} $ $ 5.26\, (5.32)\cdot 10^{-3} $
$\tau $ $ 0.088 $ $ 2.78\, (2.69)\cdot 10^{-3} $ $ 7.14\, (6.79)\cdot 10^{-4} $
$\ln[10^{10} A_s] $ $ 3.09784 $ $ 7.32\, (8.26)\cdot 10^{-1} $ $ 2.44\, \, (2.44)\cdot 10^{-2} $
--------------------------- ----------------------------- ----------------------------------- --------------------------------------- --
Omniscope $z>7$ {#sec:om}
---------------
redshift slices $B_W$ $D$ $f_{cover}$ $t_{obs}$ $f_{sky}$
----------------- -------- ------- ------------- ----------- ----------- -- --
9/10/11/12 10 Mhz 10 km 0.1 1 yr 0.5
: Specifications of the omniscope-like experiment for which we have considered $10^6$ antennas, see also Ref. [@Clesse:2012th].[]{data-label="tab:omn"}
We provide in Tab. \[tab:omn\] the specifications of the future omniscope-like experiment explored here. Table \[tab:omn12\] shows the $1\sigma$ errors for the two fiducial models previously illustrated for the CHIME-like experiment [^4]. While the $\Omega_{\rm b} h^2$ parameter can be measured with a precision similar to the one achieved with current CMB data, the $\ln A_s$ parameter is still better constrained by the latter measurements. For the setup and the fiducial model considered here, the errors on $\tau$ and $n_s$ are significantly better than for CMB experiments, see also the discussion in Ref. [@Clesse:2012th]. The addition of Planck the and/or the BICEP2 priors does not change much the overall picture for the marginalised errors depicted in Tab. \[tab:omn\]. Let us emphasise that we did not take into account extra ionising sources that can severely damage the variances of the reionisation model parameters, see e.g. [@Mao:2008ug; @Clesse:2012th].
Concerning the dark energy parameters, the constraints on the background parameters $w_0$ and ${\Omega_{\textrm{eDE}}}$ reach high precision levels, with 2% and 7% (6%) errors, respectively, for ${\Omega_{\textrm{eDE}}}=0.01 (0.03)$. A similar precision on the measurement of the dark energy clustering parameters ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$ is obtained with future 21 cm measurements, except for the case in which $c_{vis}^2= 0$ and ${c^2_{\textrm{eff}}}= 1.0$. For this particular scenario, the constraint on ${c^2_{\textrm{eff}}}$ is very poor.
Figure \[fig:omn12-2\] shows the two-dimensional 1 and 2$\sigma$ allowed regions in a reduced number of parameters for the fiducial scenario with ${\Omega_{\textrm{eDE}}}= 0.01$, $c_{vis}^2= 0$ and ${c^2_{\textrm{eff}}}= 1.0$. The top panel of Fig. \[fig:omn12-2\] illustrates the expected correlation in the $(w,n_s)$ plane. As in the case of the analysis of Sec. \[sec:current\], ${\Omega_{\textrm{eDE}}}$ and $w_0$ are anti-correlated, and therefore there exists a mild anti-correlation between ${\Omega_{\textrm{eDE}}}$ and $n_s$, as depicted in the bottom panel of Fig. \[fig:omn12-2\]. We also depict in red solid lines the resulting contours after adding the Planck measurements.
--------------------------- ----------------------------- ----------------------------------- ----------------------------------
fiducial$_1$ (fiducial$_2$) Omniscope Omniscope
+ Planck & WP
$\Omega_{\rm b} h^2$ $ 0.02258 $ $ 2.85\, (5.75)\cdot 10^{-5} $ $ 2.64\,(4.79)\cdot 10^{-5}$
$h $ $ 0.71 $ $ 5.51\, (5.54)\cdot 10^{-3} $ $ 3.39\, (3.78)\cdot 10^{-3}$
$\Omega_{\rm c} h^2$ $ 0.1109 $ $ 2.51\, (5.73)\cdot 10^{-4} $ $ 2.44\, (4.65)\cdot 10^{-4}$
${\Omega_{\textrm{eDE}}}$ $ 0.01\, (0.03) $ $ 0.697\, (1.6)\cdot 10^{-3} $ $ 0.684\, (1.47)\cdot 10^{-3}$
${c^2_{\textrm{vis}}}$ $ 0.\, (0.33) $ $ 1.93\, (1.4)\cdot 10^{-1} $ $ 1.59\, (1.21)\cdot 10^{-1}$
$w_0 $ $ -0.9 $ $ 1.53\, (1.56)\cdot 10^{-2} $ $ 0.953 (1.09)\cdot 10^{-2}$
${c^2_{\textrm{eff}}}$ $ 1.\, (0.33) $ $ 1.78\, (0.22) $ $ 2.86\, (1.7)\cdot 10^{-1}$
$n_s $ $ 0.963 $ $ 2.89\, (4.27)\cdot 10^{-4} $ $ 2.65\, (3.96)\cdot 10^{-4}$
$\tau $ $ 0.088 $ $ 3.11\, (3.09)\cdot 10^{-5} $ $ 3.1\, (3.08)\cdot 10^{-5}$
$\log[10^{10} A_s] $ $ 3.09784 $ $ 3.34\, (3.18)\cdot 10^{-2} $ $ 1.98\, (1.94)\cdot 10^{-2}$
$\Delta_z $ $ 1.5 $ $ 8.39\, (8.8)\cdot 10^{-4} $ $ 8.38\, (8.79)\cdot 10^{-4}$
--------------------------- ----------------------------- ----------------------------------- ----------------------------------
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![$1$ and $2\sigma$ allowed regions from the Fisher analysis with an omniscope-like experiment with four redshift slices $z=9,10,11,12$ for the fiducial model 1. The addition of the Planck measurements results in the continuous red contours.[]{data-label="fig:omn12-2"}](fid1-ns-w.png "fig:"){width="6cm"}
![$1$ and $2\sigma$ allowed regions from the Fisher analysis with an omniscope-like experiment with four redshift slices $z=9,10,11,12$ for the fiducial model 1. The addition of the Planck measurements results in the continuous red contours.[]{data-label="fig:omn12-2"}](fid1-ns-ome.png "fig:"){width="6cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Summary and Conclusions {#sec:conclusions}
=======================
In the last few years Cosmic Microwave Background (CMB) measurements have reached an extremely high sensitivity, allowing for high precision cosmology and providing, therefore, very tight constraints on the basic parameters governing the standard $\Lambda$CDM model. The recent claimed detection of primordial B-modes from the BICEP2 experiment has also offered new insights in cosmology. Here we have exploited the former signal, together with the latest CMB measurements, to update the constraints on an Early Dark Energy component. We find ${\Omega_{\textrm{eDE}}}<0.009$ at $95\%$ CL when Planck, WMAP polarization and BICEP2 data are considered, assuming that the early dark energy component can be described by a perfect fluid. If the former assumption is relaxed, and the dark energy perturbation parameters ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$ are allowed to vary freely, ${\Omega_{\textrm{eDE}}}$ turns out to be less well constrained. Furthermore we find that current CMB measurements are unable to constrain ${c^2_{\textrm{eff}}}$ and ${c^2_{\textrm{vis}}}$.
In this case, future cosmological measurements of the $21$ cm line can be crucial. In the optimistic approach followed here (i.e. in the absence of foregrounds or extra ionising sources), our Fisher matrix analyses of future data from an omniscope-like experiment show that the combination of these $21$ cm cosmological probes and current CMB measurements will be able to distinguish between the canonical quintessence scenario (characterised by ${c^2_{\textrm{eff}}}=1$ and ${c^2_{\textrm{vis}}}=0$) and other possible models (with non standard clustering parameters, as, for instance, with ${c^2_{\textrm{eff}}}=0.33$ and ${c^2_{\textrm{vis}}}=0.33$) with $2\sigma$ significance, in the presence of a non-negligible early dark energy component ${\Omega_{\textrm{eDE}}}$. The errors on the energy density of the former parameter from the joint analysis of future $21$ cm data and current CMB measurements, assuming ${\Omega_{\textrm{eDE}}}\,=\,0.01 \, (0.03)$, are $0.684 \, (1.47)\cdot 10^{-3}$. Future $21$ cm probes can therefore achieve a precision below $10\%$ in the measurement of an early, non-homogeneous dark energy component.
Acknowledgements {#acknowledgements .unnumbered}
================
OM is supported by the Consolider Ingenio project CSD2007–00060, by PROMETEO/2009/116, by the Spanish Grant FPA2011–29678 of the MINECO. OM and MA are also partially supported by PITN-GA-2011-289442-INVISIBLES. We also thank the spanish MINECO (Centro de excelencia Severo Ochoa Program) under grant SEV-2012-0249. LLH is supported through an “FWO-Vlaanderen” post doctoral fellowship project number 1271513. LLH also recognizes partial support from the Strategic Research Program “High Energy Physics” of the Vrije Universiteit Brussel and from the Belgian Federal Science Policy through the Interuniversity Attraction Pole P7/37 “Fundamental Interactions”.
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[^1]: Notice that the clustering properties of a universe with $-1<w<-1/3$ deviate from those of a $\Lambda$CDM universe with $w=-1$ and therefore it can be inconsistent with observations [@effects].
[^2]: Note that $\sigma$ here is related to the variable $ \pi$ in [@Hu:1998tk] through the relation $\sigma=(2/3)\,
\pi/(1+w)$.
[^3]: $\Theta$ is taken to be lower than $1$ rad to be in agreement with the flat-sky approximation.
[^4]: In this case, we have also marginalised over the parameter $\Delta_z$ specifying the duration of the reionisation process, see [@Clesse:2012th] for more details on the background reionisation model.
| {
"pile_set_name": "ArXiv"
} |
14.85 pt
**Double commutants of multiplication operators on $C(K).$**
**A. K. Kitover**
Department of Mathematics, CCP, Philadelphia, PA 19130, USA
**Abstract.** Let $C(K)$ be the space of all real or complex valued continuous functions on a compact Hausdorff space $K$. We are interested in the following property of $K$: for any real valued $f \in C(K)$ the double commutant of the corresponding multiplication operator $F$ coincides with the norm closed algebra generated by $F$ and $I$. In this case we say that $K \in \mathcal{DCP}$. It was proved in [@Ki] that any locally connected metrizable continuum is in $\mathcal{DCP}$. In this paper we indicate a class of arc connected but not locally connected continua that are in $\mathcal{DCP}$. We also construct an example of a continuum that is not arc connected but is in $\mathcal{DCP}$.
Introduction
============
The famous von Neumann’s double commutant theorem [@N] can be stated the following way. Let $(X, \Sigma, \mu)$ be a space with measure and $f$ be a real-valued element of $L^\infty(X, \Sigma, \mu)$. Let $F$ be the corresponding multiplication operator in $L^2(X, \Sigma, \mu)$, i.e. $(Fx)(t) = f(t)x(t)$ for $x \in L^2(X, \Sigma, \mu)$ and $t$ from a subset of full measure in $X$. Then $$\{F\}^{cc} = \mathcal{A}_F$$ where $\{F\}^{cc}$ is the double commutant (or bicommutant) of $F$, i.e. $\{F\}^{cc}$ consists of all bounded linear operators on $L^2(X, \Sigma, \mu)$ that commute with every operator commuting with $F$ and $\mathcal{A}_F$ is the closure in the weak (or strong) operator topology of the algebra generated by $F$ and the identity operator $I$.
The generalization on the case of complex multiplication operators (or normal operators on a Hilbert space) is then immediate. Quite naturally arises the question of obtaining similar results for multiplication operators on other Banach spaces of functions. De Pagter and Ricker proved in [@PaR] that von Neumann’s result remains true for spaces $L^p(0,1), 1 \leq p < \infty$, and more generally for any Banach ideal $X$ in the space of all measurable functions such that $X$ has order continuous norm and $L^\infty(0,1) \subset X \subseteq L^1(0,1)$. But they also proved that the double commutant of the operator $T$, $(Tx)(t) = tx(t), x \in L^\infty, t \in [0,1]$, is considerably larger than the algebra $\mathcal{A}_T$ and consists of all operators of multiplication by Riemann integrable functions on $[0,1]$. The last result gives rise to the following question: let $C(K)$ be the space of all continuous real-valued functions on a Hausdorff compact space $K$. When is it true that for every multiplication operator $F$ on $C(K)$ its double commutant coincides with the algebra $\mathcal{A}_F$? This property is obviously a topological invariant of $K$ and we will denote the class of compact Hausdorff spaces that have it as $\mathcal{DCP}$ (short for double commutant property).
Continuums with $\mathcal{DCP}$ property
========================================
In [@Ki] the author proved that if $K$ is a compact metrizable space without isolated points then the following implications hold.
1. If $K$ is connected and locally connected then $ K \in \mathcal{DCP}$.
2. If $K \in \mathcal{DCP}$ then $K$ is connected.
In the presence of isolated points the analogues of the above statements become more complicated (see [@Ki Theorem 1.15]). To avoid these minor complications and keep closer to the essence of the problem we will assume that the compact spaces we consider have no isolated points.
A simple example (see [@Ki Example 1.16]) shows that the condition that $K$ is connected is not sufficient for $K \in \mathcal{DCP}$.
\[e1\] Let $K$ be the closure in $\mathds{R}^2$ of the set $\{(x, \sin{1/x}) : x \in (0,1]\}$. Let $f(x,y) =x, (x,y) \in K$, and let $F$ be the corresponding multiplication operator. Then it is easy to see (see details in [@Ki Example 1.16]) that the double commutant $\{F\}^{cc}$ consists of all operators of multiplication on functions from $C(K)$ but $\mathcal{A}_F$ consists of operators of multiplication on functions from $C(K)$ that are constant on the set $\{(0,y): y \in [0,1]\}$.
Therefore the next question is whether the condition that $K$ is connected and locally connected is necessary for $K \in \mathcal{DCP}$? Below we provide a negative answer to this question. In order to consider the corresponding example let us recall the following two simple facts.
\[p1\] Let $K$ be a compact Hausdorff space and $f \in C(K)$. Let $F$ be the corresponding multiplication operator. Then
1. The double commutant $\{F\}^{cc}$ consists of multiplication operators.
2. The algebra $\mathcal{A}_F$ coincides with the closure of the algebra generated by $F$ and $I$ in the operator norm.
$(1)$. Let $T \in \{F\}^{cc}$. Let $\mathbf{1}$ be the function in $C(K)$ identically equal to 1. Clearly for every $a \in C(K)$ the operator $F$ commutes with the multiplication operator $A$ where $Ax=ax, x \in C(K)$. Therefore for any $a \in C(K)$ $T$ commutes with $A$ and $T(a) = T(a\mathbf{1})= TA\mathbf{1} = AT\mathbf{1} = aT\mathbf{1} = (T\mathbf{1})a$. Hence if we take $g = T\mathbf{1}$ then $T$ coincides with the multiplication operator $G$ generated by the function $g$.
$(2)$ If $T \in \{F\}^{cc}$ then by part $(1)$ of the proof $T=G$ where $G$ is a multiplication operator by a function $g \in C(K)$. It remains to notice that $\|G\| = \|G \mathbf{1}\|_{C(K)}$ and therefore on $\{F\}^{cc}$ the convergence in strong operator topology implies convergence in the operator norm.
\[c1\] Let $f \in C(K)$ and $F$ be the corresponding multiplication operator. The following two statements are equivalent.
$(1)$ $\{F\}^{cc} = \mathcal{A}_F$.
$(2)$ For any $G \in \{F\}^{cc}$ and for any $s, t \in K$ the implication holds $$f(s) = f(t) \Rightarrow g(s) = g(t),$$ where $g \in C(K)$ is the function corresponding to the operator $G$.
In what follows our main tool will be the following lemma which was actually proved though not stated explicitly in [@Ki] (see [@Ki Proof of Theorem 1.14]).
\[l1\] Let $K$ be a compact metrizable space, $f \in C(K)$, and $F$ be the corresponding multiplication operator. Let $G \in \{F\}^{cc}$ and $g$ be the corresponding function from $C(K)$. Let $u, v \in K$ be such that
- $f(u) = f(v)$.
- The points $u$ and $v$ have open, and locally connected neighborhoods in $K$.
- For any open connected neighborhood $U$ of $u$ there is an open interval $I_U$ in $\mathds{R}$ such that $f(u) \in I_U \subset f(U)$.
Then $g(u) = g(v)$.
We will also need a simple lemma proved in [@Ki Lemma 1.13]
\[l2\] Let $K$ be a compact Hausdorff space, $F, G$ multiplication operators on $C(K)$ by functions $f$ and $g$, respectively and $G \in \{F\}^{cc}$. Let $k \in K$ be such that $Int f^{-1}(\{f(k)\}) \neq \emptyset$. Then $g$ is constant on $f^{-1}(\{f(k)\})$.
Now we are ready to give an example of a metrizable connected compact space $K$ such that $K$ is not locally connected but $K \in \mathcal{DCP}$. Let $B$ be the well known **“broom”**. $$B = cl \{(x,y) \in \mathds{R}^2 : \; x \geq 0, \; y = \frac{1}{n} x, \; n \in \mathds{N}, \; x^2 + y^2 \leq 1 \}.$$
\[p2\] $B \in \mathcal{DCP}$.
Let $f \in C(B)$ and $G \in \{F\}^{cc}$. By part $(1)$ of Proposition \[p1\] $G$ is a multiplication operator. Let $g$ be the corresponding function from $C(K)$. Let $u, v \in B$ and $f(u) = f(v)$. We can assume without loss of generality that $f \geq 0$ and $\min \limits_{k \in B} f(k) = 0$. Let $D = \{k \in B: \; k = (x,0), 0 < x \leq 1\}$. We will divide the remaining part of the proof into four steps.
$(I)$. Assume first that $u, v \in B \setminus D$ and that $0 < f(u) = f(v) < M = \max \limits_{k \in B} f(k)$. [^1] For any $m \in \mathds{N}$ let $B_m = \{(x,y) \in \mathds{R}^2 : \; x \geq 0, \; y = \frac{1}{n} x, \; n \geq m, \; x^2 + y^2 \leq 1 \}$. Then for any large enough $m$ we have $$\min \limits_{k \in B_m} f(k) < f(u) < \max \limits_{k \in B_m} f(k). \eqno{(1)}$$. Notice that for every $m \in \mathds{N}$ the set $B_m$ is a compact, connected and locally connected subset of $B$. Moreover, every point of $B_m$ is a point of local connectedness in $B$ and the set $B_m \setminus \{0,0\}$ is open in $B$. Let $B_m^1 = cl\{k \in B_m : \; f(k) < f(u)\}$ and $B_m^2 = cl\{k \in B_m : \; f(k) > f(u)\}$. There are two possibilities. $(a)$. The set $B^1_m \cap B^2_m$ is empty. In this case, because $B_m$ is connected, $f \equiv f(u)$ on some open subset of $B$ and by Lemma \[l2\] we have $g(u) = g(v)$.
$(b)$. $\exists w \in B^1_m \cap B^2_m$. Because $B$ is locally connected at $w$ the pairs of points $(u,w)$ as well as $(v,w)$ satisfy all the conditions of Lemma \[l1\] whence $g(u) = g(v)$.
$(II)$ Let $u, v \in B \setminus D$ and $f(u) = f(v) = 0$. There are two possibilities. First: $f \equiv 0$ on some open neighborhood of either $u$ or $v$. Then $g(u) = g(v)$ by Lemma \[l2\]. Second: $f$ is not constant on any open neighborhood of either $u$ or $v$. In this case, because $B \setminus D$ is locally connected, we can find sequences $u_n \mathop \rightarrow \limits_{n \to \infty} u$ and $v_n \mathop \rightarrow \limits_{n \to \infty} v$ such that $u_n, v_n \in B \setminus D$ and $0 < f(u_n) = f(v_n) < M, \; n \in \mathds{N}$. Then by the previous step $g(u_n) = g(v_n)$ whence $g(u) = g(v)$. The case $u,v \in B \setminus D$ and $f(u) = f(v) = M$ can be considered similarly.
$(III)$. Now we will assume that $u$ and $v$ are arbitrary distinct points of $B$ and that $0 < f(u) = f(v) < M$. Let again $m \in \mathds{N}$ be so large that inequalities $(1)$ hold. Like on step $(I)$ we have two alternatives $(a)$ and $(b)$. In case $(a)$ we apply again Lemma \[l2\]. In case $(b)$ we cannot apply Lemma \[l1\] directly because $B$ might be not locally connected at $u$ and/or at $v$. Therefore we fix $w \in B^1_m \cap B^2_m$ and consider two subcases. $(b1)$. $f$ is constant on some neighborhood of either $u$ or $v$. Then $f(u) = f(v)$ by Lemma \[l2\]. $(b2)$. $f$ is not constant on any open neighborhood of $u$ or $v$. Let $u_n \in B \setminus D$ be such that $u_n \mathop \rightarrow \limits_{n \to \infty} u$. Because $f(w)$ is an inner in $\mathds{R}$ point of the set $f(W)$ where $W$ is an arbitrary connected open neighborhood of $w$ in $B_m$ we can find a sequence of points $w_n$ such that $w_n \in B_m \subset B \setminus D$ and for any large enough $n \in \mathds{N}$ we have $f(u_n) = f(w_n)$. Then by step $(I)$ $g(u_n) = g(w_n)$ whence $g(u) = g(w)$. Similarly we prove that $g(v) = g(w)$.
$(IV)$. Finally assume that $u$ and $v$ are arbitrary points in $B$ and $f(u) = f(v) = 0$ (the case $f(u) = f(v) = M$ can be considered in the same way). If there is a point $w \in B \setminus D$ such that $f(w) =0$ then we can proceed as in step $(III)$. Let us assume therefore that $f > 0$ on $B \setminus D$. Let $a \in (0,1)$ be the smallest number such that $f(a,0) = 0$. Then for any $n \in \mathds{N}$ such that $n > 1/a$ the set $f(\{(x,0) : \; a - 1/n \leq x \leq a \})$ is an interval $[0, \delta_n]$ where $\delta_n > 0$. Therefore we can find $a_n \in [a - 1/n, a)$ and $u_n \in B \setminus D$ such that $u_n \mathop \rightarrow \limits_{n \to \infty} u$ and $f(u_n) = f(a_n,0), n \in \mathds{N}, \; n > 1/a$, whence by step $(III)$ $g((a_n,0) = g(u_n)$ and therefore $g((a,0) = g(u)$. Similarly, $g(v) = g(a,0) = g(u)$ and we are done.
By analyzing the steps of the proof of Proposition \[p2\] and the properties of the broom $B$ we used, we come to the following more general statement that can be proved in exactly the same way as Proposition \[p2\].
\[p5\] Let $K$ be a compact connected metrizable space. Assume that there are compact subsets $K_m, m \in \mathds{N}$ of $K$ with the properties.
1. $K_m \varsubsetneqq K_{m+1}$.
2. $K_m$ is connected and locally connected.
3. The interior of $K_m$ in $K$ is dense in $K_m$, $m \in \mathds{N}$.
4. Every point of $K_m$ is a point of local connectedness in $K$.
5. The set $\bigcup \limits_{m=1}^\infty K_m $ is dense in $K$.
6. For every point $k \in K \setminus \bigcup \limits_{m=1}^\infty K_m $ there is a path in $K$ from $k$ to a point in $\bigcup \limits_{m=1}^\infty K_m $.
Then $K \in \mathcal{DCP}$.
\[e2\] This example is somewhat similar, though topologically not equivalent to the broom. The corresponding compact subspace of $\mathds{R}^2$ is traditionally called the **“bookcase”** and is defined as follows. $$BC = cl \bigcup_{n=1}^\infty \{(x, 1/n) : \; x \in [0,1] \} \cup \{(0,y) : \; y \in [0,1]\} \cup \{(1,y) : \; y \in [0,1]\} .$$ We claim that $BC \in \mathcal{DCP}$.
For any $m \in \mathds{N}$ let $BC_m = BC \cap \{(x,y) \in \mathds{R}^2 : \; y \geq 1/m \}$. Then the compacts $BC_m$ have properties $(1) - (6)$ from the statement of Proposition \[p5\].
The conditions of Proposition \[p5\] and the arc connectedness theorem (see [@Why Theorem 5.1, page 36]) guarantee that the compact space $K$ satisfying the conditions of that proposition is arc connected. It is not known to the author if the arc connectedness of a metrizable compact $K$ is sufficient for $K \in \mathcal{DCP}$, but as our next example shows it surely is not necessary.
\[p4\] Let $K$ be the union of the square $[-1,0] \times [-1,1]$ and the set $\{(x, \sin{1/x}): \; 0 < x \leq 1 \}$. Then $K \in \mathcal{DCP}$.
Let $f \in C(K)$, $F$ be the corresponding multiplication operator, $G \in \{F\}^{cc}$, and $g \in C(K)$ the function corresponding to $G$. We can assume without loss of generality that $f(K) = [0, M]$ where $M > 0$. Let $E = \{(0,y) : \; y \in [-1,1]\}$. Notice that $K$ is locally connected at any point of $K \setminus E$. The set $K \setminus E$ is the union of two disjoint open connected subsets of $K$: $C_1 = [-1,0) \times [-1,1]$ and $C_2 = \{(x, \sin{(1/x)}: x \in (0,1]\}$. Like in the proof of Proposition \[p2\] we have to consider several possibilities.
1. If $u, v \in C_1$ or $u, v \in C_2$ and $0 < f(u) = f(v) < M$. In this case we can prove that $g(u) = g(v)$ in very much the same way as in step $(I)$ of the proof of Proposition \[p2\] by considering the sets $C_{1,m} = [-1 \times -1/m], m \in \mathds{N}$ (respectively the sets $C_{2,m} = \{(x, \sin{(1/x)}) : \; 1/m < x < 1 \}$).
2. Let now assume that $0 < f(u) = f(v) < M$, $u \in C_1$, $v \in C_2$, and at least one of the inequalities holds $f(u) < \sup \limits_{k \in C_1}f(k)$ or $f(v) < \sup \limits_{k \in C_2}f(k)$. Then like in the proof of Proposition \[p2\] we can either find an open subset of $K$ on which $f$ is identically equal to $f(u)$ and apply Lemma \[l2\], or there is a point $w \in C_1 \cup C_2 $ such that $f(w) = f(u)$ and for every open connected neighborhood $W$ of $w$ there is an open interval $I_w$ such that $f(w) \in I_w \subset f(W)$, and in this case we can apply Lemma \[l1\].
3. Let $0 < f(u) = f(v) < M$, $u \in C_1$, $v \in C_2$, $f(u) = \sup \limits_{k \in C_1}f(k)$, and $f(v) = \inf \limits_{k \in C_2}f(k)$. It follows immediately that $f \equiv f(u) $ on $E$. For any $m \in \mathds{N}$ let $U_m$ be the open disk centered at $u$ and of radius $1/m$, $V_m = \{(x, \sin{(1/x)}): x_v -1/m < x < x_v +1/m \}$ where $x_v$ is the $x$-coordinate of point $v$, and $W_m = K \cap (-1/m, 1/m) \times [-1,1]$. For any large enough $m$ the sets $U_m$, $V_m$, and $W_m$ are disjoint open neighborhoods in $K$ of $u$, $v$, and $E$, respectively. We can assume that $f$ is not identically equal to $f(u)$ on any open subset of $K$; indeed, otherwise we are done by Lemma \[l2\]. Then $f(cl U_m) = [f(u) - \alpha_m, f(u)]$, $f(cl V_m) = [f(u), f(u) + \beta_m]$, and $f(cl W_m) = [f(u) - \gamma_m , f(u) + \delta_m]$, where $\alpha_m, \beta_m, \gamma_m, \mathrm{and}\; \delta_m \searrow 0$. Therefore we can find points $u_m \in cl U_m$, $v_m \in cl V_m$, $w_m \in cl W_m \cap C_1$, and $z_m \in cl W_m \cap C_2$ such that $f(u_m) = f(w_m)$ and $f(v_m) = f(z_m)$. By part $(1)$ of the proof $g(u_m) = g(w_m)$ and $g(v_m) = g(z_m)$. Let $w$ (respectively, $z$) be a limit point of the sequence $w_m$ (respectively, $z_m$). Then $w, z \in E$, $g(u) =g(w)$, and $g(v) = g(z)$. It remains to prove that $g(w) = g(z)$. If $w = z$ there is nothing to prove, therefore assume that $w \neq z$. Let $A_m$ (respectively, $B_m$) be the intersection of the closed disk with the center $w$ (respectively, z) and of radius $1/m$ with the closure of $C_2$. Recalling our assumption that $f$ is not identically equal to $f(u)$ on any open subset of $K$ we see that we can find sequences $a_m, b_m \in C_1$ that converge to $w$ (respectively, to $z$) and such that $f(a_m) = f(b_m)$. By step 1 $g(a_m) = g(b_m)$ whence $g(w) = g(z)$.
4. The implications $f(u) = f(v) =0 \Rightarrow g(u) = g(v)$ and $f(u) = f(v) =M \Rightarrow g(u) = g(v)$ can be easily proved by using the same type of reasoning as in parts $(1)$ - $(3)$.
Finally let us state some open questions.
\[pr1\]
1. Is it possible to characterize the metrizable continua from the class $\mathcal{DCP}$ in purely topological terms not involving multiplication operators?
2. In particular, is it true that any metrizable arc connected continuum belongs to $\mathcal{DCP}$?
3. This question is a special case of the previous one. Let $C$ be the standard Cantor set and $$K = \{(x,y): \; x \in C, y \in [0,1] \} \cup \{(x,0): x \in [0,1]\}.$$ Is it true that $K \in \mathcal{DCP}$? A positive answer to question $(3)$ would be in the author’s opinion a strong indication that the answer to question $(2)$ should also be positive.
[999]{}
Kitover A.K., *Bicommutants of multiplication operators*., Positivity, Volume 14, Issue 4, 2010, pp 753-769. von Neumann, J. *Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren*, Math. Ann. **102** (1929), 370-427. de Pagter B., Ricker W. J., *Bicommutants of algebras of multiplication operators*. Proc. London Math. Soc. (3) **72**(2) (1996), 458–480. Whyburn G.T., *Analytic topology*, AMS Colloquium Publications, v. 28, 1942.
[^1]: We can assume of course that $M > 0$ because otherwise $F = 0$ and the statement $\{F\}^{cc} = \mathcal{A}_F$ becomes trivial.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
[*Abstract*]{}. – Using the Quantum Inverse Scattering Method we construct an integrable Heisenberg-XXZ-model, or equivalently a model for spinless fermions with nearest-neighbour interaction, with defects. Each defect involves three sites with a fine tuning between nearest-neighbour and next-nearest-neighbour terms. We investigate the finite size corrections to the ground state energy and its dependence on an external flux as a function of a parameter $\nu$, characterizing the strength of the defects. For intermediate values of $\nu$, both quantities become very small, although the ground state wavefunction remains extended.
$(^*)$ Electronic mail: Peter.Schmitteckert@physik.uni-augsburg.de
author:
- |
[P. Schmitteckert$(^*)$, P. Schwab]{} and [U. Eckern]{}\
[*Institut für Physik, Universität Augsburg*]{}\
[*D-86135 Augsburg, Germany*]{}
date: '18.5.1995'
title: 'Quantum Coherence in an Exactly Solvable One-dimensional Model with Defects'
---
\
------- ---------- ---------------------------------------
PACS: 71.27.+a Strongly correlated electron systems.
71.30.+h Metal-insulator transitions.
75.10.Jm Quantized spin models.
------- ---------- ---------------------------------------
\
[*Introduction*]{}. – Three recent experiments have demonstrated that persistent currents, periodic in the magnetic flux, exist in mesoscopic metal [@Levy] and semiconductor [@Benoit93] rings at very low temperatures. Surprisingly, though the current is found to be small, of the order of $\sim
ev_F/L$ for single rings ($v_F$ is the Fermi velocity, and $L$ the circumference), it is still two orders of magnitude larger than expected theoretically, at least for the metal rings studied in [@Levy]. In the latter, the electron motion is diffusive, i.e. the elastic mean free path is much smaller than the circumference. While it is well established that the Coulomb interaction gives an important contribution to the current for a measurement on an ensemble of rings [@Ambegaokar91], the interaction effect in single rings, is far from being understood theoretically.
In this article, we consider a one-dimensional, interacting model in the presence of a magnetic flux, or equivalently, with twisted boundary conditions. We introduce very special “defects” into the model describing spinless fermions with nearest-neighbour interaction. Despite this inhomogeneity, the model remains integrable and we present exact results for the finite size corrections to the ground state energy, and its dependence on the magnetic flux, as a function of a parameter $\nu$ characterizing the strength of the defects. Clearly, our investigation does not provide an answer to the questions raised by the experiments (there, the number of transverse channels is much larger than one). Instead, our work is closely related to, and an extension of, various recent theoretical studies \[4–9\] of quantum coherence in strongly interacting electron systems.
[*Construction of the model*]{}. – Using the Quantum Inverse Scattering Method (QISM), we construct our model from the ${\cal R}$ and ${\cal L}$ matrices of the Heisenberg-XXZ-model on an inhomogeneous lattice as, for example, described in [@Korepin93]. The central equation of the QISM is the Yang-Baxter equation, which guarantees that a scattering process factorizes in two-particle scattering processes and does not depend on the order of these. In order to construct a model with defects, we allow that the local ${\cal
L}_n$ matrix depends, in addition to the spectral parameter $\lambda$, on a parameter $\nu_n$, ${\cal L}_n(\lambda) = {\cal L}(\lambda+\nu_n)$. The transfer matrix is given by $T(\lambda)= \mbox{Tr} \prod_{n=1}^{M} {\cal
L}(\lambda + \nu_n)$, where $M$ denotes the number of lattice sites. To include twisted boundary conditions, we multiply the ${\cal L}_M$ matrix of the Heisenberg-XXZ-ring with $\exp{\left( {i}\phi\,\hat{\sigma}^z /2\right)}$.
The Hamiltonian is then given as the logarithmic derivative of the transfer matrix with respect to $\lambda$, at a specific value [@Korepin93]. In particular, in the special case in which all $\nu_n=0$, we obtain the usual XXZ-model, which can be transformed to a spinless fermion model by a Jordan-Wigner transformation. For a general set of parameters, $\{\nu_n\}$, it is difficult to determine the Hamiltonian explicitly, with one exception, namely where there are no defects on neighbouring sites, i.e. $\nu_n\nu_{n+1}=0$ for all $n$. This is the situation we study in the following. As an illustration, consider a vanishing nearest-neighbour interaction, and a single defect at the site $n_1$ characterized by the parameter $\nu$. The resulting Hamiltonian is given by $$\begin{aligned}
{\cal H} &=& {\cal H}^0 \;+\;{\cal H}^I_{n_1}(\nu) \;=\;
- \sum_{n=1}^M \Big( c^{+}_n c^{}_{n+1} \;+\; c^{+}_{n+1}
c^{}_{n}\Big)\;+\; {\cal H}^{I}_{n_1}(\nu)\\
{\cal H}^I_{n_1}(\nu) &=& \big( 1 - \frac{1}{\cosh \nu}\Big)\,\Big(
c^{+}_{n_1-1} c^{}_{n_1} \;+\; c^{+}_{n_1} c^{}_{n_1+1} \Big)
\;-\; e^{{i}\pi/2}\,\tanh(\nu)\, c^{+}_{n_1-1} c^{}_{n_1+1}\;+\;
\mbox{h.c.} \label{def:HI}\end{aligned}$$ where the $\{c^{+}_n\}$ and $\{c^{}_n\}$ are the standard fermion creation and annihilation operators. The generalization to $r$ defects is straightforward (assuming $\nu_n\nu_{n+1}=0$), $${\cal H}={\cal H}^0+\sum_{\ell=1}^{r} {\cal H}^I_{n_\ell}(\nu_{n_\ell})$$ where $n_\ell$ denotes the location of a defect with strength $\nu_{n_\ell}$. An illustration is given in Fig. \[fig:Defect\]. The expression for the Hamiltonian in the presence of a finite nearest-neighbour interaction is more lengthy but similar in structure, i.e. a defect located at $n_\ell$ affects the lattice sites $n_\ell-1$, $n_\ell$, and $n_\ell+1$ only [@Schmitteckert].
[*Single defect, no interaction*]{}. – As is apparent from Eq. (\[def:HI\]), for $\nu=0$, the Hamiltonian reduces to ${\cal H}^0$, i.e. the standard single-band tight-binding model (the hopping matrix element is chosen to be unity). In the opposite limit, $\nu=\infty$, the lattice site $n_1$ is cut out of the ring. As a result, the model represents free fermions on a ring of $M-1$ sites, however, with an additional phase factor $e^{{i}\delta_1}$, $\delta_1=\pi/2$, for the hopping between $n_1-1$ and $n_1+1$, plus one uncoupled site. We emphasize that the parameters, $\cosh^{-1}(\nu)$ and $\tanh{(\nu)}$, as well as the phase factor $\delta_1=\pi/2$, are fine-tuned in the following sense: a generic impurity breaks translational invariance and lifts the degeneracies of the single-particle spectrum, which are found at $\phi=0,\pm\pi$. While our defects also break translational invariance, this symmetry is replaced by another, of not as clear physical origin. As a result we find that even when changing $\nu$, no degeneracies are lifted — they only occur at different, $\nu$-dependent values of $\phi$. The corresponding $\nu$-dependent symmetry operators can be constructed [@Korepin93].
The localization of electronic states is another, well established phenomenon, in one-dimensional disordered systems. In Fig. \[plot:WF\], we plot the squared modulus of the wavefunction for the single-particle level lowest in energy. Clearly, for the integrable case, the wavefunction is extended though reduced at the defect. Allowing, however, the phase $\delta_1$ to be different from $\pi/2$, which corresponds to the non-integrable case, we find a drastically different behaviour with a clear localization of the wave-function near the defect.
[*Several defects, finite interaction*]{}. – The results for a finite nearest-neighbour interaction, and in the presence of several defects, are obtained from the Bethe equations, which we derive from the algebraic Bethe ansatz, with the result $$\label{eq:BE}
\left[\frac{ \cosh(\lambda_j-{i}\eta)}{\cosh(\lambda_j+{i}\eta)}
\right]^{M-r}
\prod^{r}_{\ell=1}
\frac{\cosh(\lambda_j+\nu_{n_\ell}-{i}\eta)
}{\cosh(\lambda_j+\nu_{n_\ell}+{i}\eta)} \;=\; e^{{i}\phi}\,
\prod^{N}_{ k=1 \atop k\ne j}
\frac{\sinh(\lambda_j-\lambda_k - 2{i}\eta)
}{\sinh(\lambda_j-\lambda_k + 2{i}\eta)}.$$ Here, $N$ is the number of fermions and $M$ and $r$ denote the number of lattice sites and defects, respectively. The nearest-neighbour interaction, in units of the hopping matrix element, is parametrized as $\Delta=\cos{(2\eta)}$ (we consider $-1 < \Delta < 1$ only, $\Delta>0$ corresponds to an attractive interaction). For the spin model and for the fermion model with an odd number of fermions, $\phi=0$ ($\pi$) represents periodic (antiperiodic) boundary conditions, and vice versa for an even number of fermions. The phase $\phi$ is directly related to the magnetic flux $\Phi$, for odd $N$ according to the relation $\phi = 2\pi \Phi/\Phi_0$, $\Phi_0 = h/e$. (For even $N$, $\phi = 2\pi \Phi/\Phi_0 -\pi$.)
The equations (\[eq:BE\]) have a remarkable consequence: given a set of defect parameters $\{\nu_n\}$ the roots $\{ \lambda_j\}$ of the Bethe equations do not depend on the distribution of the defects over the sites. This implies that the defects can be moved or permutated without any change in the energy spectrum, which is quite different from the observation that in mesoscopic physics, thermodynamic and transport properties can vary considerably by moving only one impurity by a few lattice constants.
In the following, we simplify our defect model further, by assuming that $|\nu_{n_\ell}|=\nu$, and choosing an equal number of defects of strength $+\nu$ and $-\nu$. With this choice, the ground state energy is an even function of the phase $\phi$. We define $x= \lim_{r,M\rightarrow\infty} (r/M)$ to be the density of defects. Assuming in addition $N=M/2$, i.e. a half filled band, we calculate in a first step the ground state energy per lattice site in the thermodynamic limit, $e_\infty = \lim_{M\rightarrow\infty} e_M$, $e_M \equiv E_M/M$. Using the standard method [@Korepin93; @YangYang66], we find $$\begin{aligned}
e_\infty &=& -\sin^{2}(2\eta) \int^{\infty}_{-\infty} \frac{
\rho_{\infty}(\lambda) }{\cosh(\lambda+{i}\eta)\cosh(\lambda-{i}\eta)}
\mbox{d}\!\lambda \;-\;\frac{\cos(2\eta)}{2} \label{eq:ETDL}\\
\rho^0_\infty(\lambda)&=& \Big\{\, 2(\pi-2\eta)\,\cosh [
\pi\lambda/(\pi-2\eta) ] \,\Big\}^{-1}\\
\rho_\infty(\lambda) &=& (1-x) \rho^0_\infty(\lambda) \;+\; x\,\Big[
\rho^0_\infty(\lambda+\nu)\;+\; \rho^0_\infty(\lambda-\nu)\Big]/2.
\label{eq:rhoinfty} \label{eq:rhoinftynull}\end{aligned}$$ The last equation relates the density of roots for the infinite system, $\rho_\infty$, to the corresponding density of roots of the homogeneous system, $\rho^0_\infty$ . For $\nu=0$, clearly, $\rho_\infty=\rho^0_\infty$, and we recover the results of [@YangYang66]. On the other hand, for $\nu\rightarrow\infty$, it follows from (\[eq:rhoinftynull\]) that $\rho_\infty = (1-x) \rho^0_\infty$; as a direct consequence, the integral in Eq. (\[eq:ETDL\]) is simply multiplied by the factor $(1-x)$ compared to the homogeneous case, and $e_\infty$ is easily calculated. Apparently, for all values of $\nu$, the ground state energy $e_\infty$ is a linear function of the defect concentration (and it is independent of $\phi$, i.e. the boundary conditions).
The leading order finite size corrections can be determined with the help of the Wiener-Hopf technique, as described in [@Hamer87], with the result $$\begin{aligned}
\label{eq:FSC}
e_M(\phi) - e_\infty&=&-\frac{(\pi^2/6) \sin{(2\eta)}/(\pi-2\eta)}{1-x +
x\cosh{[\pi \nu/(\pi-2\eta)]}}
\left( 1 - \frac{ 3 \phi^2}{4\pi\eta} \right) \frac{1}{M^2},\end{aligned}$$ which for $x=0$ is in agreement with [@Hamer87]. In contrast to $e_\infty$, the finite size corrections depend in a nonlinear way on the defect density, and they are exponentially suppressed, $\sim \exp{[ -\pi\nu/(\pi-2\eta)]}$, in the limit of large $\nu$. The finite size corrections, $(e_M-e_\infty)M^2$, obtained by solving numerically the Bethe equations (\[eq:BE\]), are plotted in Fig. \[plot:FSC\] as a function of the defect parameter $\nu$, for half filling ($N=M/2$), and $\Delta=0.5$, $x=0.2$. The system size is varied from $10^2$ to $10^4$. The numerical result is in perfect agreement with the analytical expression (\[eq:FSC\]), but we also observe that the limits $M\rightarrow\infty$ and $\nu\rightarrow\infty$ do not commute. Taking $\nu\rightarrow\infty$ first, we obtain a system with $r$ sites cut out; however, the occupancy of these sites still appears in the Hamiltonian, but as a good quantum number. The finite size corrections thus correspond to a system of $M-r$ sites, i.e. they are enhanced in magnitude by the factor $(1-x)^{-1} = 5/4$ as compared to a system with $x=0$. For large values of $\nu$, the asymptotic result (\[eq:FSC\]) (applicable for $\nu$ finite, $M\rightarrow\infty$) is only reached for extremely large systems.
Finally, we consider the phase sensitivity of the ground state energy, similiar to Refs. \[4-6\], where the homogeneous system was studied. Here we choose $M=10^3$ and $x=0.2$, and we plot $[e_M(\pi)-e_M(0)]M^2$ as a function of $\nu$ in Fig. \[plot:PS\] for different fillings $N/M$. Close to half filling ($N=440, 490$), the dependence on $\nu$ is very similar to the result described above for the finite size corrections, i.e. the phase sensitivity is very small between $\nu\approx4$ and $\nu\approx8$. In this context, note that Eq. (\[eq:FSC\]) implies that $[e_M(\pi)-e_M(0)]/[e_M(0)-e_\infty] = -3\pi/(4\eta)$; but note also that different $\Delta$’s have been chosen in Figs. \[plot:FSC\] and \[plot:PS\].
The asymptotic (large $\nu$) results shown in Fig. \[plot:PS\] can be explained as follows: in this limit (and for the given parameters), 200 sites are cut out from the system of 1000 sites. But, as discussed above, the occupancy of these sites still enters into the Hamiltonian. The defect phases, $\delta_\ell$, are found to be $\pm 2\eta$, and as a result, we find additional cusps in the energy-phase relation, for example, for $\eta$ slightly less than $\pi/4$, at $\phi=\pm 4 \eta$. Consequently, the number of discontinuities in the persistent current, $I=-\partial E_M/\partial\Phi$, increases, implying a reduction of the phase average of the current over half a period, i.e. $e_M(\pi)-e_M(0)$, compared to its $\nu=0$ value.
For low filling, $N\ll M$, on the other hand, the interaction is unimportant and we may apply the free electron result, $I=-(ev_F/L)\phi/\pi$, which implies that $E_M(\pi)-E_M(0) \approx \pi^2N/M^2$ for small $N$. (In our units, $L=M$ and $\hbar v_F=2 \sin(\pi N/M)$.) Comparing the limits $\nu=0$ and $\nu\rightarrow\infty$, the phase sensitivity is thus approximately enhanced in the latter case by the factor $(1-x)^{-2}$, which is apparent in Fig. \[plot:PS\].
[*Conclusions*]{}. – We have presented exact results for the finite size corrections to the ground state energy and its flux sensitivity for an one-dimensional, interacting model in the presence of defects. Through its construction, the model remains integrable. The defects are “magnetic” in the sense that a defect triangle, compare Fig. \[fig:Defect\], encloses a finite magnetic flux. This means that time reversal symmetry is broken and, for general $\{ \nu_n\}$, the energy is not an even function of $\phi$. Surprisingly, the energy spectrum is independent of the spatial distribution of defects, and we find neither level repulsion nor localized states, which are considered to be generic properties of a “real” impurity. We believe that the absence of these effects is strongly related to the integrability of the model. Nevertheless, the finite size corrections and the phase sensitivity, i.e. the persistent current, can become very small.
\*\*\*
We thank [K.-H. Höck]{} for interesting discussions and useful comments on the manuscript. P. Schmitteckert acknowledges financial support through a graduate student fellowship by the state of Bavaria.
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Figure \[fig:Defect\]:
: \[fig:Defect\] Graphical representation of a defect at the site $n_\ell$, compare Eq. (\[def:HI\]). Note that for $\nu \rightarrow 0$ ($\nu \rightarrow \infty$) the dashed (dotted) lines representing the corresponding hopping contributions are effectively cut.
Figure \[plot:WF\]:
: \[plot:WF\] Squared modulus of the wavefunction of the lowest energy eigenstate for the non-interacting limit in the presence of a single defect ($\nu=1$) at the site $n_1=50$ ($M=100$). For the integrable case ($\delta_1=\pi/2$), the eigenstate is extended, though reduced at the defect, while a detuning of the defect phase ($\delta_1\ne \pi/2$) leads to a localization of the wavefunction.
Figure \[plot:FSC\]:
: \[plot:FSC\] Finite size corrections to the ground state energy, ($e_M - e_\infty)\,M^2$, as a function of the defect parameter $\nu$ for half filling, $N=M/2$. The system size is varied from $M=10^2$ to $M=10^4$; the results for large systems fit perfectly with the analytical expression, Eq. (\[eq:FSC\]) for $\phi=0$. The nearest-neighbour interaction is chosen to be $\Delta=0.5$, and the defect density is $x=0.2$.
Figure \[plot:PS\]:
: \[plot:PS\] Phase sensitivity of the ground state energy, $[ e_M(\phi=\pi) - e_M(0) ]
M^2$, as a function of $\nu$ for different fillings, i.e. numbers of fermions, $N$ ($M=10^3$, $\Delta=0.2$).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Richardson-Lucy unfolding approach is reviewed. It is extremely simple and excellently performing. It efficiently suppresses artificial high frequency contributions and permits to introduce known features of the true distribution. An algorithm to optimize the number of iterations has been developed and tested with five different types of distributions. The corresponding unfolding results were very satisfactory independent of the number of events, the number of bins in the observed and the unfolded distribution, and the experimental resolution.'
address: 'Universität Siegen, D-57068 Siegen, Germany'
author:
- 'G. Zech'
title: 'Iterative unfolding with the Richardson-Lucy algorithm'
---
unfolding; Richardson-Lucy; iterative unfolding
Introduction
============
In many experiments the measurements are deformed by limited acceptance, sensitivity or resolution of the detectors. To be able to compare and combine results from different experiments and to compare the published data to a theory, the detector effects have to be unfolded. While acceptance losses can be corrected for, unfolding resolution effects is quite involved. Naive methods produce oscillations in the unfolded distribution that have to be suppressed by regularization schemes.
Various unfolding methods have been proposed in particle physics [@any91; @cernworkshop; @cowan]. The data are usually treated in form of histograms. This is also the case in the Richardson-Lucy (R-L) method [@rich72; @lucy74] which is especially simple, reliable, independent of the dimension of the histogram and independent of the underlying metric.
Iterative unfolding with the R-L algorithm has initially been used for picture restoration. Shepp and Vardi [@shepp82; @vardi85], and independently Kondor, [@kondor83] have introduced it into physics. It corresponds to a gradual unfolding. Starting with a first guess of the smooth true distribution, this distribution is modified in steps such that the difference between its smeared version and the observed distribution is reduced. With increasing number of steps, the iterative procedure develops oscillations. These are avoided by stopping the iterations as soon as the unfolded distribution, when folded again, is compatible with the observed data within the uncertainties. We will discuss the details below. The R-L algorithm originally was derived using Bayesian arguments [@rich72] but it can also be interpreted in a purely mathematical way [@muelthei86; @muelthei2005]. It became finally popular in particle physics after it had been promoted by D’Agostini [@dago] with the label Bayesian unfolding. In Ref. [@lindemann] it was adapted to unbinned unfolding. In Ref. [@na38] the R-L algorithm was applied to a 4-dimensional distribution.
The present situation in particle physics is unsatisfactory for two reasons: i) There is a lack of comparative systematic studies of the different unfolding methods and ii) the way to fix the degree of smoothing, the regularization strength, is usually only vaguely defined.
In the following section we introduce the notation and formulate the mathematical relations. In Section 3 we discuss regularization and the problem of assigning errors to the unfolded distribution. In Section 4 the R-L iterative approach is described. A criterion is developed to fix the number of iterations that have to be applied and which determine the degree of regularization. Section 5 contains examples. We conclude with a summary and recommendations.
Definitions and basic relations
===============================
An event sample with variables $\{x_{1},\ldots,x_{n}\}$, the *input sample* is produced according to a statistical distribution $f(x)$. It is observed in a detector. The *observed sample* $\{x_{1}^{\prime},\ldots,x_{n^{\prime}}^{\prime}\}$ is distorted due the finite resolution of the detector and reduced because of acceptance losses. We distinguish between four different histograms: The *true histogram* with content $\theta_{j}$, $j=1,\ldots,N$ of bin $j$. $\theta_{j}\propto
\int_{bin\text{ }j}f(x)dx$ corresponds to $f(x)$. The *input histogram* contains the input sample. The content of its bin $j$ is drawn from a Poisson distribution with mean value $\theta_{j}$. The *observed histogram* contains the observed sample with $d_{i}$ events in bin $i$, $i=1,\ldots,M$. The expected number of events $t_{i}$ in bin $i$ is given by $t_{i}\propto
\int_{bin\text{ }i}f^{\prime}(x^{\prime})dx^{\prime}$ where the functions $f^{\prime}$ and $f$ are related through $f^{\prime}(x^{\prime})=\int
g(x^{\prime},x)f(x)dx$ with the response function $g(x^{\prime},x)$. We choose $M>N$ to constrain the problem. The result of the unfolding procedure is again a histogram, the *unfolded histogram*, with bin content $\hat{\theta}_{j}$. We are confronted with a standard inference problem where the wanted parameters are the bin contents $\theta_{j}$ of the true histogram. It is to be solved by a least square (LS) or a maximum likelihood (ML) fit. We discuss only one-dimensional histograms but the corresponding array may represent a multi-dimensional histogram with arbitrarily numbered cells as well.
The numbers $t_{i}$ and $\theta_{j}$ are related by the linear relation $$t_{i}=\sum_{j=1}^{N}A_{ij}\theta_{j} \label{transfer}%$$ with the response matrix $A_{ij}$ $$A_{ij}=\frac{\int_{bin\text{ }i}f^{\prime}(x^{\prime})dx^{\prime}}%
{\int_{bin\text{ }j}f(x)dx}\;.\;$$
$A_{ij}$ is the probability to observe an event in bin $i$ that belongs to the true bin $j$. We calculate $A_{ij}$ by a Monte Carlo simulation, but as we do not know $f(x)$, we have to use a first guess of it. If the size of the bins is smaller than the experimental resolution, the elements of the response matrix show little dependence on the distribution that is used to generate the events.
We assume that the observed values $d_{i}$ fluctuate according to the Poisson distribution with the expectation $t_{i}$ and the variance $\delta_{i}%
^{2}=t_{i}$.
The representation of the unfolded distribution by a histogram is a first smoothing step. We call it *implicit regularization*. With wide enough bins, strong oscillations in the unfolded histogram are avoided. LS or ML fits will produce the parameter estimates $\hat{\theta}_{j}$ together with reliable error estimates. With the prediction $t_{i}$ for $d_{i}$ we can define $\chi^{2}$,
$$\chi^{2}=%
%TCIMACRO{\dsum \limits_{i=1}^{M}}%
%BeginExpansion
{\displaystyle\sum\limits_{i=1}^{M}}
%EndExpansion
\frac{\left[ d_{i}-t_{i}\right] ^{2}}{t_{i}}\;,$$
and the log-likelihood $\ln L$ derived from the Poisson distribution, $$\ln L=\sum_{i=1}^{M}\left[ d_{i}\ln t_{i}-t_{i}\right] \;.\label{likstat}%$$ Minimizing $\chi^{2}$ or maximizing $\ln L$ determines the estimates of the parameters $\hat{\theta}_{j}$. The ML fit is applicable also with small event numbers $d_{i}$ and suppresses negative estimates of the parameter values. Negative values can occur in rare cases.
The regularization and the error assignment
===========================================
In particle physics the data are often distorted by resolution effects. This means that without regularization the number of events in neighboring bins of the unfolded histogram are negatively correlated and as a consequence local fluctuations are observed. More precisely, the fitted parameters $\hat{\theta}_{j}%
,\hat{\theta}_{j^{\prime}}$ in two true bins $j,j^{\prime}$ are anti-correlated if their events have sizable probabilities $A_{ij}%
,A_{ij^{\prime}}$ to fall into the same observed bin $i$. These specific correlations are taken into account in most unfolding methods. An exception is entropy regularization [@nara86; @sch94; @maga98] which also penalizes fluctuations between distant bins.
The $\chi^{2}$ surface of the unregularized fit near its minimum $\chi_{0}%
^{2}$ is rather shallow and large correlated parameter changes produce only small changes $\Delta\chi^{2}$ of $\chi^{2}$ of the fit. The location of the true parameter point in the parameter space is badly known but the surfaces of $\chi_{0}^{2}+\Delta\chi^{2}$ for not too small values of $\Delta\chi^{2}$ are well defined and fix the error intervals which should not be affected by the regularization. We are allowed to move the point estimate but the error intervals should not be shifted. The regularization should lead only to a small increase of $\chi^{2}$. The increase $\Delta\chi^{2}=\chi^{2}-\chi_{0}^{2}$ defines an $N$ dimensional error interval around the fitted point in the parameter space. It can be converted to a $p$-value $$p=\int_{\Delta\chi^{2}}^{\infty}u_{N}(z)dz \label{pvalue}%$$ where $u_{N}$ is the $\chi^{2}$ distribution for $N$ degrees of freedom. Strictly speaking, $p$ is a proper $p$-value only in the limit where the test quantity $\chi^{2}$ is described by a $\chi^{2}$ distribution. Fixing $p$ fixes the regularization strength. A large value of $p$ corresponds to a weak regularization and means that the unfolding result is well inside the commonly used error interval of the likelihood fit. The optimal value of a cut in $p$ depends on the unfolding method. Remark that here the value of $\chi_{0}^{2}$ of the fit is irrelevant; what is relevant is its change due to the regularization. A large value $\chi_{0}^{2}$ could indicate that something is wrong with the model.
In most applications outside physics, like picture restoration, the uncertainties of the unfolded distribution are of minor concern. Of interest are mainly the point estimates which are obtained with a regularization that the user chooses according to his personal experience. In physics problems, the error bounds are as important as the point estimates. The manipulations related to the regularization in most methods constrain the fit and therefore reduce the errors of the unfolded histogram as provided by the unconstrained fit [@hoecker; @truee]. As a consequence, these errors depend on the regularization strength and do not cover the true distribution with a fixed probability. Distributions with narrow structures that are compatible with the data may be excluded. An example for such a behavior is presented in Appendix 1. It is not possible to associate classical confidence intervals to explicitly regularized solutions. As stated above, standard error intervals are provided by fits without regularization.
In the iterative method the errors could in principle be calculated by error propagation but these errors would not be constrained and therefore usually be large and strongly correlated. Furthermore their interpretation would be difficult. Therefore it does not make sense to include them in the graphical representation. A very qualitative way to indicate the errors is presented in Appendix 2.
To document quantitatively the precision of the data, a fit with a small number of bins and without explicit regularization of the unfolded histogram should be done, such that by a wide enough binning artificial oscillations are sufficiently suppressed. The result together with the corresponding error matrix[^1] estimate contain the information that is necessary for a comparison with theoretical predictions or other experiments. An example is given in Appendix 2. Alternatively, the data vector and the response matrix could be kept. These items, however, require some explanation to non-experts.
In case we have a theoretical prediction in analytic form, depending on unknown parameters, we should avoid unfolding and the regularization problem and estimate the parameters directly [@bohm]. A direct fit does not require the construction of a response matrix and is independent of assumptions about the shape of the distribution used to simulate the experiment, parameter inference is possible even with very low event numbers where unfolding is problematic, the results are unbiased and the full information contained in the experimental data can be explored.
The Richardson-Lucy iteration
=============================
The method
----------
Replacing the expected number $t_{i}$ in relation (1) by the observed number $d_{i}$, the corresponding matrix relation $d=A\hat{\theta}$ can be solved iteratively for the estimate $\hat{\theta}$. The idea behind the iteration algorithm is the following: Starting with a preliminary guess $\hat{\theta
}^{(0)}$of $\theta$, the corresponding prediction for the observed distribution $d^{(0)}$ is computed. It is compared to $d$ and for a bin $i$ the ratio $d_{i}/d_{i}^{(0)}$ is formed which ideally should be equal to one. To improve the agreement, all true components are scaled proportional to their contribution $A_{ij}\hat{\theta}_{j}^{(0)}$ to $d_{i}^{(0)}$. This procedure when iterated corresponds to the following steps:
The prediction $d^{(k)}$ of the iteration $k$ is obtained in a *folding step* from the true vector $\hat{\theta}^{(k)}$: $$d_{i}^{(k)}=\sum_{j=1}^{N}A_{ij}\hat{\theta}_{j}^{(k)}\;. \label{folding}%$$
In an *unfolding step*, the components $A_{ij}\hat{\theta}_{j}^{(k)}$ are scaled with $d_{i}/d_{i}^{(k)}$ and added up into the bin $j$ of the true distribution from which they originated: $$\hat{\theta}_{j}^{(k+1)}=\sum_{i=1}^{M}A_{ij}\hat{\theta}_{j}^{(k)}\frac
{d_{i}}{d_{i}^{(k)}}\left/ \alpha_{j}\right. \;. \label{unfolding}%$$
Dividing by the acceptance $\alpha_{j}=\sum_{i}A_{ij}$ corrects for acceptance losses.
The result of the iteration converges to the maximum likelihood solution as was proven by Vardi et al. [@vardi85] and Mülthei and Schorr [@muelthei86] for Poisson distributed bin entries. Since we start with a smooth initial distribution, the artifacts of the unregularized ML estimate (MLE) occur only after a certain number of iterations.
The regularization is performed simply by interrupting the iteration sequence. As explained above, the number of applied iterations should be based on a $p$-value criterion which measures the compatibility of the regularized unfolding solution with the MLE.
To this end, first the number of iterations is chosen large enough to approach the asymptotic limit with the ML solution which provides the best estimate of the true histogram if we put aside our prejudices about smoothness. Folding the result and comparing it to the observed histogram, we obtain $\chi_{0}^{2}$ of the fit.
\[ptb\]
[oscil.EPS]{}
Of course, the MLE does not depend on the starting distribution but the regularized solution obtained by stopping the iteration does. We may choose it according to our expectation. In most cases the detailed shape of it does not matter, and a uniform starting distribution will provide reasonable results.
As may be expected, the speed of convergence decreases with the spatial frequency of the true distribution if we consider a Gaussian type of smearing described by a point spread function. This is shown in Fig. \[oscil\]. Here the true distributions consisting of a superposition of a uniform distribution of $1000$ events and a squared sine/cosine distributions of $9000$ events with $1$ to $6$ oscillations is smeared and distributed into $40$ bins. The corresponding histogram is unfolded to a $20$ bin histogram starting with a uniform histogram. The statistic $\Delta\chi^{2}$ for $20$ degrees of freedom is plotted as a function of the number of iterations. The discrete points are connected by a line. The horizontal line corresponds to a $p$-value of $0.5$. As expected, the number of required iteration steps that are needed to reach the $p=0.5$ value increases with the frequency of the distribution. This means that high frequency contributions and artificial fluctuations of correlated bins are strongly suppressed in the R-L approach. The reason can be inferred from Relation (\[unfolding\]): The parameters $\theta_{j},\theta_{j^{\prime
}}$ of bins $j,j^{\prime}$ that are correlated in that they have similar values $A_{ij}$, $A_{ij^{\prime}}$ are scaled in a similar way and relative fluctuations develop only slowly with increasing number of iterations.
*Remark*: By construction, the R-L method is invariant against an arbitrary re-ordering of the bins. A multidimensional histogram can be transformed to a one-dimensional histogram. A rather general class of distortions can be treated. This is also true for entropy regularization and methods based on truncation of the eigenvalue sequence in singular value decomposition (SVD) [@hoecker] but not for local regularization schemes like curvature suppression [@tikhonov] which is difficult to apply in higher dimensions.
\[ptb\]
[iter2b2040.EPS]{}
The regularization strength
---------------------------
Without recipes how to fix the regularization strength, unfolding methods are incomplete and the results are to a certain extent arbitrary. In most of the proposed methods a recommendation is missing or rather vague. In the iterative method, we have to find a criterion, based on a $p$-value, when to stop the iteration process. The optimum way may depend on several parameters: the number of events, the number of bins, the resolution and the shape of the true distribution. Not all combinations of these parameters can be investigated in detail. We will study some specific Monte Carlo examples to derive a stopping criterion and then test it with further distributions. It will be shown that a general prescription works reasonably well for all studied examples.
The unfolded histogram is compared to the input histogram. In all examples we take care that the estimates of the elements of the response matrix have negligible statistical uncertainties. If not stated differently, the iteration starts with a uniform distribution as a first guess for the true distribution. The observed histogram has, with two exceptions, $40$ bins and the unfolded histogram usually comprises $20$ bins. With the standard settings the value of $\chi_{0}^{2}$ should be compatible with the $\chi^{2}$ distribution with $20$ degrees of freedom because we have $40$ measurements and $20$ estimated parameters.
### Example 1: Two peaks {#example-1-two-peaks .unnumbered}
We start with a two-peak distribution, a superposition of two normal distributions $N(x|0.3;0.10)$, $N(x|0.75;0.08)$ and a uniform distribution $U(x)$ in the interval $0<x<1$. Here $N(x|\mu;\sigma$) is the normal distribution of $x$ with the mean value $\mu$ and the standard deviation $\sigma$. The number of events attributed to the three distributions is $25,000$, $15,000$ and $10,000$, respectively. The experimental distribution is observed with a Gaussian resolution $\sigma=0.07.$ It is of the same order as the width of the peaks. Events are accepted in the interval $0<x,x^{\prime}<1$.
In Fig. \[iter2b2040\] unfolding results for different values of the number of iterations are shown. The shaded histograms (input histograms) correspond to the observation with an ideal detector and are close to the true histogram. The left top plot displays the observed histogram as squares. With increasing number of iterations the unfolded histogram (squares) quickly approaches the true histogram. The agreement is quite good in a wide range of the number of iterations. It deteriorates slowly when increasing the number of iterations beyond $32$. At $1000$ iterations oscillations are visible and after $100,000$ iterations the sequence has approached the maximum likelihood solution with strong fluctuations and no explicit regularization. We find $\chi_{0}%
^{2}=23.4$ for $20$ degrees of freedom.
\[ptb\]
[iterbumbprob.EPS]{}
The variation of $\chi^{2}$ as a function of the number of iterations is shown in Fig. \[bumbprob\] top, left hand scale. The corresponding $p$-value (right hand scale) jumps within a few iterations from a negligible value to a value close to one. To judge the quality of the unfolding, we compute the quantity $X^{2}=\Sigma_{i}(\hat{\theta}_{i}-\theta_{i})^{2}/\theta_{i}$ which is available in toy experiments. It is difficult to estimate the range of values of $X^{2}$ that correspond to acceptable solutions, but qualitatively the agreement of the unfolded histogram with the true histogram improves with decreasing $X^{2}$. The dependence of $X^{2}$ from the iteration number is displayed at the top center of the same figure. The minimum is reached at $14$ iterations with a $p$-value of $0.98$ but there is little change between $8$ and $16$ iterations. The corresponding unfolding result is shown on the right hand side. Repeating the same experiment with ten times less events, i.e. $5,000$, we obtain the results displayed at the bottom of Fig. \[bumbprob\]. Here the best agreement is reached after $9$ iterations.
\[ptb\]
[probcut.EPS]{}
The study is repeated for $5$ different samples. The $p$-values are shown as a function of the number of iterations in Fig. \[probcut\]. All curves start rising nearly at the same iteration, remain close to each other at the beginning but separate at large $p$-values. With $5,000$ events the lowest value of the test quantity $X^{2}$ is always obtained for $8$ or $9$ iterations, while the corresponding $p$-values vary because of the small slopes near $p$-values of one. Therefore, we should base the cut of the chosen number of iterations on a lower $p$-value. The following choice has proven to be quite stable and efficient: We stop the iteration at twice the value at which the $p$-value crosses the $0.5$ line. For the left hand plot with $5,000$ events the crossing is close $4.5$ and thus $9$ iterations should be performed. With $50,000$ events this criterion leads to a choice of $15$ iterations. Actually, from the $X^{2}$ variation, acceptable values are located between $11$ and $16$ iterations. In Table 1 the results for the same distribution but different number of bins of the observed and the unfolded histogram and for different resolutions $\sigma$ are summarized. From left to right the columns contain the number of generated events, the number of bins in the observed and the true histograms, the standard deviation of the Gaussian response function, the number of applied iterations as based on the stopping criterion, $X^{2}$, the corresponding $p$-value, the number of iterations that minimizes $X^{2}$ and the minimal value of $X^{2}$. In each case two independent toy experiments have been performed. The results from the second one are given in parentheses. They are close to those of the first one. In all cases the recipe for the choice of the number of iterations leads in most cases to very sensible results. The $p$-values are close to $1$ in most cases and always above $0.95$.
For the resolution $0.1$ the optimal number of iterations and also the $X^{2}$ values differ considerably from the those found by the stopping criterion. The visual inspection shows however that the unfolded distributions that correspond to the stopping prescription agree qualitatively well with the true distributions. For comparison, the example with $50,000$ events and resolution $0.1$ has also been repeated with a likelihood fit and entropy penalty regularization. The regularization constant was varied until the minimum of $X^{2}$ was obtained. The results was $X^{2}=159$ significantly larger than the value $91$ obtained with iterative unfolding. With the prescription $\Delta\chi^2=1$ [@sch94], $X^{2}=873$ was obtained. Regularization with a curvature penalty is not suited for this example. Here the best value of $X^2$ is $700$.
\[c\][|l|l|l|l|l|l|l|l|]{}events & bins & $\sigma$ & $\#$ & $X^{2}$ & $p$-value & $\#_{best}$ & $X_{best}^{2}$\
50000 & 40/20 & 0.07 & 15 (15) & 33 (40) & 0.989 (0.986) & 15 (14) & 33 (40)\
5000 & 40/20 & 0.07 & 9 (8) & 25 (39) & 0.958 (0.980) & 9 (9) & 25 (39)\
50000 & 40/14 & 0.07 & 18 (16) & 25(32) & 0.978 (0.989) & 16 (17) & 25 (32)\
5000 & 40/14 & 0.07 & 9 (10) & 27 (40) & 0.997 (0.971) & 10 (8) & 26 (38)\
50000 & 40/30 & 0.07 & 13 (13) & 44 (45) & 1.000 (1.000) & 14 (15) & 44 (44)\
5000 & 40/30 & 0.07 & 7 (7) & 28 (39) & 0.997 (1.000) & 8 (8) & 27 (39)\
50000 & 40/20 & 0.05 & 8 (8) & 31 (21) & 1.000 (1.000) & 7 (11) & 31 (21)\
5000 & 40/20 & 0.05 & 5 (6) & 9 (22) & 0.997 (0.971) & 6 (5) & 9 (20)\
50000 & 40/20 & 0.10 & 33 (33) & 143 (148) & 1.000 (1.000) & 205 (176) & 91 (108)\
5000 & 40/20 & 0.10 & 15 (18) & 100 (57) & 1.000 (0.985) & 23 (23) & 77 (52)\
50000 & 80/20 & 0.7 & 15 (15) & 32 (37) & 0.991 (0.985) & 14 (15) & 32 (37)\
5000 & 80/20 & 0.7 & 8 (8) & 26 (36) & 0.970 (0.999) & 7 (8) &26 (36)\
### Interpolation for fast converging iterations
In situations where the response function is narrow, usually the iteration sequence converges quickly to a reasonable unfolded histogram, sometimes after a single iteration. Then one might want to stop the sequence somewhere between two iterations. This is possible with a modified unfolding function. We just have to introduce a parameter $\beta>0$ into (\[unfolding\]): $$\hat{\theta}_{j}^{(k+1)}=\left[ \sum_{i=1}^{M}A_{ij}\hat{\theta}_{j}%
^{(k)}\frac{\hat{d}_{i}}{d_{i}^{(k)}}\left/ \alpha_{j}\right. +\beta
\hat{\theta}_{j}^{(k)}\right] \;\left/ (1+\beta)\right. .
\label{unfoldingsmooth}%$$
The value $\beta=0$ produces the original sequence (\[unfolding\]), with $\beta=1$ the convergence is slowed down by about a factor of two and in the limit where $\beta$ approaches infinity, there is no change. It is proposed to choose $\beta$ such that at least $5$ iteration steps are performed.
\[ptb\]
[iterpub.EPS]{}
Subjective elements
-------------------
Unfolding is not an entirely objective procedure. The choice of the method and the kind of regularization depend at least partially on personal taste. For a given value of $\chi^{2}$ there exist an infinite number of unfolded histograms. There is no objective criterion which would allow us to choose the best solution. Given the R-L iterative unfolding, with the stopping criterion as defined above and a uniform starting distribution all parameters are fixed, but in some rare situations it may make sense to modify the standard method.
### Choice of the starting distribution
Instead of a uniform histogram we may choose a different starting histogram. As long as the corresponding distribution shows little structure, the unfolding result will not be affected very much. If we start in our Example 1 ($50,000$ events) with an exponential distribution $f(x)=e^{-x}$ the unfolded histogram is hardly distinguishable from that with a uniform starting distribution. The difference is less than $1\%$ in all bins except for the two border bins with only about $60$ entries where it amounts to $2\%$. In both cases $15$ iterations are required.
For an input distribution that is close to the true distribution, the results are in most cases again very similar to those of the uniform input distribution, but of course the number of required iterations is reduced to one ore two. The situation is different for distribution with sharp structures, for instance, if there is a narrow peak with a small smooth background. Starting with a uniform distribution a large number of iterations is required which may lead to oscillations in the background region. This unpleasant effect is avoided if we start with a distribution that includes a peak structure and where only few iterations are necessary.
We have to be careful when choosing a starting distribution different from a monotone function. Only statistically well established structures should be modeled in the starting distribution.
The starting distribution can be obtained by fitting a polynomial, spline functions or another sensible parametrization to the data with the method described in Ref. [@bohm].
### Manual smoothing
In the specific example with a narrow peak which we discuss below, starting with a uniform distribution we can also avoid the oscillations if we replace the oscillating part in the true input histogram by a smooth distribution before the last iteration step[^2].
Examples with various distributions
===================================
We test the R-L unfolding and the stopping criterion with four different distributions, a single peak distribution, an exponential distribution, a step distribution and a uniform distribution. The results are displayed in Fig. \[iterpub\]. The number of events and the number of iterations are indicated in the plots. The starting true function is uniform, except for the last column where a rough guess of the true distribution is used. The input histogram is shaded, the unfolded histogram is indicated by squares and the observed histogram is plotted as circles in the left hand graphs.
### Example 2: Single narrow peak {#example-2-single-narrow-peak .unnumbered}
We turn now to a more difficult problem and consider a distribution of $40,000$ events distributed according to $N(x|0.6;0.05)$ and $10,000$ events distributed uniformly. The Gaussian response function with $\sigma=0.07$ is wider than the peak. There is a problem because for the flat region we would be satisfied with few iterations while the peak region requires many iterations. Here $\operatorname{about}$ $60$ iterations are needed because relatively high frequencies are required to model the narrow peak. We get $\chi^{2}=27$ while the value of $\chi_{0}^{2}$ after $100,000$ iterations is $20.6$. The unfolded histogram is shown in Fig. \[iterpub\] top left together with the smeared histogram and the true histogram. The peak is well reproduced. The corresponding results for $5,000$ events is shown at the center of the first row. The right hand plot is obtained with a modified input distribution for the last iteration. The unfolding result after $18$ iterations is used as input, but the flat region is replaced by a uniform distribution and one additional iteration is applied. In this way the artificial oscillations in the background region are reduced.
To test the effect of an improved starting distribution, a superposition of a quadratic basic spline function (b-spline) and a uniform distribution was fitted to the data. Four parameters were adjusted, two normalization parameters, the location and the width of the b-spline bump. With this starting distribution, after a single iteration the input distribution is almost perfectly reproduced. The test quantity $X^{2}$ is $47$ compared to $216$ with a uniform starting distribution.
### Example 3: Exponential distribution {#example-3-exponential-distribution .unnumbered}
$50,000$ events are generated in the interval $1<x<5$ according to an exponential distribution $f(x)=e^{-x}$ and $\sqrt{x}$ is smeared with a Gaussian resolution of $\sigma=0.1$ which means that the smearing of $x$ increases proportional to $\sqrt{x}$. The events are observed in the interval $0.5<x^{\prime}<5$ and distributed into $40$ bins. The convergence is rather fast because the distribution is smooth even though we start with a uniform true distribution. We stop after $5$ iterations and get $\chi^{2}=31.5$ which corresponds to a $p$-value of $0.996$. The results are shown in the second row of Fig. \[iterpub\]. In fact the agreement of the unfolded distribution improves slightly with additional iterations and is optimum after $7$ iterations. With $5,000$ events the convergence is faster and a reasonable agreement is obtained after $3$ iterations. Starting with a first guess of an exponential distribution the result slightly improves (right hand plot).
### Example 4: Step function {#example-4-step-function .unnumbered}
A step function is rather exotic. The sharp edge is not easy to reconstruct. We locate the edge at the center of the interval and superpose two uniform distributions containing $40,000$ events in the interval $0<x<0.5$ and $10,000$ events in the interval $0.5<x<1$ with the resolution $\sigma=0.05$. The unfolding results shown in the third row of Fig. \[iterpub\] are disappointing. The $p$-value of $\ 0.99$ is reached after $25$ iterations with $\chi^{2}=20.63$ ($\chi_{0}^{2}=12.42$). A problem is that to model the sharp edge, many iterations are required while for the flat regions oscillations start after a few iterations. However if we replace the uniform starting distribution by the result displayed in the left hand plot replacing the $16$ bins of the flat region by uniform distributions the result (right hand plot) near the edge is not improved
### Example 5: Uniform distribution {#example-5-uniform-distribution .unnumbered}
A uniform distribution is easy to unfold. $50,000$ events generated in the interval $0<x<1$ with a Gaussian resolution of $\sigma=0.1$ and observed in the same interval are unfolded. As the iteration starts with a uniform distribution, no iteration is necessary and the result is optimal with a $p$-value close to one. The initial value of $\chi^{2}$ is $26.4$ and the minimum value is $19.3$ corresponding to the strongly oscillating ML solution. In the case of $5,000$ events $1$ iteration is applied.
\[c\][|l|rrr|rrr|rrr|rrr|rrr|]{}case & & & & &\
& $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \#\
50,000 & 27 & 216 & 60 & 31 & 33 & 15 & 29 & 20 & 10 & 24 & 600 & 14 & 26 & 3 & 0\
50,000 best & 29 & 209 & 51 & 31 & 33 & 15 & 30 & 19 & 7 & 18 & 488 & 48 & 26 & 3 & 0\
5,000 & 32 & 167 & 18 & 37 & 25 & 9 & 43 & 7 & 2 & 37 & 104 & 3 & 45 & 6 & 1\
5,000 best & 29 & 71 & 70 & 37 & 25 & 9 & 43 & 7 & 2 & 33 & 96 & 6 & 45 & 6 & 1\
### Test of the stopping criterion
In Table 2 we compare the result obtained with the stopping criterion to the result obtained with the optimal number of iterations (denoted by *best* in the table). In all cases the iteration starts with a uniform distribution. The agreement with the observed distribution, indicated by $\chi^{2}$, the compatibility of the unfolded distribution with the input distribution, measured with $X^{2}$ and the number of applied iterations are given. The stopping criterion produces very satisfactory results in all cases. With the exception of the single peak distribution with $5,000$ events, it is close to the optimum. Here the observed discrepancy between the number of iterations from the stopping criterion and the number derived from the minimum of $X^{2}$ is due to the fact that the distribution consists of a flat region where few iteration are needed and the peak region which requires many iteration to converge to an optimal result. Nevertheless also the solution with $18$ iteration is satisfactory.
Summary, conclusions and recommendations
========================================
Iterative unfolding with the R-L approach is extremely simple, independent of the number of dimensions, efficiently damps oscillations of correlated histogram bins and needs little computing time. A general stopping criterion has been introduced that fixes the number of iterations, e.g. the regularization strength, that should be applied. It has a simple statistical interpretation. Its stability has been demonstrated for five different distributions, two different event numbers, two different experimental resolutions and three binnings. The results are very satisfactory. The present study should be extended to more distributions with varying statistics and binning and also be applied to higher dimensions.
In most problems a uniform distribution should be used as starting distribution, but the dependence on its shape is negligible as long as this distribution does not contain pronounced structures. In cases where the observed distribution indicates that there are sharp structures in the true distribution, the iterative method permits to implement these in the input distribution. In this way the number of iterations is reduced and oscillations are avoided.
Standard errors, as we associate them commonly in particle physics to measurements, cannot be attributed to explicitly regularized unfolded histograms. We propose to indicate the precision of the graphical representation of the result qualitatively in a way that is independent of the regularization strength. For a quantitative documentation, the unfolding results without explicit regularization should be published together with an error matrix or its inverse. The widths of the bins of the corresponding histogram have to be large enough to suppress excessive fluctuations.
A quantitative comparison of the R-L unfolding with other unfolding methods is difficult, because in most of them a clear prescriptions for the choice of the regularization strength is missing or doubtful. A sensible comparison requires similar binning and regularization strengths in all methods. The latter could be measured with the $p$-value. Independent of the unfolding method that is used, in publications the values of $\chi^{2}$ obtained with and without regularization should be given to indicate the regularization strength and the reliability of the unfolded distribution.
Whenever it is possible to parametrize the true distribution, the parameters should be fitted directly.
Acknowledgment {#acknowledgment .unnumbered}
==============
I thank Gerhard Bohm for many valuable comments.
Appendix 1: The problem of the error assignment {#appendix-1-the-problem-of-the-error-assignment .unnumbered}
===============================================
\[ptb\]
[svderror.EPS]{}
In most unfolding schemes the oscillations are suppressed, either by introduction of a penalty term in the fit, or by reduction of the effective number of parameters [@blobel]. Both approaches constrain the fit and thus reduce the errors. As a consequence the assigned uncertainties do not necessarily cover the true distribution. An example is shown in Fig. \[svderror\] right hand side. The parameters of the LS fit have been orthogonalized with a singular value decomposition (SVD) [@hoecker]. The left hand plot shows the significance of the parameters which is defined as the ratio of the parameter and its error as assigned by the fit. The $20$ parameters are ordered with decreasing eigenvalues. A smooth cut is applied at parameter $\varepsilon_{c}=11$. Contributions are then weighted by $\phi(\varepsilon)=\varepsilon/(\varepsilon+\varepsilon_{c)}$. In this way oscillations are suppressed that might be caused by an abrupt cut, similar to Gibbs oscillations as observed with Fourier approximations [@hoecker; @blobel]. Obviously the number of $11$ effectively used parameters is insufficient to reproduce the peak and the true distribution is excluded. With the addition of further parameters oscillations start to develop. The problem is especially severe with low event numbers. With $10$ times more events the discrepancy between the true distribution and the unfolded one is considerably reduced.
Regularization with a curvature penalty reduces the statistical errors even in the limit where the resolution is perfect. The errors presented by an experiment that suffers from a limited resolution may be smaller than those of a corresponding experiment with an ideal detector where unfolding is not required.
Appendix 2: The documentation of the results {#appendix-2-the-documentation-of-the-results .unnumbered}
============================================
\[ptb\]
[ilustrate.EPS]{}
In the following we present a possible way to document unfolding results such that they can be compared to theoretical predictions and to other experiments.
The left hand plot of Fig. \[ilustrate\] shows the result of a ML fit of the content of the $10$ bins of a histogram without explicit regularization for Example 1 with $5,000$ events. The errors are indicated. They are large due to the strong negative correlation between adjacent bins which amounts to $80 \%$. The fitted values together with the error matrix can be used for a quantitative comparison with predictions. Instead of the error matrix its inverse could be presented. The inverse is in fact the item that is required for parameter fitting. Even more information is contained in the combination of the data vector and the response matrix. These items, however, require some explanation to non-experts.
The right hand side of Fig. \[ilustrate\] shows a possibility to indicate the precision of an explicitly regularized unfolded histogram. The plot is based on the same data as in the left hand plot. The vertical error bar corresponds to the uncertainty of the bin content neglecting correlations and the horizontal bars indicate the uncertainty in the location of the events. In the absence of acceptance corrections the vertical error of bin $i$ is simply equal to the square root of the bin content, $\sqrt{\theta_i}$. If the average acceptance of the events in the bin is $\alpha_i$, the error is $\theta_i /\sqrt{\alpha_i\theta_i}$. The horizontal bar indicates the experimental resolution. Such a graph is intended to show the likely shape of the distribution but is not to be used for a quantitative comparison with other results or predictions. It usually overestimates the uncertainties but for an experienced scientist it indicates quite well the precision of a result.
[99]{}
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[^1]: Instead of the error matrix its inverse could be published. The inverse is needed if data are combined or if parameters are estimated.
[^2]: A similar but more drastic proposal has been made in Ref. [@dago1].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A *Bagnera-de Franchis variety* $X = A/G$ is the quotient of an abelian variety $A$ by a free action of a finite cyclic group $G \subset \operatorname{Bihol}(A)$, which does not contain only translations. Constructing explicit polarizations and using a method introduced by F. Catanese, we classify split Bagnera-de Franchis varieties up to complex conjugation in dimensions $\leq 4$.'
address: |
Lehrstuhl Mathematik VIII\
Mathematisches Institut der Universität Bayreuth, NW II\
Universitätsstr. 30\
D-95447 Bayreuth\
Germany
author:
- Andreas Demleitner
title: 'Classification of Bagnera-de Franchis Varieties in Small Dimensions'
---
Introduction
============
This work studies free group actions of finite groups $G$ on abelian varieties $A$ and the corresponding quotients. Here, the group $G$ is a group of affine transformations of $A$, but not a subgroup of the group of translations (else, the quotient would be again an abelian variety). A quotient of an abelian variety by such a group $G$ is called a *generalized hyperelliptic variety*. More generally, one defines a *generalized hyperelliptic manifold* to be the quotient of a complex torus by a group $G$ as above.\
The study of these dates back to the beginning of the 20th century, when Bagnera and de Franchis as well as Enriques and Severi published their seminal works [@BdF] and [@Enr-Sev], respectively. In the surface case, the classification result of Bagnera and de Franchis shows that there are no non-projective hyperelliptic manifolds. Since then, several authors have studied hyperelliptic manifolds, as well as related areas that contributed a lot to today’s understanding of this topic. To name only a few works: [@Uchida-Yoshihara], [@Fujiki], [@BGL], [@Catanese-Ciliberto]. In 2001, Lange ([@Lange]) gave a method to classify BdF-varieties up to dimension $4$, using heavily the tables of linear automorphisms of abelian surfaces and threefolds (loc. cit), although he omitted some calculations in his work. It does not seem that this method can be used for the classification in dimension $> 4$ (because tables of linear automorphisms are - as far as we know - currently only known up to dimension $3$). Instead, Catanese [@Fabrizio] introduced a method for the classification based on elementary linear algebra and number theory which will be explained and used in this paper for the classification in higher dimensions.
Let us explain how this work is organized. The first chapter mainly recalls some basic facts we will need and establishes some elementary results concerning combinatorics of automorphisms of complex tori. In section \[bdf-chapter\] we introduce Bagnera-de Franchis varieties as quotients of an abelian variety $A$ by a free action of a finite cyclic group $G$ which is not a subgroup of the group of translations and state a characterization for them: a BdF-variety $X=A/G$ splits as $A = (B_1 \times B_2)/(G \times T_r)$, where $T_r$ is a finite group of translations, such that suitable properties are satisfied (cf. Theorem \[charac\]). Here, $G$ acts on $B_1$ by translation and linearly on $B_2$.\
In Chapter 2 we follow [@Fabrizio] and introduce the *Hodge type* of a $G$-Hodge decomposition, an invariant attached to a faithful representation $G \to \operatorname{GL}({\Lambda})$, where ${\Lambda}$ is a free abelian group of even rank.\
Catanese’s method (loc. cit.) for the classification of BdF-varieties will be discussed in Chapter 3. We will assume here that the lattice ${\Lambda}_2$ of $B_2$ is a module over a direct sum of cyclotomic rings (in this case we call $X$ *split*). This yields a decomposition of our abelian variety $B_2$ into $G$-invariant abelian subvarieties $B_{2,k}$, on which $G$ acts with eigenvalues of order $k$. We go on classifying complex tori which admit a linear automorphism acting only with eigenvalues of order $k$ to be able to list all possible decompositions for $B_2$. In Chapter 4 and Chapter 5, we put all pieces together (such that the conditions in the characterization of BdF-varieties are satisfied) and obtain the following classification result:
\[class-result\] The following classification results hold.
1. There are no BdF-curves.
2. Families of split BdF-varieties $X$ of dimension $\leq 4$ are fully classified up to complex conjugation in <span style="font-variant:small-caps;">Tables 5-7</span>.
3. Families of complex tori of dimension $\leq 5$, which admit a linear automorphism of order $m := |G|$ whose eigenvalues are only primitive $m$-th roots of unity are fully classified up to complex conjugation in sections \[surface-case\] to \[5dim\].
4. Each family of complex tori of dimension $\leq 5$ as in iii) contains an abelian variety.
Moreover, except possibly for the cases listed in <span style="font-variant:small-caps;">Table 7</span>, every family listed in iii) contains a principally polarized abelian variety.
The one-dimensional case i) is an easy consequence of the Riemann-Hurwitz formula, while the classification result for two-dimensional BdF-varieties is exactly the classification result of Bagnera-de Franchis, Enriques-Severi ([@BdF], [@Enr-Sev]). The threefold case was treated by Lange ([@Lange]). However, the result ii) in $\dim(X) = 4$ is new, as well as iii) and iv) are (as far as we know).\
The problem we face during the classification of BdF-varieties is that we do not know whether the classified families of complex tori in iii) really contain abelian varieties. This question will be dealt with in the last chapter: we find explicit polarizations for these, which turn out to be principal in most cases. We also investigate the problem of projectivity from another point of view, explaining how the following result by T. Ekedahl (for a detailed proof, see [@Demleitner]) applies to our situation.
\[Ekedahl\] Let $({\ensuremath{T}},G)$ be a rigid group action of a finite group $G$ on a complex torus ${\ensuremath{T}}$. Then ${\ensuremath{T}}/G$ is projective (or equivalently, $T = A$ is an abelian variety).
**Notation**: We fix the following notation throughout the whole work. We will work over the field ${\ensuremath{\mathbb{C}}}$ of complex numbers. By an *abelian variety*, we will therefore mean a complex abelian variety. The notion of a *ring* will always mean a commutative ring with unit element. The set of natural numbers ${\ensuremath{\mathbb{N}}}$ will denote the set of all non-negative integers. The dual space of a vector space $V$ is denoted $V^\vee$.
[^1]
Preliminaries
=============
In this section, we recall some basic facts which we will need in the sequel. Let ${\ensuremath{T}}= V/{\Lambda}$ be a complex torus. It is well-known (cf. for instance [@Cpl-Ab-Var] ) that ${\ensuremath{T}}$ is an abelian variety if and only if there is an alternating ${\ensuremath{\mathbb{Z}}}$-bilinear form ${\ensuremath{E}}$ on ${\Lambda}$ such that the associated ${\ensuremath{\mathbb{R}}}$-bilinear form $H \colon V \times V \to {\ensuremath{\mathbb{R}}}$ given by $H(v,w) = {\ensuremath{E}}({\ensuremath{\iota}}v,w)+ {\ensuremath{\iota}}{\ensuremath{E}}(v,w)$ is Hermitian and positive definite. These conditions are publicly known as the two *Riemann Bilinear Relations*. The Riemann Bilinear Relations can also be expressed in the following way. The form ${\ensuremath{E}}$ extends ${\ensuremath{\mathbb{C}}}$-linearly to a form ${\ensuremath{E}}$ on ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= V \oplus \overline{V}$. We have $$\begin{aligned}
{\ensuremath{E}}\in (V^\vee \otimes V^\vee) \oplus (V^\vee \otimes \overline{V}^\vee) \oplus (\overline{V}^\vee \otimes V^\vee) \oplus (\overline{V}^\vee \otimes \overline{V}^\vee).
\end{aligned}$$ Hence, ${\ensuremath{E}}$ splits as a sum ${\ensuremath{E}}= {\ensuremath{E}}_1 + H_1 + H_2 + {\ensuremath{E}}_2$ (where ${\ensuremath{E}}_1$ is in the first direct summand, $H_1$ is in the second one, and so on). Now we have (cf. [@Griffiths-Harris p. 327]):
The following statements hold:
1. The first Riemann Bilinear Relation holds if and only if ${\ensuremath{E}}_1 = 0$.
2. The second Riemann Relation holds if and only if the first Riemann Relation holds and the form $H_1(-{\ensuremath{\iota}}z,\overline{w})$ is positive definite.
Combinatorics of Automorphisms of Complex Tori
----------------------------------------------
In this section we develop combinatorical and group-theoretical restrictions concerning automorphisms of complex tori. These results will be very useful to determine all possible classes of BdF-varieties (cf. Chapters 3 and 4).
Let ${\ensuremath{T}}= V/{\Lambda}$ be a complex torus of dimension $n$. Let ${\alpha}\in \operatorname{Aut}({\ensuremath{T}})$ and $\rho \colon \operatorname{Aut}({\ensuremath{T}}) \to \operatorname{GL}(V)$ be the complex representation. This yields a representation $\rho' \colon \operatorname{Aut}({\ensuremath{T}}) \to \operatorname{GL}({\Lambda})$, called the rational representation. An easy observation is the following lemma (cf. [@Cpl-Ab-Var]).
The representations $\rho' \otimes 1$ and $\rho \oplus \overline{\rho}$ are equivalent.
Before we start discussing combinatorial restrictions on automorphisms of complex tori, let us first fix some notation. We define $\operatorname{Aut}({\ensuremath{T}})$ to be the group of all (not necessarily linear) automorphisms of ${\ensuremath{T}}$ and $\operatorname{Aut}({\ensuremath{T}},0)$ to be the subgroup of all linear automorphisms of ${\ensuremath{T}}$, and analogously $\operatorname{End}({\ensuremath{T}},0)$. For ${\alpha}\in \operatorname{Aut}({\ensuremath{T}},0)$, we define $\operatorname{Fix}({\alpha})$ as the set consisting of all $a \in {\ensuremath{T}}$ which are fixed under ${\alpha}$.
The following result can be found, together with its proof, in [@BGL]; we give a different, more elementary proof here.
\[p-pow\] Let ${\alpha}\in \operatorname{Aut}({\ensuremath{T}})$ be a linear automorphism of order $m = p^k$ for a prime $p$ and assume that ${\Lambda}$ is a free ${\ensuremath{\mathbb{Z}}}[\zeta_m]$-module. Suppose furthermore that the eigenvalues of $\rho({\alpha})$ are all primitive $m$-th roots of unity. Then $\operatorname{Fix}({\alpha}) \cong ({\ensuremath{\mathbb{Z}}}/p{\ensuremath{\mathbb{Z}}})^{2\cdot \dim({\ensuremath{T}}) /{\varphi}(m)}$. Here, ${\varphi}$ denotes the Euler totient function.
Denote by $x$ the class of $X$ in ${\ensuremath{\mathbb{Z}}}[\zeta_m] = {\ensuremath{\mathbb{Z}}}[X]/(\phi_m(X))$ and note that ${\alpha}$ acts as multiplication with $x$. We know that $v \in \operatorname{Fix}({\alpha})$ iff $v \in {\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{Q}}}$ and $(x-1)\cdot v \in {\Lambda}$. It is well known that the $p^k$-th cyclotomic polynomial is $\phi_p(x^{p^{k-1}}) = \sum_{i=0}^{p-1} x^{p^{k-1}\cdot i}$. Since ${\Lambda}\cong {\ensuremath{\mathbb{Z}}}[\zeta_m]^{2\cdot \dim(A) /{\varphi}(m)}$, it suffices to show that $\{w \in {\ensuremath{\mathbb{Z}}}[\zeta_m] \otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{Q}}}\, | \, (x-1)\cdot w \in {\ensuremath{\mathbb{Z}}}[\zeta_m]\} \cong {\ensuremath{\mathbb{Z}}}/p{\ensuremath{\mathbb{Z}}}$.\
For this, we abbreviate $l = {\varphi}(p^k) = \deg(\phi_{p^k})$ and we write $w = a_0 + a_1x + ... + a_{l-1}x^{l-1}$ with $a_i \in {\ensuremath{\mathbb{Q}}}$ for all $i$. Then a straightforward calculation shows the claim: $$\begin{aligned}
(x-1)w= -\sum_{i=0}^{l-1} a_ix^i + \sum_{i=1}^{l-1} a_{i-1}x^i - \sum_{i=0}^{p-2} a_{l-1} x^{p^{k-1} \cdot i},
\end{aligned}$$ where we have used the equality $x^l = -\sum_{i=0}^{p-2}x^{p^{k-1}\cdot i}$.\
The condition is now that all the coefficients of powers of $x$ must be integers, so the sum of the coefficients has to be an integer. Adding the coefficients, we find that $-p \cdot a_{l-1}$ has to be an integer. This shows the assertion, as all $a_i$ are congruent to $a_{l-1}$ modulo ${\ensuremath{\mathbb{Z}}}$.
\[torsionpoints\] Let ${\ensuremath{T}}\cong_{\text{top.}} (S^1)^{2n}$ be a torus of real dimension $2n$. Then the preceding homeomorphism gives ${\ensuremath{T}}[m] \cong ({\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}})^{2n}$.
\[el-of-order-m\] It follows from standard combinatorics that the number of elements of order exactly $m$ in $({\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}})^{r}$ is precisely $$\begin{aligned}
\sum_{k=0}^{r-1} {\varphi}(m)\cdot(m-{\varphi}(m))^k\cdot m^{r-k-1}.
\end{aligned}$$
An Introduction to Bagnera-de Franchis Varieties {#bdf-chapter}
------------------------------------------------
We start this section with a few preliminaries on generalized hyperelliptic varieties.
\[hyper\] A *generalized hyperelliptic variety* (resp. *generalized hyperelliptic manifold*) $X = A/G$ of dimension $n$ is the quotient of an abelian variety (resp. a complex torus) $A$ by a finite group $G$ of affine transformations of $A$, such that $G$ acts freely and is not a subgroup of the group of translations. If $G$ is cyclic, $X$ is called a *Bagnera-de Franchis variety* (resp. Bagnera-de Franchis manifold); for short: BdF-variety (BdF-manifold).
\[trans\] Let $X = A/G$ be a generalized hyperelliptic variety. By virtue of our assumption that, in the above situation, $G$ is not a subgroup of the group of translations, we can assume without loss of generality that $G$ does not contain any translation:\
Let $G_T$ be the subgroup of translations in $G$, which is a normal subgroup. Hence, we get an abelian variety $A' = A/G_T$ and a group $G' = G/G_T$ without translations such that $A'/G' \cong A/G$.
Also note that there are no hyperelliptic varieties of dimension $1$ in the above sense, since the Riemann-Hurwitz-formula holds.
\[eigenvalue\] Let $X = A/G$ be a generalized hyperelliptic variety, and $A = V/{\Lambda}$. Write $\operatorname{id}\neq g \in G$ as $g(x) = {\alpha}x + b$, where ${\alpha}\in \operatorname{GL}(V)$ and $b \in V$. Then the property of $g$ acting freely on $A$ is equivalent to the fact that no pair of $(x,{\lambda}) \in V \times {\Lambda}$ solves the equation $$({\alpha}- \operatorname{id})x = {\lambda}- b.$$ Hence, the freeness of the action of $g \in G$ implies that ${\alpha}$ has the eigenvalue $1$.
A BdF-variety (resp. BdF-manifold) $X = A/G$ is said to be of *product type*, if $A = B_1 \times B_2$ is a product of abelian varieties (resp. complex tori) and $G \cong {\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}}$ is generated by an automorphism $g$ acting as $g(a_1,a_2) = (a_1 + b', {\alpha}'a_2)$, where $b' \in B_1$ is an element of order $m$ and ${\alpha}' \in \operatorname{Aut}(B_2)$ is a linear automorphism of order $m$, not admitting the eigenvalue $1$.
We have the following characterization of BdF-varieties:
\[charac\] The following two statements are equivalent:
1. $X = A/G$ is a BdF-variety.
2. $X$ is the quotient of a BdF-variety $(B_1 \times B_2)/G$ of product type by a finite group ${\ensuremath{T_r}}$ of translations such that the following conditions hold.
1. ${\ensuremath{T_r}}$ is the graph of an isomorphism $T_{r,1} \to T_{r,2}$, where $T_{r,i}$ is a finite group of translations of $B_i$.
2. If $g \in G$ is given by $g(a_1,a_2) = (a_1 + b', {\alpha}'a_2)$, then $({\alpha}'-\operatorname{id})T_{r,2} = \{0\}$. \[2nd\]
3. If $g$ is as above of order $m$, then $\langle b' \rangle \cap T_{r,1} = \{ 0 \}$. \[3rd\]
In particular, we may write $X = (B_1 \times B_2)/(G \times {\ensuremath{T_r}})$.
See for instance [@Fabrizio Proposition 15].
Actions of Finite Groups on Complex Tori
========================================
We give the most important definitions for the classification of BdF-varieties.
Let ${\Lambda}$ be a free abelian group of even rank and $G \to \operatorname{GL}({\Lambda})$ be a faithful representation of a finite group $G$. A *$G$-Hodge decomposition* is a decomposition $$\begin{aligned}
{\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= H^{1,0} \oplus H^{0,1}, \, \, \, \overline{H^{1,0}} = H^{0,1},
\end{aligned}$$ into $G$-invariant linear subspaces.
Splitting ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}$ using the canonical decomposition, we can write $$\begin{aligned}
{\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= \bigoplus_\chi U_\chi,
\end{aligned}$$ where the sum runs over all characters belonging to irreducible representations of $G$. Thus, $U_\chi = W_\chi \otimes M_\chi$. Here, $W_\chi$ is the corresponding irreducible representation and $M_\chi$ is a trivial representation. Write $$\begin{aligned}
V := H^{1,0} = \bigoplus_\chi V_\chi
\end{aligned}$$ with $V_\chi = W_\chi \otimes M_\chi^{1,0}$. We define
The *Hodge type* of a $G$-Hodge decomposition is the collection of the dimensions $\nu(\chi) = \dim_{{\ensuremath{\mathbb{C}}}} M_\chi^{1,0}$. Here, $\chi$ runs over all non-real characters.
1. Note that for a non-real irreducible character $\chi$, one has $\nu(\chi) + \nu(\overline{\chi}) = \dim_{\ensuremath{\mathbb{C}}}M_\chi$.
2. All $G$-Hodge decompositions of a fixed Hodge type are parametrized in an open set of a product of Grassmanians: for a real irreducible character $\chi$, one simply chooses a $\frac12\dim(M_\chi)$-dimensional subspace of $M_\chi$, and for a non-real irreducible character, one chooses a $\nu(\chi)$-dimensional subspace of $M_\chi$. Then the condition is that $M_\chi^{1,0}$ and $M_\chi^{0,1} = \overline{M_{\overline{\chi}}^{1,0}}$ do not intersect.
Let $\rho \colon G \to \operatorname{GL}(V)$ be the linear representation which sends $g \in G$ to its linear part. Denote by $G^\vee$ the group of characters of $G$, i.e., the group of group homomorphisms from $G$ to ${\ensuremath{\mathbb{C}}}^*$. For simplicity, we write $\rho_g$ instead of $\rho(g)$. Since $G$ was assumed to be abelian, all the irreducible representations of $G$ have degree $1$. The $\chi$-eigenspace of $G$ is denoted $V_\chi$, i.e., $$\begin{aligned}
V_\chi = \{ v \in V \, | \, \rho_g(v) = \chi(g)\cdot v, \, \forall g \in G\}.
\end{aligned}$$ \[unity\_eigenspaces\] Thus, we can split $V = \bigoplus_{\chi} V_\chi$. Denote by $M$ the set of all characters such that $V_\chi \neq \{0\}$. We then have $V = \bigoplus_{\chi \in M} V_\chi$.\
Of course, we want to apply these considerations to generalized hyperelliptic varieties. In particular, we will have $G$ satisfying the following two properties:
- $G$ acts freely.
- $G$ contains no translations, i.e., $\rho$ is faithful.
We have the following elementary result if $G$ is even cyclic:
Let $\colon G \to \operatorname{GL}({\Lambda})$ be a representation of a finite cyclic group $G = \langle g \rangle$, ${\Lambda}$ a free abelian group of rank $2n$ and ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= \bigoplus_\chi U_\chi$ be the canonical decomposition. If two characters $\chi$ and $\chi'$ have the same order, the spaces $U_\chi$ and $U_{\chi'}$ have the same dimension.
We have $G \to \operatorname{GL}({\Lambda}) \hookrightarrow \operatorname{GL}({\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}})$. Denote by $\rho^0$ the representation $G \to \operatorname{GL}({\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}})$. Let $\chi$ and $\chi'$ be two characters of the same order $k$. Since $\chi$ and $\chi'$ have order $k$ and the characteristic polynomial $f_g$ of $\rho^0_g$ has integral coefficients, a power of the $k$-th cyclotomic polynomial divides $f_g$. This yields that the multiplicities of $\chi(g)$ and $\chi'(g)$ as zeros of $f_g$ are equal. We are now finished, because $\rho_g^0$ is diagonalizable.
Primary BdF-Varieties
=====================
In this section, we want to classify split BdF-varieties in small dimensions. We use Theorem \[charac\], where we saw that a BdF-variety is the quotient of a BdF-variety of product type by a finite group of translations such that the conditions in the quoted theorem are satisfied. We follow [@Fabrizio] for the presentation of the method used for the classification.
1. Assume that a cyclic group $G$ generated by a linear automorphism $g$ of order $m$ acts on an abelian variety (resp. a complex torus) $A$. Then $(A,G)$ is called *primary*, if a generator $g$ of $G$ has only primitive $m$-th roots of unity as eigenvalues. By abuse of notation, we will call an abelian variety (resp. a complex torus) $A$ primary if the group $G$ is clear from the context.
2. A BdF-variety (resp. BdF-manifold) $(B_1 \times B_2)/G$ of product type is said to be *primary*, if the abelian variety $B_2$ (resp. the complex torus $B_2$) is primary.
Let $X = (B_1 \times B_2)/G$ a BdF-variety of product type and write as usual $B_i = V_i/{\Lambda}_i$. In this situation, ${\Lambda}_2$ is a $G$-module, thus a module over the ring ${\ensuremath{\mathbb{Z}}}[G] \cong {\ensuremath{\mathbb{Z}}}[X]/(X^m-1)$. By the Chinese Remainder theorem, we find that ${\ensuremath{\mathbb{Z}}}[G]$ embeds into a direct sum of cyclotomic rings, i.e., we have $$\begin{aligned}
{\ensuremath{\mathbb{Z}}}[X]/(X^m-1) \subset \bigoplus_{k|m} {\ensuremath{\mathbb{Z}}}[X]/(\phi_k(X)),
\end{aligned}$$ where $\phi_k(X)$ is the $k$-th cyclotomic polynomial.
Fabrizio Catanese (private communication) pointed out that the inclusion above is not an isomorphism. In [@Fabrizio], he inadvertently claimed that exactly this was the case, while in fact, this is only true over ${\ensuremath{\mathbb{Q}}}$. Throughout this paper we only deal with the case where ${\Lambda}_2$ is as well a module over the direct sum $\bigoplus_{k|m} {\ensuremath{\mathbb{Z}}}[X]/(\phi_k(X))$. In the general case, one obtains tori which are isogenous to ones with splitting lattices as above (we shall deal with this question with more precision in a future paper).
We write $R := {\ensuremath{\mathbb{Z}}}[x]/(x^m-1)$ and $R_k$ for the cyclotomic ring ${\ensuremath{\mathbb{Z}}}[x]/(\phi_k(x))$. We assume that ${\Lambda}_2$ splits according to the order of the eigenvalues, ${\Lambda}_2 = \bigoplus_{k|m} {\Lambda}_{2,k}$ (note that ${\Lambda}_{2,k}$ is an $R_k$-module). The vector space $V_2$ splits accordingly, $V_2 = \bigoplus_{k|m} V_{2,k}$. We find that $$\begin{aligned}
\label{splitting}
B_2 = \bigoplus_{k|m} B_{2,k},
\end{aligned}$$ where $B_{2,k}$ is a $G$-invariant abelian subvariety of $B_2$ such that a generator $g \in G$ (as in Theorem \[charac\]) acts on $B_{2,k}$ with eigenvalues of order $k$.
We call the abelian variety $B_2$ *split* if it admits a splitting as in \[splitting\]. A BdF variety $X = (B_1 \times B_2)/(G \times T)$ is called *split* if $B_2$ is split.
Let $X$ be a BdF-variety of dimension $n>1$. We furthermore assume that $X = (B_1 \times B_2)/G$ is a primary BdF-variety. In this case, ${\Lambda}_2$ is a module over the ring $R_m = {\ensuremath{\mathbb{Z}}}[x]/(\phi_m(x))$, which is a Dedekind domain (see for example theorem 2.6 in [@Washington]). In fact, ${\Lambda}_2$ is a projective $R_m$-module. We therefore get a splitting ${\Lambda}_2 = R_m^r \oplus I$ of ${\Lambda}_2$ into a free part and an ideal $I \subset R_m$ (see [@Milnor]). Indeed, ${\Lambda}_2$ is free if $R_m$ is a PID, i.e., if the class number $h(R_m) := \# \operatorname{Cl}(R_m)$ is equal to $1$. We know that $R_m$ is a free ${\ensuremath{\mathbb{Z}}}$-module of rank ${\varphi}(m)$ (where ${\varphi}$ denotes the Euler totient function), hence we have $$\begin{aligned}
\label{eqrank}
{\Lambda}_2 = R_m^r \cong {\ensuremath{\mathbb{Z}}}^{{\varphi}(m)\cdot r}
\end{aligned}$$ for all $m$ satisfying $h(R_m) = 1$. As $B_1$ is an abelian variety of positive dimension, we have the following important observation: $$\begin{aligned}
\label{ineq}
{\varphi}(m) \leq 2(n-1).
\end{aligned}$$
Let ${\ensuremath{T}}= V/{\Lambda}$ be a complex torus. If ${\Lambda}= R_m^k$, then ${\ensuremath{T}}$ is called *elementary*.
We have the following well-known table of values ${\varphi}(m)$:\
[lr]{}
${\varphi}(m)$ $m$
---------------- -------------
$1$ $1,2$
$2$ $3,4,6$
$4$ $5,8,10,12$
&
${\varphi}(m)$ $m$
---------------- ------------------
$6$ $7,9,14,18$
$8$ $15,16,20,24,30$
$10$ $11,22$
According to the table on page 353 of [@Washington], we know in particular that $h(R_m) = 1$ for all values of ${\varphi}(m)$ listed above.
In the following sections, we will classify BdF-varieties in small dimensions. To obtain a satisfying classification, one first determines all possibilities for $B_2$. To give a complex structure to $({\Lambda}_2 \otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}})/{\Lambda}_2$ it is sufficient and necessary to give a decomposition ${\Lambda}_2 \otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= V \oplus \overline{V}$ (see [@Griffiths-Harris], pages 326 and 327). This amounts to determining all possible $G$-Hodge decompositions corresponding to ${\Lambda}_2$.
1. Note that, by the above inequality \[ineq\], the case $m=2$ obviously occurs for all dimensions $n>1$. The only automorphism of order $2$ of an abelian variety with finitely many fixed points is multiplication with $-1$, so the case $m=2$ can be easily classified. In what follows, this trivial case will be omitted during the calculations, but will be listed in the tables. (Note that the number of parameters for this case is given by the dimension of the Siegel upper half space.)
2. In the following sections, we classify families of complex tori admitting a faithfully acting linear automorphism of a certain order up to dimension $5$ *up to complex conjugation*. Usually, we will drop the phrase ’up to complex conjugation’ in the statement of our results.
3. We refrain from listing the well-known table of elliptic curves admitting automorphisms of finite order.
The Surface Case {#surface-case}
----------------
In this and the upcoming sections, we classify families of abelian varieties admitting a linear automorphism of certain order acting faithfully. To abbreviate, we denote by ${\ensuremath{(\ast_{n,m})}}$ the following condition to be satisfied by a torus $T$:
------------------------------- --------------------------------------------------------------------------------------------------------------------------------------
${\ensuremath{(\ast_{n,m})}}$ $\dim(T) = n$, and $T$ admits a linear automorphism of some order $m > 2$ having only primitive $m$-th roots of unity as eigenvalues
------------------------------- --------------------------------------------------------------------------------------------------------------------------------------
For now, we assume that ${\ensuremath{T}}$ satisfies ${\ensuremath{(\ast_{2,m})}}$. We write ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= V \oplus \overline{V}$. Using inequality \[ineq\], we find that we only have two possibilities in equation \[eqrank\]:
*Case 1:* ${\varphi}(m) = 2$ (hence $m \in \{3,4,6\}$) and $r = 2$, or\
*Case 2:* ${\varphi}(m) = 4$ (hence $m \in \{5,8,10,12\}$) and $r = 1$.
In the first case, $R_m$ has rank $2$ as a ${\ensuremath{\mathbb{Z}}}$-module. We have the decomposition ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= W_\chi \oplus W_{\overline{\chi}}$ into $2$-dimensional isotypical components. Recall that we have a decomposition ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= V \oplus \overline{V}$. As we have seen in chapter 2, we also get a decomposition for $V$ (i.e., we have ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= V_\chi \oplus \overline{V_{\overline{\chi}}} \oplus V_{\overline{\chi}} \oplus \overline{V_\chi}$). We will always (i.e., in any dimension) distinguish between the cases where the following assumption holds (or does not hold). $$\label{assump}
\text{Only pairwise non-conjugate characters appear in } V.$$
The result [@BGL Proposition 1.8, a)] together with \[eqrank\] now give the following
\[assump-cor\] If ${\ensuremath{T}}= V/{\Lambda}$ is elementary (of arbitrary dimension) such that the rank of ${\Lambda}$ as an $R_m$-module is $1$, then \[assump\] holds.
Note that the case where \[assump\] holds is dealt with in [@Catanese-Ciliberto]; according to the authors, the classification of two-dimensional tori which admit an automorphism of order $m$ acting faithfully (such that \[assump\] holds) is as follows. There are exactly
- $m=3,6$: one isomorphism class, namely $E_\rho \times E_\rho$ ($E_\rho$ being the equianharmonic elliptic curve).
- $m=4$: one isomorphism class, namely $E_{{\ensuremath{\iota}}} \times E_{{\ensuremath{\iota}}}$ ($E_{{\ensuremath{\iota}}}$ being the harmonic elliptic curve).
- $m=5, 10$: one isomorphism class $S_{10}$.
- $m=8,12$: two isomorphism classes $S_m'$, $S_m''$.
Moreover Catanese and Ciliberto prove:
Let ${\ensuremath{T}}$ satisfy ${\ensuremath{(\ast_{n,m})}}$ and $\ref{assump}$. Then the following hold.
1. If $m=3,6$, then $T \cong E_\rho^n$.
2. If $m=4$, then $T \cong E_{{\ensuremath{\iota}}}^n$.
3. If $m=5,10$, then $T \cong S_{10}^{n/2}$.
4. If $m=8,12$, then there are $k, l \in {\ensuremath{\mathbb{N}}}$ such that $T \cong (S_m')^k \times (S_m'')^l$.
Now we analyze the case where \[assump\] does not hold.
\[conj-case-dim2\] Assume that $V$ splits into the two one-dimensional spaces $V_\chi$ and $V_{\overline{\chi}}$, i.e., \[assump\] does not hold. Then there is a two-parameter family of tori $T$ satisfying ${\ensuremath{(\ast_{2,m})}}$.
We denote these families by $S_4$ for $m=4$ and $S_6$ for $m \in \{3,6\}$.
It suffices to show that under the given conditions, one has a two-parameter family of complex tori modulo a group $G$ with the desired properties. This is clear, as we have only the possibility $V = V_\chi \oplus V_{\overline{\chi}}$ (up to complex conjugation) to decompose $V$. But $(V_\chi, V_{\overline{\chi}})$ is contained in the open set $\{(W,W') \in \operatorname{Gr}(1,2) \times \operatorname{Gr}(1,2)|W \cap W' = \{0\} \}$, which is two-dimensional.
We sum up our results in the following
\[surface\_case\] There are exactly $19$ families (up to complex conjugation) of two-dimensional complex tori admitting a non-trivial linear automorphism of finite order acting faithfully. These families are listed in <span style="font-variant:small-caps;">Table 2</span> below. By $p$, we denote the number of moduli of the corresponding family. The rest of the notation used is explained below the table.
<span style="font-variant:small-caps;">Table 1</span>
The table is organized as follows. By $E$ resp. $S$ we denote an arbitrary elliptic curve resp. abelian surface, while $S_4$, $S_6$, $S_{10}$, $S_m'$, $S_m''$ denote the (families of) abelian surfaces introduced above. We decided to list the same torus ${\ensuremath{T}}$ more than once if their entries in $\operatorname{Fix}(g)$ are different. Finally, $\operatorname{Fix}(g)$ denotes the fixed locus of a linear automorphism $g$ of ${\ensuremath{T}}$ of order $m$.
The result of Theorem \[surface\_case\] was already achieved by Fujiki [@Fujiki] with different methods, see also [@MTW] for a study in the case where $m$ is prime.
The Threefold Case
------------------
Let ${\ensuremath{T}}$ satisfy ${\ensuremath{(\ast_{3,m})}}$. The only possibilies are:
*Case 1:* ${\varphi}(m) = 2$ (hence $m \in \{3,4,6\}$) and $r = 3$, or\
*Case 2:* ${\varphi}(m) = 6$ (hence $m \in \{7,9,14,18\}$) and $r=1$.
We first deal with case 1. In the same way as in the surface case there is the decomposition ${\Lambda}\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}= W_\chi \oplus W_{\overline{\chi}} = V_\chi \oplus \overline{V_{\overline{\chi}}} \oplus V_{\overline{\chi}} \oplus \overline{V_\chi}$. We have the following proposition:
\[conj-case-dim3\] Let ${\ensuremath{T}}$ satisfy ${\ensuremath{(\ast_{3,m})}}$ and ${\varphi}(m) = 2$. Then the following statements hold:
1. If \[assump\] holds, then ${\ensuremath{T}}$ is isomorphic to a product of elliptic curves. More precisely, if $m \in \{3,6\}$, then ${\ensuremath{T}}= E_\rho \times E_\rho \times E_\rho$. If $m = 4$, then ${\ensuremath{T}}= E_{{\ensuremath{\iota}}} \times E_{{\ensuremath{\iota}}} \times E_{{\ensuremath{\iota}}}$.
2. If \[assump\] does not hold, there is a four parameter family of complex tori admitting a linear automorphism of order $m$ acting faithfully.
We denote the respective families by $A_m$, $m \in \{4,6\}$. We also use the notation $A_6$ for the family obtained for $m=3$.
The first assertion is clear. For the second assertion, note that the only possible Hodge type is $(1,2)$, so the $G$-Hodge decompositions are parametrized by an open set in $\operatorname{Gr}(1,3) \times \operatorname{Gr}(2,3)$, which has dimension $4$.
Case 2 is easy to analyze. Note that by Proposition \[assump-cor\] we can assume that the eigenvalues of a generator $g \in G$ are distinct and pairwise non-complex conjugate.
\[ab-threefolds\] There are exactly two isomorphism classes of tori satisfying ${\ensuremath{(\ast_{3,m})}}$ for any $m \in \{7,9,14,18\}$. We denote them by $A_7'$, $A_7''$ for $m \in \{7,14\}$ and by $A_9'$, $A_9''$ for $m \in \{9, 18\}$.
We only list one element of each orbit for $m=7,9$. For $m=7$, there are exactly two orbits corresponding to the tuples of pairwise non-complex conjugate characters $(1,2,3)$, $(1,2,4)$. For $m=9$, the two orbits correspond $(1,2,4)$ and $(1,4,7)$.
Now, our assumption \[splitting\] guarantees that we can give a list of all isomorphism classes of abelian threefolds admitting an automorphism of finite order. Here, ${\ensuremath{T}}$ denotes an arbitrary abelian threefold. Again, we refrain from listing the same tori more than once unless their entries in the $\operatorname{Fix}(g)$-column are different. The rest of the notation is as in <span style="font-variant:small-caps;">Table 1</span>.
There are exactly $56$ families (up to complex conjugation) of three-dimensional complex tori admitting a non-trivial linear automorphism of finite order acting faithfully. These families are listed in <span style="font-variant:small-caps;">Table 2</span> below. Again, $p$ denotes the number of moduli.
<span style="font-variant:small-caps;">Table 2</span>\
The Fourfold Case {#4dim}
-----------------
Let ${\ensuremath{T}}$ satisfy ${\ensuremath{(\ast_{4,m})}}$. As in the previous sections, we determine all possibilities for the order $m$ of $G$:
*Case 1:* ${\varphi}(m) = 2$ (hence $m \in \{3,4,6\}$) and $r = 4$, or\
*Case 2:* ${\varphi}(m) = 4$ (hence $m \in \{5,8,10,12\}$) and $r = 2$, or\
*Case 3:* ${\varphi}(m) = 8$ (hence $m \in \{15,16,20,24,30\}$) and $r = 1$.
Most of the proofs of this section are very similar to the proofs given in the previous sections; therefore, we will omit most of them. We start with the first case.
Let ${\ensuremath{T}}$ satisfy ${\ensuremath{(\ast_{4,m})}}$ such that ${\varphi}(m) = 2$. Then the following statements hold:
1. If \[assump\] holds, then ${\ensuremath{T}}$ is isomorphic to a product of elliptic curves. More precisely, if $m \in \{3,6\}$, then ${\ensuremath{T}}= E_\rho^4$. If $m = 4$, then ${\ensuremath{T}}= E_{{\ensuremath{\iota}}}^4$.
2. If \[assump\] does not hold, there are exactly two families of four-dimensional complex tori admitting an automorphism of order $m$, which have $6$ resp. $8$ parameters, respectively.
We denote the respective families in ii) by $X_m^6$, $X_m^8$, $m \in \{4,6\}$. We also use the notation $X_6^6$, $X_6^8$ for the families obtained for $m=3$.
For case 2, we have
Let ${\ensuremath{T}}$ satisfy ${\ensuremath{(\ast_{4,m})}}$ such that ${\varphi}(m) = 4$. Then the following statements hold:
1. If \[assump\] holds, then ${\ensuremath{T}}$ is isomorphic to one of the following complex tori: $(S_k')^2, S_k' \times S_k'', (S_k'')^2$.
2. If \[assump\] does not hold, there are exactly two families of complex tori of dimension $4$ admitting an automorphism of order $m$, which have $2$ resp. $4$ parameters.
We denote the respective families in ii) by $X_m^2$, $X_m^4$, $m \in \{8,10,12\}$. We also use the notation $X_{10}^2$, $X_{10}^4$ for the families obtained for $m=5$.
Finally, the third case is analyzed. Again, we have that \[assump\] holds by Proposition \[assump-cor\].
There are exactly four isomorphism classes of $4$-dimensional complex tori satisfying ${\ensuremath{(\ast_{4,m})}}$ for each $m \in \{15, 16, 20, 30\}$ and exactly five isomorphism classes when $m = 24$.
We only list one element of each orbit for $m=15,16,20,24$: for $m=15$, the orbits correspond to the $4$-tuples of pairwise non-complex conjugate characters $(1,2,4,7)$, $(1,2,4,8)$, $(1,2,7,11)$, $(1,4,7,13)$. For $m=16$, the orbits correspond to $(1,3,5,9)$, $(1,3,5,7)$, $(1,3,9,11)$, $(1,5,9,13)$, while for $m=20$, they correspond to $(1,3,7,9)$, $(1,3,7,11)$, $(1,3,11,13)$, $(1,9,13,17)$. Finally, for $m=24$, the tuples $(1,5,7,11)$, $(1,5,7,13)$, $(1,5,13,17)$, $(1,7,13,19)$, $(1,11,17,19)$ give the five orbits.
We denote representatives of the respective isomorphism classes for $m \in \{15,30\}$ by $X_{30}^{(i)}$, $i \in \{1,2,3,4\}$. For $m \in \{16,20\}$, the representatives are denoted $X_m^{(i)}$, $i \in \{1,2,3,4\}$. Finally, for $m = 24$, the representatives of the five isomorphism classes are denoted $X_{24}^{(j)}$, with $j \in \{1,2,3,4,5\}$.
We now could - in principle - give a list of all $4$-dimensional complex tori admitting a linear automorphism of finite order which have finite fixed locus. This will be omitted, as it is very tedious, yet not very enlightening. In lieu thereof, we will only list those $4$-dimensional complex tori which admit a linear automorphism of order $m$ with eigenvalues of order $m$.
The table below contains $4$-dimensional complex tori which admit a linear automorphism $g$ of order $m$, whose eigenvalues are only primitive $m$-th roots of unity. Conversely, every such complex torus belongs (up to complex conjugation) to one of the families listed below.\
Here, $\tilde{\ensuremath{T}}$ denotes an arbitrary $4$-dimensional complex torus. The remaining notation is as introduced above.
<span style="font-variant:small-caps;">Table 3</span>\
The Fivefold Case {#5dim}
-----------------
Let $T$ satisfy ${\ensuremath{(\ast_{5,m})}}$. We have the following possibilities:
*Case 1:* ${\varphi}(m) = 2$ (hence $m \in \{3,4,6\}$) and $r = 5$, or\
*Case 2:* ${\varphi}(m) = 10$ (hence $m \in \{11,22\}$) and $r = 1$.\
Let $T$ satisfy ${\ensuremath{(\ast_{5,m})}}$ such that ${\varphi}(m) = 2$. Then:
1. If \[assump\] holds, then ${\ensuremath{T}}$ is isomorphic to a product of elliptic curves.
2. If \[assump\] does not hold, then there are exactly two families of such abelian varieties, which have $8$ resp. $12$ parameters.
The analysis of the second case yields
There are exactly four isomorphism classes of tori satisfying ${\ensuremath{(\ast_{5,m})}}$ for $m \in \{11, 22\}$.
It suffices to deal with the case $m=11$. The four orbits correspond to the $5$-tuples of pairwise non-complex conjugate characters $(1,2,3,4,5)$, $(1,2,3,4,6)$, $(1,2,3,5,7)$, $(1,3,4,5,9)$.
Writing a table containing all families of $5$-dimensional complex tori which admit a linear automorphism of order $m$ acting faithfully is omitted, as the table would need to much space.
Classification of BdF-Varieties in Small Dimensions
===================================================
In this chapter, we list all candidates for split BdF-varieties of dimensions $\leq 4$. It suffices to give a list of $m$, $B_1$, $B_2$ and $T_{r,i}$ with our desired properties (cf. Theorem \[charac\]). As we saw, $B_2$ has to admit an automorphism of order $m$: all possibilities for $B_2$ up to dimension $3$ with corresponding $m$ were listed in the last section. An arbitrary abelian variety of a suitable dimension can be chosen for $B_1$. To compute all possibilities for $T$, we use the combinatorical restrictions, namely [@BGL Corollary 1.7, Proposition 1.8] (note that part (c) of 1.8 in *loc. cit.* equals Proposition \[p-pow\] in this paper), Remarks \[torsionpoints\] and \[el-of-order-m\]. We abbreviate ${\ensuremath{\mathbb{Z}}}_m := {\ensuremath{\mathbb{Z}}}/m{\ensuremath{\mathbb{Z}}}$. The notation is explained before the tables.
At this point, it is not clear that all the BdF-manifolds listed in the tables below are really projective, at least in dimension $> 2$. We will deal with this question in the next chapter.
We start out with the surface case. We list all BdF-surfaces of product type together will all possibilities of $T_{r,i}$.
There are exactly seven families of BdF-surfaces. They are listed in <span style="font-variant:small-caps;">Table 4</span> below.
<span style="font-variant:small-caps;">Table 4</span>
Here, $p$ denotes the number of moduli.
This is the classification result of Bagnera-de Franchis [@BdF] and Enriques-Severi [@Enr-Sev].
Every threedimensional BdF-variety belongs to a family which is listed in <span style="font-variant:small-caps;">Table 5</span> below.
<span style="font-variant:small-caps;">Table 5</span>\
Every fourdimensional BdF-variety belongs to a family which is listed in <span style="font-variant:small-caps;">Table 6</span> below.
<span style="font-variant:small-caps;">Table 6</span>\
We leave it to the interested reader to write down all the possibilities for $T_{r,i}$ in <span style="font-variant:small-caps;">Table 6</span>.
One could, in theory, write tables with all candidates for BdF-varieties up to dimension $11$ using the presented method; in fact, the first case where ${\Lambda}_2$ is not a free $R_m$-module is $m=23$, meaning $\dim(B_2) = 11$.
Projectivity of BdF-manifolds
=============================
In the last section, we listed all candidates for BdF-varieties. However, the answer to the following question still remains.
When do the families of BdF-manifolds listed in the tables above contain BdF-varieties?
In the sequel, we will prove that each family of BdF-manifolds contains a projective member using different methods. The first method is a theorem of T. Ekedahl, which briefly can be stated as ’rigid group actions on tori are projective’, and whose proof was sketched to F. Catanese at an Oberwolfach conference. In the subsequent section, we find explicit forms for the polarization. These methods overlap somehow - nevertheless we think that it is of interest to present both methods.
Rigid Group Actions on Tori
---------------------------
We state a result by Ekedahl; a detailed proof of it can be found in the preprint [@Demleitner] by F. Catanese and the author.
\[ekedahl\] Let $({\ensuremath{T}},G)$ be a rigid group action of a finite group $G$ on a complex torus ${\ensuremath{T}}$. Then ${\ensuremath{T}}$ (or, equivalently, ${\ensuremath{T}}/G$) is projective.
Writing $T = V/{\Lambda}$, the rigidity of a pair $({\ensuremath{T}},G)$ amounts to requiring that each character $\chi$ of $G$ appears in at most one of $V^{1,0}$ and $V^{0,1}$, so we can easily determine which complex tori in the above tables are projective:
The complex tori admitting a linear automorphism of order $m > 2$ acting faithfully for which assumption \[assump\] holds are projective. In particular, they give rise to BdF-varieties.
Explicit Forms for the Polarization
-----------------------------------
In the following we find directly an explicit elementary form of the polarization, showing that each family of BdF-manifolds contains a projective member. Moreover, we determine the type of the polarization whenever possible.
\[m-odd\] The following statements hold: There is no non-degenerate $G$-invariant alternating bilinear form on $R = {\ensuremath{\mathbb{Z}}}[X]/(X^m-1)$ for each $m > 2$. But there is such a form ${\ensuremath{E}}$ on ${\ensuremath{\mathbb{Z}}}[X]$ satisfying $$\begin{aligned}
\ker({\ensuremath{E}}) = \left\{
\begin{array}{l l}
(X^{m-1} + X^{m-3} ... + 1), & m \text{ odd} \\
(X^{m-2} + X^{m-4} + ... + 1), & m \text{ even}.
\end{array}
\right.
\end{aligned}$$
We only prove the lemma in the case where $m$ is odd, the other case is similar. Let ${\ensuremath{E}}$ be a form on ${\ensuremath{E}}$ as in the statement of the lemma. Then we have, by $G$-invariance,
${\ensuremath{E}}(X^i,X^j) = \begin{cases} 0, & i = j, \\
{\ensuremath{E}}(1,X^{j-i}), & j > i,
\end{cases}$
and defining the rest by ${\ensuremath{\mathbb{Z}}}$-bilinearity and alternation. Setting ${\lambda}_i = {\ensuremath{E}}(1,X^i)$ for $1 < i < m$, we find $$0 = {\ensuremath{E}}(X^i, X^m-1) = {\ensuremath{E}}(X^i,X^m) - {\ensuremath{E}}(X^i,1) = {\lambda}_{m-i} + {\lambda}_i.$$ Hence, ${\ensuremath{E}}$ is uniquely determined by ${\lambda}_1, ..., {\lambda}_{\lfloor m/2 \rfloor}$, which are not all equal to $0$. By slight abuse of notation, we use the letter ${\ensuremath{E}}$ not only for the alternating form, but also for its matrix ${\ensuremath{E}}= \left(E(X^i,X^j)\right)_{ij}$. One sees that $X^{m-1}+...+1$ is always in the kernel of ${\ensuremath{E}}$, independent of the choice of the ${\lambda}_i$.\
Setting ${\lambda}_1 := 1$, ${\lambda}_{m-1} := -1$ and ${\lambda}_i := 0$ for $i \in \{2, ..., m-2\}$, one obtains a desired form as stated in the lemma.
\[cor-n-even\] The alternating form ${\ensuremath{E}}$ of the previous lemma induces a non-degenerate $G$-invariant bilinear form on the ${\ensuremath{\mathbb{Z}}}$-module $R_k = {\ensuremath{\mathbb{Z}}}[X]/(\phi_k(X))$ for a divisor $k$ of $n+1 = m$ for each $n > 1$. More precisely, ${\ensuremath{E}}$ can be uniquely written as a sum of non-degenerate $G$-invariant bilinear forms ${\ensuremath{E}}_k$ on $R_k$ for $k$ dividing $n+1$.
Write $R' = R_k \oplus R_k' = {\ensuremath{\mathbb{Z}}}[X]/(\phi_k(X)) \oplus {\ensuremath{\mathbb{Z}}}[X]/(\psi_k(X))$. It suffices to show that ${\ensuremath{E}}(R_k, R_k') = 0$. Note that $R_k$ and $R_k'$ decompose in direct sums of character-eigenspaces; first, let $r \in R_k, r' \in R_k'$ such that $r,r'$ belong to different character-eigenspaces. Hence the condition that ${\ensuremath{E}}$ is $G$-invariant yields $$\begin{aligned}
g\cdot {\ensuremath{E}}(r,r') = \chi(g) \overline{\psi}(g) {\ensuremath{E}}(r,r') = {\ensuremath{E}}(r,r'),
\end{aligned}$$ with different characters $\chi \neq \psi$. Hence ${\ensuremath{E}}(r,r') = 0$. If $r \in R_k$, $r' \in R_k'$ are general elements (i.e., a sum of elements belonging to the respective character eigenspaces), we get ${\ensuremath{E}}(r,r') = 0$ by bilinearity.
\[cplstructure\]
- Recall that, if ${\Lambda}= {\ensuremath{\mathbb{Z}}}[X]/(X^n+...+1)$ (for even $n$ and $m = n+1$), we have decompositions $${\Lambda}\otimes {\ensuremath{\mathbb{C}}}= V \oplus \overline{V} = \bigoplus_{\operatorname{id}\neq \chi \in G^\vee} W_{\chi} = \bigoplus_{j < \frac{m}{2}} W_{\chi_j} \oplus W_{\overline{\chi_j}}$$ and $V = \bigoplus_\chi V_\chi$, such that, for each $\chi$, either $W_\chi = V_\chi$ or $W_{\overline{\chi}} = V_{\overline{\chi}}$. We fix the indices such that $\chi_j$ corresponds to the eigenvalue ${\epsilon}^j$ (where ${\epsilon}= \exp\left(\frac{2\pi i}{m}\right)$). One can do the same if ${\Lambda}= {\ensuremath{\mathbb{Z}}}[X]/(X^{n-1} + X^{n-3}+...+1)$ in the case where $n$ is odd. Unless otherwise stated, we choose the complex structure such that $W_{\chi_j} = V_{\chi_j}$ for every $j < \frac{m}{2}$.
- From the above discussion it follows that a general alternating form ${\ensuremath{E}}$ as in Lemma \[m-odd\] takes the following shape $$\begin{aligned}
\label{genform_A}
{\ensuremath{E}}_{ij} = \begin{cases} 0, & i = j, \\
{\lambda}_{j-i}, & j > i, \\
-{\lambda}_{i-j}, & j < i.
\end{cases}
\end{aligned}$$
\[case-odd-form\] For each $n > 1$, there is a non-degenerate $G$-invariant alternating bilinear form ${\ensuremath{E}}$ on $R' = {\ensuremath{\mathbb{Z}}}[X]/(X^n+...+1)$ resp. $R' = {\ensuremath{\mathbb{Z}}}[X]/(X^n+X^{n-2}...+1)$ (depending on the parity of $n$), such that the associated form $H \colon {\ensuremath{\mathbb{C}}}^n \times {\ensuremath{\mathbb{C}}}^n \to {\ensuremath{\mathbb{R}}}$ defined by $H(v,w) = {\ensuremath{E}}({\ensuremath{\iota}}\cdot v,w) + {\ensuremath{\iota}}{\ensuremath{E}}(v,w)$ and ${\ensuremath{\mathbb{R}}}$-bilinear extension is Hermitian and positive definite.
Let $m = n+1$. We only prove the lemma in the case where $n$ is even. Write $R = {\ensuremath{\mathbb{Z}}}\oplus R'$, where $R$ is as in Lemma \[m-odd\] and let ${\ensuremath{E}}$ be given as in \[genform\_A\]. It remains to show that the Riemann Bilinear Relations are satisfied. The first Riemann Bilinear Relation is satisfied if and only if ${\ensuremath{E}}(V,V) = 0$. Due to the previous corollary, it suffices to check this condition on eigenvectors: it is well-known that $V_{\chi_j}$ (for $j < \frac{m}{2}$) is generated by the element $v_j = \sum_{i = 0}^{n} {\epsilon}^{-ji} X^i \in R$. Then one computes:
$$\begin{aligned}
{\ensuremath{E}}(v_h,v_k) &= \sum_{j=1}^n \sum_{i=0}^{n} {\lambda}_j \left((X^i)^\vee \otimes (X^{i+j})^\vee - (X^{i+j})^\vee \otimes (X^i)^\vee \right)(v_h,v_k)\\
& = \sum_{j=1}^n {\lambda}_j ({\epsilon}^{-kj} - {\epsilon}^{-hj}) \sum_{i=0}^{n} {\epsilon}^{-hi-ki}.
\end{aligned}$$
The last sum is $0$ unless $k+h \equiv 0 \pmod m$. But if $k + h \equiv 0 \pmod m$, we find that $v_k$ and $v_{-k} = \overline{v_k}$ do not both belong to $V$. Restricting the form to $R'$, this shows that the first Riemann Bilinear Relation is satisfied, independently of the choice of the ${\lambda}_i$.\
Now consider the special form ${\ensuremath{E}}$ obtained by setting ${\lambda}_1 = 1$, ${\lambda}_j = 0$ for every other $j$, and restricting to $R'$. By abuse of notation, we denote the images of the $v_k$ in $R'$ again by $v_k$. We prove the second Riemann Bilinear Relation for this form: we have to check that ${\ensuremath{E}}(- {\ensuremath{\iota}}v_k, \overline{v_k}) > 0$ for every $k < \frac{m}{2}$. A simple calculation gives $$\begin{aligned}
&{\ensuremath{E}}(- {\ensuremath{\iota}}v_k,\overline{v_k}) = 2n \cdot \sin\left(\frac{2 \pi k}{m}\right) > 0.
\end{aligned}$$
Summarizing our results, we have the following:
Let ${\ensuremath{T}}= V/{\Lambda}$ be a complex torus and $G$ a cyclic group of order $m \geq 2$ of order $m$ acting freely on ${\ensuremath{T}}$ such that ${\Lambda}$ is a cyclotomic submodule of ${\ensuremath{\mathbb{Z}}}[X]/(X^{m-1}+...+1)$ or ${\ensuremath{\mathbb{Z}}}[X]/(X^{m-2}+X^{m-4}+...+1)$ (depending on the parity of $m$). Assume furthermore that $V$ splits as in Remark \[cplstructure\] i). Then there is a non-degenerate $G$-invariant alternating bilinear form ${\ensuremath{E}}$ on ${\Lambda}$ whose associated Hermitian form is positive definite, such that the polarization on ${\ensuremath{T}}$ is principal.
The matrix of ${\ensuremath{E}}$ as chosen in the proofs above has determinant $1$.
Indeed, we have shown in this sub-chapter that all of the following families of BdF-varieties $A= (B_1 \times B_2)/(T \times G)$ contain a projective member:
- The families where \[assump\] is not satisfied (i.e., the non-rigid cases): In this case, a product of lower-dimensional abelian varieties for which \[assump\] holds is always in the same family as $B_2$.
- The families where the complex structure is chosen as in Remark \[cplstructure\] i) (or the one conjugate to it).
We briefly treat the exceptional cases which we not dealt with in the previous discussion, i.e., the ones where the complex structure is different from the one in Remark \[cplstructure\]. Note that all these cases are rigid, hence projective by Ekedahl’s Theorem \[ekedahl\]; nevertheless, it is also of interest to give an explicit form of a polarization for these cases. One checks computationally that these values of ${\lambda}_i$ give rise to a positive definite form on ${\ensuremath{\mathbb{Z}}}[X]/(X^{m-1}+...+1)$ resp. ${\ensuremath{\mathbb{Z}}}[X]/(X^{m-2}+X^{m-4}+...+1)$.
<span style="font-variant:small-caps;">Table 7</span>
Here we used the shorthand notation $a^k = \underbrace{(a,...,a)}_{k \text{ times}}$ for an integer $a$.
Hence we have proved
Let $X = (B_1 \times B_2)/(G \times T)$ be a BdF-manifold. Then the family containing $X$ contains a BdF-variety.
This concludes the proof of Theorem \[class-result\].
Quite interesting is also the datum of explicit equations for the given variety, or one isogenous to it. This problem requires a lot of time and effort and is therefore not dealt with in this paper. See for instance [@Catanese-Ciliberto] for results in this direction.
[xxxxxxxxxxxxx]{}
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[^1]: **Acknowledgements:** The author cordially thanks Prof. Fabrizio Catanese for a lot of useful advice regarding this topic. This work was generously supported by the ERC Advanced Grant, n. 340258, ’TADMICAMT’.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present the results from a monitoring campaign of the Narrow-Line Seyfert 1 galaxy PG 1211+143. The object was monitored with ground-based facilities (UBVRI photometry; from February to July, 2007) and with [[*Swift*]{}]{} (X-ray photometry/spectroscopy and UV/Optical photometry; between March and May, 2007). We found PG 1211+143 in a historical low X-ray flux state at the beginning of the [[*Swift*]{}]{} monitoring campaign in March 2007. It is seen from the light curves that while violently variable in X-rays, the quasar shows little variations in optical/UV bands. The X-ray spectrum in the low state is similar to other Narrow-Line Seyfert 1 galaxies during their low-states and can be explained by a strong partial covering absorber or by X-ray reflection onto the disk. With the current data set, however, it is not possible to distinguish between both scenarios. The interband cross-correlation functions indicate a possible reprocessing of the X-rays into the longer wavelengths, consistent with the idea of a thin accretion disk, powering the quasar. The time lags between the X-ray and the optical/UV light curves, ranging from $\sim$2 to $\sim$18 days for the different wavebands, scale approximately as $\sim \lambda^{4/3}$, but appear to be somewhat larger than expected for this object, taking into account its accretion disk parameters. Possible implications for the location of the X-ray irradiating source are discussed.'
title: 'Studying X-ray reprocessing and continuum variability in quasars: PG 1211+143'
---
\[firstpage\]
quasars: individual: PG 1211+143; quasars: general; galaxies:active, photometry; accretion, accretion disks
Introduction
============
Powered by accretion, supposedly onto a supermassive black hole, quasars (Active Galactic Nuclei, AGN) are long known mostly as highly energetic, exotic objects in the hearts of the galaxies. Not until recently was their key role in galaxy evolution realized, revealed mostly as a strong correlation between the properties of the central black hole and that of the host galaxy (Magorrian et al. 1998, Ferrarese & Merritt 2000). Studying quasars, therefore, is not only important to understand the underlying physics; it can also help to shed some light on the strange interplay between the accreting matter from the host and outflows from the center, which ultimately shape both – the black hole and the galaxy.
Although a general picture of the structure of a typical quasar seems to be widely accepted (e.g. Elvis 2000; see also Krolik 1999), there are still many details in this picture that are not fully understood. Many of the problems to be solved concern AGN continuum variability – a rather common property, observed in practically all energy bands. Its universality indicates perhaps that variability should be an intrinsic property of the processes, responsible for continuum generation. The optical/UV to X-ray part of the continuum spectrum, as typically assumed, originates from an accretion disk around the central supermassive black hole.
Generally, X-ray variability can be caused by several factors: a change in the accretion rate; variable absorption (e.g. Abrassart & Czerny 2000); variable reflection (e.g. through a change of the height of the irradiating source, Miniutti & Fabian 2004; see also Gallo 2006, Done & Nayakshin 2007); some combination of reflection and absorption (e.g. Chevalier et al. 2006; Turner & Miller 2009); hot spots orbiting the central black hole (Turner et al. 2006; Turner & Miller 2009); local flares (Czerny et al. 2004), etc.
The AGN type with the strongest X-ray variability is the class of Narrow-Line Seyfert 1 galaxies (NLS1s, e.g. Osterbrock & Pogge, 1985). In addition, NLS1s show the steepest X-ray spectra seen among all AGN (e.g. Boller et al. 1996, Brandt et al. 1997, Leighly 1999a, b, Grupe et al. 2001). Most of their observed properties, like spectral slopes, FeII and \[OIII\] line ratios, CIV shifts, etc., appear to be driven by the relatively high Eddington ratio $L/L_{\rm Edd}$ in these objects (e.g. Sulentic et al. 2000, Boroson 2002, Grupe 2004, Bachev et al. 2004).
What concerns the optical/UV variability, the picture there is even more puzzling. There are many factors that can contribute to the variations of the optical flux, but most of them can account for the long-term (months to years) changes. There are often reported in many objects, however, short-term (day to week) optical/UV variations, simultaneous with or shortly lagging behind the X-ray variations. An interesting idea that can explain such a behaviour is reprocessing of the highly variable X-ray emission into optical/UV bands.
In this paper we address the question of the relations between the X-ray and the optical/UV emission by studying the variability from X-rays to I-band of the NLS1 PG 1211+143. This NLS1 has been the target of almost all major X-ray observatories since EINSTEIN (Elvis et al. 1985). The X-ray continuum displays a strong and variable soft X-ray excess (Pounds & Reeves 2007). From XMM-Newton RGS data, Pounds et al. (2003) suggested the presence of high-velocity outflows in PG 1211+143, a result that was questioned by Kaspi & Behar (2006). However, high-velocity outflows seen in X-rays have been repeatedly reported (e.g. Leighly et al. 1997) and new XMM-Newton data of PG 1211+143 (Pounds & Page 2006) seem to confirm the previous claims made by Pounds et al. (2003).
Our primary goal is to find out if and how the X-ray variations are transferred into the longer-wavelength continuum. Time delays between the flaring X-ray emission, presumably coming from a compact, central source and the optical/UV light curves are expected, provided the X-rays are reprocessed in the outer, colder part of an accretion disk. Such a study may have implications on two important problems – the radial temperature distribution of an accretion disk (and hence – the type of the disk) and the location of the X-ray source, based on how much the disk “sees” it.
This paper is organized as follows: In Section 2 we describe the [[*Swift*]{}]{} and ground-based optical monitoring observations. Section 3 focuses on presenting the results of this study and in Section 4 we discuss these results in the context of the general picture of AGN. Throughout the paper spectral indexes are denoted as energy spectral indexes with $F_{\nu} \propto \nu^{-\alpha}$. Luminosities are calculated assuming a $\Lambda$CDM cosmology with $\Omega_{\rm M}$=0.27, $\Omega_{\Lambda}$=0.73 and a Hubble constant of $H_0$=75 km s$^{-1}$ Mpc$^{-1}$.
Observations and reductions
===========================
Swift data
----------
The [[*Swift*]{}]{} Gamma-Ray Burst (GRB) explorer mission (Gehrels et al. 2004) monitored PG 1211+143 between 2007 March 08 and May 20. Note, that scheduled observations were twice bumped by detections of Gamma-Ray-Bursts[^1], explaining the absence of segments 15 and 20 (Table A1). After our monitoring campaign in 2007, PG 1211+143 was re-observed by [[*Swift*]{}]{} in February 2008 (segment 24) However, this observation was used to slew between two targets. Therefore, this observation is very short (188s) and no X-ray spectra or UVOT photometry data were obtained. This observation only allows to measure a count rate. A summary of all [[*Swift*]{}]{} observations is given in Table\[obs\_log\]. The [[*Swift*]{}]{} X-Ray Telescope (XRT; Burrows et al. 2005) was operating in photon counting mode (PC mode; Hill et al. 2004) and the data were reduced by the task [*xrtpipeline*]{} version 0.10.4, which is included in the HEASOFT package 6.1. Source photons were selected in a circular region with a radius of 47$^{''}$ and background region of a close by source-free region with $r=188^{''}$. Photons were selected with grades 0–12. The photons were extracted with [*XSELECT*]{} version 2.4. The spectral data were re-binned by using [*grppha*]{} version 3.0.0 having 20 photons per bin. The spectra were analyzed with [*XSPEC*]{} version 12.3.1 (Arnaud 1996). The ancillary response function files (arfs) were created by [*xrtmkarf*]{} and corrected for vignetting and bad columns/pixels using the exposure maps. We used the standard response matrix [*swxpc0to12s0\_20010101v010.rmf*]{}. Especially during the low-state the number of photons during one segment is too small to derive a spectrum with decent signal-to-noise. Therefore we co-added the data of several segments to obtain source and background spectra. In order to examine spectral changes at different flux/count rate levels, we created spectral for the low, intermediate, and high states with count rates CR $<$0.12 counts s$^{-1}$, 0.13$<$CR$<$0.2, and CR $>$0.2 counts s$^{-1}$. This lead to high state source and background spectra co-adding the data from 2007 April 22, May 09 and 14 (segments 018, 021, and 022), 2007 March 26 and April 02 (segments 13 and 14) for the intermediate state, and all other for the low state. As for the arfs, we created an arf for each segment and coadded them by using the ftool [*addarf*]{} weighted by the exposure times. Due to the low number of photons in the February 2008 observation (segment 024) no spectra could be derived. Fluxes in the 0.2–2.0 and 2–10 keV band for this segment were determined from the count rates in these bands by comparing the fluxes during the high state during segments 018, 021, and 022, assuming no spectral changes. All spectral fits were performed in the observed 0.3–10.0 keV energy band. In order to compare the observations from different missions we use the HEASARC tool [*PIMMS*]{} version 3.8.
Data were also taken with the UV/Optical Telescope (UVOT; Roming et al. 2005), which operates between 1700–6500 Å using 6 photometry filters. Before analyzing the data, the exposures of each segment were co-added by the UVOT task [*uvotimsum*]{}. Source counts were selected with the standard 5$^{''}$ radius in all filters (Poole et al. 2008) and background counts in a source-free region with a radius r=20$^{''}$. The data were analyzed with the UVOT software tool [*uvotsource*]{} assuming a GRB like power law continuum spectrum. The magnitudes were all corrected for Galactic reddening $E_{\rm B-V}=0.035$ given by Schlegel et al. (1998) using the extinction correction in the UVOT bands given in Roming et al. (2009).
[[*XMM-Newton*]{}]{} data analysis
----------------------------------
In order to compare the results derived from the [[*Swift*]{}]{} observations we also analyzed the [[*XMM-Newton*]{}]{} data of PG 1211+143. [[*XMM-Newton*]{}]{} observed PG 1211+143 on 2001 June 15 and 2004 June 21 for 53 and 57 ks, respectively (Pounds & Reeves 2007). Because our paper focuses on the [[*Swift*]{}]{} and ground-based monitoring campaigns in 2007 we reduced only the [[*XMM-Newton*]{}]{} EPIC pn data. A complete analysis of these [[*XMM-Newton*]{}]{} data sets can be found in Pounds et al. (2003), Pounds & Page (2006) and Pounds & Reeves (2007). The [[*XMM-Newton*]{}]{} EPIC pn data were reduced in the standard way as described e.g in Grupe et al. (2004).
Ground-based observations
-------------------------
Additional broad-band monitoring in UBVRI bands was performed on 3 telescopes: the 2-m RCC and the 50/70-cm Schmidt telescopes of Rozhen National Observatory, Bulgaria and the 60-cm telescope of Belogradchik Observatory, Bulgaria. All telescopes are equipped with CCD cameras: the 2-m telescope with a VersArray CCD, while the smaller telescopes – with SBIG ST-8. Identical (U)BVR$_{\rm c}$I$_{\rm c}$ filters are used in all telescopes. The ground-based monitoring covered a period of about 5 months (February – July, 2007), during which the object was observed in more than 40 epochs in BVRI bands, and occasionally – in U. The photometric errors varied significantly depending on the telescope, the filter, the camera in use and the atmospheric conditions, but were typically 0.02 – 0.03 mag. (rarely up to $\sim$0.1 mag in some filters for the smaller instruments).
Results
=======
Long-term X-ray Light Curve
---------------------------
Figure\[pg1211\_xray\_lt\_lc\] displays the long-term 0.2–2.0 and 2.0–10.0 keV light curves. Most of the data prior 2000 were taken from Janiuk et al. (2001). The ROSAT All-Sky Survey point at 1990.9 was taken from Grupe et al. (2001). The XMM 2001 and 2004 and the [[*Swift*]{}]{} fluxes were from our analysis as presented in this paper. PG 1211+143 has become fainter over the last decades in both soft and hard bands with the strongest changes in the soft band. Historically, in the early 1980s, PG 1211+143 had been in a much brighter X-ray state than over the last decade. During the beginning of the [[*Swift*]{}]{} monitoring campaign in March 2007, PG 1211+143 appeared to be in the lowest state seen so far. At the end of our monitoring campaign in May 2007 PG 1211+143 was back in the high state that was previously known from the XMM observations. The latest data point is from February 2008. The 0.2–2.0 and 2–10 keV fluxes during that observation are comparable with the XMM observations. As for the [[*Swift*]{}]{} data taken in 2007 and 2008 we used the flux values obtained from the low and high state spectra. As for the February 2008 data flux points we converted the count rates into fluxes assuming an X-ray spectrum as seen during the [[*Swift*]{}]{} high states.
[[*Swift*]{}]{} XRT and UVOT light curves
-----------------------------------------
The [[*Swift*]{}]{} XRT count rates and hardness ratios[^2], and UVOT magnitudes are listed in Table\[xrt\_uvot\_res\]. These values are plotted in Figure\[swift\_lc\]. At the beginning of the [[*Swift*]{}]{} monitoring campaign in March 2007, PG 1211+143 was found in a very low state. Compared to previous ROSAT and [[*XMM-Newton*]{}]{} observations, reported by e.g. Grupe et al. (2001) and Pounds et al. (2003, 2006, 2007), PG 1211+143 appeared to be fainter by a factor of about 10. At the end of the monitoring campaign in May 2007, PG 1211+143 reached a flux level that was expected from the previous ROSAT and [[*XMM-Newton*]{}]{} observations. A later observation by [[*Swift*]{}]{} on 2008 February 17 found it at a level of 0.375[$\pm$]{}0.045 counts s$^{-1}$ and confirmed that it returned back in a high state. The low-state, found in March 2007, seems to be just a short temporary event. A behaviour like this is not unseen in AGN and has been recently reported for the NLS1 Mkn 335 by Grupe et al. (2007b, 2008a), which had been found in a historical low state by [[*Swift*]{}]{} and [[*XMM-Newton*]{}]{}.
Besides the X-ray variability, PG 1211+143 also displays some variability at optical/UV wavelengths, although on a much smaller level than in X-rays. Table\[xrt\_uvot\_res\] lists the magnitudes measured in all 6 UVOT filters. All 6 light curves are also plotted in Figure\[swift\_lc\]. The most significant drop occurred in all 6 filters during the 2007 April 17 observation. During the next observation on 2007 April 22, PG 1211+143 not only became brighter again in the optical/UV, but also showed an increase in count rate by a factor of almost 4 in X-rays.
Ground-based monitoring
-----------------------
### Secondary standards
In order to facilitate future photometric studies of PG 1211+143, we calibrated secondary standards in the field of the object, shown as stars “A” (USNO B1 1039-0200330) and “B” (USNO B1 1040-0199800) in Figure 3. The magnitudes with the errors (due primarily to the errors of the calibration) are given in Table 1. Mostly Landolt standard sequences (Landolt 1992) were used for the calibration (PG 1633+099) and in some occasions – M67 (Chevalier & Ilovaisky 1991).
Star B V R I
------ -------------- -------------- -------------- --------------
A 11.80 (0.08) 11.35 (0.05) 10.97 (0.06) 10.71 (0.05)
B – 15.34 (0.10) 14.80 (0.07) 14.35 (0.07)
: Field standards
### Magnitude adjustments
The light curve (LC) of PG 1211+143 (Sect. 3.4) is built by measuring its differential magnitude in respect to the adjacent field stars, none of which showed signs of variability (with the exception of a known RR Lyr type star, CI Com, located very close to the quasar). Since the data are collected on different instruments using different cameras (even with identical filters) it is not unusual for an object with strong emission lines to show a differential magnitude, slightly depending on the instrument. The reason for this complication is mostly related to the nature of the quasar spectrum: if a strong emission line falls in a wavelength region where the cameras have different sensitivities, or the filter transparency is slightly different, one may get a broad-band differential magnitude depending on the instrument. In our case all R-band magnitudes of the quasar had to be adjusted by 0.1 mag for one of the instruments (the 50/70cm Schmidt telescope), probably due to the reasons described above. The adjustment corrections were easily obtained through comparison of the light curves, which cover each other on many occasions. We should note, however, that the exact quasar magnitudes are not of importance for this study, as the variations are only considered.
Additionally, the ground-based UBV magnitudes were similarly adjusted to match the corresponding Swift magnitudes. Actually, after the correction for the Galactic reddening, the adjustments for the ground-based B and V magnitudes were very minor, typically less than 0.03 mag, which is smaller than the uncertainties of the calibrated field standards (Sect. 3.3.1).
A log of the ground-based observations, including the obtained UBVRI magnitudes of the quasar after the corrections for the Galactic reddening is presented in Table A3.
Combined light curves
---------------------
The combined continuum light curves of PG 1211+143 for the time of monitoring are presented in Figs. 4 and 5. Fig. 4 compares the X-ray with the optical (UBVRI) variations, all transformed into arbitrary magnitudes for presentation purposes. The optical data are combined from all participating instruments. One sees that the erratic X-ray variations (almost 2.5 magnitudes) hardly influence the optical flux, which shows only minor variations on a generally decaying trend. Figure 5 presents the most intense period of the monitoring, comparing X-ray and V-band magnitudes. The V-band LC for that period stays remarkably stable, with a RMS smaller or comparable to the typical photometric errors.
Time delays
-----------
In order to study the time delay dependence of the wavelength we performed a linear-interpolation cross-correlation analysis (Gaskell & Sparke 1986) between the X-ray and the other bands’ light curves. The interpolation between the photometric points is needed due to unevenly sampled data and is one of the frequently used methods. Other methods applied in the literature do not seem to obtain significantly different results (e.g. discrete CCF, Edelson & Krolik 1986; z-transformed CCF, Alexander 1997) when compared.
Figure 6 shows the interpolation cross-correlation functions, ICCF($\tau$), between the X-ray LC and the other band LCs. A maximum of an ICCF($\tau$) for a positive $\tau$ indicates a time delay behind the X-ray changes and is a signature of a possible reprocessing. Although the ICCFs are mostly negative, due to the different overall trends of the X-ray and optical/UV wave-bands, a clear maximum for a positive $\tau$ is evident for most wave-bands.
Since the X-ray points distribution was far from a Gaussian, even on a magnitude scale, a rank correlation was attempted, but the resulting ICCFs appeared to be very similar.
Table 2 and Figure 7 show the wavelength dependence of the time lag. The wavelengths are taken from the corresponding transmission curve of the filters used (with uncertainties associated with the band widths) and the time delays are from the ICCF maxima.
One sees that for the I-band the highest peak of the ICCF (Figure 6) is at $\tau \simeq -2$ days, indicating a possible short lag of the X-rays behind the near-infrared emission. Another, lower peak at $\tau \simeq +18$ days is also evident. This maximum could probably be associated with reprocessing and is plotted in Figure 7 mostly to demonstrate its consistence with the fit (see below). However, the I-band response to the X-ray variations seems to be more complicated than the simple reprocessing model suggests, as we discuss later in Sect. 4
Uncertainties of the maximum of the ICCFs are difficult to assess. Although there are methods, described in the literature (Gaskell & Peterson 1987), one can hardly rely completely on so computed uncertainties, since the true behaviour of the light curve at the places where it is interpolated is anyway impossible to predict. That is why we accepted the width of ICCF profile at an appropriate level around the peak as an indicative of the uncertainty. This approach is very simple and in addition incorporates into the errors such unknowns as the inclination of the disk in respect to the observer, the spatial size of the irradiating source, etc. See Bachev (2009) for more discussions on these issues.
A clear relation between $\tau$ and $\lambda$ is seen and an acceptable (nonlinear) fit to the data is $\tau_{\lambda}\simeq 9\lambda_{\rm 5000}^{4/3}$ \[days\][^3], where $\lambda_{\rm 5000}$ is $\lambda / 5000$Å. Section 4 discusses possible implications of this result and how it fits into the model of reprocessing from a thin accretion disk.
Filter $\lambda_{\rm 0}$ (Å) FWHM (Å) $\tau$ (days) $\Delta \tau$ (days)
-------- ----------------------- ------------ --------------- ----------------------
UVW2 1928 657 2.5 3
UVM2 2246 498 2.5 3.5
UVW1 2600 693 2 3
U 3465 785 4 3.5
B 4392 975 5 3.5
V 5468 769 13 3
R $\sim$6500 $\sim$700 14 4
I $\sim$8300 $\sim$1000 $-$2 (18) 3 (4)
: Wavelength dependence of the time delays with uncertainties. For I-band, the highest and the first positive (in parentheses) ICCF maxima are shown (see the text).
X-ray Spectroscopy
------------------
As described in Section 2.1, the data were combined to derive low, intermediate and high-state spectra of PG 1211+143. These data were first fitted by a single absorbed power law model with the absorption column density fixed to the Galactic value (2.47$\times 10^{20}$ cm$^{-2}$; Kalberla et al. 2005). Table\[xrt\_xspec\] lists the spectral fit parameters. Obviously a single power law model does not represent the observed spectrum. Figure\[xrt\_xspec\_plot\] displays this fit simultaneously to the low and high state [[*Swift*]{}]{} spectra. As a comparison, Table\[xrt\_xspec\] also lists the results for the fits to the 2001 and 2004 [[*XMM-Newton*]{}]{} EPIC pn data. As the next step we fitted the spectra with a broken power law model. Although this model significantly improves the fits and describes the spectra quite well, it is not a physical model. Especially in the low-state the hard X-ray spectral slope appears to be very flat with [$\alpha_{\rm X}$]{}=–0.18. This behaviour is typical when the X-ray spectrum is either affected by partial covering absorption or reflection (e.g. Turner & Miller 2009, Grupe et al. 2008a).
Next the spectra were fitted with a power law model with a partial covering absorber. These fits suggest a strong partial covering absorber in the low-state spectrum with an absorption column density of the order of 9$\times 10^{22}$ cm$^{-2}$ and a covering fraction of 95%. During the intermediate state the column density of the absorber decreases to 8$\times 10^{22}$ cm$^{-2}$ with a covering fraction of 93% and drops down to $N_{\rm H, pc}=3.5\times 10^{22}$ cm$^{-2}$ and $f_{\rm pc}$=78% during the high state. In order to check whether the data can be self-consistently fit, we fitted the low and high-state [[*Swift*]{}]{} data simultaneously in [*XSPEC*]{}. In this case we tied the column densities of the partial covering absorber and the X-ray spectral slopes of both spectra but left the covering fractions and the normalisations free. Here we found an absorption column density of $N_{\rm H,pc}=8.1\times10^{22}$ cm$^{-2}$ with covering fractions of 91% and 86% for the low and high states, respectively. In all cases, [[*Swift*]{}]{} as well as [[*XMM-Newton*]{}]{}, the X-ray spectral slope remains around [$\alpha_{\rm X}$]{}=2.1 and does not show any significant changes within the errors. The partial covering absorber model can explain the variability seen in X-rays in PG 1211+143. The results obtained from the [[*Swift*]{}]{} data during the high state are consistent with those derived from the 2001 and 2004 [[*XMM-Newton*]{}]{} EPIC pn data.
The spectra were also fitted with a blurred reflection model (Ross & Fabian 2005). Such models, where the primary emission (i.e. the power law component) illuminates the accretion disk producing a reflection spectrum that is blurred by Doppler and relativistic effects close to the black hole (e.g. Fabian et al. 1989) have been successfully applied to several NLS1 X-ray spectra (e.g. Fabian et al. 2004; Gallo et al. 2007a, 2007b; Grupe et al. 2008a; Larsson et al. 2008). As shown in Figures\[pg1211\_refl\_mod\] and \[pg1211\_refl\_ratio\], the reflection model broadly describes the high- and low-flux states of PG 1211+143. In the simplest case, the blurring parameters and disk ionisation are linked between the two epochs. The disk inclination ($i$) and outer radius ($R_{\rm out}$) are fixed to 30$^{\circ}$ and $400 R_{\rm g}$, respectively; $R_{\rm g}$ is the Schwarzschild radius. The continuum shape ($\Gamma$) and normalisations of the reflection and power law were free to vary independently at each epoch. The resulting fit is reasonable ($\chi^{2}_{\nu}/dof = 1.30/105$), considering the obvious over-simplification of our model. The inner disk radius and emissivity index were found to be $R_{\rm in}=1.76^{+0.27}_{-0.35} R_{\rm g}$ and $q=5.57^{+0.54}_{-0.71}$, respectively. The disk ionisation was $\xi = 116 \pm 10$. The intrinsic power law shape was significantly harder during the low-flux state, $\alpha_{\rm x,low}=0.55^{+0.09}_{-0.06}$, compared to $\alpha_{\rm x,high}=0.96^{+0.13}_{-0.10}$ during the high-state. The primary difference between the high and low state is the relative contribution of the power law component to the total 0.3–10 keV flux, being approximately 0.42 and 0.12, respectively. During the low-flux state the reflection component dominates the spectrum.
[lcccccccc]{}
Model & $\alpha_{\rm x,soft}$ & $E_{\rm Break}$ & $\alpha_{\rm x,hard}$ & $N_{\rm H,pc}$ & $f_{\rm pc}$ & $\chi^2/\nu$ & $F_{\rm 0.2-2.0 keV}$ & $F_{\rm 2-10 keV}$\
\
Powl & 1.73[$\pm$]{}0.10 & — & — & — & — & 312/60 & 1.59[$\pm$]{}0.10 & 0.39[$\pm$]{}0.02\
Bknpo & 2.29$^{+0.14}_{-0.13}$ & 1.42$^{+0.13}_{-0.11}$ & –0.18$^{+0.17}_{-0.18}$ & — & — & 68/58 & 1.81[$\pm$]{}0.16 & 2.45[$\pm$]{}0.21\
Zpcfabs \* powl & 2.18$^{+0.10}_{-0.12}$ & — & — & 9.45$^{+1.87}_{-1.62}$ & 0.95[$\pm$]{}0.02 & 82/58 & 1.76[$\pm$]{}0.26 & 1.54[$\pm$]{}0.23\
\
Powl & 1.71[$\pm$]{}0.15 & — & — & — & — & 103/26 & 3.65[$\pm$]{}0.36 & 0.92[$\pm$]{}0.09\
Bknpo & 2.44[$\pm$]{}0.26 & 1.15$^{+0.27}_{-0.14}$ & 0.31$^{+0.26}_{-0.34}$ & — & — & 30/24 & 4.34[$\pm$]{}0.76 & 3.85[$\pm$]{}0.67\
Zpcfabs \* powl & 2.22[$\pm$]{}0.20 & — & — & 7.88$^{+3.21}_{-2.33}$ & 0.93$^{+0.03}_{-0.04}$ & 35/24 & 4.07[$\pm$]{}1.47 & 2.84[$\pm$]{}1.04\
\
Powl & 1.47[$\pm$]{}0.06 & — & — & — & — & 168/62 & 7.04[$\pm$]{}0.29 & 2.76[$\pm$]{}0.12\
Bknpo & 2.05$^{+0.16}_{-0.15}$ & 1.04$^{+0.13}_{-0.12}$ & 0.84[$\pm$]{}0.14 & — & — & 91/60 & 7.79[$\pm$]{}0.78 & 5.06[$\pm$]{}0.51\
Zpcfabs \* powl & 1.98[$\pm$]{}0.14 & — & — & 3.48$^{+1.99}_{-1.02}$ & 0.78$^{+0.05}_{-0.07}$ & 101/60 & 7.50[$\pm$]{}1.76 & 4.02[$\pm$]{}0.94\
\
Powl & 1.56[$\pm$]{}0.05 & — & — & — & — & 488/123 & 1.49/7.24 & 0.50/2.41\
Bknpo & 2.14[$\pm$]{}0.10 & 1.30[$\pm$]{}0.12 & 0.34$^{+0.12}_{-0.12}$ & — & — & 215/121 & 1.79/7.65 & 1.62/6.92\
Zpcfabs \* Powl & 2.00[$\pm$]{}0.09 & — & — & 8.11$^{+1.64}_{-1.33}$ & 0.91[$\pm$]{}0.02/0.86[$\pm$]{}0.03 & 209/120 & 1.65/7.68 & 1.53/4.52\
\
Powl & 1.97[$\pm$]{}0.02 & — & — & — & — & 13092/994 & 5.56[$\pm$]{}0.03 & 0.86[$\pm$]{}0.01\
Bknpo & 2.20[$\pm$]{}0.01 & 1.28[$\pm$]{}0.05 & 0.71[$\pm$]{}0.05 & — & — & 4860/992 & 5.57[$\pm$]{}0.02 & 3.04[$\pm$]{}0.02\
Zpcfabs \* powl & 2.15[$\pm$]{}0.01 & — & — & 6.41[$\pm$]{}0.15 & 0.87[$\pm$]{}0.02 & 5424/992 & 5.50[$\pm$]{}0.10 & 2.64[$\pm$]{}0.05\
\
Powl & 1.62[$\pm$]{}0.01 & — & — & — & — & 4122/940 & 5.63[$\pm$]{}0.05 & 1.06[$\pm$]{}0.02\
Bknpo & 1.75[$\pm$]{}0.01 & 1.58[$\pm$]{}0.05 & 0.85[$\pm$]{}0.03 & — & — & 1828/938 & 5.58[$\pm$]{}0.05 & 3.12[$\pm$]{}0.03\
Zpcfabs \* powl & 1.73[$\pm$]{}0.01 & — & — & 8.73[$\pm$]{}0.34 & 0.70[$\pm$]{}0.02 & 1958/938 & 5.58[$\pm$]{}0.17 & 2.89[$\pm$]{}0.09\
Spectral energy distribution
----------------------------
Figure\[pg1211\_sed\] displays the spectral energy distributions (SED) during the low state observation on 2007 April 17 (blue squares) and the high state observation on April 22 (red triangles). The optical/UV slope [$\alpha_{\rm UV}$]{} slightly changes from $-$0.67[$\pm$]{}0.12 to $-$0.56[$\pm$]{}0.10 between the low and high states. Most significant, however, is the change in the optical-to-X-ray spectral slope [$\alpha_{\rm ox}$]{}[^4] from [$\alpha_{\rm ox}$]{}=1.84 during the low state to [$\alpha_{\rm ox}$]{}=1.48 during the high state. This low-state [$\alpha_{\rm ox}$]{} almost makes it an X-ray weak AGN according to the definition by Brandt et al. (2000), who defines AGN with [$\alpha_{\rm ox}$]{}$>$2.0 as X-ray weak. The luminosities in the Big-Blue-Bump are log $L_{\rm BBB}$=38.42 and 38.53 \[W\] for the low and high states, respectively. These luminosities correspond to Eddington ratios of $L/L_{\rm Edd}$=0.26 and 0.33, respectively, assuming a mass of the central black hole of 9$\times
10^7$[$M_{\odot}$]{} (Vestergaard & Peterson 2006).
Discussion
==========
Based on the results from this study of the NLS1 PG 1211+143 we found that the short-time (and perhaps – even the long-time) variations of the X-ray and optical/UV continuums do not seem to correlate well (Smith & Vaughan 2007, and the references within). While the X-ray continuum varied rapidly (more than 5 times during [[*Swift*]{}]{} monitoring campaign) with a general trend of brightness increase, the optical/UV continuum showed minimal changes with a general trend of brightness decrease. This is not unusual and is in fact reported for other objects (e.g. NGC 5548, Uttley et al. 2003). Such a behaviour sets constraints on different reprocessing scenarios. We are going to discuss briefly several possibilities, assuming that the optical/UV emission is produced by a standard, thin accretion disk, the central black hole mass is $M_{\rm BH}\simeq 9 \times 10^{7}$ [$M_{\odot}$]{} and the accretion rate (in Eddington units) is $\dot m \simeq 0.3$ (Sect. 3.7). The accretion rate is also consistent within the errors with the one found by Kaspi et al. (2000) and Loska et al. (2004).
Cause of the X-ray weakness
---------------------------
As shown in Figure\[longterm\] over the last 20 years PG 1211+143 appears to be fainter in X-rays compared with the X-ray observation during the 1980th. Especially during our [[*Swift*]{}]{} observations during the beginning of our monitoring in March 2007, PG 1211+143 was found to be in an historical X-ray low state especially in the 0.2–2.0 keV band. The X-ray spectrum during the low state is somewhat similar to that found during the historical low state in Mkn 335 (Grupe et al. 2007b, 2008a). Also, here the low state could be explained by a strong partial covering absorber. Later monitoring with [[*Swift*]{}]{} suggests that the absorber has disappeared again and that Mkn 335 is back in a high-state (Grupe et al. 2009, in prep.). Similarly, the low-state here was just a temporary event that lasted for a maximum of about a year. Partial covering absorbers, however, can last significantly longer. One example is the X-ray transient NLS1 WPVS 007 (Grupe et al. 1995), which has developed strong broad absorption line features in the UV (Leighly et al. 2009) and a strong partial covering absorber in X-rays (Grupe et al. 2008b). It has been in an extreme low X-ray state for more than a decade (Grupe et al. 2007a, 2008b).
Statistically we cannot distinguish between the partial covering absorber or the reflection models. Both models result in similar $\chi^2/\nu$. Both models can also be fit self-consistently leaving the intrinsic X-ray spectrum fixed and only affected by either partial covering absorption or reflection.
In the case of a partial covering absorber we can expect that the observed light is polarized like it has been seen in the NLS1 Mkn 1239 which is highly optically polarized and shows a strong partial covering absorber in X-rays (Grupe et al. 2004). However, both PG 1211+143 and Mkn 335 do not show any sign of optical continuum polarization (Berriman et al. 1999, Smith et al. 2002). Note, however, that all these polarimetry measurements were done when the objects were in their high-state. There are no polarimetry measurements (at least not to our knowledge) that were performed during their low states. Therefore the non polarization in the optical does not exclude the partial covering model.
Compact central X-ray source or extended medium?
------------------------------------------------
It is commonly assumed that the most part of the AGN X-ray emission is produced close to the center, within the inner few tens $R_{\rm g}$. The X-rays in radio-quiet objects may come from the inner part of an accretion disk, perhaps operating there in a mode of a very hot, geometrically thick, low-efficient accretion flow (e.g. ADAF) or from an active corona, sandwiching the disk. In either case, when studying the X-ray irradiation of the *optically* emitting outer parts of the disk (at $\sim 100-1000 R_{\rm g}$), the X-ray producing region could be considered as a point source, elevated slightly above the center of the disk and illuminating the periphery. Thus, a part of this highly variable by presumption X-ray emission could be reprocessed into optical/UV emission, which variations will lag behind the X-ray variations.
The temperature of a thin (Shakura-Sunyaev) accretion disk scales with the radial distance $r$ (in Schwarzschild radii) as $T(r) \simeq 6~10^{5}(\dot m /M_{\rm 8})^{1/4} r^{-3/4}$ \[K\] (Frank et al. 2002). If a point-like X-ray source, located at some distance $H$ above the disk, close to the center, irradiates the outer parts, it causes a temperature increase by a certain factor, but the radial temperature dependence happens to be the same (at least for $r>>H$). Since most of the visual/UV light is presumably due to viscous heating, not to irradiation, the temperature increase could be considered as a small addition to the usual disk temperature. Nevertheless, the X-ray variations should transform into *some* optical/UV variations with a time delay, due to the light crossing time. If each disk ring emits mostly wavelengths close to the maximum of the *Planck* curve for the corresponding temperature, one expects the time delays to scale with the wavelength as $\tau_{\rm \lambda} \simeq 5 (\dot m M_{\rm 8}^{2})^{1/3} \lambda_{\rm 5000}^{4/3}$ \[days\], which transforms to $\tau_{\rm \lambda} \simeq 5 \lambda_{\rm 5000}^{4/3}$ \[days\] for the accretion parameters accepted above.
The delay obtained from the fit in Sect. 3.5, however, appears to be $\sim$2 times longer than expected, yet consistent with the expected dependence of $\lambda^{4/3}$. Although different fits to the data are possible, due to the uncertainties, the general offset seems to indicate a time inconsistency. A possible explanation may be searched in the spatial location of the X-ray source. Even if the very center produces most of the X-ray emission, the outer disk may not “see” much of it. Instead, a large-scale back-scattering matter may be located at significant height above the disk, thus increasing the light-crossing time 2 – 3 times (Loska et al. 2004; Czerny & Janiuk 2007, see also their Fig. 6). In fact, the presence of such a back-scattering matter, in a form of a high-velocity outflow or a warm absorber is suggested by independent studies of this object (Pounds et al. 2003, Pounds & Page 2006). One is to note however, that for a similar otherwise object – Mkn 335, the delays were found to be consistent with a direct irradiation from a compact central source (Czerny & Janiuk 2007). On the other hand, two recent studies by Arévalo et al. (2008) and Breedt et al. (2009) found significant correlation between the X-ray and optical bands at essentially zero time lag (yet, not necessarily inconsistent with the direct reprocessing, considering the uncertainties) for MR 2251–178 and Mkn 79, respectively. Finally, for a broad-line radio galaxy (3C 120), an otherwise different type of object, but with similar to PG 1211+143 black hole mass and accretion rate, Marshall et al. (2009) found a $\sim$28 days delay of R-band behind the X-rays, being longer than expected and similar to our findings. Since the results from these and other studies have not shown a systematic inter-band behaviour, one approach to resolve the problem could be to study separately different types of objects, grouped by their intrinsic characteristics, like central masses, accretion rates, line widths, X-ray and radio properties, etc. (e.g. Bachev, 2009), in order to reveal how the presence of disk reprocessing might be related to overall quasar appearance.
On the other hand, instead from around the central black hole, the variable X-ray emission could come from many active regions (flares) in the corona, covering the optically emitting parts of the disk (Czerny et al., 2004). Such active regions (or hot spots) can irradiate the disk locally, producing almost instantaneous optical/UV continuum changes. The later can be significant enough to be observable under some conditions. The observed wavelength-dependent time delay however, seems to make this possibility unlikely.
Why is the optical/UV continuum poorly responding to the X-ray changes?
-----------------------------------------------------------------------
A few factors can contribute to the apparent lack of significant correlation between the optical/UV and X-ray continua. First, the possibility that the X-ray emission is highly anisotropic (Papadakis et al. 2000), or the disk geometry is far from flat (bumpy surface, Cackett et al. 2007; or warps, Bachev 1999), leading only to a minor optical response to the huge otherwise X-ray changes, cannot entirely be ruled out. Yet, the observed $\sim$2 times longer lags than expected are difficult to explain in terms of such an assumption, since in the presence of a large-scale backscattering medium, needed to account for the extra light travel path, the unevenness of the surface should be of little significance.
Another possibility, which might explain the large X-ray variations and the absence of optical/UV such is the presence of absorbing matter along the line of sight. If located close enough to the center, it can partially obscure the compact X-ray source from the observer, but not too much from the larger, optically emitting part of the accretion disk. In such a case, indeed, the large X-ray variations observed would hardly be transferred into the optical bands. Unfortunately, the X-ray spectral fitting cannot distinguish well enough between reflection and absorption models, to be able to determine which one shapes the X-ray continuum most.
X-ray emission – leading or trailing?
-------------------------------------
Taking into account the position of the highest maximum of the I-band ICCF (Figure 6), the I-band changes appear to lead the X-ray ones. One way to explain this result, if real at all, is invoking the synchrotron-self Compton (SSC) mechanism to account for part of the produced X-rays. SSC assumes that some of the near-IR photons might have a synchrotron origin, and could later be scattered by the same relativistic electrons to produce the X-ray flares, lagging behind the infrared. However, the available data set, based merely on the light curves information, cannot justify undisputedly such an explanation. Furthermore, no strong jet or significant radio emission is present in this object (where SSC is typically assumed to play a significant role). So, if SSC is to account for the delay of the X-ray behind the I-band, this process has to take place in a the base of a possible failed jet (Czerny et al., 2008, and the references therein) or in the central parts of the disk, where the disk could operate as a very hot flow, and where hot electrons and perhaps strong magnetic fields could be present.
Summary
=======
In this paper we presented the results of a continuum (X-ray – to optical I-band) monitoring campaign of PG 1211+143, performed with [[*Swift*]{}]{} and ground-based observatories. The main results are summarized below:
1. In spite of being in a very low X-ray state, the quasar PG 1211+143 showed significant X-ray variations (up to 5 times) on daily basis, with only minor optical/UV flux changes. This behaviour indicates that a rather small amount of the hard radiation is reprocessed into longer wavelengths. Since both, reflection and absorption models fit equally well the X-ray spectrum, we are unable to determine the exact cause of the X-ray weakness of PG 1211+143 during its 2007 minimum.
2. Interband cross-correlation functions suggest that a wavelength-dependent time delay between the X-ray and the optical/UV bands is present, indicating that at least a part of the X-rays is reprocessed into longer wavelengths.
3. Although the $\tau - \lambda$ dependence followed the general trend expected for a thin accretion disk (i.e. $\tau_{\rm \lambda} \sim \lambda^{4/3}$), the delays are $\sim$2 times longer, implying the possible existence of a large-scale back-scattering matter above the disk (wind/warm absorber), rather than a central point-like X-ray source, directly irradiating the disk.
4. Even if the object is radio-quiet, with no strong jet known, we found indications that the SSC mechanism may play some role in the X-ray production.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Neil Gehrels for approving our ToO request and the [[*Swift*]{}]{} team for performing the ToO observations of PG 1211+143. We would also like to thank the anonymous referee for his/her helpful comments and suggestions which significantly improved this paper. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration. [[*Swift*]{}]{} is supported at PSU by NASA contract NAS5-00136. This research was supported by NASA contract NNX07AH67G (D.G.).
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Log of observations
===================
Segment $T_{\rm start}$ $T_{\rm end}$ JD-2454000 $T_{\rm XRT}$ $T_{\rm V}$ $T_{\rm B}$ $T_{\rm U}$ $T_{\rm W1}$ $T_{\rm M2}$ $T_{\rm W2}$
--------- ------------------- ------------------- ------------ --------------- ------------- ------------- ------------- -------------- -------------- --------------
001 2007 Mar 08 00:48 2007 Mar 08 02:35 167 1111 — — — — — —
002 2007 Mar 09 01:03 2007 Mar 09 02:54 168 1781 148 148 148 298 373 596
003 2007 Mar 10 17:20 2007 Mar 10 22:17 169 1695 154 154 154 309 203 621
004 2007 Mar 11 19:03 2007 Mar 11 23:59 170 1711 154 154 154 310 217 621
005 2007 Mar 12 15:55 2007 Mar 12 20:52 171 1666 149 149 149 300 219 601
006 2007 Mar 13 20:50 2007 Mar 13 22:34 172 844 74 74 74 149 117 301
007 2007 Mar 14 09:10 2007 Mar 14 12:34 173 1995 168 168 168 336 403 670
008 2007 Mar 15 11:07 2007 Mar 15 13:02 174 1769 145 145 145 292 399 583
009 2007 Mar 16 00:02 2007 Mar 16 17:54 175 1566 142 145 145 295 82 593
010 2007 Mar 17 08:21 2007 Mar 17 19:12 176 2689 208 231 231 467 478 932
011 2007 Mar 18 10:05 2007 Mar 18 16:39 177 1738 149 149 149 300 292 601
012 2007 Mar 19 10:11 2007 Mar 19 16:45 178 1683 144 144 144 292 282 582
013 2007 Mar 26 07:09 2007 Mar 26 08:28 185 2006 166 166 166 331 444 664
014 2007 Apr 02 20:57 2007 Apr 02 22:36 192 2450 201 201 201 403 559 808
016 2007 Apr 11 09:12 2007 Apr 11 14:09 201 1756 157 157 157 318 233 633
017 2007 Apr 17 00:13 2007 Apr 17 06:46 207 2043 168 168 168 338 417 678
018 2007 Apr 22 11:53 2007 Apr 22 15:18 212 2063 171 171 171 342 434 684
019 2007 Apr 30 04:44 2007 Apr 30 06:32 220 1356 114 114 114 226 289 454
021 2007 May 09 15:13 2007 May 09 17:04 229 1398 70 139 139 277 183 539
022 2007 May 14 04:04 2007 May 14 05:59 234 2257 185 186 186 372 518 744
023 2007 May 20 01:28 2007 May 20 03:12 240 857 — 176 176 354 — 120
024 2008 Feb 17 07:56 2008 Feb 17 07:59 513 188 ... ... ... ... ... ...
Segment CR HR V B U UV W1 UV M2 UV W2
--------- --------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- --------------------
001 0.050[$\pm$]{}0.007 –0.19[$\pm$]{}0.13 ... ... ... ... ... ...
002 0.057[$\pm$]{}0.006 +0.02[$\pm$]{}0.11 14.32[$\pm$]{}0.02 14.59[$\pm$]{}0.01 13.33[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.16[$\pm$]{}0.01 13.18[$\pm$]{}0.01
003 0.049[$\pm$]{}0.006 –0.27[$\pm$]{}0.12 14.28[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.25[$\pm$]{}0.01 13.17[$\pm$]{}0.01 13.12[$\pm$]{}0.01
004 0.085[$\pm$]{}0.008 –0.14[$\pm$]{}0.09 14.26[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.29[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.15[$\pm$]{}0.01 13.12[$\pm$]{}0.01
005 0.119[$\pm$]{}0.009 –0.21[$\pm$]{}0.08 14.26[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.08[$\pm$]{}0.01 13.11[$\pm$]{}0.01
006 0.059[$\pm$]{}0.009 –0.17[$\pm$]{}0.15 14.29[$\pm$]{}0.02 14.52[$\pm$]{}0.01 13.33[$\pm$]{}0.01 13.27[$\pm$]{}0.01 13.07[$\pm$]{}0.03 13.11[$\pm$]{}0.01
007 0.071[$\pm$]{}0.006 –0.09[$\pm$]{}0.08 14.26[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.23[$\pm$]{}0.01 13.10[$\pm$]{}0.01 13.09[$\pm$]{}0.01
008 0.060[$\pm$]{}0.006 +0.08[$\pm$]{}0.06 14.26[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.37[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.17[$\pm$]{}0.01 13.18[$\pm$]{}0.01
009 0.071[$\pm$]{}0.007 –0.16[$\pm$]{}0.10 14.29[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.29[$\pm$]{}0.01 13.25[$\pm$]{}0.01 13.22[$\pm$]{}0.04 13.09[$\pm$]{}0.01
010 0.059[$\pm$]{}0.005 –0.10[$\pm$]{}0.09 14.27[$\pm$]{}0.02 14.54[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.12[$\pm$]{}0.01 13.12[$\pm$]{}0.01
011 0.072[$\pm$]{}0.007 –0.15[$\pm$]{}0.09 14.29[$\pm$]{}0.02 14.60[$\pm$]{}0.01 13.33[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.12[$\pm$]{}0.01 13.15[$\pm$]{}0.01
012 0.045[$\pm$]{}0.006 –0.05[$\pm$]{}0.12 14.29[$\pm$]{}0.02 14.60[$\pm$]{}0.01 13.38[$\pm$]{}0.01 13.27[$\pm$]{}0.01 13.14[$\pm$]{}0.01 13.13[$\pm$]{}0.01
013 0.139[$\pm$]{}0.009 –0.14[$\pm$]{}0.06 14.27[$\pm$]{}0.02 14.55[$\pm$]{}0.01 13.30[$\pm$]{}0.01 13.20[$\pm$]{}0.01 13.08[$\pm$]{}0.01 13.09[$\pm$]{}0.01
014 0.171[$\pm$]{}0.008 –0.22[$\pm$]{}0.05 14.27[$\pm$]{}0.02 14.57[$\pm$]{}0.01 13.30[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.10[$\pm$]{}0.01 13.11[$\pm$]{}0.01
016 0.037[$\pm$]{}0.005 –0.18[$\pm$]{}0.13 14.28[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.32[$\pm$]{}0.01 13.29[$\pm$]{}0.01 13.15[$\pm$]{}0.01 13.15[$\pm$]{}0.01
017 0.060[$\pm$]{}0.006 –0.17[$\pm$]{}0.09 14.36[$\pm$]{}0.02 14.68[$\pm$]{}0.01 13.43[$\pm$]{}0.01 13.37[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.32[$\pm$]{}0.01
018 0.332[$\pm$]{}0.013 –0.21[$\pm$]{}0.04 14.26[$\pm$]{}0.02 14.57[$\pm$]{}0.01 13.34[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.10[$\pm$]{}0.01 13.10[$\pm$]{}0.01
019 0.095[$\pm$]{}0.009 –0.16[$\pm$]{}0.09 14.33[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.30[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.11[$\pm$]{}0.01 13.16[$\pm$]{}0.01
021 0.293[$\pm$]{}0.015 –0.11[$\pm$]{}0.05 14.26[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.32[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.11[$\pm$]{}0.02 13.13[$\pm$]{}0.01
022 0.294[$\pm$]{}0.012 –0.15[$\pm$]{}0.04 14.27[$\pm$]{}0.02 14.60[$\pm$]{}0.01 13.39[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.17[$\pm$]{}0.01 13.17[$\pm$]{}0.01
023 0.095[$\pm$]{}0.011 –0.18[$\pm$]{}0.11 ... 14.61[$\pm$]{}0.01 13.35[$\pm$]{}0.01 13.30[$\pm$]{}0.01 ... 13.18[$\pm$]{}0.01
024 0.375[$\pm$]{}0.045 –0.10[$\pm$]{}0.12 ... ... ... ... ... ...
JD (2454..) U B V R I Instr.
------------- -------------------- -------------------- -------------------- -------------------- --------------------- --------
116.52 ... 14.79[$\pm$]{}0.10 14.29[$\pm$]{}0.03 13.96[$\pm$]{}0.01 13.75 [$\pm$]{}0.02 AOB
153.52 ... ... 14.22[$\pm$]{}0.04 ... ... AOB
157.39 ... 14.49[$\pm$]{}0.04 14.24[$\pm$]{}0.02 13.91[$\pm$]{}0.02 13.72 [$\pm$]{}0.02 RSh
171.63 13.28[$\pm$]{}0.07 14.50[$\pm$]{}0.03 14.25[$\pm$]{}0.02 13.91[$\pm$]{}0.02 13.73 [$\pm$]{}0.03 RSh
172.42 13.33[$\pm$]{}0.05 14.51[$\pm$]{}0.02 14.27[$\pm$]{}0.02 13.93[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 RSh
174.34 ... 14.58[$\pm$]{}0.07 14.28[$\pm$]{}0.02 13.94[$\pm$]{}0.01 13.73 [$\pm$]{}0.01 AOB
175.35 ... 14.57[$\pm$]{}0.07 14.28[$\pm$]{}0.02 13.93[$\pm$]{}0.01 13.74 [$\pm$]{}0.01 AOB
176.31 ... 14.47[$\pm$]{}0.10 14.28[$\pm$]{}0.02 13.94[$\pm$]{}0.01 13.75 [$\pm$]{}0.01 AOB
176.45 13.32[$\pm$]{}0.03 14.49[$\pm$]{}0.03 14.26[$\pm$]{}0.02 13.93[$\pm$]{}0.02 13.72 [$\pm$]{}0.02 RSh
177.32 ... ... 14.28[$\pm$]{}0.03 13.96[$\pm$]{}0.02 13.73 [$\pm$]{}0.03 AOB
178.32 ... 14.55[$\pm$]{}0.10 14.28[$\pm$]{}0.02 13.92[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 AOB
179.47 ... 14.57[$\pm$]{}0.10 14.26[$\pm$]{}0.02 13.94[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 AOB
199.49 ... 14.50[$\pm$]{}0.01 14.25[$\pm$]{}0.01 13.98[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 R2m
200.43 ... 14.51[$\pm$]{}0.01 14.23[$\pm$]{}0.01 13.97[$\pm$]{}0.01 13.76 [$\pm$]{}0.01 R2m
201.44 ... 14.56[$\pm$]{}0.03 14.22[$\pm$]{}0.03 13.94[$\pm$]{}0.03 ... R2m
201.45 ... 14.57[$\pm$]{}0.02 14.23[$\pm$]{}0.01 13.97[$\pm$]{}0.01 13.75 [$\pm$]{}0.01 RSh
203.32 ... 14.57[$\pm$]{}0.01 14.23[$\pm$]{}0.03 14.00[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 RSh
204.34 ... 14.63[$\pm$]{}0.01 14.23[$\pm$]{}0.01 13.97[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 RSh
208.49 13.39[$\pm$]{}0.03 14.56[$\pm$]{}0.02 14.28[$\pm$]{}0.02 13.97[$\pm$]{}0.02 13.77 [$\pm$]{}0.02 RSh
210.41 13.35[$\pm$]{}0.02 14.56[$\pm$]{}0.02 14.29[$\pm$]{}0.02 13.97[$\pm$]{}0.02 13.76 [$\pm$]{}0.02 RSh
211.38 13.39[$\pm$]{}0.03 14.59[$\pm$]{}0.02 14.31[$\pm$]{}0.01 13.99[$\pm$]{}0.02 13.77 [$\pm$]{}0.02 RSh
213.29 ... 14.60[$\pm$]{}0.05 14.31[$\pm$]{}0.02 13.94[$\pm$]{}0.01 13.76 [$\pm$]{}0.02 AOB
217.29 ... 14.56[$\pm$]{}0.05 14.36[$\pm$]{}0.03 13.99[$\pm$]{}0.02 13.77 [$\pm$]{}0.02 AOB
230.35 ... 14.62[$\pm$]{}0.05 14.28[$\pm$]{}0.02 13.96[$\pm$]{}0.01 13.76 [$\pm$]{}0.01 AOB
231.30 ... 14.61[$\pm$]{}0.05 14.27[$\pm$]{}0.02 13.95[$\pm$]{}0.01 13.74 [$\pm$]{}0.01 AOB
232.31 ... 14.63[$\pm$]{}0.06 14.29[$\pm$]{}0.02 13.95[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 AOB
233.42 ... 14.59[$\pm$]{}0.05 14.38[$\pm$]{}0.02 13.98[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 AOB
234.30 ... 14.66[$\pm$]{}0.06 14.23[$\pm$]{}0.02 13.98[$\pm$]{}0.01 13.77 [$\pm$]{}0.01 AOB
236.32 ... 14.60[$\pm$]{}0.07 14.31[$\pm$]{}0.02 13.98[$\pm$]{}0.01 13.77 [$\pm$]{}0.01 AOB
238.37 ... 14.57[$\pm$]{}0.01 14.23[$\pm$]{}0.01 13.91[$\pm$]{}0.01 ... R2m
247.45 ... 14.56[$\pm$]{}0.02 14.29[$\pm$]{}0.03 13.99[$\pm$]{}0.03 13.81 [$\pm$]{}0.03 RSh
260.39 13.35[$\pm$]{}0.05 14.60[$\pm$]{}0.03 14.31[$\pm$]{}0.02 14.00[$\pm$]{}0.02 13.82 [$\pm$]{}0.02 RSh
261.36 .... 14.62[$\pm$]{}0.15 14.38[$\pm$]{}0.02 14.02[$\pm$]{}0.02 13.83 [$\pm$]{}0.02 AOB
262.40 ... 14.65[$\pm$]{}0.03 ... 14.01[$\pm$]{}0.02 13.79 [$\pm$]{}0.03 RSh
263.36 ... 14.61[$\pm$]{}0.03 14.33[$\pm$]{}0.02 13.97[$\pm$]{}0.02 13.81 [$\pm$]{}0.03 RSh
265.35 ... 14.64[$\pm$]{}0.10 14.39[$\pm$]{}0.02 14.05[$\pm$]{}0.02 13.87 [$\pm$]{}0.02 AOB
265.42 ... 14.68[$\pm$]{}0.02 14.35[$\pm$]{}0.02 14.00[$\pm$]{}0.03 13.84 [$\pm$]{}0.02 RSh
266.40 ... 14.66[$\pm$]{}0.02 14.33[$\pm$]{}0.02 14.01[$\pm$]{}0.02 13.81 [$\pm$]{}0.02 RSh
289.31 ... 14.72[$\pm$]{}0.13 14.36[$\pm$]{}0.02 14.01[$\pm$]{}0.02 13.84 [$\pm$]{}0.02 AOB
290.35 ... 14.76[$\pm$]{}0.13 14.35[$\pm$]{}0.02 14.01[$\pm$]{}0.02 13.85 [$\pm$]{}0.03 AOB
291.34 ... 14.69[$\pm$]{}0.02 14.32[$\pm$]{}0.02 14.06[$\pm$]{}0.01 13.89 [$\pm$]{}0.01 RSh
292.31 ... 14.68[$\pm$]{}0.02 14.28[$\pm$]{}0.02 14.03[$\pm$]{}0.01 13.87 [$\pm$]{}0.01 RSh
292.36 ... ... 14.32[$\pm$]{}0.06 13.95[$\pm$]{}0.06 ... AOB
293.32 ... 14.67[$\pm$]{}0.01 14.38[$\pm$]{}0.02 14.04[$\pm$]{}0.01 13.85 [$\pm$]{}0.01 RSh
296.33 ... 14.73[$\pm$]{}0.03 14.38[$\pm$]{}0.02 14.02[$\pm$]{}0.01 ... RSh
298.30 ... 14.62[$\pm$]{}0.03 14.37[$\pm$]{}0.02 14.03[$\pm$]{}0.03 13.85 [$\pm$]{}0.03 RSh
299.30 13.51[$\pm$]{}0.09 14.66[$\pm$]{}0.03 14.36[$\pm$]{}0.02 14.02[$\pm$]{}0.02 13.86 [$\pm$]{}0.03 RSh
303.30 ... 14.81[$\pm$]{}0.02 14.38[$\pm$]{}0.01 14.07[$\pm$]{}0.01 13.89 [$\pm$]{}0.02 R2m
\[lastpage\]
[^1]: Although [[*Swift*]{}]{} has turned into a multi-purpose observatory, its main focus is still on observing GRBs and therefore GRBs will supersede scheduled ToO observations.
[^2]: The XRT hardness ratio is defines as HR=(H-S)/(H+S) with S and H are the counts in the 0.3–1.0 keV and 1.0–10.0 keV bands, respectively.
[^3]: All the calculations here are performed using the observer’s frame measurements. Due to the similar way the times and the wavelengths are affected by the redshift, for the quasar rest frame the delay in the $\tau - \lambda$ dependence [*increases*]{} only by $(1+z)^{-1/3}$, Sect. 4, which is only $\sim 3\%$ and is much less than the expected errors.
[^4]: The X-ray loudness is defined by Tananbaum et al. (1979) as [$\alpha_{\rm ox}$]{}=–0.384 log($f_{\rm 2keV}/f_{2500\rm \AA}$).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Trapped ultracold neutrons (UCN) have for many years been the mainstay of experiments to search for the electric dipole moment (EDM) of the neutron, a critical parameter in constraining scenarios of new physics beyond the Standard Model. Because their energies are so low, UCN preferentially populate the lower region of their physical enclosure, and do not sample uniformly the ambient magnetic field throughout the storage volume. This leads to a substantial increase in the rate of depolarization, as well as to shifts in the measured frequency of the stored neutrons. Consequences for EDM measurements are discussed.'
author:
- 'P.G. Harris'
- 'J.M. Pendlebury'
- 'N.E. Devenish'
bibliography:
- 'neutron\_edm.bib'
title: 'Gravitationally enhanced depolarization of ultracold neutrons in magnetic field gradients, and implications for neutron electric dipole moment measurements '
---
Introduction
============
Ultracold neutrons (UCN) are neutrons of extremely low energy, typically less than or of the order of 200 neV, which therefore have wavelengths that are long compared with the spacing between atomic nuclei in solids. The surfaces of many materials then appear as a positive potential barrier (the so-called Fermi potential) from which these neutrons reflect. This allows the storage of such neutrons in material bottles, typically for several minutes at a time, which in turn permits the study of their fundamental properties. One such study is the ongoing search for the electric dipole moment (EDM) of the neutron, of which the most recent measurement was carried out at the Institut Laue-Langevin, Grenoble, by a collaboration led by the University of Sussex and the Rutherford Appleton Laboratory,[@baker06] using apparatus at room temperature (in contrast to its cryogenic successor, now under development).
The internal volume of the neutron trap used in the room-temperature EDM experiment (RT-nEDM) was an upright cylinder 12 cm high, with quartz walls 37 cm in diameter and a roof and floor of aluminium coated with diamond-like carbon. Crucial to the analysis of the experimental data was the fact that the UCN, being of very low energy, tended to populate preferentially the lower part of the storage volume, whereas the cohabiting mercury ($^{199}$Hg) magnetometer[@green98] filled the volume uniformly. Any vertical magnetic-field gradient $\dBzdz$ applied to the volume would affect the two species differently, such that the gyromagnetic-ratio-corrected ratio of the neutron and mercury Larmor precession frequencies $$\label{eqn:R}
R = \left| \frac{\nu_n}{\nu_{\rm Hg}}\cdot \frac{\gamma_{\rm Hg}}{\gamma_{n}} \right|$$ would, to first order, be shifted by $$\label{eqn:DeltaR}
\Delta R = \pm \Delta h \cdot \frac{\dBzdz}{B_{0_z}},$$ where $\Delta h$ is the (always positive) difference in height between the centre of mass of the mercury and that of the UCN, and the $\pm$ sign depends upon the relative directions of $B_{0_z}$ and $\dBzdz$: $R$ increases (i.e. $\Delta R$ becomes more positive) as the absolute magnitude of the field at the bottom of the storage cell (sampled preferentially by the neutrons) increases relative to that at the top of the cell.
The Larmor precession frequency of the UCN was measured by means of the Ramsey method of separated oscilliatory fields, for which a time $T$ = 130 s between the two r.f. pulses was used consistently. During this period, the UCN would suffer some loss of their (transverse) polarization. This study looks at some of the mechanisms and consequences of this so-called $T_2$ relaxation. For the sake of example, all values of the various parameters used in modelling the phenomenon (storage cell size, Fermi potential, $B_{0_z}$ magnitude etc.) are those appropriate to the RT-nEDM experiment.
UCN density distributions
=========================
It is convenient to refer to the energy of UCN in terms of the maximum height attainable within Earth’s gravitational field. Phase space arguments can be used to demonstrate[@golub_UCN_book; @pendlebury94] that a population of trapped UCN each of energy $\epsilon$ has a density distribution with height $h$ of the form $$\label{eqn:height_dist}
n(h) = \left(1-\frac{h}{\epsilon}\right)^{1/2}n(0).$$ Integration and inversion of this function shows that the height distribution of such UCN within a storage cell of height $H$ may be generated from numbers $X$ distributed uniformly between 0 and 1 via the equation $$\label{eqn:h_generator}
h = \epsilon\left[1-\left(1-kX\right)^{2/3}\right].$$ The constant $k=1$ when $\epsilon<H$, and $k=1-\left(1-H/\epsilon\right)^{3/2}$ otherwise. We note that the average height of the UCN within this population is $$\label{eqn:average_height}
\left<h\right> = -\frac{\epsilon}{k}\left[0.6-k-0.6\left(1-k\right)^{5/3}\right].$$
UCN may be generated by capturing the very low-energy tail of the Maxwell-Boltzmann distribution within a thermal source, or else by downscattering from e.g. liquid helium in a superthermal source. In either case, the energy distribution tends to be close to $$\label{eqn:energy_distn}
n(\epsilon)d\epsilon \propto \epsilon^{1/2}d\epsilon.$$ By the time the UCN are stored, this distribution can change: for example, allowing the UCN to fall under gravity will shift the entire energy distribution upwards; or passage through a polarising foil can remove those of low energy. The top of the energy distribution tends to have a fairly sharp cut-off, corresponding to the Fermi potential of the storage vessel. In the case of RT-nEDM, the UCN rose under gravity after passage through a polarising foil, and the bottom of the storage cell was positioned such that those with just enough energy to pass through the foil would also have just enough energy to reach the cell. Here, therefore, the energy distribution is modelled with the simple function of \[eqn:energy\_distn\], using the 93 cm equivalent height Fermi potential of the quartz walls of the vessel as the cutoff energy. As above, integration and inversion yields a generating function $$\label{eqn:E_generator}
\epsilon = \epsilon_{F}Y^{2/3},$$ where $\epsilon_{F}$ is the (Fermi potential) cut-off energy, and the numbers $Y$ are distributed uniformly between 0 and 1.
The distribution of average heights of a population of UCN with such an energy distribution is shown in \[fig:height\_dist\]. The centre of mass of this modelled population of UCN is 3.0 mm below the centre of the storage trap, in good agreement with the 2.81 mm reported in Baker et al.[@baker06]. Some 4.6% of the UCN are not sufficiently energetic to reach the top of the trap. There is a small but extended tail, amounting to some 4.6% of the total population, of neutrons that do not have sufficient energy to reach the lid of the bottle. Over time, in a vertical magnetic-field gradient, two processes contributing to $T_2$ depolarisation come into play: (a) There is an energy dependence to the natural depolarisation rate in a magnetic-field gradient, because of the different rates at which the neutrons sample the measurement volume. This will be referred to as the [*intrinsic*]{} component, and is modelled in this study by means of a simulation. It is applicable even in the absence of a gravitational field. (b) Under gravity, UCN at different average heights effectively sample different magnetic fields, and therefore on average precess at different rates. This will be referred to as the [*enhanced*]{} component, and is here modelled by means of the analytic distributions described above.
\[ht\]
Simulation of UCN depolarisation in magnetic-field gradients
============================================================
Other studies have considered $T_1$ (longitudinal) and $T_2$ (transverse) relaxation rates of atoms in various configurations of electromagnetic fields and storage trap geometries.[@gamblin65; @schearer65; @cates88; @cates88b; @mcgregor90; @schmid08] An approach often adopted is that of the autocorrelation function, as outlined by McGregor.[@mcgregor90] In this instance, however, the situation is complicated by the parabolic nature of the orbits of the UCN moving under gravity. For this study, therefore, a Monte Carlo simulation has been developed, in which the UCN move in ballistic trajectories within the RT-nEDM cylindrical trap described above, and their spins evolve classically according to the solution $$\begin{aligned}
\vec{\sigma}(t) &= \left(\vec{\sigma}_0-\frac{\left(\vec{\sigma}_0\cdot \vec{B}\right)\vec{B}}{B^2}\right)\cos\left(\omega t\right) \\
&+ \frac{\vec{\sigma}_0\times\vec{B}}{B}\sin\left(\omega t\right) \\
&+ \frac{\left(\vec{\sigma}_0\cdot\vec{B}\right)
\vec{B}}{B^2}\end{aligned}$$ of the equation of motion $$\dot{\vec{\sigma}} = \gamma\vec{\sigma}\times\vec{B}$$ of the spin $\vec{\sigma}$ in a magnetic field $\vec{B}$. A vertical holding field $B_{0_z}$ of 1 $\mu$T was applied.
Intuitively, since the polarisation is the ensemble average of projections $\cos(\delta \theta)\sim1-\delta\theta^2/2$, one can argue that the depolarisation rate should depend upon the variance of this quantity, and hence on the variance of $B$. Thus, one expects that the (intrinsic) $T_2$ should depend inversely upon $\left(\partial B_z/\partial z\right)^2$. \[fig:T2\_energy\] shows the values of $T_2\cdot(\partial B_z/\partial z)^2$, for a variety of different gradients, as a function of the UCN energy (represented by the maximum achievable height). For convenience, the gradients are in nT/m, which is appropriate for the magnitudes of gradients to be expected in such experiments. The scatter of the data points is representative of the uncertainty. The dependence upon $(\partial B_z/\partial z)^2$ provides an extremely good match across several orders of magnitude. The minimum $T_2$ corresponds to the point at which the UCN just have sufficient energy to reach the roof of the trap. For reference, the case in which gravity provides no influence is also shown.
\[ht\]
In the simulations underlying \[fig:T2\_energy\], completely specular reflections were assumed to occur 80% of the time. In fact, $T_2$ also has a significant dependence upon the specularity because of the inclination of specular reflections to lead to individual UCN lingering in particular orbits within the trap rather than sampling the volume uniformly. Representative sample points are shown (based upon a field gradient of 100 nT/m) from the equivalent curves with specularities ranging from 0% (perfectly diffuse reflections at each wall collision) to 60%.
Effect upon frequency and polarization
======================================
In EDM measurements, the Larmor spin precession frequency is normally determined by the Ramsey method of separated oscillating fields. This allows a precise measurement of the ensemble average difference in accumulated phase (per unit time) between the spins of the UCN population and the reference oscillator providing the spin-manipulating r.f. fields. Over time, as different UCN sample different regions of the trap, the distribution of phases spreads out, and polarization is lost.
In the absence of a gravitational field, all of the neutrons would normally sample all regions of the trap with equal probability. The distribution of accumulated phases would therefore be expected to be Gaussian, with the frequency determined by the phase at the peak of the distribution. In the case of UCN, however, the distribution is skewed by the low-energy tail. \[fig:phase\_dist\] shows this distribution of phases for a measurement of 130 s duration (as used in RT-nEDM) in a magnetic-field gradient of 5 nT/m. The numerical values of these phases are relative to that appropriate to the volume-averaged magnetic field, i.e. the field at the geometric centre of the trap. The solid curve shows the distribution excluding the intrinsic contribution: the latter provides a relatively small additional spreading of the phases. The reference phase $\hat{\phi}$, from which the frequency is determined, is given by $$\hat{\phi} = \tan^{-1}\left( \frac{\left<\sin\phi\right>}{\left<\cos\phi\right>} \right),$$ where $\cos\phi$ and $\sin\phi$ are averaged over all of the individual phases $\phi$. $\hat\phi$ is represented on the figure by the central (solid) vertical line, and it is clearly not at the peak of the distribution. Also indicated (by dashed vertical lines) are the phases $\pm \pi$ away from the reference phase.
\[ht\]
It will be noted that, as the phase distribution of \[fig:phase\_dist\] spreads, the population within the low-energy tail passes beyond $\pi$ radians from the reference phase. Since the Ramsey technique is sensitive only to phase modulo $2\pi$, these neutrons effectively reappear on the other side of the distribution, which pulls the reference phase back up towards the peak again, thus effectively [*reducing*]{} the frequency shift from the value (c.f. \[eqn:DeltaR\]) $$\Delta \nu = \gamma \Delta h \cdot \frac{\partial B_z}{\partial z}$$ that would naively be expected from the 3 mm height reduction of the centre of mass in combination with the applied magnetic field gradient. This effect is shown in \[fig:freq\_shift\].
\[ht\]
The frequency shift also changes with time, as shown in \[fig:freq\_vs\_time\]. This effect is likewise a direct consequence of the asymmetric nature of the distribution of phases in \[fig:phase\_dist\]. If the reference phase $\hat{\phi}$ were simply the average accumulated phase, the frequency would be constant for a given gradient. The frequency change is initially fairly rapid, but then slows down as the tail of the distribution “wraps around” and starts to pull the reference phase back towards the peak.
\[ht\]
There is an additional second-order frequency-shift effect. Since the intrinsic contribution to the depolarization causes different parts of the energy spectrum to relax at different rates, and since depolarized UCN cannot contribute to the frequency measurement, the energy distribution of contributing UCN changes over time. This in turn changes the effective centre of mass of the polarised UCN, and thus, via the applied field gradient, produces a second-order frequency shift. The depolarization times $T_2$ in this scenario are however sufficiently long that this effect is negligible – about two orders of magnitude smaller than the shifts that we have been considering thus far.
The polarization $\alpha$ is the average projection onto the reference phase of all of the spin vectors. $\alpha$ as a function of the applied gradient, again for a measurement period of 130 s, is shown in \[fig:alpha\_peak\]. The intrinsic contribution to the depolarization is shown explicitly as a separate set of points. The structure that is apparent at an applied gradient of 4-5 nT/m is another effect of the tail of the distribution wrapping around and moving towards the peak from the other side: temporarily at least, it reduces the average spread of the distribution, and thus moderates the fall in polarization. $\alpha$ as a function of time, for an applied gradient of 10 nT/m, is presented in \[fig:alpha\_vs\_time\]; once again, the intrinsic contribution (which falls off exponentially) is shown explicitly.
\[ht\]
\[ht\]
Effect upon EDM systematic error calculations
=============================================
Whilst any depolarization will reduce the sensitivity of an EDM measurement, it does not of itself bias the result. Likewise, a frequency shift that is independent of the electric field is not necessarily a cause for concern. Effective changes in the velocity spectrum due to differential depolarization could in principle influence the geometric-phase (GP) contribution, although any such effect is liable to be completely negligible: in RT-nEDM, the GP contribution arising from the UCN was approximately fifty times smaller than that from the cohabiting mercury magnetometer.
A more significant concern is in the interpretation of GP-induced false-EDM signals. These are proportional to the applied field gradient $\partial B_z/\partial z$. In a real experiment such as RT-nEDM, this gradient can be most easily inferred from the ratio $R$ (see \[eqn:R\]) of the neutron frequency to the frequency of the cohabiting mercury magnetometer, which samples the volume uniformly. The measured EDM signals as a function of this ratio are shown in 13 of [@pendlebury04], and are fitted to the straight lines anticipated from \[eqn:DeltaR\] above. However, the frequency shifts due to the enhanced depolarization mean that the appropriate frequency ratio is more complex than this, and a function similar to that shown in \[fig:freq\_shift\] is required instead. Fitting to these lines should therefore be carried out with due care and attention, and only after careful modelling. It is clearly far preferable to undertake EDM measurements in conditions of very low magnetic-field gradients.
Conclusion
==========
UCN are of very low energy, and preferentially populate the lower regions of any trap within which they are contained. It has been shown that these gravitational effects result in a significant enhancement of the $T_2$ relaxation of the UCN, and can also lead to shifts in the measured Larmor precession frequency. Although there are potential impacts upon systematic-error calculations for EDM measurements, these are at a very manageable level; nonetheless, they underline the importance both of careful and precise modelling of the system, and also of keeping to an absolute minimum any magnetic-field gradients within the measurement apparatus.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by grant no. ST/K001329/1 from the UK Science and Technology Facilities Council.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We have recently identified metal-sandwich (MS) crystal structures and shown with [*ab initio*]{} calculations that the MS lithium monoboride phases are favored over the known stoichiometric ones under hydrostatic pressure \[Phys. Rev. B 73, 180501(R) (2006)\]. According to previous studies synthesized lithium monoboride tends to be boron-deficient, however the mechanism leading to this phenomenon is not fully understood. We propose a simple model that explains the experimentally observed off-stoichiometry and show that compared to such boron-deficient phases the MS-LiB compounds still have lower formation enthalpy under high pressures. We also investigate stability of MS phases for a large class of metal borides. Our [*ab initio*]{} results suggest that MS noble metal borides are less unstable than the corresponding AlB$_2$-type phases but not stable enough to form under equilibrium conditions.'
author:
- 'Aleksey N. Kolmogorov and Stefano Curtarolo'
title: Theoretical study of metal borides stability
---
1. Introduction {#section.introduction}
===============
The interest in the AlB$_2$ family of metal diborides re-emerged after the discovery of superconductivity in MgB$_2$ with a surprisingly high transition temperature of 39 K[@origin]. Boron $p$-states have been shown to be key for both stability and superconductivity in these compounds[@Kortus; @Shein; @Oguchi]. MgB$_2$ is a unique metal diboride because it has a significant density of boron $p\sigma$-states at the Fermi level which give rise to the high T$_c$ superconductivity, and yet enough of them are filled for the compound to be structurally stable[@Kortus; @Shein; @Oguchi]. The effectively hole-doped noble- and alkali-metal diborides would have higher $p\sigma$ density of states (DOS) at E$_F$, but they have been demonstrated to be unstable under normal conditions[@Oguchi]. The effort to achieve higher T$_c$ has thus primarily focused on doping magnesium diboride with various metals; however, doping this material has proven to be difficult[@dope_review] and no improvement on T$_c$ has yet been reported. According to a recent theoretical study of nonlocal screening effects in metals, MgB$_2$ may already be optimally doped[@peihong]. Lithium borocarbide with a doubled AlB$_2$ unit cell has been suggested as a possible high-T$_c$ superconductor under hole-doping[@LiBC], but disorder in the heavily doped Li$_x$BC appears to forbid superconductivity above 2K[@LixBC].
In this work we investigate whether there could be stable high-T$_c$ superconducting metal borides in configurations beyond the standard AlB$_2$ prototype. We have recently proposed metal-sandwich (MS) structures MS1 and MS2, which also have $sp^2$ layers of boron but separated by two metal layers[@MGB]. Despite their rather simple unit cells these structures have apparently never been considered before. As we demonstrate below, identification of the MS structures is not straightforward because they represent a local minimum not usually explored with current compound prediction strategies[@MGB]. We reveal trends in the cohesion of MS phases by calculating formation energies for a large class of metal borides and show that some monovalent-metal borides benefit from having additional layers of metal. The MS noble-metal borides still have positive formation energy, but they are less unstable than the AlB$_2$-type phases. This result helps resolve the question of what phases would form first in the noble-metal boride systems under non-equilibrium conditions[@AgxB2; @hype; @PF; @Ag_laser].
Our main finding concerns the Li-B system, in which the MS lithium monoboride is stable enough to compete against the known stoichiometric phases. According to our previous [*ab initio*]{} calculations the MS lithium monoboride is comparable in energy to these phases under normal conditions, but it becomes the ground state at 50% concentration under moderate hydrostatic pressures[@MGB]. Here we extend the analysis to non-stoichiometric Li-B phases which could potentially intervene in the synthesis of the MS phases. In particular, synthesized lithium monoboride with linear chains of boron is known to be boron-deficient for reasons not fully understood so far. We simulate the incommensurate LiB$_y$ phases (notation explained in Ref. [@xy]) by constructing a series of small periodic Li$_{2n}$B$_m$ structures and show that the minimum formation energy is achieved for $y\approx0.9$, in very good agreement with the experimentally observed values. Using this simple model of the off-stoichiometry phases with linear chains of boron we demonstrate that relative to them MS-LiB still has lower formation enthalpy under high pressures. Simulations of other alkali-metal borides, MB$_y$ (M = Na, K, Rb, Cs), suggest that that these nearly stoichiometric phases might form under moderate pressures.
The paper is divided in the sections describing: 2) simulation details; 3) construction of the MS prototypes; 4) stability of MS phases for a large class of metal borides; 5) detailed investigation of the Li-B system; 6) simulations of other monovalent and higher-valent metal borides; 7) summary of the electronic and structural properties of the MS phases.
2. Computation details {#section.methods}
======================
Present [*ab initio*]{} calculations are performed with Vienna Ab-Initio Simulation Package [VASP]{} [@kresse1993; @kresse1996b] with Projector Augmented Waves (PAW) [@bloechl994] and exchange-correlation functionals as parametrized by Perdew, Burke, and Ernzerhof (PBE)[@PBE] for the Generalized Gradient Approximation (GGA). Because of a significant charge transfer between metal and boron in most structures considered we use PAW pseudopotentials in which semi-core states are treated as valence. This is especially important for the Li-B system as discussed in Refs. [@MGB; @US_PAW]. Simulations are carried out at zero temperature and without zero-point motion; spin polarization is used only for magnetic alloys. We use an energy cutoff of 398 eV and at least 8000/(number of atoms in unit cell) ${\bf k}$-points distributed on a Monkhorst-Pack mesh [@MONKHORST_PACK]. We also employ an augmented plane-wave+local orbitals (APW+lo) code [WIEN2K]{} to plot characters of electronic bands[@WIEN2K]. All structures are fully relaxed. Our careful tests show that the relative energies are numerically converged to within 1$\sim$2 meV/atom.
Construction of binary phase diagrams A$_x$B$_{1-x}$ is based on the calculated formation enthalpy $H_{f}$, which is determined with respect to the most stable structures of pure elements. For boron there are two competing phases $\alpha$-B and $\beta$-B[@bB]; we use $\alpha$-B (Ref. [@Oguchi]), theoretically shown to be the more stable phase at low temperatures and high pressures [@bB]. A structure at a given composition $x$ is considered stable (at zero temperature and without zero-point motion) if it has the lowest formation enthalpy for any structure at this composition and if on the binary phase diagram $H_{f}(x)$ it lies below a [*tie-line*]{} connecting the two stable structures closest in composition to $x$ on each side.
3. Identification of MS prototypes {#section.identification}
==================================
Data-mining of quantum calculations (DMQC), introduced in our previous work[@SC1], is a theoretical method to predict the structure of materials through efficient re-use of [*ab initio*]{} results. The DMQC iteratively determines correlations in the calculated energies on a chosen library of binary alloys and structure types. The last work has demonstrated that for a set of 114 crystal structures and 55 binary metallic alloys the method gives an almost perfect prediction of the ground states (within the library) in a fraction of all possible computations[@SC1; @Morgan]. The speed-up (commonly by a factor from 3 to 4) is achieved by the method’s rational strategy for suggesting the next phase to be evaluated. An essential feature of these calculations is the full relaxation of the considered structures, which ensures an accurate determination of the correlations in the chosen library[@DMQC].
We have recently begun expanding the $114\times55$ library of [*ab initio*]{} energies of binary alloys[@SC1] to include metal borides. Boron tends to form covalent bonds in intermetallic compounds; to have this correlation in future predictions with the DMQC we needed first to add a few compound-forming metal-boride systems into the library. Introduction of a new system involves calculations of energies for all the prototype entries in the library. Surprisingly, in the very first system considered, Mg-B, one of the fcc structures with 4-atom unit cell at 50% concentration, A$_2$B$_2$ fcc-(111) (or V2 [@Zunger1]), relaxed almost all the way down to the AlB$_2$-MgB$_2$$\leftrightarrow$hcp-Mg tie-line. Significant relaxations are not uncommon in our simulations; they usually correspond to the transformation from a starting configuration to a known stable prototype and are automatically detected by the change in the symmetry. The magnesium monoboride phase, however, retained its original space group R$\bar{3}$m (\#166).
Having examined the relaxation process we found that there is a continuous symmetry-conserving path from V2 to a new structure MS1[@MGB; @MS1]. V2 has 4 atoms per unit cell with 4 free parameters $a$, $c$, $d_B$, and $d_M$ (the last two are fractional distances between boron and metal layers), so that atoms are constrained only to the vertical lines (see Fig. 1). The perfect fcc lattice corresponds to $c/a$=4$\sqrt{2/3}$ and $d_B=d_M$=0.25, but this special case does not grant additional symmetry operations and local relaxation have been seen in some metallic systems[@Zunger1]. In metal borides a more dramatic transformation leads to a much more stable configuration: boron atoms rearrange themselves to form covalent bonds in a hexagonal layer ($d_B\rightarrow 0$) rather then share electrons in close-packed triangular layers, while metal atoms remain in a close-packed bilayer. We have checked other alkali, alkaline and transition metal borides not present in the DMQC library and confirmed that they all benefit from this transformation; however, some electron-rich systems might not escape from the local fcc-type minimum, as shown in Fig. 1 for FeB. This could be a reason why the MS1 prototype has apparently been overlooked so far. We would like to point out that identification of new prototypes is not an intended function of the DMQC. This interesting accidental result should be credited to the exhaustive consideration of all candidates (regardless of how unlikely they seem to be a stable phase - an fcc supercell is hardly a suitable configuration for a magnesium boride phase) and the careful structural relaxation in the calculation of their ground state energies.
We proceed by constructing a library of related MS prototypes. Structures, where the metal atoms closest to boron sit directly above the center of boron hexagons, are uniquely specified by the positions of the metal layers (such as $\alpha$, $\beta$, $\gamma$, in Fig. 2). The MS1 structure can thus also be labeled as $|\alpha\beta|$: the Greek letters show the positions of the two metal layers and vertical bars correspond to boron layers. A hexagonal supercell for this phase[@MS1] is obtained when the last metal layer matches the first: $|\alpha\beta|\beta\gamma|\gamma\alpha|$. The fourth metal layer can alternatively be shifted back to site $\alpha$ (see Fig. 2(d)), resulting in another structure at the same stoichiometry MS2 ($|\alpha\beta|\beta\alpha|$)[@MGB; @MS2]. Examples of more metal-rich structures are $|\alpha\beta\alpha|$ (MS3)[@MS3] and $|\alpha\beta\gamma|$. Various stoichiometries can also be achieved by combination of smaller cells, i.e. $|\alpha\beta|\beta|$ (MS4) and $|\alpha|\alpha\beta|\beta\alpha|$. In this notation the AlB$_2$-prototype is labeled simply as $|\alpha|$, which we will use henceforth to avoid confusion with the aluminum diboride compound. Positioning metal atoms above boron hexagon centers is not the only possibility. Stacking faults have been experimentally observed in MgB$_2$[@stack_exp], though such defects have been shown to be energetically costly for this compound[@stack_theory]. We have constructed a periodic structure ($\delta$-MB$_2$) with 3 atoms per unit cell where the metal atoms in $|\alpha|$ are shifted along ([**a**]{}+[**b**]{}) to be above the middle of a boron-boron bond[@deltaAu].
4. Stability of the MS phases {#section.stability}
=============================
Formation of a particular compound in systems with a few competing phases is determined by a number of factors, i.e. the ground state energy, synthesis conditions, thermodynamic and kinetic effects. Comparison of high-throughput [*ab initio*]{} results[@Morgan; @CALPHAD; @MGA] with experimental databases has shown that the calculated total energies alone allow to identify the correct phases observed in the experiment in about 96.7% of investigated cases (Eq. 3 in Ref. [@CALPHAD]). In this section we use the total energy criterium to narrow down the set of systems in which the MS phases might occur.
We first calculate the formation energy for a large library of alkali, alkaline, and transition metals in the $|\alpha|$ and MS1 configurations (Fig. \[mb\]). Our results for $|\alpha|$-MB$_2$ are consistent with the previous calculation by Oguchi[@Oguchi] (note that we put noble metals in the first valence group). The MS1 phases exhibit a similar trend in cohesion: they are most stable for tetra-valent metal borides. It is convenient to analyze the stability of the MS phases by comparing them to the corresponding $|\alpha|$-phases because the MS structures are effectively a combination of the $|\alpha|$-structure and additional layers of metal. Three immediate effects can be expected from the insertion of an extra metal layer: 1) different strain conditions between the boron network and the triangular layers of metal; 2) different doping level of the boron layer; 3) significant reduction of interlayer overlaps between $p$-orbitals of boron due to the increase in the interlayer distance. To decouple these effects we calculate [*relative stability*]{} of the MS1 phase with respect to phase separation into $|\alpha|$ and pure element for the large library of metal borides. The relative stability for compound M$_x$B$_{1-x}$ ($\equiv X$) is defined as $$\begin{aligned}
\Delta E_{X} \equiv E_{X}-\frac{3}{2}\left[(1-x)E_{|\alpha|}+\left(x-\frac{1}{3}\right)E_{pure}\right]\end{aligned}$$ (all energies are per atom), and reflects whether a metal layer prefers to be in a layered boride environment or stay in pure bulk structure. To illustrate the amount of strain in the system we plot this energy difference versus equilibrium intralayer distance in pure fcc or hcp bulk metal structure, whichever is more stable at zero temperature[@bcc]. Figure 4 shows that monoborides of metals in the same valence group and similar dimensions (for example Zn, Cd, and Mg) have close relative stability. Metal layers mismatched with the boron layer (for example Be, Ca, Na, and K) cause a significant energy penalty when inserted in the respective diborides.
Another general trend captured in Fig. 4 is a consistent decrease of the MS1 phase relative stability with the increase of valence electrons (up to four). Relative stability depends on the binding mechanisms in all the three phases in Eq. 1 and the analysis of its variation with the metal valence is not straightforward. For monovalent metal borides the observed gain in binding for the MS1 phase is consistent with the fact that diborides of low-valent metals have available $p\sigma$ bonding states and stabilize as the metal valence increases[@Kortus; @Oguchi]. However, charge redistribution in these phases may follow different scenarios: in $|\alpha|$ metal atoms are exposed to boron and become almost entirely ionized[@Kortus], while in MS1 boron extracts charge through the surface of the metal bilayer and likely leaves more charge in the metal system (Section 7). With increasing valence in the transition metal series of diborides bonding $p\sigma$ and $d$-$p\pi$ bands become occupied, so that the binding reaches its maximum for Group IV metals and eventually goes down[@Oguchi]. Consequently, we observe a noticeable increase in the relative stability of the MS1 structure for higher-valent metals (in fact, all metal diborides with at least five valence electrons benefit from insertion of an extra metal layer with the largest gain of -340 meV/atom obtained for RhB). However, these electron-rich systems allow other phases with significantly lower energies (prototypes NiAs, NaCl, FeB-b, etc.[@PF]). Hence, we focus on low-electron systems that have been shown to stabilize through incorporation of extra metal layers and could compete with existing phases.
5. Li-B system {#section.Li-B}
==============
[*Overview.*]{} A few compounds at different stoichiometries have been reported for the Li-B system[@PF; @aLi; @bLi; @Alkali; @B-Li; @Li3B14; @LiB3]. On the boron-rich side the experimentally reproducible compounds Li$_3$B$_{14}$ and LiB$_3$ have large unit cells with fractional occupancies[@Li3B14; @LiB3; @PF] and cannot be presently simulated with [*ab initio*]{} methods with desired degree of accuracy. The composition of the most lithium-rich LiB$_y$ compounds (near 50% concentration) apparently depends on synthesis conditions and post-synthesis treatment, as the reported values for $y$ range from 0.8 to 1 (notation explained in Ref. [@xy]). In the early experiments the formed compounds were ascribed compositions Li$_5$B$_4$[@Wan78; @Wan79] or Li$_7$B$_6$[@Dal79]; Wang [*at al.*]{} used a rhombohedral model to explain the observed x-ray patterns[@Wan78; @Li5B4]. However, a more consistent interpretation of the available x-ray data on nearly stoichiometric lithium monoboride has been recently given by Liu [*et al.*]{}[@aLi]. The authors demonstrated that the main x-ray peaks can be indexed with a four-atom hexagonal unit cell $\alpha$-LiB (Fig. 5(a)), which consists of linear chains of boron embedded in hexagonal lithium shells[@aLi].
While the simple $\alpha$-LiB sheds light on what the structure of the lithium monoboride is, an important question remains open as to why the LiB$_y$ compounds are boron-deficient. Wörle and Nesper have offered an insightful model of LiB$_y$, in which the boron chains are uncorrelated and incommensurate with the lithium sublattice[@Wor00]. By using a large unit cell containing 32,000 atoms the authors reproduced a kink at $2\theta\approx 60^{\circ}$ in the x-ray pattern and attributed it to the average boron-boron distance of 1.59 Å. They also suggested that the boron chains could be dimerized or have vacancies[@Wor00]. According to a recent theoretical study, boron chains in lithium monoboride are not expected to dimerize but might indeed be able to slide freely along the lithium sublattice[@bLi].
[*Model of the LiB$_y$ compounds.*]{} Simulation of the disordered LiB$_y$ compounds is essential for finding a possible stability region of the MS-LiB phases. While the large unit cells with thousands of atoms are needed to reproduce the x-ray data, such sizes prohibit the use of [*ab initio*]{} methods for ground state energy calculations. Therefore, we simulate the incommensurate LiB$_y$ compounds by constructing a series of relatively small commensurate Li$_{2n}$B$_m$ phases. The number of lithium atoms in a unit cell must be even since they occupy alternating sites along the $c$-axis. To determine the optimal relative position of the two sublattices we fix only the $z$-components of one lithium and one boron atoms and allow all the other degrees of freedom to relax. We find that the relative placement matters only for the smallest Li$_2$B$_2$ unit cells: as we have shown in Ref. [@MGB] the energy difference between $\alpha$-LiB and $\beta$-LiB is 10 meV/atom. For all other periodic structures the barriers to sliding for the two sublattices are below 1 meV per unit cell. The situation is similar to the relative motion in multiwalled carbon nanotubes, where the rigid layers interact weakly with one another: in long-period commensurate nanotubes the barriers to intertube sliding are extremely small, and in incommensurate ones the intertube sliding mode is gapless[@AL]. Local relaxations in Li$_{2n}$B$_m$ ($m>2$, $m\not=2n$) are insignificant due to the rigidity of the boron-boron chains and a smooth charge density distribution along the chains[@bLi].
[*Stability and structure of the LiB$_y$ compounds.*]{} Figure 6 shows an immediate benefit for the lithium monoboride to change composition: as the level of lithium concentration increases by a few percent the phase undergoes stabilization by over 20 meV/atom at zero pressure. The points on the enthalpy versus concentration plot for the Li$_{2n}$B$_m$ series nicely fit to a parabola (with minimum at $y\approx0.894$ at zero pressure). This leads to an interesting situation, in which the LiB$_y$ phases are actually stable over a [*range*]{} of concentrations. The lower boron concentration limit corresponds to $y_{min}\approx0.874$ (the tangent to the parabola going through $x=1$, shown in Fig. 6), while the higher one depends on the location of LiB$_3$ on the phase diagram and should be around $y_{max}\approx0.9$. The allowed concentrations are in excellent agreement with the Wörle and Nesper’s value of $y=0.9$ inferred from the analysis of the x-ray data[@Wor00]. Considering that the lithium and boron sublattices are nearly independent, it seems possible to manipulate the stoichiometry with active solutions by removing the alkali metal through the surface of the sample. By using tetrahydrofuran-naphtalene solution Liu [*et al.*]{} may have extracted not only the free lithium, but also the lithium from the LiB$_y$ compound pushing the concentration of boron towards the higher limit of the stability region ($y_{min}$,$y_{max}$), and maybe beyond it.
To help determine the composition of the LiB$_y$ compounds from experimental data we plot the fully relaxed lattice parameters in Fig. 7. We observe that the $c$-axis undergoes an almost linear expansion with the increase of the boron to lithium ratio: $c= 0.365 +
2.746 y$. Because of the 1$\sim$2% systematic errors in the bond length calculations within the GGA these results cannot be used to pinpoint the absolute value of $y$. However, the [*variation*]{} of the lattice parameters as a function of $y$ is expected to be much more accurate and allows one to estimate the range of concentrations for the experimentally observed compounds. For example, the measured $c=2.875(2)$ Å and $c=2.792(1)$ Å values corresponded to nominal compositions $y=1.0$ and $y=0.82$, respectively[@Wor00]; the slope from Fig. 7 indicates that the difference in $y$ in the synthesized compounds was, in fact, about 6 times smaller ($\Delta y=0.03$). The discrepancy in the measured (2.796 Å, Ref. [@aLi]) and calculated (3.102 Å, Ref. [@bLi]) $c$-axis values pointed out by Rosner and Pickett[@bLi] can be explained as that the synthesized compound was not a stoichiometric lithium monoboride but rather LiB$_{y\approx 0.89}$.
It should be noted that the determination of the concentration ($y=a_{B-B}/c_{Li-Li}$, $c_{Li-Li}=c/2$) presents difficulty only because the length of the boron-boron bond is hard to extract from the experiment[@Wor00]. However, it is the covalent boron-boron bond, being very rigid, that determines the $c$-axis dimension. This can be best illustrated by simulating linear chains of boron in hypothetical structures B$^{-1/y}$ where lithium is replaced with the equivalent number of extra electrons $1/y$ (a uniform positive background is used here to impose charge neutrality). The B$^{-1/y}$ structures with one boron atom per unit cell keep the basal lattice vectors of the corresponding LiB$_y$ compounds while the $c$-axis is optimized. Fig. 7 shows that the charge transfer from lithium to boron alone can explain the variation of the bond length for $y=0.8\sim1.0$. The origin of the nearly constant offset between the two curves becomes evident when the lithium sublattice is, in turn, simulated without boron. A hypothetical Li$^{+1}$ structure with the same in-plane dimensions as in $\alpha$-LiB has a much shorter equilibrium $c$-axis of 2.22 Å. The $E_{Li^{+1}}(c)$ dependence in the 2.5$\sim$3.1 Å range of $c$ is almost linear with a slope of 0.76 eV/Å. As a result, the lithium sublattice in the LiB$_y$ phases exerts a small stress on the linear chains of boron for all $y$ from 1 to 0.8. The stress induces the shortening of the boron-boron bond by 0.029 Å ($y=1$) and 0.033 Å ($y=0.8$)[@stress] and thus turns out to be the main reason for the 0.035 Å ($y=1$) and 0.032 Å ($y=0.8$) bond length mismatches in the simulated phases with and without lithium (Fig. 7). The result illustrates why LiB$_y$ can be represented well as a superposition of the two electronically independent doped sublattices. However, our next test shows that lithium does play an important role in defining the optimal composition of LiB$_y$ by affecting the electronic states of boron near the Fermi level.
To further investigate the mechanism leading to the existence of the off-stoichiometry lithium borides we plot the total and partial DOS for several Li$_{2n}$B$_m$ phases in Figs. 8 and 9 and the band structure in a representative $\alpha$-LiB phase in Fig. 10. The states near the Fermi level are hybridized $p\pi$-B and Li states[@bLi]. The average presence of the Li character in the DOS in the -10 to 3 eV energy range is small in $\alpha$-LiB ($N^{Li}/N^B_{p\pi}\approx 0.3$), but becomes more significant in Li$_4$B$_3$ ($N^{Li}/N^B_{p\pi}\approx 0.5$). The $p\pi$ boron states extend into the lithium-filled interstitials the furthest, so they are affected by the electrostatic potential from the lithium ions the most (in fact, the van Hove singularity in $\alpha$-LiB at $E=0$ is not present in the corresponding B$^{-1}$ structure described above, see dotted curves in Figs. 8 and 10). Note that the bonding and antibonding boron states are not well separated in the LiB$_y$ phases: in the case of $\alpha$-LiB both types are present in the 0-1.3 eV energy range (Fig. 10). Since the Fermi level in $\alpha$-LiB (Fig. 8) already catches the edge of the antibonding $p\pi$-B states, one would naturally expect for the compound to benefit from [*losing*]{} lithium. However, the system does not follow the rigid band scenario as the concentration of lithium increases: the van Hove singularity is pushed away from the Fermi level (in $\alpha$-LiB) to the right by over 2 eV (in Li$_4$B$_3$) (see Fig. 8). The optimal position of the Fermi level near the bottom of the $p\pi$ pseudogap is achieved in Li$_{10}$B$_9$ and Li$_8$B$_7$, which is a part of the reason why the LiB$_y$ phases have the minimum formation energy at $y\approx
0.9$. The correlation between the position of the Fermi level in the pseudogap and the maximum stability has been observed in various systems[@Oguchi; @EF].
[*Comparison of the MS phases with the known metal borides phases under pressure.*]{} To check whether there are more optimal charge transfer and strain conditions than those in MS1 and MS2 we simulate MS phases at other concentrations (Fig. 6, top panel). In the Li-B system the MS3 and MS4 phases have energies well above the $\alpha$-B$\leftrightarrow$$\alpha$-LiB and $\alpha$-LiB$\leftrightarrow$fcc-Li tie-lines, so they will be unstable against phase separation into the known compounds (these MS phases remain metastable under hydrostatic pressure as well). This test confirms our earlier finding that MS1-LiB and MS2-LiB are particularly stable because of the near-optimal occupation of the binding boron states[@MGB].
Our conjecture that the hydrostatic pressure would favor the MS lithium monoboride phases over the off-stoichiometric LiB$_y$ compounds[@MGB] is also supported by the results shown in Fig. 6. As far as the possibility of the MS-LiB formation is concerned, it is rather unfortunate that the LiB$_y$ phases additionally stabilize by becoming more lithium rich under pressure. The information in Fig. 6 can be used to evaluate the minimum pressure required to stabilize MS2-LiB with respect to $\alpha$-B and LiB$_y$. We estimate by interpolation that at $P_{min}\approx 5$ GPa the MS2-LiB phase lies on the line that goes through $x=0$ and is tangent to the LiB$_y$ parabola. Therefore, MS2-LiB could appear in the experiment at $P>P_{min}$ (not accounting for the possible systematic errors and the thermodynamics effects[@MGB]) if all other phases in the Li-B system were metastable under such pressures. Using the parabola coefficients for the three pressures given in Fig. 6 we also find by interpolation that MS2-LiB has the same formation enthalpy as the most stable LiB$_{y\approx 0.82}$ phase at about 12 GPa. Finally, we observe that the line connecting MS2-LiB and fcc-Li crosses the parabolas in all the cases for pressures below 30 GPa[@H]. This implies that if the MS2-LiB phase was synthesized, LiB$_y$ might still be present in the sample as a by-product. For analysis of the Li-B system at higher pressures one needs to take into account that pure lithium undergoes phase transformations from fcc to $hR1$ and eventually to $cI16$ near 40 GPa[@Li_pure].
The chances for the formation of the MS-LiB phases depend on where they are located on the phase diagram relative to LiB$_y$ and the most lithium rich stable phase below 50% concentration. The known phases in this region have small atomic volume ($V_{Li_3B_{14}}=6.4$ and $V_{LiB_3}=7.5$ Å$^{3}$/atom under ambient conditions[@Li3B14; @LiB3; @PF]) compared to MS2-LiB (11.2 and 6.7 Å$^{3}$/atom at 0 and 30 GPa, respectively). The boron-rich phases could potentially bar the formation of MS-LiB under pressure, however they would need to have a very low formation enthalpy; for example, at $P=12$ GPa it would need to be below $H_{MS2-LiB}= H_{LiB_{y\approx
0.82}}=$ -0.38 eV/atom. Simulation of a known phase with CaB$_6$ prototype, present in the K-B system[@KB6; @Alkali], could give information on how boron-rich phases respond to hydrostatic pressure. However, the ordered CaB$_6$-LiB$_6$ has a large atomic volume and actually becomes less stable under pressure: $H_{f}=0.016$ eV/atom, $V=10.1$ Å$^{3}$/atom at zero pressure and $H_{f}=0.303$ eV/atom, $V=8.6$ Å$^{3}$/atom at 30 GPa pressure. Rosner and Pickett pointed out that a compact pseudodiamond phase B32 (NaTl prototype) might appear under pressure[@bLi]. Using enthalpy versus pressure curves for the $\alpha$-LiB and B32 phases we find the crossover pressure to be 22 GPa. MS2-LiB stays below B32-LiB until about 65 GPa. Overall, our simulations suggest that there might be a window of pressures at which the MS-LiB phases can be synthesized.
6. Other monovalent and some higher-valent metal borides {#section.Noble}
========================================================
[*Alkali and transition metal borides.*]{} According to the experimental databases and the latest review of alkali-metal borides[@PF; @Alkali] there are no stable sodium or potassium borides above 15% concentration of metal and no stable rubidium or cesium borides in the whole concentration range. Na$_3$B$_{20}$ (Pearson symbol $oS46$) and KB$_6$ (Pearson symbol $cP7$) compounds, made out of boron polyhedra intercalated with alkali atoms, are considered the most metal-rich known borides in the Na-B and K-B systems, respectively[@Na3B20; @KB6; @PF; @Alkali]. Our simulations confirm that these compounds have negative formation enthalpies of -58 meV/atom for Na$_3$B$_{20}$ and -29 meV/atom for KB$_6$ with respect to $\alpha$-B and fcc-M[@bcc]. The formation Gibbs free energies for these compounds might be less negative if they were evaluated with respect to $\beta$-B at finite temperature[@bB]. This could be the reason why there is no conclusive evidence of potassium hexaboride synthesis[@Alkali]. Complete theoretical investigation of the boron-rich compounds such as Na$_3$B$_{29}$ (Ref. [@Na3B29]) is beyond the scope of this study but it is interesting to see where the nearly stoichiometric MB$_y$ phases place with respect to the known phases in these systems.
As in the LiB$_y$ compounds, the metal and boron sublattices in the alkali boride phases MB$_y$ (M = Na, K, Rb, Cs) are found to be very weakly correlated. In fact, the larger alkali atoms push the boron chains farther apart (see Table I) weakening the boron interchain bonds (note the lower energy difference between $\alpha$-MB and $\beta$-MB listed in Table I). This again leads to the situation when MB$_y$ (M = Na, K, Rb, Cs) compounds can easily adapt to an optimal composition by having incommensurate metal and boron sublattices. Figures 11 and 12 demonstrate that in all Na-B, K-B, Rb-B, Cs-B systems the stoichiometric phases with linear chains of boron prefer to lose some metal, the opposite tendency compared to the Li-B system. While the most stable LiB$_y$ composition appears to be determined primarily by the optimal level of boron doping, in the larger alkali-metal borides, MB$_y$, the lattice mismatch between the metal and boron sublattices must be playing a more significant role. Note that the formation enthalpy points for these M$_{2n}$B$_m$ phases are not symmetric, curving up more rapidly in the metal rich region.
The stabilization from losing a few percent of alkali metal is noticeable but not enough for NaB$_y$ and KB$_y$ to have a negative formation enthalpy at zero pressure. Because the interchain spacing is determined mostly by the alkali cations the $C_{11}+C_{12}$ force constant in $\beta$-MB decreases as one moves down the periodic table (see Table I). $C_{33}$ also becomes smaller as the boron-boron bond length gets longer. The softness of the MB$_y$ phases (M = Na, K, Rb, Cs) invites the use of hydrostatic pressure for their synthesis. Moreover, in the Na-B and K-B systems the MB$_y$ phases stabilize more rapidly than the MS2-MB phases, which makes it unlikely for the latter to form under the pressures considered. Synthesis of MB$_y$ (M = Na, K, Rb, Cs) would provide valuable information on the ways the linear chains of boron could be stabilized. Because the alkali-metal borides are not fully explored, it would not be surprising if a not considered here or a completely unknown phase appeared in such an experiment.
---------- ------- ------- ----------------- ---------- ------------------------------
compound $a_0$ $c_0$ $C_{11}+C_{12}$ $C_{33}$ $E_{\beta-MB}-E_{\alpha-MB}$
(Å) (Å) (GPa) (GPa) (meV/atom)
LiB 4.013 3.120 139 542 -10
NaB 4.697 3.196 111 379 -3.1
KB 5.390 3.240 79 268 -1.6
RbB 5.662 3.267 77 251 -1.7
CsB 5.976 3.325 69 252 -2.0
RhB 3.382 4.185 498 309 -267
PtB 3.765 3.655 546 293 -108
---------- ------- ------- ----------------- ---------- ------------------------------
: Calculated properties of $\beta$-MB metal borides (NiAs prototype).
\[NiAs\]
While the $\beta$-MB phases are only metastable for the borides in the alkali-metal series, there are two reported stable transition-metal monoborides in this configuration: RhB and PtB (Ref. [@PF; @RhB_PtB]). Our fully relaxed unit cell parameters (see Table I) agree well with experiment for $\beta$-RhB ($a=3.309$ Å, $c=4.224$ Å), but they disagree by over 10% with the measured values ($a=3.358$ Å, $c=4.058$ Å) for $\beta$-PtB[@PF; @RhB_PtB]. Identification of the source of this discrepancy requires additional study of this system. Nevertheless, the data on $\beta$-RhB and $\beta$-PtB in Table I give an idea about what difference the $d$-electrons cause in the boron-boron binding compared to the case of the alkali-metal monoborides. For example, $\beta$-RhB and $\beta$-PtB no longer have the optimally doped double-bonded boron chains: the boron-boron bond is so overstretched that it exceeds the $sp^2$ bond length in the AlB$_2$-type compounds, resulting in the increase of the $c$-axis compressibility compared to the alkali-metal monoborides. The significant reduction of the interchain distances leads to an over 300% increase in the $C_{11}+C_{12}$ force constants. One more important consequence of the more compact arrangement of atoms in the lateral direction and the hybridization of the $d$-orbitals of metal with the valence states of boron is the much larger energy difference between the $\beta$-MB and $\alpha$-MB structures. This makes the formation of the off-stoichiometry phases with linear chains of boron in the transition-metal monoborides energetically unfavorable.
[*Noble- and divalent-metal borides.*]{} AgB$_2$ and AuB$_2$ have been shown to have big positive formation energies[@Oguchi], so they should not form at ambient pressure. Recent experiments suggest that some superconducting Ag-B phase was formed by pulsed laser deposition[@Ag_laser]. Because of the synthesis conditions, the thin-film samples were inhomogeneous and did not produce new x-ray peaks. It was assumed that the observed phase was $|\alpha|$-prototype, although the $T_c$ turned out to be much lower than the anticipated value[@hype]. Formation of Ag vacancies in $|\alpha|$-Ag$_x$B$_2$ was suggested by Shein [*et al.*]{} as a possible explanation of the observed data[@AgxB2]. Current simulations offer other possibilities for formation of phases under non-equilibrium conditions: the proposed phases still have positive formation energies but they are less unstable and below the respective $|\alpha|\leftrightarrow$metal tie-lines (Fig. 13). The intermediate phases in the concentration range from 33% to 60% generally stay below the tie-lines. If formed, MS2-AgB would be even more dynamically stable than MS2-LiB: we find that the former represents a stable equilibrium with the frequencies of the softest interlayer modes $\omega_{x,y}$ three times higher than those in the latter[@MGB]. This is an expected result, considering that the silver bilayer in MS2-AgB remains bound by the $d$-electrons even if it donates most of the charge from the $s$-orbital to boron as lithium does. The electronic properties and the stability of the MS noble-metal borides are further discussed in Section 7.
We find that the $\delta$-MB$_2$ phase[@deltaAu] is surprisingly much more stable than the $|\alpha|$-MB$_2$ phase for several metals: Au, Ca, K, Pd, and Pt (by 215, 28, 214, 28, and 272 meV/atom, respectively). Apparently, metal atoms prefer to hybridize their valence states with $p$-orbitals of boron more strongly by shifting to a boron-boron bond, rather than simply donate their valence electrons. Note that the five metals are either large in size ($d_{Ca}$=3.88 Å, $d_K$=4.71 Å, see Fig. 4) or have a big work function (bulk values: $\phi_{Au}$=5.1 eV, $\phi_{Pd}$=5.12 eV, $\phi_{Pt}$=5.65 eV[@WF; @monolayer]). The discovery of the lower-symmetry $\delta$-AuB$_2$ phase[@deltaAu] rules out the possibility of $|\alpha|$-AuB$_2$ synthesis. The formation energy of $\delta$-AuB$_2$ remains positive (0.45 eV/atom, see Fig. 13), which makes this phase unlikely to form as well. In the Ca-B system $\delta$-CaB$_2$ is still unstable against phase separation into CaB$_6$ and fcc-Ca [@Oguchi; @PF] by 144 meV/atom.
The MS phases in the Mg-B system stay at least a few meV/atom above the $|\alpha|$-MgB$_2$$\leftrightarrow$hcp-Mg tie-line. Hydrostatic pressure is insignificant to their relative stability because they compete against similar phases. Therefore, the MS magnesium boride phases are not likely to form and could possibly exist only in the form of a defect in $|\alpha|$-MgB$_2$. We calculate the following series: $|\alpha\beta|$, $|\alpha|\alpha\beta|$, $|\alpha|\alpha|\alpha\beta|$, and so on up to 8 $|\alpha|$ unit cells, and find by extrapolation that the energy required to insert a single magnesium layer into the $|\alpha|$-MgB$_2$ matrix is quite high: over 25 meV per atom in the additional layer of magnesium.
7. Summary {#section.summary}
==========
phase Li Ag Au Cu Mg
------------ ------------------- -------- -------- -------- -------- --------
$E_{f}$ 0.003 0.520 0.667 0.371 -0.131
$a_{B-B}$ 1.717 1.744 1.737 1.721 1.776
$|\alpha|$ $c_{B-B}$ 3.469 4.081 4.260 3.382 3.521
(33%) $E_{\Gamma}$ 1.48 1.29 1.26 1.14 0.39
$E_A$ 1.70 1.01 0.95 0.65 0.77
$N_{p_{xy}}^B(0)$ 0.076 0.090 0.093 0.106 0.049
$E_{f}$ -0.162 0.374 0.488 0.266 -0.089
$\Delta E_{MS2}$ -0.164 -0.016 -0.012 -0.013 0.009
MS2 $a_{B-B}$ 1.765 1.739 1.734 1.734 1.805
(50%) $c_{B-B}$ 5.522 6.369 6.589 4.915 5.989
[@MS2] $z_M$ 0.496 0.368 0.356 0.388 0.440
$E_{\Gamma}$ 0.99 1.19 1.17 0.69 0.15
$E_A$ 0.99 1.19 1.17 0.69 0.15
$N_{p_{xy}}^B(0)$ 0.059 0.086 0.091 0.091 0.043
$E_{f}$ -0.117 0.305 0.394 0.277 -0.061
$\Delta E_{MS3}$ -0.118 -0.007 -0.006 0.054 0.018
MS3 $a_{B-B}$ 1.745 1.731 1.722 1.695 1.816
(60%) $c_{B-B}$ 8.318 8.785 9.097 6.903 8.511
[@MS3] $z_M$ 0.178 0.231 0.237 0.228 0.194
$E_{\Gamma}$ 1.23 1.22 1.25 0.88 0.03
$E_A$ 1.23 1.22 1.25 0.89 0.03
$N_{p_{xy}}^B(0)$ 0.066 0.090 0.098 0.074 0.039
: Metal boride phases: formation energy ($E_{f}$, eV/atom), relative stability ($\Delta E_X$, eV/atom, Eq. 1), in-plane boron-boron bond ($a_{B-B}$, Å), interplanar boron-boron distance ($c_{B-B}$, Å), internal coordinate of the metal atom ($z_M$)[@MS2; @MS3], position of $p\sigma$-band in boron at $\Gamma$ and $A$ k-points ($E_{\Gamma}$, $E_A$, eV), and PDOS at $E_F$ for B-$p_{xy}$ ($N_{p_{xy}}^B(0)$, states/(eV$\cdot$spin$\cdot$ boron atom)).
\[table2\]
[*Electronic properties.*]{} It is illustrative to compare the important features of the electronic structure in the low-valent metal borides to those in the lithium borides, which were discussed in our previous study[@MGB]. We focus mainly on the $p$-states of boron, important for the stability and the superconductivity in these compounds. For convenience, we calculate the band structure and PDOS for $|\alpha|$, MS2, and MS3 phases because all three have a hexagonal unit cell. The key characteristics of the boron states, along with parameters of the unit cells for these phases are given in Table II.
For all MS2 metal borides the $p\sigma$ band along $\Gamma$-A is practically flat because of the large separation between boron layers, as shown in Fig. 4 of Ref.[@MGB] for MS2-LiB and in Fig. 14 for MS2-AgB. For Ag and Au this band does not move much from the respective average positions in $|\alpha|$ and the PDOS of $p\sigma$ states in boron at the Fermi level, $N_{p_{xy}}^B(0)$, stays nearly the same (Table II and Fig. 15). In fact, these boron properties remain the same in the more metal-rich structure MS3. The results suggest that the level of doping of the boron layers in the MS noble metal borides is nearly independent of the number of metal layers. Considering this and the fact that the MS phases in the 33%-100% range closely follow the $|\alpha|\leftrightarrow$fcc lines on the Ag-B and Au-B phase diagrams these MS phases can be viewed as a mixture of weakly interacting building blocks: the $|\alpha|$ unit cell with an established charge redistribution within it and the closed-packed layers of pure metal. Therefore, one could expect the superconducting properties of the boron layer in the hypothetical phases $|\alpha|$-AgB$_2$ and the MS silver borides in the 33%-100% concentration range to be similar. However, the nonequilibrium conditions necessary to synthesize such compounds[@Ag_laser] may introduce disorder destroying the superconducting states[@LixBC].
Compared to the MS gold and silver borides, the MS copper borides deviate more from the $|\alpha|$-CuB$_2$$\leftrightarrow$fcc-Cu line (Fig. 13) and the level of boron doping is more nonmonotonic as a function of the metal concentration (Table II), even though the work functions of Cu and Ag are close (bulk values: $\phi_{Cu}$ = 4.65 eV, $\phi_{Ag}$ = 4.26 eV, Ref. [@WF]). The different behavior must be the result of the more pronounced strain between the copper and boron layers caused by the smaller size of copper ($d_{Cu}$ = 2.563 Å, $d_{Ag}$ = 2.937 Å, $d_{Au}$ = 2.949 Å, see Fig. 4). In the more electron-rich Mg-B system the level of boron doping increases as additional magnesium layers are added. In MS3-Mg$_3$B$_2$ the $p\sigma$ states are almost completely occupied.
Lithium turns out to be a special case among the monovalent metals: it has the right size and can easily donate electrons to stabilize the boron layers in the MS phases. Fig. 6 and Table II show that the stabilization effect is largest for the 50% concentration MS1 and MS2 lithium borides. No less importantly, by giving up most of their charge the lithium layers interact only weakly with one another, which makes the compound very soft along the $c$-axis and gives the opportunity to stabilize the compound even more with hydrostatic pressure[@MGB]. The addition of extra layers does not result in further stabilization, as boron in MS3-Li$_3$B$_2$ is doped less than in MS2-LiB, judging by the location of the $p\sigma$ band along $\Gamma$-A (Table II and Fig. 15). The present PAW calculations confirm our previous result that MS2-LiB has a higher PDOS of boron $p\sigma$ states at the Fermi level than that in MgB$_2$ (Ref. [@MGB]). Note that the PDOS in our simulations is found by decomposition of the wavefunction within a sphere of fixed radius and can slightly vary with this parameter, as well as with the approximation used. In the APW+lo calculation we obtained a 12% increase of $N_{p_{xy}}^B(0)$ in MS2-LiB compared to that in MgB$_2$ using $R^B_{MT}$ = 1.6 a.u.[@MGB], while in the present PAW calculations we observe a 20% increase using the default PAW radius of 1.7 a.u.
The ample amount of the boron $p\sigma$ states at the Fermi level in MS2-LiB holds great promise for this compound to be a good superconductor. However, a more thorough calculation of the electron-phonon coupling, such as in Ref. [@Calandra], is required to say with certainty whether the new lithium monoboride can compete with the record-holder MgB$_2$. Such calculation is underway[@MGC].
[*X-ray reflections.*]{} As we pointed out previously[@MGB], the resulting structure in metal borides at 1:1 composition could be a random mixture of MS1 and MS2, because they differ only by a long-period shift in stacking order and, therefore, are nearly degenerate. The two phases also have very close nearest neighbor distances in most systems and random structures would still have a constant separation between boron layers. For magnesium and silver MS monoborides these periods correspond to $2\theta=16.4^\circ$ and $2\theta=16.8^\circ$, respectively (for $\lambda=1.5418$ Å).
The largest difference (3%) in the c-axis for MS1 and MS2 is actually found for Li-B, resulting in $2\theta=16.6^\circ$ (d$_{MS1-LiB}=5.35$ Å) and $2\theta=16.1^\circ$ (d$_{MS2-LiB}=5.52$ Å); the more lithium-rich phase MS3-Li$_3$B$_2$ would produce a reflection at $2\theta=10.6^\circ$ (d$_{MS3-Li_3B_2}=8.32$ Å). The most pronounced peaks in the published x-ray data for lithium monoboride are at $2\theta$ = $25.5^\circ$, $41.3^\circ$, and $45.0^\circ$ which fit well to the calculated $\alpha$-LiB x-ray pattern[@aLi]. Interestingly, two reflections at low angles $2\theta$ = $12.2^\circ$ and $20.9^\circ$ were observed at 40-50% of lithium concentration[@Wan78]. A low-angle reflection was also detected at 12.8$^\circ$ in Al$_{1-x}$Li$_x$B$_2$ under heavy Li doping[@AlLiB]. However, none of the observed peaks in the samples prepared at ambient pressure match the calculated x-ray reflections in the MS lithium borides.
8. Conclusions {#section.conclusions}
==============
The main results of the present study can be summarized as follows:
[**i.**]{} We have identified a previously unknown class of metal-rich layered phases that are comparable in energy to existing metal borides. This interesting accidental result should be credited to the exhaustive consideration of all candidates in the DMQC method and the careful structural relaxation in the calculation of their ground state energies.
[**ii.**]{} Our [*ab initio*]{} results suggest that the MS phases are most suitable for electron-deficient metal boride systems. In the Ag-B and Au-B systems the MS phases are less unstable than the corresponding diborides with the AlB$_2$ prototype but they still have positive formation energies. In the Mg-B system the MS phases are metastable and could possibly exist only as a defect in MgB$_2$.
[**iii.**]{} The MS-LiB phases present a special case among the MS metal borides: lithium has the right size and valence to stabilize the hexagonal layers of boron at 1:1 composition. The MS lithium monoboride phases are shown to have lower formation enthalpy with respect to the experimentally observed nearly stoichiometric LiB$_y$ phases under hydrostatic pressure. This encouraging result suggests that the new superconducting MS-LiB phases might form under proper conditions. The lowest required pressure depends on the position of boron-rich phases in the Li-B phase diagram and could be as low as several GPa.
[**iv.**]{} For a more complete description of the Li-B system we introduce a simple model of the off-stoichiometric LiB$_y$ phases which explains the available experimental data. We demonstrate that because of the weak correlation between the boron and lithium sublattices the compound can easily adapt to an optimal composition, which corresponds to an optimal level of boron doping with the Fermi level lying near the bottom of the pseudogap. Interestingly, these phases turn out to be stable over a [*range*]{} of concentrations around $y=0.9$, in excellent agreement with experiment. We list the relaxed unit cell parameters which should be helpful in the determination of the LiB$_y$ composition.
[**v.**]{} We consider the MB$_y$ phases for other alkali-metal borides and find that these compounds also benefit from going off-stoichiometry, only in this case they prefer to lose some metal. The ensuing gain in enthalpy does not make them stable under ambient conditions, however these phases might form under hydrostatic pressure. Synthesis of the MB$_y$ phases (M = Na, K, Rb, Cs) would provide valuable information on how linear chains of boron could be stabilized.
We thank M. Calandra, F.H. Cocks, V. Crespi, P. Lammert, E. Margine, and J. Sofo for valuable discussions. We acknowledge the San Diego Supercomputer Center for computational resources.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The paper introduces an extension of the proposal according to which conceptual representations in cognitive agents should be intended as *heterogeneous proxytypes*. The main contribution of this paper is in that it details how to reconcile, under a heterogeneous representational perspective, different theories of typicality about conceptual representation and reasoning. In particular, it provides a novel theoretical hypothesis - as well as a novel categorization algorithm called DELTA - showing how to integrate the representational and reasoning assumptions of the theory-theory of concepts with the those ascribed to the prototype and exemplars-based theories.'
author:
- Antonio Lieto
bibliography:
- 'bibliography.bib'
title: |
Heterogeneous Proxytypes Extended:\
Integrating Theory-like Representations and Mechanisms with Prototypes and Exemplars
---
Introduction
============
The proposal of characterizing the representational system of cognitive artificial agents by considering conceptual representations as *heterogeneous proxytypes* was introduced in [@lieto2014computational][^1] and has been recently employed and successfully tested in systems like DUAL-PECCS [@lieto2016dual; @lieto15ijcai; @lieto2016towards], later integrated with diverse cognitive architectures such as ACT-R [@anderson2004integrated], CLARION [@sun06clarion], SOAR [@laird2012soar] and Vector-LIDA [@snaider2014vector]. The main contribution of this work is in that it offers a proposal to reconcile, under a heterogeneous representational perspective, not only prototype and exemplars based representations and reasoning procedures, but also the representational and reasoning assumptions ascribed to the so called theory-theory of concepts [@murphy2002big]. In doing so, the paper proposes a novel categorization algorithm, called *DELTA* (i.e. unifie**D** Cat**E**gorization a**L**gorithm for he**T**erogeneous represent**A**tions) able to unify and integrate, in a cognitively oriented perspective, all the common-sense categorization mechanisms available in the cognitive science literature. The rest of the paper is organized as follows: the Section 2 provides an overview of the main representational paradigms proposed by the Cognitive Science and the Cognitive Modelling communities. Section 3, briefly synthesize the representational framework intending concepts as *heterogeneous proxytypes* by showing how such theoretical proposal has been actually implemented and successfully tested in the DUAL-PECCS system. Section 4, proposes a more close analysis of the findings of the theory-theory of concepts, while, Section 5, proposes a novel and extended categorization algorithm integrating the theory-theory representational and reasoning mechanisms with those involving both exemplars and prototypes.
Prototypes, Exemplars, Theories and Proxytypes {#prototypes_exemplars_proxytypes}
==============================================
In the Cognitive Science literature, different theories about the nature of concepts have been proposed. According to the so called classical theory, concepts can be simply defined in terms of sets of necessary and sufficient conditions. Such theory was dominant until the mid ’$70$s of the last Century, when Rosch’s experimental results demonstrated the inadequacy of such a theory for ordinary –or common-sense – concepts [@rosch75cognitive]. Rosch’s results suggested, on the other hand, that ordinary concepts are characterized and organized in our mind in terms of *prototypes*. Since then, different theories of concepts have been proposed to explain different representational and reasoning aspects concerning the problem of typicality: the prototype theory, the exemplars theory and the theory-theory. According to the *prototype* view, knowledge about categories is stored in terms of prototypes, i.e., in terms of some representation of the “best” instance of the category. In this view, the concept *bird* should coincide with a representation of a typical bird (e.g., a robin). In the simpler versions of this approach, prototypes are represented as (possibly weighted) lists of typical features. According to the *exemplar* view, a given category is mentally represented as set of specific exemplars explicitly stored in memory: the mental representation of the concept *bird* is a set containing the representation of (some of) the birds we encountered during our past experience. Another well known typicality-based theory of concepts is the so called the *theory-theory* [@murphy2002big]. Such approach adopts some form of holistic point of view about concepts. According to some versions of the theory-theories, concepts are analogous to theoretical terms in a scientific theory. For example, the concept *cat* is individuated by the role it plays in our mental theory of zoology. In other versions of the approach, concepts themselves are identified with micro-theories of some sort. For example, the concept *cat* should be identified with a mentally represented microtheory about cats.
Although these approaches have been largely considered as competing ones (since they propose different models and predictions about how we organize and reason on conceptual information), they turned out to be not mutually exclusive [@malt1989line]. Rather, they seem to succeed in explaining different classes of cognitive *phenomena*, such as the fact that human subjects use different representations to categorize concepts. In particular, it seems that we can use - in different situations - exemplars, prototypes or theories [@smith1998prototypes; @murphy2002big; @keil1989concepts]. Such experimental evidences led to the development of the so called “heterogeneous hypothesis” about the nature of concepts: this approach assumes that concepts do not constitute a unitary phenomenon, and hypothesizes that different types of conceptual representations may co-exist: prototypes, exemplars, theory-like or classical representations [@machery2009doing]. All such representations, in this view, constitute different *bodies of knowledge* and contain different types of information associated to the the same conceptual entity. Furthermore, each body of conceptual knowledge is assumed to be featured by specific processes in which such representations are involved (e.g., in cognitive tasks like recognition, learning, categorization, *etc.*). In particular prototypes, exemplars and theory-like default representations are associated with the possibility of dealing with non-monotonic strategies of reasoning and categorization, while the classical representations (i.e. that ones based on necessary and/or sufficient conditions) are associated with standard deductive mechanism of reasoning [^2].
In recent years an alternative theory of concepts has been proposed: the *proxytype theory*. It postulates a biological localization and interaction between different brain areas for dealing with conceptual structures. Such localization have a direct counterpart in the well known distinction between *long term* and *working memory* [@prinz2002furnishing]. In addition, such characterization is particularly interesting for the explanation of phenomena such as, for example, the activation (and the retrieval) of conceptual information. In this setting, concepts are seen as *proxytypes*. A *proxytype* is any element of a complex representational network *stored in long-term memory* corresponding to a particular *category* that can be tokenized in working memory to ‘go proxy’ for that category [@prinz2002furnishing]. In other terms, the proxytype theory, inspired by the work of Barsalou [@barsalou1999perceptual], considers concepts as *temporary constructs* of a given category, activated (tokenized) in working memory as a result of conceptual processing activities, such as concept identification, recognition and retrieval.
Heterogeneous Proxytypes
========================
In the original formulation of the proxytypes theory, however, proxytypes have been depicted as monolithic conceptual structures, primarily intended as prototypes [@de2005prinz]. A revised view of this approach has been recently proposed, hypothesizing the availability of a wider range of representation types than just prototypes [@lieto2014computational]. They correspond to the kinds of representations hypothesized by the above mentioned heterogeneous approach to concepts. In this sense, proxytypes are assumed to be heterogeneous in nature (i.e., they are assumed to be composed by heterogeneous networks of conceptual representations and not only by a monolithic one)[^3].
In this renewed formulation, heterogeneous representations (such as *prototypes*, *exemplars*, *theory-like* structures, *etc.*) for each conceptual category are assumed to be *stored in long-term memory*. They can be activated and accessed by resorting to different categorization strategies. In this view, each representation has its associated accessing procedures. In the following, I will briefly present how such theoretical hypothesis has been implemented in the DUAL-PECCS categorization system, and I will use the latter system as a computational referent for showing how the proposals presented in this paper can extend both the system itself and, more importantly, its underlying theoretical framework.
Heterogeneous Proxytypes in DUAL-PECCS
--------------------------------------
DUAL-PECCS [@lieto15ijcai; @lieto2016dual], is a cognitive categorization system explicitly designed and implemented under the heterogeneous proxytypes assumption[^4] for both the representational level (that is: it is equipped with a hybrid knowledge base composed of heterogeneous representations, each endowed with specific reasoning mechanisms) and for the ‘proxyfication’ mechanisms (i.e.: the set of procedures implementing the tokenization of the different representations in working memory). The heterogeneous conceptual architecture of DUAL PECCS includes prototypes, exemplars and classical representations. All these different bodies of knowledge point to the same conceptual entity (the anchoring for these different types of representations is obtained via the Wordnet, see again [@lieto2016dual]). An example of the heterogeneous conceptual architecture of DUAL PECCS is provided in the Figure \[fig:dual\]. Such figure shows how it is represented the concept *dog*. In this case, the prototypical representation grasps information such as that dogs are usually conceptualized as domestic animals, with typically four legs, a tail *etc.*; the exemplar-based representations grasp information on individuals. For example, in Fig. \[fig:dual\] it is represented the individual of *Lessie*, which is a particular exemplar of *dog* with white and brown fur and with a less domestic attitude w.r.t. the prototypical dog (e.g. its typical location is lawn). Within the system, both the exemplar and prototype-based representations make use of non classical (or typical) information and are represented by using the framework of the conceptual spaces [@gardenfors2004conceptual; @lieto16conceptual]: a particular type of vector space model adopting standard similarity metrics to determine the distance between instances and concepts within the space. The representation of classical information (e.g. the fact that $\textsl{Dog} \sqsubseteq \textsl{Animal}$, that is to say that “$Dogs$ are also $Animals$”) is, on the other hand, demanded to standard ontological formalisms. In the current version of the system the classical knowledge component is grounded in the OpenCyc ontology [@lenat1985cyc].
By assuming the heterogeneous hypothesis, in DUAL-PECCS, different kinds of reasoning strategies are associated to these different bodies of knowledge. In particular, the system combines non-monotonic common-sense reasoning (associated to the *prototypical* and *exemplars-based*, conceptual spaces, representations) and standard monotonic categorization procedures (associated to the classical, ontological, body of knowledge). These different types of reasoning are harmonized according to the theoretical tenets coming from the dual process theories of reasoning and rationality [@evans2009two; @kahneman2011thinking]. As emerges from the figure \[fig:dual\], a missing part of the current conceptual architecture in the DUAL-PECCS system (and in its underlying theoretical hypothesis) concerns the representation of the default knowledge in terms of theory-like representational structures (while it already integrates classical, prototypical and exemplars based knowledge reprentation and processing mechanisms). In the next section we will show how Theory-like representations can be considered *dual* in nature (at least from a formal point of view) and therefore may deserve a dual treatment also form a computational point of view.
![Heterogeneous representational architecture for the concept DOG in DUAL-PECCS.[]{data-label="fig:dual"}](heterog_dog.pdf){width="103.00000%"}
The Duality of Theory-Like Representations
==========================================
As mentioned in Section 2, Theory-theory approaches [@murphy1985role; @murphy2002big] assume that concepts consists of more or less complex mental structures representing (among other things) causal and explanatory relations between them (including folk psychology connections). During the 80’s, these approaches stemmed from a critique to the formerly dominant theory of concepts as prototypes. Consider, for example, the famous [@keil1989concepts] transformation experiments, in which subjects were asked to make categorization judgments about the biological membership of an animal that had undergone unusual transformations. In such experiment, Keil showed that people relies on theory-like representation (instead of prototypes) in order to execute their categorization task. In particular, it was shown that the type of knowledge retrieved by the subjects to solve these tasks belongs to their “default common-sense theory” associated to a given concept.
The idea that for most of our categories, our default knowledge includes a common-sense theory of that category (and that theory-like default bodies of knowledge are associated with a distinct kind of categorization process) is, however, only one of the available interpretations about the theory-like representational structures [@machery2009doing]. Another kind of interpretation, in fact, assumes that theory-like structures do not constitute our typical default knowledge but that, on the other hand, they are constitutive of our classical background knowledge [@blanchard2010default]. In order to better explain this difference, and thus the *duality* of the theory-like representations, let us consider the case of DUAL-PECCS. As mentioned above, the current version of the system does not allow to represent the type of theory-like default knowledge belonging to the typical conceptual component of the architecture (see footnote 1 for an example of the non-monotonic reasoning that could be enabled by this kind of knowledge). On the other hand, it allows to represent (in terms of IF-THEN rules enabling monotonic inferences), the kind of theory-like knowledge structures which are compliant with the ontological semantics of the classical conceptual component. In other words: only certain types of theories, i.e. causal theories, belonging to the background knowledge of a cognitive agents, are currently covered by the integration of the current state of the art ontology languages and rules [@frixione14towards] in the DUAL-PECCS system. However, as already pointed out before, common sense knowledge is mostly characterized in terms of “theories” which are based on arbitrary, i.e. experience-based, rules. Therefore, in order to represent, within an artificial system, more realistic (from a cognitive standpoint) “theories”, i.e. including common-sense default theories as intended in the theory-theory approaches, there is the need of going beyond classical logic rules. Recently, graphical models (in particular Bayesian networks) have been proposed as a computational framework able to represent [@danks2007theory; @danks2004psychological] knowledge networks of theory-like common-sense default representation. The integration of such framework within the DUAL-PECCS system represent a current and future area of development, not yet concluded. In the remaining of this paper, such integration will not be discussed and, in a certain sense, will be taken for granted. I shall focus, instead, on the presentation of a novel unifying categorization algorithm - named DELTA - able to harmonize all the different types of typicality-based representations and reasoning mechanisms associated with the common-sense knowledge: exemplars, prototypes and default theory-like representations. I will leave aside the discussion concerning the integration of such common-sense categorization mechanisms with those concerning the classical monotonic ones. As above mentioned, in fact, such integration is already provided in DUAL-PECCS [@lieto2016dual; @lieto15ijcai] and is tackled by recurring to the dual process theory of reasoning (i.e.: the non monotonic reasoning results of the heterogeneous common-sense conceptual components are then checked and integrated with the monotonic reasoning strategies executed in the classical conceptual component). Therefore, the underlying *heterogeneous-proxytypes* assumption, integrated with the dual process theory of reasoning, has been already proven to be effective to harmonize non monotonic and monotonic categorization strategies associated to heterogeneous body of knowledge. I will not report here the details of such harmonization procedure because it already documented elsewhere [@lieto2016dual]. I will focus, instead, on the harmonization procedures concerning the non monotonic categorization processes of the typical conceptual components.
A Unified Categorization Algorithm for Exemplars, Prototypes and Theory-Like Representations
============================================================================================
In the following, I propose a novel categorization algorithm that, given a certain stimulus $d$, must select the most appropriate typicality-based representation available in the declarative memory of a cognitive agent (i.e. a prototype, an exemplar or a theory-like structure). According to what introduced in the previous sections, such declarative memory is assumed to rely on the *heterogeneous proxytypes* hypothesis.
The implemented procedure works as follows: when the input *stimulus* is similar enough to an exemplar representation (a threshold has been fixed to these ends), the corresponding exemplar of a given category is retrieved. Otherwise, the prototypical representations are also scanned and the representation (prototype or exemplar) that is closest to the input is returned. By following a preference that has been experimentally observed in human cognition [@medin1978context], this algorithm favors the results of the exemplars-based categorization if the knowledge-base stores any exemplars similar to the input being categorized. As an additional constraint, I have hypothesized a mechanism in which theory-like structures of default knowledge can also override the categorization based on prototypes. Such mechanism has been devised based on the fact that theory-theorists have shown that, in some categorical judgments tasks (e.g. assessing the situation where a dog is made to look like a raccoon), categorization is driven by the possession of a rudimentary biological theory and by theory-like representations [@atran2008native]. In other words: being a dog isn’t just a matter of looking like a dog. It seems, in fact, that it is more important to have a network of appropriate hidden properties of dogs: the dog “essence”[@atran2008native]. In the proposed algorithm, I have taken into account this element by hypothesizing: i) to measure the similarity between the theory-like representation of the first retrieved prototype with the stimulus $d$ [^5] and ii) to compare the obtained result with a *Conceptual Coherence Threshold* that should measure how much the considered stimulus $d$ shares, i.e. is *conceptually coherent*, with the corresponding theory-like representation of the retrieved prototype. The analysis of the conceptual coherence can be solved as a constraint satisfaction problem as shown in [@thagard1998coherence].
In this setting, if the distance between the stimulus $d$ and theory-like representation of the originally retrieved prototype is above the considered threshold, it means that the retrieved prototype is assumed to be representative enough of the common-sense “essence’ of $d$ (i.e it is “coherent enough’). In this case, the prototypical answer is maintained, otherwise it is overridden by the theory-like representation which is closer to $d$.
Let us assume, for example, that the stimulus to categorize is represented by an atypical Golden Zebra (which is almost totally white) and that in our agent’s long-term memory there is no exemplar similar enough to this entity. This means that there will be no exemplar-based representation selected by our algorithm, and that the most similar representation to $d$ will be searched among the prototypical representations in the agent knowledge base. Now: if we assume that the retrieved prototype is a typical white horse, we could discard such representation by simply relying on some additional information coming from the comparison of the stimulus $d$ (e.g. the fact that lives in the Savannah, etc.) with the default and common-sense theory associated to a horse (i.e. the category associated to the original prototypical choice). In this case the categorical assignment to the class Golden Zebra would be obtained by exploiting theory-like representational networks. A synthetic representation of the proposed procedure is presented in the Algorithm 1.
{width="105.00000%"} \[fig:algo\]
Conclusions and Future Work
===========================
In this paper I have proposed a categorization algorithm able to unify all the common-sense categorization strategies proposed in the cognitive science literature: exemplars, prototypes and theory-like common-sense knowledge structures. To the best of my knowledge, this proposal represents the first attempt of providing a unifying categorization strategy by assuming a heterogeneous representational hypothesis. In particular, the proposed algorithm relies and extends both the representational and the reasoning framework considering concepts as heterogeneous proxytypes [@lieto2014computational]. The current theoretical proposal needs to be tested on the empirical ground in order to show both its feasibility with psychological data and its efficacy in the area of artificial cognitive systems. Also, there are additional elements, only sketched in this paper, requiring a more precise characterization. For example: the design of a method to calculate which is the most appropriate theory-like representations to select (line 12, Algorithm 1). On this point, however, it is worth-noticing that, since the most promising computational candidates for representing the theory-like body of knowledge are graphical models and probabilistic semantic networks, it seems plausible to imagine that such calculation could be performed with standard heuristics search on graph structures. Similarly, the individuation and the construction of a plausible *Conceptual Coherence Threshold* represents an issue that should be faced and solved empirically.\
**Acknowledgements**\
The topics presented in this paper have been discussed in these years with a number people in international conferences, symposia, panels and workshops. I thank all them for the received comments. In particular, I am indebted to Marcello Frixione, Leonardo Lesmo, Paul Thagard, David Danks, Ismo Koponen and Christian Lebiere for their feedback and suggestions. I also thank Valentina Rho for her comments on an earlier version of this paper.
[^1]: The expression *heterogeneous proxytypes* refers to both a theoretical and computational hypothesis combining the proxytype theory of concepts with the so called heterogeneity approach to concept representation. The Section 3 of this paper contains a brief discussion of the proposal.
[^2]: In order to explain the different categorization strategies associated to different kinds of representations, let us consider the following examples: if we have to categorize a stimulus with the following features: “it has fur, woofs and wags its tail”, the result of a *prototype-based categorization* would be *dog*, since these cues are associated to the prototype of *dog*. Prototype-based reasoning, however, is not the only type of reasoning based on typicality. In fact, if an exemplar corresponding to the stimulus being categorized is available, too, it is acknowledged that humans use to classify it by evaluating its similarity w.r.t. the exemplar, rather than w.r.t. the prototype associated to the underlying concepts [@frixione2013representing]. For example, a penguin is rather dissimilar from the prototype of *bird*. However, if we already know an exemplar of penguin, and if we know that it is an instance of *bird*, it is easier to classify a new penguin as a *bird* w.r.t. a categorization process based on the similarity with the prototype of that category. This type of common-sense categorization is known in literature as *exemplars-based categorization*. An example of theory-like common sense reasoning is when we typically associate to a light switch the learned rule that if we turn it “on" then the light will be provided (this is a non-monotonic inference with a defeasible conclusion). Finally, the classical representations (i.e. those based on necessary and/or sufficient conditions) are associated with standard deductive mechanism of reasoning. An example of standard deductive reasoning is the categorization as *triangle* of a stimulus described by the features: “it is a polygon, it has three corners and three sides”. Such cues, in fact, are necessary and sufficient for the definition of the concept of triangle. All these representations, and the corresponding reasoning mechanisms, are assumed to be potentially co-existing according to the heterogeneity approach.
[^3]: The heterogeneity assumption has been recently pointed out as one of the problems to face in order to address the problems affecting the knowledge level in cognitive systems and architecture [@lieto2017knowledge; @AAAIStandardModel].
[^4]: The characterization in terms of “heterogeneous proxytypes”, among the other things, enables the system to deal with the problem of the “contextual activation” of a given information based on the external stimulus being considered. In particular (by following the idea that, when we categorize a stimulus, we do not activate the whole network of knowledge related to its assigned category but, conversely, we only activate the knowledge that is “contextually relevant” in its respect), DUAL-PECCS *proxyfyes* only the type of representation that minimizes the distance w.r.t. the percept (see [@lieto2014computational] for further details).
[^5]: Since all the different bodies of knowledge are assumed to be co-referring representational structure pointing to the same conceptual entity, it is possible to recover the theory-like representation associated, for example, to a given prototypical or exemplar based representation.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider Vlasov-type scaling for the Glauber dynamics in continuum with a positive integrable potential, and construct rescaled and limiting evolutions of correlation functions. Convergence to the limiting evolution for the positive density system in infinite volume is shown. Chaos preservation property of this evolution gives a possibility to derive a non-linear Vlasov-type equation for the particle density of the limiting system.'
author:
- 'Dmitri Finkelshtein[^1]'
- 'Yuri Kondratiev[^2]'
- 'Oleksandr Kutoviy[^3]'
title: Vlasov scaling for the Glauber dynamics in continuum
---
Introduction
============
Kinetic equations are a useful approximation for the description of dynamical processes in multi-body systems, see, e.g., the reviews by H.Spohn [@Spo1980], [@Spo1991]. Among them, the Vlasov equation has important role in physics (in particular, physics of plasma). It describes the Hamiltonian motion of an infinite particle system in the mean field scaling limit when the influence of weak long-range forces is taken into account. The convergence of the Vlasov scaling limit was shown rigorously by W.Braun and K.Hepp [@BH1977] (for the Hamiltonian dynamics) and by R.L.Dobrushin [@Dob1979] (for more general deterministic dynamical systems). However, the resulting Vlasov-type equations for particle densities are considered in classes of integrable functions (or, in the weak form, of finite measures). This, in fact, restricts us to the case of finite volume systems or systems with zero mean density in an infinite volume. Detailed analysis of Vlasov-type equations for integrable functions is presented in the recent paper by V.V.Kozlov [@Koz2008].
In [@FKK2010a], we proposed a general approach to study the Vlasov-type scaling for some classes of stochastic evolutions in the continuum, in particular, for spatial birth-and-death Markov processes. The approaches mentioned above are not applicable to these dynamics (even in a finite volume) due to essential reasons (see [@FKK2010a] for details). One of them is a possible variation of the particle number during the evolution. More essentially is that for these processes the possibility of their descriptions in terms of proper stochastic evolutional equations for particle motion is, generally speaking, absent. There are only few works concerning general spatial birth-and-death evolutions, see [@Pre1975], [@HS1978], [@GK2006], [@GK2008], [@Pen2008], [@Qi2008]. However, the conditions for the existence (in different senses) of the evolutions considered therein are quite far from the general form.
Therefore, we looked for an alternative approach to the derivation of kinetic Vlasov-type equations from stochastic dynamics. The correct Vlasov limit can be easily guessed from the BBGKY hierarchy for the Hamiltonian system, see, e.g., [@Spo1980]. Such a heuristic derivation does not assume the integrability condition for the density, but until now, it could not be made rigorously due to the lack of detailed information about the properties of solutions to the BBGKY hierarchy. Our approach is based on this observation applied in a new dynamical framework. Note that we already know that many stochastic evolutions in continuum admit effective descriptions in terms of hierarchical equations for correlation functions which generalize the BBGKY hierarchy from Hamiltonian to Markov setting, see, e.g., [@FKO2009] and the references therein. Even more, these hierarchical equations are often the only available technical tools for a construction of considered dynamics [@KKM2008], [@KKZ2006], [@FKK2009].
Developing this point of view, our scheme for the Vlasov scaling of stochastic dynamics is based on the proper scaling of the hierarchical equations. This scheme has also a clear interpretation in the terms of scaled Markov generators. An application of the considered scaling leads to the limiting hierarchy which posses a chaos preservation property. Namely, if we start from a Poissonian (non-homogeneous) initial state of the system, then during the time evolution this property will be preserved. Moreover, a special structure of the interaction in the resulting virtual Vlasov system gives a non-linear evolutional equation for the density of the evolving Poisson state.
The control of the convergence of Vlasov scalings for the considered hierarchies is a quite difficult technical problem which should be analyzed for any particular model separately. In the present paper, we solve this problem for the Glauber dynamics in continuum. These dynamics have given reversible states which are grand canonical Gibbs measures. The corresponding equilibrium dynamics which preserve the initial Gibbs state in the time evolution were considered in, e.g., [@KL2005], [@KLR2007], [@KMZ2004], [@FKL2007]. Note that, in applications, the time evolution of initial state is the subject of the primary interest. Therefore, we understand the considered stochastic (non-equilibrium) dynamics as the evolution of initial distributions for the system. Actually, the corresponding Markov process (provided it exists) itself gives a general technical equipment to study this problem. Moreover, using the techniques developed in [@GK2006], it is possible to construct this Markov process as a solution of a stochastic differential equation. Unfortunately, this approach does not give any information about the properties of the corresponding correlation functions which we need for the study of Vlasov scaling as was mentioned above.
However, we note that the transition from the micro-state evolution corresponding to the given initial configuration to the macro-state dynamics is the well developed concept in the theory of infinite particle systems. This point of view appeared initially in the framework of the Hamiltonian dynamics of classical gases, see, e.g., [@DSS1989]. Again, the lack of the general Markov processes techniques for the considered systems makes it necessary to develop alternative approaches to study the state evolutions in the Glauber dynamics. Such approaches we realized in [@KKM2008], [@KKZ2006], [@FKKZ2010], [@FKK2010]. The description of the time evolutions for measures on configuration spaces in terms of an infinite system of evolutional equations for the corresponding correlation functions was used there. The latter system is a Glauber evolution’s analog of the famous BBGKY-hierarchy for the Hamiltonian dynamics.
Here we extend the approximation approach proposed in [@FKKZ2010], [@FKK2010] to the Vlasov scaling for the Glauber dynamics in continuum. We construct and study semigroups corresponding to properly rescaled Markov generator of the Glauber dynamics (Propositions \[descsemigroupexist\] and \[sun-inv\]). We prove for the integrable and bounded potential the convergence of these semigroups to the limiting semigroup which describe Vlasov evolution (Theorem \[maintheorem\]). We derive the corresponding Vlasov-type equation from this evolution (Theorem \[Vlasovscheme\]). Note that the stationary solution of this equation will satisfied the well-known Kirkwood–Monroe equation in the freezing theory (Remark \[RemarkKirkwoodMonroe\]).
Glauber dynamics in continuum
=============================
Basic facts and notation
------------------------
Let ${\mathcal{B}}({{\mathbb{R}}^d})$ be the family of all Borel sets in ${{\mathbb{R}}^d}$, $d\geq 1$; ${\mathcal{B}}_{\mathrm{b}}
({{\mathbb{R}}^d})$ denotes the system of all bounded sets in ${\mathcal{B}}({{\mathbb{R}}^d})$.
The configuration space over space ${{\mathbb{R}}^d}$ consists of all locally finite subsets (configurations) of ${{\mathbb{R}}^d}$, namely, $$\label{confspace}
\Gamma =\Gamma_{{\mathbb{R}}^d} :=\Bigl\{ \gamma \subset
{{\mathbb{R}}^d} \Bigm| |\gamma _\Lambda |<\infty, \ \mathrm{for \ all } \ \Lambda \in {\mathcal{B}}_{\mathrm{b}} ({{\mathbb{R}}^d})\Bigr\}.$$ Here $\gamma_\Lambda:=\gamma\cap\Lambda$, and $|\cdot|$ means the cardinality of a finite set. The space $\Gamma$ is equipped with the vague topology, i.e., the minimal topology for which all mappings $\Gamma\ni\gamma\mapsto
\sum_{x\in\gamma}
f(x)\in{\mathbb{R}}$ are continuous for any continuous function $f$ on ${{\mathbb{R}}^d}$ with compact support; note that the summation in $\sum_{x\in\gamma} f(x)$ is taken over finitely many points of $\gamma$ which belong to the support of $f$. In [KK2006]{}, it was shown that $\Gamma$ with the vague topology may be metrizable and it becomes a Polish space (i.e., complete separable metric space). Corresponding to this topology, the Borel $\sigma
$-algebra ${\mathcal{B}}(\Gamma )$ is the smallest $\sigma $-algebra for which all mappings $\Gamma \ni \gamma \mapsto |\gamma_ \Lambda |\in{\mathbb{N}}_0:={\mathbb{N}}\cup\{0\}$ are measurable for any $\Lambda\in{\mathcal{B}}_{\mathrm{b}}({{\mathbb{R}}^d})$.
The space of $n$-point configurations in an arbitrary $Y\in{\mathcal{B}}({{\mathbb{R}}^d})$ is defined by $$\Gamma^{(n)}_Y:=\Bigl\{ \eta \subset Y \Bigm| |\eta |=n\Bigr\} ,\quad n\in {\mathbb{N}}.$$ We set also $\Gamma^{(0)}_Y:=\{\emptyset\}$. As a set, $\Gamma^{(n)}_Y$ may be identified with the symmetrization of $$\widetilde{Y^n} = \Bigl\{ (x_1,\ldots ,x_n)\in Y^n \Bigm| x_k\neq
x_l \ \mathrm{if} \ k\neq l\Bigr\} .$$
Hence one can introduce the corresponding Borel $\sigma $-algebra, which we denote by ${\mathcal{B}}(\Gamma^{(n)}_Y)$. The space of finite configurations in an arbitrary $Y\in{\mathcal{B}}({{\mathbb{R}}^d})$ is defined by $$\Gamma_{0,Y}:=\bigsqcup_{n\in {\mathbb{N}}_0}\Gamma^{(n)}_Y.$$ This space is equipped with the topology of disjoint unions. Therefore, one can introduce the corresponding Borel $\sigma
$-algebra ${\mathcal{B}} (\Gamma _{0,Y})$. In the case of $Y={{\mathbb{R}}^d}$ we will omit the index $Y$ in the notation, namely, $\Gamma_0:=\Gamma_{0,{{\mathbb{R}}^d}}$, $\Gamma^{(n)}:=\Gamma^{(n)}_{{{\mathbb{R}}^d}}$.
The restriction of the Lebesgue product measure $(dx)^n$ to $\bigl(\Gamma^{(n)}, {\mathcal{B}}(\Gamma^{(n)})\bigr)$ we denote by $m^{(n)}$. We set $m^{(0)}:=\delta_{\{\emptyset\}}$. The Lebesgue–Poisson measure $\lambda $ on $\Gamma_0$ is defined by $$\label{LP-meas-def}
\lambda :=\sum_{n=0}^\infty \frac {1}{n!}m^{(n)}.$$ For any $\Lambda\in{\mathcal{B}}_{\mathrm{b}}({{\mathbb{R}}^d})$ the restriction of $\lambda$ to $\Gamma _\Lambda:=\Gamma_{0,\Lambda}$ will be also denoted by $\lambda $. The space $\bigl( \Gamma,
{\mathcal{B}}(\Gamma)\bigr)$ is the projective limit of the family of spaces $\bigl\{( \Gamma_\Lambda, {\mathcal{B}}(\Gamma_\Lambda))\bigr\}_{\Lambda \in {\mathcal{B}}_{\mathrm{b}} ({{\mathbb{R}}^d})}$. The Poisson measure $\pi$ on $\bigl(\Gamma ,{\mathcal{B}}(\Gamma
)\bigr)$ is given as the projective limit of the family of measures $\{\pi^\Lambda \}_{\Lambda
\in {\mathcal{B}}_{\mathrm{b}} ({{\mathbb{R}}^d})}$, where $\pi^\Lambda:=e^{-m(\Lambda)}\lambda $ is the probability measure on $\bigl( \Gamma_\Lambda, {\mathcal{B}}(\Gamma_\Lambda)\bigr)$. Here $m(\Lambda)$ is the Lebesgue measure of $\Lambda\in {\mathcal{B}}_{\mathrm{b}} ({{\mathbb{R}}^d})$.
For any measurable function $f:{{\mathbb{R}}^d}\rightarrow
{\mathbb{R}}$ we define a *Lebesgue–Poisson exponent* $$\label{LP-exp-def}
e_\lambda(f,\eta):=\prod_{x\in\eta} f(x),\quad \eta\in\Gamma_0;
\qquad e_\lambda(f,\emptyset):=1.$$ Then, by , for $f\in L^1({{\mathbb{R}}^d},dx)$ we obtain $e_\lambda(f)\in L^1(\Gamma_0,d\lambda)$ and $$\label{LP-exp-mean}
\int_{\Gamma_0}
e_\lambda(f,\eta)d\lambda(\eta)=\exp\Biggl\{\int_{{\mathbb{R}}^d}
f(x)dx\Biggr\}.$$
A set $M\in {\mathcal{B}} (\Gamma_0)$ is called bounded if there exists $\Lambda \in {\mathcal{B}}_{\mathrm{b}} ({{\mathbb{R}}^d})$ and $N\in {\mathbb{N}}$ such that $M\subset \bigsqcup_{n=0}^N\Gamma
_\Lambda^{(n)}$. The set of bounded measurable functions with bounded support we denote by $B_{\mathrm{bs}}(\Gamma_0)$, i.e., $G\in B_{\mathrm{bs}}(\Gamma_0)$ if $G\upharpoonright_{\Gamma_0\setminus M}=0$ for some bounded $M\in {\mathcal{B}}(\Gamma_0)$. Any ${\mathcal{B}}(\Gamma_0)$-measurable function $G$ on $\Gamma_0$, in fact, is a sequence of functions $\bigl\{G^{(n)}\bigr\}_{n\in{\mathbb{N}}_0}$ where $G^{(n)}$ is a ${\mathcal{B}}(\Gamma^{(n)})$-measurable function on $\Gamma^{(n)}$. We consider also the set ${{\mathcal{F}}_{\mathrm{cyl}}}(\Gamma )$ of *cylinder functions* on $\Gamma$. Each $F\in {{\mathcal{F}}_{\mathrm{cyl}}}(\Gamma )$ is characterized by the following relation: $F(\gamma
)=F\upharpoonright_{\Gamma_\Lambda
}(\gamma_\Lambda )$ for some $\Lambda\in {\mathcal{B}}_{\mathrm{b}}({{\mathbb{R}}^d})$.
There is the following mapping from $B_{\mathrm{bs}} (\Gamma_0)$ into ${{\mathcal{F}}_{\mathrm{cyl}}}(\Gamma )$, which plays the key role in our further considerations: $$KG(\gamma ):=\sum_{\eta \Subset \gamma }G(\eta ), \quad \gamma \in
\Gamma, \label{KT3.15}$$ where $G\in B_{\mathrm{bs}}(\Gamma_0)$, see, e.g., [@KK2002; @Len1975; @Len1975a]. The summation in is taken over all finite subconfigurations $\eta\in{\Gamma}_0$ of the (infinite) configuration $\gamma\in{\Gamma}$; we denote this by the symbol, $\eta\Subset\gamma $. The mapping $K$ is linear, positivity preserving, and invertible, with $$K^{-1}F(\eta ):=\sum_{\xi \subset \eta }(-1)^{|\eta \setminus \xi
|}F(\xi ),\quad \eta \in \Gamma_0. \label{k-1trans}$$ We denote the restriction of $K$ onto functions on $\Gamma_0$ by $K_0$.
A measure $\mu \in {\mathcal{M}}_{\mathrm{fm} }^1(\Gamma )$ is called locally absolutely continuous with respect to (w.r.t. for short) the Poisson measure $\pi$ if for any $\Lambda \in
{\mathcal{B}}_{\mathrm{b}} ({{\mathbb{R}}^d})$ the projection of $\mu$ onto $\Gamma_\Lambda$ is absolutely continuous w.r.t. the projection of $\pi$ onto $\Gamma_\Lambda$. By [@KK2002], in this case, there exists a *correlation functional* $k_{\mu}:\Gamma_0 \rightarrow
{\mathbb{R}}_+$ such that for any $G\in B_{\mathrm{bs}} (\Gamma_0)$ the following equality holds $$\label{eqmeans}
\int_\Gamma (KG)(\gamma) d\mu(\gamma)=\int_{\Gamma_0}G(\eta)
k_\mu(\eta)d\lambda(\eta).$$ The restrictions $k_\mu^{(n)}$ of this functional on $\Gamma_0^{(n)}$, $n\in{\mathbb{N}}_0$ are called *correlation functions* of the measure $\mu$. Note that $k_\mu^{(0)}=1$.
We recall now without a proof the partial case of the well-known technical lemma (cf., [@KMZ2004]) which plays very important role in our calculations.
\[Minlos\] For any measurable function $H:\Gamma_0\times\Gamma_0\times\Gamma_0\rightarrow{\mathbb{R}}$ $$\label{minlosid}
\int_{\Gamma _{0}}\sum_{\xi \subset \eta }H\left( \xi ,\eta
\setminus \xi ,\eta \right) d\lambda \left( \eta \right)
=\int_{\Gamma _{0}}\int_{\Gamma _{0}}H\left( \xi ,\eta ,\eta \cup
\xi \right) d\lambda \left( \xi \right) d\lambda \left( \eta \right)$$ if only both sides of the equality make sense.
Non-equilibrium Glauber dynamics in continuum {#dualconstraction}
---------------------------------------------
Let $\phi:{{{{\mathbb R}}^d}}{\rightarrow}{{\mathbb R}}_+:=[0;+\infty)$ be an even non-negative function which satisfies the following integrability condition $$\label{weak_integrability}
C_\phi := \int_{{{{\mathbb R}}^d}}\bigl(1-e^{-\phi(x)}\bigr) dx < +\infty.$$ For any ${\gamma}\in{\Gamma}$, $x\in{{{{\mathbb R}}^d}}\setminus{\gamma}$ we set $$\label{relativeenergy}
E^\phi(x,{\gamma}) :=\sum_{y\in{\gamma}} \phi(x-y) \in [0;\infty].$$
Let us define the (pre-)generator of the Glauber dynamics: for any $F\in{{{{\mathcal F}}_{\mathrm{cyl}}}}({\Gamma})$ we set $$\begin{aligned}
(LF)({\gamma}):=&\sum_{x\in{\gamma}} \bigl[F({\gamma}\setminus x) -F({\gamma})\bigr]
\label{genGa}
\\&+ z \int_{{{{{\mathbb R}}^d}}} \bigl[F({\gamma}\cup x)
-F({\gamma})\bigr]\exp\bigl\{-E^\phi(x,{\gamma})\bigr\} dx, \qquad
{\gamma}\in{\Gamma}.\nonumber\end{aligned}$$ Here $z>0$ is the [*activity*]{} parameter. Note that for any $F\in{{{{\mathcal F}}_{\mathrm{cyl}}}}({\Gamma})$ there exists a ${\Lambda}\in{{{\mathcal B}}_{\mathrm{b}}}({{{{\mathbb R}}^d}})$ such that $F({\gamma}\setminus x)=F({\gamma})$ for all $x\in{\gamma}_{{\Lambda}^c}$ and $F({\gamma}\cup
x)=F({\gamma})$ for all $x\in{\Lambda}^c$; note also that $\exp\bigl\{-E^\phi(x,{\gamma})\bigr\}\leq 1$, therefore, sum and integral in are finite.
For any fixed $C>1$ we consider the following Banach space of ${\mathcal{B}} (\Gamma_0)$-measurable functions $${{\mathcal L}}_C:=\biggl\{ G:\Gamma_0\rightarrow{\mathbb{R}} \biggm| \|G\|_C:=
\int_{\Gamma_0} |G(\eta)| C^{|\eta|} d\lambda(\eta) <\infty\biggr\}.$$
In [@FKKZ2010 Proposition 3.1], it was shown that the mapping ${{\hat{L}}}:=K^{-1}LK$ given on ${B_{\mathrm{bs}}}({\Gamma}_0)$ by $$\begin{aligned}
\label{Lhat}
({{\hat{L}}}G)(\eta) =& - |\eta| G(\eta) \\& + z
\sum_{\xi\subset\eta}\int_{{{{\mathbb R}}^d}}e^{-E^\phi(x,\xi)} G(\xi\cup x)e_{\lambda}(e^{-\phi (x - \cdot)}-1,\eta\setminus\xi) dx \nonumber\end{aligned}$$ is a linear operator on ${{\mathcal L}}_C$ with the dense domain ${{\mathcal L}}_{2C}\subset{{\mathcal L}}_C$. If additionally, $$\label{verysmallparam}
z\leq \min\bigl\{Ce^{-CC_{\phi }} ; 2Ce^{-2CC_{\phi }}\bigr\},$$ then $\bigl({{\hat{L}}}, {{\mathcal L}}_{2C}\bigr)$ is closable linear operator in ${{\mathcal L}}_C$ and its closure $\bigl({{\hat{L}}}, D({{\hat{L}}})\bigr)$ generates a strongly continuous contraction semigroup ${{\hat{T}}}(t)$ on ${{\mathcal L}}_C$ (see [@FKKZ2010 Theorem 3.8] for details).
Let us set $d{\lambda}_C:= C^{|\cdot|} d{\lambda}$; then the dual space $({{\mathcal L}}_C)'=\bigl(L^1({\Gamma}_0, d{\lambda}_C)\bigr)'=L^\infty({\Gamma}_0, d{\lambda}_C)$. The space $({{\mathcal L}}_C)'$ is isometrically isomorphic to the Banach space $${{\mathcal K}}_{C}:=\left\{k:{\Gamma}_{0}{\rightarrow}{{\mathbb R}}\,\Bigm| k\cdot C^{-|\cdot|}\in
L^{\infty}({\Gamma}_{0},{\lambda})\right\}$$ with the norm $\|k\|_{{{\mathcal K}}_C}:=\|C^{-|\cdot|}k(\cdot)\|_{L^{\infty}({\Gamma}_{0},{\lambda})}$ where the isomorphism is provided by the isometry $R_C$ $$\label{isometry}
({{\mathcal L}}_C)'\ni k \longmapsto R_Ck:=k\cdot C^{|\cdot|}\in {{\mathcal K}}_C.$$
In fact, one may consider the duality between the Banach spaces ${{\mathcal L}}_C$ and ${{\mathcal K}}_C$ given by the following expression $${{\left\langle}\!{\left\langle}}G,\,k {{\right\rangle}\!{\right\rangle}}:= \int_{{\Gamma}_{0}}G\cdot k\, d{\lambda},\quad G\in{{\mathcal L}}_C, \
k\in {{\mathcal K}}_C \label{duality}$$ with ${\left\vert}{{\left\langle}\!{\left\langle}}G,k {{\right\rangle}\!{\right\rangle}}{\right\vert}\leq \|G\|_C \cdot\|k\|_{{{\mathcal K}}_C}$. It is clear that $k\in {{\mathcal K}}_C$ implies $|k(\eta)|\leq \|k\|_{{{\mathcal K}}_C} \,
C^{|\eta|}$ for ${\lambda}$-a.a. $\eta\in{\Gamma}_0$.
Let $\bigl( {\hat{L}} ^{\prime }, D({\hat{L}} ^{\prime })\bigr)$ be an operator in $({{\mathcal L}}_C)^{\prime }$ which is dual to the closed operator $\bigl( {\hat{L}} , D({\hat{L}} )\bigr)$. We consider also its image on ${\mathcal{K}}_C$ under the isometry $R_C$, namely, let ${\hat{L}}^{*}=R_C{\hat{L}} ^{\prime }R_{C^{-1}}$ with the domain $D({\hat{L}} ^{*})=R_C D({\hat{L}} ^{\prime })$. It was noted in [@FKK2010] that ${{\hat{L}}}^\ast$ is the dual operator to ${{\hat{L}}}$ w.r.t. the duality and that for any $k\in D({\hat{L}}^\ast)$ $$\begin{aligned}
\label{dual-descent}
({{\hat{L}}}^* k)(\eta)=&-\vert \eta \vert k(\eta)\\&+z \sum_{x\in
\eta}e^{-E^\phi (x,\eta\setminus x)} \int_{{\Gamma}_0}e_{\lambda}(e^{-\phi (x
- \cdot)}-1,\xi) k((\eta\setminus x)\cup\xi)\,d{\lambda}(\xi).\nonumber\end{aligned}$$
Under condition , we consider the adjoint semigroup ${{\hat{T}}}'(t)$ in $({{\mathcal L}}_C)'$ and its image ${{\hat{T}}}^\ast(t)$ in ${{\mathcal K}}_C$. By the general results from [@vNee1992 Sections 1.2, 1.3], the restriction ${{\hat{T}}}^\odot(t)$ of the semigroup ${{\hat{T}}}^\ast(t)$ onto its invariant Banach subspace $\overline{D({{\hat{L}}}^\ast)}$ is a contraction strongly continuous semigroup. By [@FKK2010 Proposition 3.1], for any ${{\alpha}}\in(0;1)$ we have ${{\mathcal K}}_{{{\alpha}}C}\subset D({{\hat{L}}}^\ast)$ and, moreover, by [@FKK2010 Proposition 3.3], there exists ${{\alpha}}_0={{\alpha}}_0(z,\phi,C)\in (0;1)$ such that for any ${{\alpha}}\in ({{\alpha}}_0;1)$ the set ${{\mathcal K}}_{{{\alpha}}C}$ will be also a ${{\hat{T}}}^\ast(t)$-invariant linear subspace. As a result, for any $\overline{D({{\hat{L}}}^\ast)}$ the Cauchy problem in ${{\mathcal K}}_C$ $$\begin{cases}
\dfrac{\partial}{\partial t} k_t = {{\hat{L}}}^\ast k_t \\[2mm] k_t \bigr|_{t=0}
= k_0
\end{cases}$$ is well-defined and solvable: $k_t= {{\hat{T}}}^\ast (t)k_0= {{\hat{T}}}^\odot
(t)k_0\in\overline{D({{\hat{L}}}^\ast)}$; moreover, $k_0\in{{\mathcal K}}_{{{\alpha}}C}$ implies $k_t\in{{\mathcal K}}_{{{\alpha}}C}$.
Vlasov-type scaling
===================
Description of scaling {#scalingdescr}
----------------------
We start from the explanation of the idea of the Vlasov-type scaling. We want to construct some scaling of the generator $L$, say, $L_{\varepsilon}$, ${\varepsilon}>0$, such that the following scheme holds. Suppose that we have a semigroup ${{\hat{T}}}_{\varepsilon}(t)$ with generator ${{\hat{L}}}_{\varepsilon}$ in some ${{\mathcal L}}_{C_{\varepsilon}}$. Consider the dual semigroup ${{\hat{T}}}_{\varepsilon}^\ast(t)$. Let us choose an initial function of the corresponding Cauchy problem with a big singularity by ${\varepsilon}$, namely, $k_0^{{({\varepsilon})}}(\eta) \sim {\varepsilon}^{-|\eta|} r_0(\eta)$, ${\varepsilon}{\rightarrow}0$, $\eta\in{\Gamma}_0$ with some function $r_0$, independent of ${\varepsilon}$. Our first demand to the scaling $L\mapsto L_{\varepsilon}$ is that the semigroup ${{\hat{T}}}_{\varepsilon}^\ast(t)$ preserves the order of the singularity: $$\label{ordersing}
({{\hat{T}}}_{\varepsilon}^\ast(t)k_0^{{({\varepsilon})}})(\eta) \sim {\varepsilon}^{-|\eta|} r_t(\eta), \quad
{\varepsilon}{\rightarrow}0, \ \ \eta\in{\Gamma}_0.$$ And the second one is that the dynamics $r_0 \mapsto r_t$ should preserve Lebesgue–Poisson exponents, namely, if $r_0(\eta)=e_{\lambda}(\rho_0,\eta)$ then $r_t(\eta)=e_{\lambda}(\rho_t,\eta)$ and there exists explicit (nonlinear, in general) differential equation for $\rho_t$: $$\label{V-eqn-gen}
\dfrac{\partial}{\partial t}\rho_t(x) = \upsilon(\rho_t)(x)$$ which we will call the Vlasov-type equation.
Now let us explain an informal way for the realization of this scheme. Let us consider for any ${\varepsilon}>0$ the following mapping (cf. ) on functions on ${\Gamma}_0$ $$(R_{\varepsilon}r)(\eta):={\varepsilon}^{{|\eta|}}r(\eta).$$ This mapping is “self-dual” w.r.t. the duality , moreover, $R_{\varepsilon}^{-1}=R_{{\varepsilon}^{-1}}$. Then we have $k^{{({\varepsilon})}}_0\sim
R_{{\varepsilon}^{-1}} r_0$, and we need $r_t \sim R_{\varepsilon}{{\hat{T}}}_{\varepsilon}^\ast(t)k_0^{{({\varepsilon})}}\sim R_{\varepsilon}{{\hat{T}}}_{\varepsilon}^\ast(t)R_{{\varepsilon}^{-1}}
r_0$. Therefore, we have to show that for any $t\geq 0$ the operator family $R_{\varepsilon}{{\hat{T}}}_{\varepsilon}^\ast(t)R_{{\varepsilon}^{-1}}$, ${\varepsilon}>0$ has limiting (in a proper sense) operator $U(t)$ and $$\label{chaospreserving}
U(t)e_{\lambda}(\rho_0)=e_{\lambda}(\rho_t).$$ But, informally, ${{\hat{T}}}^\ast_{\varepsilon}(t)=\exp{\{t{{\hat{L}}}^\ast_{\varepsilon}\}}$ and $R_{\varepsilon}{{\hat{T}}}_{\varepsilon}^\ast(t)R_{{\varepsilon}^{-1}}=\exp{\{t R_{\varepsilon}{{\hat{L}}}_{\varepsilon}^\ast
R_{{\varepsilon}^{-1}} \}}$. Let us consider the “renormalized” operator $$\label{renorm_def}
{{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}:= R_{\varepsilon}{{\hat{L}}}_{\varepsilon}^\ast R_{{\varepsilon}^{-1}}.$$ In fact, we need that there exists an operator ${{\hat{L}}}_V^\ast$ such that $\exp{\{t R_{\varepsilon}{{\hat{L}}}_{\varepsilon}^\ast R_{{\varepsilon}^{-1}} \}}{\rightarrow}\exp{\{t{{\hat{L}}}_V^\ast\}=:U(t)}$ for which holds. Therefore, a heuristic way to produce such a scaling $L\mapsto L_{\varepsilon}$ is to demand that $$\lim_{{\varepsilon}{\rightarrow}0}\left(\dfrac{\partial}{\partial
t}e_{\lambda}(\rho_t,\eta)-{{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}e_{\lambda}(\rho_t,\eta)\right)=0, \qquad
\eta\in{\Gamma}_0$$ if only $\rho_t$ is satisfied . The point-wise limit of ${{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}$ will be natural candidate for ${{\hat{L}}}_V^\ast$.
Note that implies ${{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}=R_{{\varepsilon}^{-1}}{{\hat{L}}}_{\varepsilon}R_{\varepsilon}$. Hence, we will use the following scheme to give rigorous meaning to all considerations above. We consider, for a proper scaling $L_{\varepsilon}$, the “renormalized” operator ${{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}$ and prove that it is a generator of a strongly continuous contraction semigroup ${{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ in ${{\mathcal L}}_C$. Next, we show that the formal limit ${{\hat{L}}}_V$ of ${{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}$ is also a generator of a strongly continuous contraction semigroup ${{\hat{T}}}_V(t)$ in ${{\mathcal L}}_C$ also. Then, we consider the dual semigroups ${{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast(t)$ and ${{\hat{T}}}_V^\ast(t)$ in the proper Banach subspace of the space ${{\mathcal K}}_C$. Finally, we prove that ${{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast(t)\rightarrow{{\hat{T}}}_V^\ast(t)$ strongly on this subspace and explain in which sense ${{\hat{T}}}_V^\ast(t)$ satisfies the properties above. Below we try to realize this scheme.
Construction and convergence of the evolutions in ${{\mathcal L}}_C$ {#subsectiondescevolution}
--------------------------------------------------------------------
Let us consider for any $F\in{{{{\mathcal F}}_{\mathrm{cyl}}}}({\Gamma})$, ${\varepsilon}>0$ $$\begin{aligned}
(L_{\varepsilon}F)({\gamma}):=&\sum_{x\in{\gamma}} \bigl[F({\gamma}\setminus x)
-F({\gamma})\bigr] \label{genGa-eps}
\\& + {\varepsilon}^{-1}z \int_{{{{{\mathbb R}}^d}}} \bigl[F({\gamma}\cup x)
-F({\gamma})\bigr]\exp\bigl\{-{\varepsilon}E^{\phi}(x,{\gamma})\bigr\} dx, \quad
{\gamma}\in{\Gamma}.\nonumber\end{aligned}$$ We define also for any $G\in{B_{\mathrm{bs}}}({\Gamma}_0)$, ${\varepsilon}>0$ $${{\hat{L}}}_{\varepsilon}G:=K^{-1}L_{\varepsilon}K G; \qquad {{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}G:=R_{{\varepsilon}^{-1}}{{\hat{L}}}_{\varepsilon}R_{\varepsilon}G.$$
Let $\phi$ be integrable function on the whole ${{{{\mathbb R}}^d}}$, namely, $$\label{integrability}
\beta :=\int_{{{{{\mathbb R}}^d}}}\phi(x)dx<+\infty.$$ We fix this notation for our considerations below.
Then, by the elementary inequality $$1-e^{- t}\leq t, \quad t\geq 0 \label{ineq_exp}$$(which we will use often), $\phi$ will satisfy and $C_\phi\leq\beta$.
For any $G\in B_{bs}\left( {\Gamma}_{0}\right)$ $$( {{\hat{L}}}_{{\varepsilon},\mathrm{ren}}G) \left( \eta \right) =\left(
L_{1}G\right) \left( \eta \right) +\left( L_{2,{\varepsilon}}G\right) \left(
\eta \right), \label{expleps}$$ where $$\begin{aligned}
\left( L_{1}G\right) \left( \eta \right) &=-{\left\vert}\eta {\right\vert}G\left( \eta \right) , \\
\left( L_{2,{\varepsilon}}G\right) \left( \eta \right) &=z\sum_{\xi \subset
\eta }\int_{{{{\mathbb R}}^d}}e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right)
},\xi \right) \\&\qquad\times e_{{\lambda}}\left( \frac{e^{-{\varepsilon}\phi
\left( x-\cdot \right) }-1}{{\varepsilon}},\eta \setminus \xi \right)
G\left( \xi \cup x\right) dx.\nonumber\end{aligned}$$ Moreover, the expression defines a linear operator in ${{\mathcal L}}_C$ with dense domain ${{\mathcal L}}_{2C}$.
By , for any $G\in{B_{\mathrm{bs}}}({\Gamma}_0)$ we have $$\begin{aligned}
\label{Lhateps}
({{\hat{L}}}_{\varepsilon}G)(\eta) =&- |\eta| G(\eta) \\&+ {\varepsilon}^{-1}z
\sum_{\xi\subset\eta}\int_{{{{\mathbb R}}^d}}e^{-{\varepsilon}E^\phi(x,\xi)} G(\xi\cup
x)e_{\lambda}(e^{-{\varepsilon}\phi (x - \cdot)}-1,\eta\setminus\xi) dx
\nonumber.\end{aligned}$$ Then $$\begin{aligned}
({{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}G)(\eta) &= (R_{{\varepsilon}^{-1}}{{\hat{L}}}_{\varepsilon}R_{\varepsilon}G)(\eta)\\ & =-{\varepsilon}^{-{{|\eta|}}} |\eta|{\varepsilon}^{{|\eta|}}G(\eta) \\&\qquad +
{\varepsilon}^{-{{|\eta|}}}{\varepsilon}^{-1}z \sum_{\xi\subset\eta}\int_{{{{\mathbb R}}^d}}e^{-{\varepsilon}E^\phi(x,\xi)} {\varepsilon}^{|\xi\cup x|}G(\xi\cup
x)e_{\lambda}(e^{-{\varepsilon}\phi (x - \cdot)}-1,\eta\setminus\xi) dx\\
& =\left( L_{1}G\right) \left( \eta \right) +\left( L_{2,{\varepsilon}}G\right) \left( \eta \right).\end{aligned}$$ Next, for any $G\in {{\mathcal L}}_{2C}$ we obtain $$\begin{aligned}
{\left\Vert}L_{1}G{\right\Vert}_{C}&=\int_{{\Gamma}_{0}}{\left\vert}\eta {\right\vert}{\left\vert}G\left( \eta
\right) {\right\vert}C^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right)
\nonumber\\&\leq \int_{{\Gamma}_{0}}2^{{\left\vert}\eta {\right\vert}}{\left\vert}G\left( \eta
\right) {\right\vert}C^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right) ={\left\Vert}G{\right\Vert}_{2C}. \label{bound1}\end{aligned}$$ From and the estimate $e^{-\phi}\leq 1$ we get $$\begin{aligned}
&{\left\Vert}L_{2,{\varepsilon}}G{\right\Vert}_{C} \nonumber \\ \leq &z\int_{{\Gamma}_{0}}\sum_{\xi \subset \eta }\int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi \cup x\right)
{\right\vert}e_{{\lambda}}\left( \frac{1-e^{-{\varepsilon}\phi \left( x-\cdot \right)
}}{{\varepsilon}},\eta \setminus \xi \right) dxC^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right) \nonumber\\
\leq &z\int_{{\Gamma}_{0}}\sum_{\xi \subset \eta }\int_{{{{\mathbb R}}^d}}{\left\vert}G\left(
\xi \cup x\right) {\right\vert}e_{{\lambda}}\left( \phi \left( x-\cdot \right)
,\eta \setminus \xi \right) dxC^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta
\right),\end{aligned}$$ then, by Lemma \[Minlos\], one may continue, $$\begin{aligned}
\leq &z\int_{{\Gamma}_{0}}\int_{{\Gamma}_{0}}\int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi \cup
x\right) {\right\vert}e_{{\lambda}}\left( \phi \left( x-\cdot \right) ,\eta
\right) dxC^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right) C^{{\left\vert}\xi
{\right\vert}}d{\lambda}\left( \xi \right)\end{aligned}$$ and yields $$\begin{aligned}
=&z\exp \left\{ C\beta
\right\} \int_{{\Gamma}_{0}}\int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi \cup x\right) {\right\vert}dxC^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right),\end{aligned}$$ then, using Lemma \[Minlos\] again, $$\begin{aligned}
=&z\exp
\left\{ C\beta \right\} C^{-1}\int_{{\Gamma}_{0}}{\left\vert}G\left( \xi \right)
{\right\vert}\cdot{\left\vert}\xi {\right\vert}C^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right)
\nonumber\\ \leq & z\exp \left\{ C\beta \right\} C^{-1}{\left\Vert}G{\right\Vert}_{2C}. \label{bound2}\end{aligned}$$ The estimates and provide the statement.
Let for any $G\in{B_{\mathrm{bs}}}({\Gamma}_0)$ $$( {{\hat{L}}}_V G) \left( \eta \right) :=\lim_{{\varepsilon}{\rightarrow}0} ( {{\hat{L}}}_{{\varepsilon},\mathrm{ren}}G) \left( \eta \right)=\left( L_{1}G\right) \left(
\eta \right) +( L_{2}^{V}G) \left( \eta \right) ,\quad \eta\in{\Gamma}_0,
\label{explV}$$where$$\begin{aligned}
( L_{2}^{V}G) \left( \eta \right) =&z\sum_{\xi \subset
\eta }\int_{{{{\mathbb R}}^d}}G\left( \xi \cup x\right) e_{{\lambda}}\left( -\phi \left(
x-\cdot \right) ,\eta \setminus \xi \right) dx.\end{aligned}$$Then, the expression defines a linear operator in ${{\mathcal L}}_C$ with dense domain ${{\mathcal L}}_{2C}$.
Since, by the definition, $${\left\Vert}L_2^VG{\right\Vert}_{C} \leq z\int_{{\Gamma}_{0}}\sum_{\xi \subset \eta
}\int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi \cup x\right) {\right\vert}e_{{\lambda}}\left( \phi
\left( x-\cdot \right) ,\eta \setminus \xi \right) dxC^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right)$$ the statement follows from and .
Let us set (cf. [@FKKZ2010 (3.12)]) for any $\delta \in \left( 0;1\right)$, ${\varepsilon}>0$, $G\in
B_{bs}\left( {\Gamma}_{0}\right)$, $\eta\in{\Gamma}_0$ $$\begin{aligned}
\bigl( {{\hat{P}}}_{\delta ,{\varepsilon}}G\bigr) \left( \eta \right) :=&\sum_{\xi
\subset \eta }\left( 1-\delta \right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}}\left( z\delta \right)^{{\left\vert}\omega
{\right\vert}}G\left( \xi \cup \omega \right) \label{contreps} \\
&\qquad \times e_{{\lambda}}\left( e^{-{\varepsilon}E^{\phi }\left( \cdot
,\omega \right) },\xi \right) e_{{\lambda}}\left( \frac{e^{-{\varepsilon}E^{\phi
}\left( \cdot ,\omega \right) }-1}{{\varepsilon}},\eta \setminus \xi \right)
d{\lambda}\left( \omega \right) . \nonumber\end{aligned}$$ and $$\begin{aligned}
\bigl( {{\hat{Q}}}_{\delta }G\bigr) \left( \eta \right) :=&\sum_{\xi \subset
\eta }\left( 1-\delta \right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}}\left(
z\delta \right)^{{\left\vert}\omega {\right\vert}}G\left( \xi
\cup \omega \right) \label{contrV} \\
&\qquad \times e_{{\lambda}}\left( -E^{\phi }\left( \cdot ,\omega
\right) ,\eta \setminus \xi \right) d{\lambda}\left( \omega \right).
\nonumber\end{aligned}$$
\[contrmaps\] Let $$ze^ { \beta C } \leq C. \label{smallz}$$Then ${{\hat{P}}}_{\delta ,{\varepsilon}}$ and ${{\hat{Q}}}_{\delta }$ given by and are well defined linear contractions on ${{\mathcal L}}_C$.
By , Lemma \[Minlos\], and , we get for any $G\in {{\mathcal L}}_C$ $$\begin{aligned}
&\max\Bigl\{ \bigl\Vert {{\hat{P}}}_{\delta ,{\varepsilon}}G\bigr\Vert _{C} ; \
\bigl\Vert
{{\hat{Q}}}_{\delta }G\bigr\Vert _{C} \Bigl\}\\
\leq &\int_{{\Gamma}_{0}}\sum_{\xi \subset \eta }\left( 1-\delta
\right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}}\left( z\delta \right)^{{\left\vert}\omega {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right) {\right\vert}e_{{\lambda}}\left(
E^{\phi }\left( \cdot ,\omega \right) ,\eta \setminus \xi \right)
d{\lambda}\left( \omega \right) C^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta
\right) \\ = &\int_{{\Gamma}_{0}}\int_{{\Gamma}_{0}}\left( 1-\delta
\right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}}\left( z\delta \right)^{{\left\vert}\omega {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right) {\right\vert}e_{{\lambda}}\left(
E^{\phi }\left( \cdot ,\omega \right) ,\eta \right) d{\lambda}\left(
\omega \right) C^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right) C^{{\left\vert}\xi
{\right\vert}}d{\lambda}\left( \xi
\right) \\
= &\int_{{\Gamma}_{0}}\int_{{\Gamma}_{0}}\left( 1-\delta
\right)^{{\left\vert}\xi {\right\vert}}\left( z\delta \right)^{{\left\vert}\omega {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right) {\right\vert}\exp \left\{ C\beta {\left\vert}\omega
{\right\vert}\right\} d{\lambda}\left( \omega \right) C^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right) \\ = &\int_{{\Gamma}_{0}}\int_{{\Gamma}_{0}}\left(
1-\delta \right)^{{\left\vert}\xi {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right)
{\right\vert}\left( z\delta \exp \left\{ C\beta \right\} C^{-1}\right)^{{\left\vert}\omega {\right\vert}}C^{{\left\vert}\omega {\right\vert}}C^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi
\right) d{\lambda}\left( \omega \right) \\
= &\int_{{\Gamma}_{0}}{\left\vert}G\left( \xi \right) {\right\vert}\left( 1-\delta
+z\delta \exp \left\{ C\beta \right\} C^{-1}\right)^{{\left\vert}\xi {\right\vert}}C^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right) \leq {\left\Vert}G{\right\Vert}_{C},\end{aligned}$$ that proves the contraction property; then, in particular, $$\bigl( {{\hat{P}}}_{\delta ,{\varepsilon}}G\bigr) \left( \eta \right)<+\infty,
\qquad \bigl( {{\hat{Q}}}_{\delta }G\bigr) \left( \eta \right)<+\infty$$ for ${\lambda}$-a.a. $\eta\in{\Gamma}_0$.
Now let us construct the approximations for the operators ${{\hat{L}}}_V $ and ${{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}$.
\[apprgenGl\] Let for $\delta \in \left( 0;1\right) $$${{\hat{L}}}_{\delta , V }:=\frac{1}{\delta }\bigl( {{\hat{Q}}}_{\delta }-1\bigr) ;~~~{{\hat{L}}}_{\delta ,{\varepsilon}}:=\frac{1}{\delta }\bigl( {{\hat{P}}}_{\delta ,{\varepsilon}}-1\bigr) , \ {\varepsilon}>0.$$Let holds, then $${\left\Vert}\bigl( {{\hat{L}}}_{\delta , V }-{{\hat{L}}}_V \bigr) G{\right\Vert}_{C}<3\delta {\left\Vert}G{\right\Vert}_{2C}$$and for any ${\varepsilon}>0$$${\left\Vert}\bigl( {{\hat{L}}}_{\delta ,{\varepsilon}}-{{\hat{L}}}_{{\varepsilon},\mathrm{ren}}\bigr) G{\right\Vert}_{C}\leq 3\delta {\left\Vert}G{\right\Vert}_{2C}.$$
Let us denote$$\begin{aligned}
\bigl( {{\hat{Q}}}_{\delta }^{\left( 0\right) }G\bigr) \left( \eta \right)
:=&\sum_{\xi \subset \eta }\left( 1-\delta \right)^{{\left\vert}\xi {\right\vert}}G\left( \xi \right) 0^{{\left\vert}\eta \setminus \xi {\right\vert}}=\left(
1-\delta \right)^{{\left\vert}\eta {\right\vert}}G\left(
\eta \right) , \\
\bigl( {{\hat{Q}}}_{\delta }^{\left( 1\right) }G\bigr) \left( \eta \right)
:=&z\delta \sum_{\xi \subset \eta }\left( 1-\delta \right)^{{\left\vert}\xi
{\right\vert}}\int_{{{{\mathbb R}}^d}}G\left( \xi \cup x\right) e_{{\lambda}}\left( -\phi \left(
x-\cdot \right) ,\eta \setminus \xi \right) dx,\end{aligned}$$and $${{\hat{Q}}}_{\delta }^{\left( \geq 2\right) }:={{\hat{Q}}}_{\delta }-\bigl( {{\hat{Q}}}_{\delta }^{\left( 0\right) }+{{\hat{Q}}}_{\delta }^{\left( 1\right) }\bigr)
.$$Clearly, we have $$\begin{aligned}
{\left\Vert}\bigl( {{\hat{L}}}_{\delta , V }-{{\hat{L}}}_V \bigr) G{\right\Vert}_{C} \leq &{\left\Vert}\frac{1}{\delta }\bigl( {{\hat{Q}}}_{\delta }^{\left( 0\right)
}-1\bigr) G-L_{1}G{\right\Vert}_{C} \\
&+{\left\Vert}\frac{1}{\delta }{{\hat{Q}}}_{\delta }^{\left( 1\right)
}G-L_{2}^{V} G{\right\Vert}_{C}+{\left\Vert}\frac{1}{\delta }{{\hat{Q}}}_{\delta }^{\left( \geq 2\right) }G{\right\Vert}_{C}.\end{aligned}$$It follows from the simple inequality $$\label{usefulineq}
0\leq n-\frac{1-\left( 1-\delta \right)^{n}}{\delta }<\delta \cdot
2^{n},\ n\in {{\mathbb N}},\ \delta >0,$$ that $${\left\Vert}\frac{1}{\delta }\bigl( {{\hat{Q}}}_{\delta }^{\left( 0\right)
}-1\bigr) G-L_{1}G{\right\Vert}_{C}={\left\Vert}\frac{1}{\delta }\left( \left(
1-\delta \right)^{{\left\vert}\cdot {\right\vert}}-1\right) G+{\left\vert}\cdot {\right\vert}G{\right\Vert}_{C}<\delta {\left\Vert}G{\right\Vert}_{2C}$$and$$\begin{aligned}
&{\left\Vert}\frac{1}{\delta }{{\hat{Q}}}_{\delta }^{\left( 1\right)
}G-L_{2}^{V}G{\right\Vert}_{C} \\
\leq & z\int_{{\Gamma}_{0}}{\left\vert}\sum_{\xi \subset \eta }\left[
\left( 1-\delta \right)^{{\left\vert}\xi {\right\vert}}-1\right] \int_{{{{\mathbb R}}^d}}G\left(
\xi \cup x\right) e_{{\lambda}}\left( -\phi \left( x-\cdot \right) ,\eta
\setminus \xi \right) dx{\right\vert}C^{{\left\vert}\eta
{\right\vert}}d{\lambda}\left( \eta \right) \\
\leq & z\int_{{\Gamma}_{0}}\int_{{\Gamma}_{0}}\left[ 1-\left(
1-\delta \right)^{{\left\vert}\xi {\right\vert}}\right] \int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi
\cup x\right) {\right\vert}e_{{\lambda}}\left( \phi \left( x-\cdot \right) ,\eta
\right) dxC^{{\left\vert}\eta {\right\vert}}C^{{\left\vert}\xi
{\right\vert}}d{\lambda}\left( \eta \right) d{\lambda}\left( \xi \right) \\
= & z\exp \left\{ C\beta \right\} \int_{{\Gamma}_{0}}\left[ 1-\left(
1-\delta \right)^{{\left\vert}\xi {\right\vert}}\right] \int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi
\cup x\right) {\right\vert}dxC^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right)
\\\leq
& \delta z\exp \left\{ C\beta \right\}
\int_{{\Gamma}_{0}}{\left\vert}\xi {\right\vert}\int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi \cup x\right)
{\right\vert}dxC^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right) \\
= &\delta z\exp \left\{ C\beta \right\}
C^{-1}\int_{{\Gamma}_{0}}{\left\vert}\xi {\right\vert}\left( {\left\vert}\xi {\right\vert}-1\right) {\left\vert}G\left( \xi \right) {\right\vert}C^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right)
<\delta {\left\Vert}G{\right\Vert}_{2C},\end{aligned}$$since, $n\left( n-1\right) \leq 2^{n}$, $n\in {{\mathbb N}}$. And, if we denote $${\Gamma}_{0}^{\left( \geq 2\right) }:=\bigsqcup\limits_{n\geq 2}{\Gamma}_{0}^{\left( n\right) },$$ we obtain $$\begin{aligned}
&{\left\Vert}\frac{1}{\delta }{{\hat{Q}}}_{\delta }^{\left( \geq 2\right)
}G{\right\Vert}_{C} \\
\leq &\frac{1}{\delta }\int_{{\Gamma}_{0}}\sum_{\xi \subset \eta }\left(
1-\delta \right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}^{\left( \geq 2\right)
}}\left( z\delta \right)^{{\left\vert}\omega {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right) {\right\vert}\\
&\qquad \times e_{{\lambda}}\left( E^{\phi }\left( \cdot ,\omega \right)
,\eta \setminus \xi \right) d{\lambda}\left( \omega \right) C^{{\left\vert}\eta
{\right\vert}}d{\lambda}\left( \eta \right) \\
=&\frac{1}{\delta }\int_{{\Gamma}_{0}}\int_{{\Gamma}_{0}}\left( 1-\delta
\right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}^{\left( \geq 2\right)
}}\left( z\delta \right)^{{\left\vert}\omega {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right) {\right\vert}\\
&\qquad \times e_{{\lambda}}\left( E^{\phi }\left( \cdot ,\omega \right)
,\eta \right) d{\lambda}\left( \omega \right) C^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right) C^{{\left\vert}\xi {\right\vert}}d{\lambda}\left(
\xi \right) \\
\leq &\delta \int_{{\Gamma}_{0}}\left( 1-\delta \right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}}\left( z\exp \left\{ C\beta \right\} \right)^{{\left\vert}\omega {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right) {\right\vert}d{\lambda}\left(
\omega \right) C^{{\left\vert}\xi {\right\vert}}d{\lambda}\left( \xi \right) \\
=&\delta \int_{{\Gamma}_{0}}\left( C-\delta C+z\exp \left\{ C\beta
\right\} \right)^{{\left\vert}\xi {\right\vert}}{\left\vert}G\left( \xi \right)
{\right\vert}d{\lambda}\left( \xi \right) \\
\leq &\delta \int_{{\Gamma}_{0}}\left( 2C-\delta C\right)^{{\left\vert}\xi {\right\vert}}{\left\vert}G\left( \xi \right) {\right\vert}d{\lambda}\left( \xi \right) <\delta {\left\Vert}G{\right\Vert}_{2C}.\end{aligned}$$
The same considerations may be done for ${{\hat{P}}}_{\delta ,{\varepsilon}}$. Namely, let$$\begin{aligned}
\bigl( {{\hat{P}}}_{\delta ,{\varepsilon}}^{\left( 0\right) }G\bigr) \left( \eta
\right) :=&\sum_{\xi \subset \eta }\left( 1-\delta \right)^{{\left\vert}\xi
{\right\vert}}G\left( \xi \right) 1^{{\left\vert}\xi {\right\vert}}0^{{\left\vert}\eta \setminus \xi
{\right\vert}}=\left( 1-\delta \right)
^{{\left\vert}\eta {\right\vert}}G\left( \eta \right) , \\
\bigl( {{\hat{P}}}_{\delta ,{\varepsilon}}^{\left( 1\right) }G\bigr) \left( \eta
\right) :=&z\delta \sum_{\xi \subset \eta }\left( 1-\delta \right)
^{{\left\vert}\xi {\right\vert}}\int_{{{{\mathbb R}}^d}}G\left( \xi \cup x\right)
\\
&\qquad \times e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right)
},\xi \right) e_{{\lambda}}\left( \frac{e^{-{\varepsilon}\phi \left( x-\cdot
\right) }-1}{{\varepsilon}},\eta \setminus \xi \right) dx,\end{aligned}$$and $${{\hat{P}}}_{\delta ,{\varepsilon}}^{\left( \geq 2\right) }:={{\hat{P}}}_{\delta ,{\varepsilon}}-\bigl( {{\hat{P}}}_{\delta ,{\varepsilon}}^{\left( 0\right) }+{{\hat{P}}}_{\delta ,{\varepsilon}}^{\left( 1\right) }\bigr) .$$Then$${\left\Vert}\frac{1}{\delta }\bigl( {{\hat{P}}}_{\delta ,{\varepsilon}}^{\left(
0\right) }-1\bigr) G-L_{1}G{\right\Vert}_{C}={\left\Vert}\frac{1}{\delta }\bigl( {{\hat{Q}}}_{\delta }^{\left( 0\right) }-1\bigr) G-L_{1}G{\right\Vert}_{C}<\delta {\left\Vert}G{\right\Vert}_{2C},$$next, by , and Lemma \[Minlos\],$$\begin{aligned}
& {\left\Vert}\frac{1}{\delta }{{\hat{P}}}_{\delta ,{\varepsilon}}^{\left(
1\right) }G-L_{2,{\varepsilon}}G{\right\Vert}_{C} \nonumber\\
\leq & z\int_{{\Gamma}_{0}}\int_{{\Gamma}_{0}}\left[ 1-\left(
1-\delta \right)^{{\left\vert}\xi {\right\vert}}\right] \int_{{{{\mathbb R}}^d}}{\left\vert}G\left( \xi
\cup x\right) {\right\vert}e_{{\lambda}}\left( \phi \left( x-\cdot \right) ,\eta
\right) dxC^{{\left\vert}\eta {\right\vert}}C^{{\left\vert}\xi
{\right\vert}}d{\lambda}\left( \eta \right) d{\lambda}\left( \xi \right) \nonumber\\
< & {\left\Vert}G{\right\Vert}_{2C}\cdot \delta e^{C\beta} C^{-1} z \leq
\delta {\left\Vert}G{\right\Vert}_{2C},\end{aligned}$$and, finally,$$\begin{aligned}
{\left\Vert}\frac{1}{\delta }{{\hat{P}}}_{\delta ,{\varepsilon}}^{\left( \geq 2\right)
}G{\right\Vert}_{C} \leq &\frac{1}{\delta }\int_{{\Gamma}_{0}}\sum_{\xi \subset
\eta }\left( 1-\delta \right)^{{\left\vert}\xi {\right\vert}}\int_{{\Gamma}_{0}^{\left(
\geq 2\right) }}\left( z\delta \right)^{{\left\vert}\omega {\right\vert}}{\left\vert}G\left( \xi \cup \omega \right) {\right\vert}\\
&\qquad \times {\left\vert}e_{{\lambda}}\left( E^{\phi }\left( \cdot ,\omega
\right) ,\eta \setminus \xi \right) {\right\vert}d{\lambda}\left( \omega \right)
C^{{\left\vert}\eta {\right\vert}}d{\lambda}\left( \eta \right) <\delta {\left\Vert}G{\right\Vert}_{2C}.\end{aligned}$$ Combining all these inequalities we obtain the assertion.
We will need the following results in the sequel.
\[EK\_res\] Let $A$ be a linear operator on a Banach space $L$ with $D\left(
A\right) $ dense in $L$, and let $|\!|\!| \cdot |\!|\!|$ be a norm on $D\left( A\right) $ with respect to which $D\left( A\right) $ is a Banach space. For $n\in \mathbb{N}$ let $T_{n}$ be a linear $\left\Vert \cdot \right\Vert $-contraction on $L$ such that $T_{n}:D\left( A\right) \rightarrow D\left( A\right) $, and define $A_{n}=n\left( T_{n}-1\right) $. Suppose there exist $\omega \geq 0$ and a sequence $\left\{ \varepsilon _{n}\right\} \subset \left(
0;+\infty \right) $ tending to zero such that for $n\in \mathbb{N}$$$\label{approperEK}
\left\Vert \left( A_{n}-A\right) f\right\Vert \leq \varepsilon
_{n}|\!|\!| f |\!|\!|,~f\in D\left( A\right)$$ and $$\label{psevdocontr}
|\!|\!| T_{n}\upharpoonright _{D(A)} |\!|\!| \leq 1+\frac{\omega
}{n}.$$ Then $A$ is closable and the closure of $A$ generates a strongly continuous contraction semigroup on $L$.
\[EK\_res-conv\] Let $L, L_n$, $n\in{{\mathbb N}}$ be Banach spaces, and $p_n: L\rightarrow L_n$ be bounded linear transformation, such that $\sup_n \|p_n\|<\infty
$. For any $n\in{{\mathbb N}}$, let $T_n$ be a linear contraction on $L_n$, let ${\varepsilon}_n>0$ be such that $\lim_{n\rightarrow \infty} {\varepsilon}_n =0$, and put $A_n={\varepsilon}_n^{-1}(T_n - {1\!\!1})$. Let $T_t$ be a strongly continuous contraction semigroup on $L$ with generator $A$ and let $D$ be a core for $A$. Then the following are equivalent:
1. For each $f\in L$, $T_n^{[t/{\varepsilon}_n]} p_n f\rightarrow p_n
T_t f$ in $L_n$ for all $t\geq0$ uniformly on bounded intervals. Here and below $[\,\cdot\,\,]$ mean the entire part of a real number.
2. For each $f\in D$, there exists $f_n\in L_n$ for each $n\in{{\mathbb N}}$ such that $f_n \rightarrow p_n f$ and $A_n f_n \rightarrow
p_n Af$ in $L_n$.
\[powersofcontractions\] Let $X$ be a Banach space with a norm $\|\cdot\|_X$; $A$ and $B$ be linear contraction mappings on $X$. Let $Y$ with a norm $\|\cdot\|_Y$ be a Banach subspace of $X$ such that $Y$ is invariant w.r.t. $B$. Suppose that the restriction of $B$ on $Y$ is also a contraction w.r.t. $\|\cdot\|_Y$. Suppose also that there exists $c>0$ such that for any $f\in Y$ $$\label{estY}
\| Af-Bf\|_X \leq c\|f\|_Y.$$ Then for any $m\in{{\mathbb N}}$ and for any $f\in Y$ $$\label{estYm}
\| A^m f-B^mf\|_X \leq cm\|f\|_Y.$$
For any $f\in Y$, $m\geq2$ we have $$\begin{aligned}
&\|A^m f -B^m f\|_X\\\leq&\|A^m f -AB^{m-1}f\|_X + \|AB^{m-1}f-B^m
f\|_X\\\leq&\|A\|\cdot\|A^{m-1} f -B^{m-1}f\|_X+\|(A-B)B^{m-1}f\|_X
\\ \intertext{(where $\|A\|$ means the norm of the operator $A$ on $X$); since
$\|A\|\leq1$ and $B^{m-1}f\in Y$, condition \eqref{estY} yields}
\leq&\|A^{m-1} f -B^{m-1}f\|_X + c\|B^{m-1}f\|_Y,
\\\intertext{
but, $B$ is a contraction on $Y$, therefore, one get}
\leq&\|A^{m-1} f -B^{m-1}f\|_X
+ c\|f\|_Y,\end{aligned}$$ that gives by induction principle.
And now one can construct the corresponding semigroups rigorously.
\[descsemigroupexist\] Let $$z\leq \min \left\{ Ce^{-C\beta },2Ce^{-2C\beta }\right\} .
\label{verysmallz}$$Then, $\bigl( {{\hat{L}}}_V ,{{\mathcal L}}_{2C}\bigr) $ and $\bigl( {{\hat{L}}}_{{\varepsilon},\mathrm{ren}},{{\mathcal L}}_{2C}\bigr) $ are closable linear operators in ${{\mathcal L}}_C$ and their closures $\bigl( {{\hat{L}}}_V , D({{\hat{L}}}_V) \bigr) $ and $\bigl( {{\hat{L}}}_{{\varepsilon},\mathrm{ren}}, D({{\hat{L}}}_{{\varepsilon},\mathrm{ren}}) \bigr) $ generate strongly continuous contraction semigroups ${{\hat{T}}}_{V}(t)$ and ${{\hat{T}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}(t)$ on ${{\mathcal L}}_C$, respectively. Moreover, for any $G\in {{\mathcal L}}_C$, ${\varepsilon}>0$ $${{\hat{Q}}}_{\frac{1}{n}}^{[nt] }G{\rightarrow}{{\hat{T}}}_{V}(t)G,\qquad {{\hat{P}}}_{\frac{1}{n},{\varepsilon}}^{[nt] }G{\rightarrow}{{\hat{T}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}(t)G,\qquad n{\rightarrow}\infty ~ \label{approx}$$for any $t\geq 0$ uniformly on bounded intervals.
Note that provides that ${{\hat{Q}}}_{\delta }$ and ${{\hat{P}}}_{\delta ,{\varepsilon}}$ are also contractions on ${{\mathcal L}}_{2C}$. Then the first part of the statement follows from Lemma \[EK\_res\]. Therefore, ${{\mathcal L}}_{2C}$ will be a core for the generators and, by Lemma \[EK\_res-conv\], we obtain the convergence .
The definition of ${{\hat{L}}}_V$ together with Proposition \[descsemigroupexist\] allow us to expect that the semigroup ${{\hat{T}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}(t)$ converges to ${{\hat{T}}}_{V}(t)$ in a proper sense. The next theorem improve this statement. However, this result is not crucial in the context of the our paper. Moreover, its proof is quite technical and, on the other hand, is very similar to the proof of the main Theorem \[maintheorem\] concerning the dual semigroups. Hence, we give the sketch of the proof only.
\[descsemigroupconv\] Let holds and suppose that $\bar{\phi}:=\sup_{{{{\mathbb R}}^d}}\phi \left( x\right) <+\infty $. Then for any $G\in {{\mathcal L}}_{2C}$ $${\left\Vert}{{\hat{T}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}(t)G-{{\hat{T}}}_{V}(t)G{\right\Vert}_{C}\leq {\varepsilon}t\,\bar{\phi}\left( 1+\beta \right) {\left\Vert}G{\right\Vert}_{2C}$$for any $t\geq 0$, ${\varepsilon}>0$. In particular, it means that ${{\hat{T}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}(t) G{\rightarrow}{{\hat{T}}}_{V}(t)G$ in ${{\mathcal L}}_C$ as ${\varepsilon}{\rightarrow}0$ for any $t\geq 0$ uniformly on bounded intervals.
By the triangle inequality,$$\begin{aligned}
{\left\Vert}{{\hat{T}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}(t)G-{{\hat{T}}}_{V}(t)G{\right\Vert}_{C} \leq
&{\left\Vert}{{\hat{T}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}(t)G-{{\hat{P}}}_{\frac{1}{n},{\varepsilon}}^{[nt] }G{\right\Vert}_{C} \label{triangleineq} \\
&+{\left\Vert}{{\hat{P}}}_{\frac{1}{n},{\varepsilon}}^{[nt] }G-{{\hat{Q}}}_{\frac{1}{n}}^{[nt] }G{\right\Vert}_{C }+{\left\Vert}{{\hat{Q}}}_{\frac{1}{n}}^{[nt] }G-{{\hat{T}}}_{V}(t)G{\right\Vert}_{C}. \nonumber\end{aligned}$$By , the first and third norms in the r.h.s. of are tend to $0$ as $n\rightarrow\infty$. Next, in a similar way as for the proof of one can show that for any $G\in {{\mathcal L}}_{2C}$ $${\left\Vert}{{\hat{P}}}_{\frac{1}{n},{\varepsilon}}G-{{\hat{Q}}}_{\frac{1}{n}}G{\right\Vert}_{C }\leq \frac{1}{n} \,
{\varepsilon}\, \bar{\phi}\left( 1+\beta \right) {\left\Vert}G{\right\Vert}_{2C}. \label{mainestq}$$By Proposition \[contrmaps\] and condition , the subspace ${{\mathcal L}}_{2C}$ is ${{\hat{Q}}}_{\frac{1}{n}}$-invariant, hence, by Lemma \[powersofcontractions\], we obtain $$\begin{aligned}
{\left\Vert}{{\hat{P}}}_{\frac{1}{n},{\varepsilon}}^{[nt] }G-{{\hat{Q}}}_{\frac{1}{n}}^{[nt] }G{\right\Vert}_{C }&\leq [nt]
\frac{\bar{\phi}\left( 1+\beta \right)}{n}{\varepsilon}{\left\Vert}G{\right\Vert}_{2C}\\ &< \bar{\phi}\left( 1+\beta \right)\left( t+\frac{1}{n}\right) {\varepsilon}{\left\Vert}G{\right\Vert}_{2C},\end{aligned}$$that fulfilled the first assertion. And, clearly, ${{\mathcal L}}_{2C}$ is a dense subspace of ${{\mathcal L}}_C$.
Convergence of the evolutions in ${{\mathcal K}}_C$
---------------------------------------------------
Let ${\varepsilon}>0$ be given. Let $\bigl( {{\hat{L}}}'_{{{\varepsilon}, \, \mathrm{ren}}}, D({{\hat{L}}}'_{{{\varepsilon}, \, \mathrm{ren}}})\bigr)$ and $\bigl( {{\hat{L}}}'_V, D({{\hat{L}}}'_V)\bigr)$ be dual operators to the closed operators $\bigl( {{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}, D({{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}})\bigr)$ and $\bigl(
{{\hat{L}}}_V, D({{\hat{L}}}_V)\bigr)$ in the Banach space $({{\mathcal L}}_C)'$. Let the operators $\bigl( {{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}, D({{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast})\bigr)$ and $\bigl( {{\hat{L}}}^\ast_V, D({{\hat{L}}}^\ast_V)\bigr)$ be their images in the space ${{\mathcal K}}_C$ under the isometry . Our aim is to transfer the previous results onto $\ast$-objects. However, similarly to the case of the operator ${{\hat{L}}}^\ast$ (see Subsection \[dualconstraction\]), the space ${{\mathcal K}}_C$ is too big. The reason is that the dual semigroup in a non-reflexive case (namely, $L^1$ case) will not be a strongly continuous semigroup on the whole dual space. Hence, we consider some Banach subspace of ${{\mathcal K}}_C$ which will be useful for the strong continuity property.
\[adjgen\] For any ${{\alpha}}\in(0;1)$, ${\varepsilon}>0$, and $k\in{{\mathcal K}}_{{{{\alpha}}C}}$ we have that $$\label{incl}
\bigl\{{{\hat{L}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}^\ast k, \,
{{\hat{L}}}_{V}^\ast k\bigr\}\subset{{\mathcal K}}_C.$$ Moreover, for any $k\in {{\mathcal K}}_{{{{\alpha}}C}}$ $$\begin{aligned}
( {{\hat{L}}}_{{{{\varepsilon}, \, \mathrm{ren}}}}^\ast k) \left( \eta \right) =&-{\left\vert}\eta {\right\vert}k\left(
\eta \right) \label{genrenadj}\\& +z\sum_{x\in\eta}\int_{{\Gamma}_0}
e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right) },\eta\setminus
x \right) \nonumber\\&\qquad\qquad \times e_{{\lambda}}\left(
\frac{e^{-{\varepsilon}\phi \left( x-\cdot \right) }-1}{{\varepsilon}},\xi \right)
k\left( \xi \cup \eta\setminus x\right) d{\lambda}(\xi)\nonumber\end{aligned}$$ and $$\begin{aligned}
({{\hat{L}}}_{V}^\ast k) \left( \eta \right) =&-{\left\vert}\eta {\right\vert}k\left( \eta
\right) \label{genVladj} \\ &+z\sum_{x\in\eta}\int_{{\Gamma}_0} e_{{\lambda}}\left(- \phi \left( x-\cdot \right),\xi \right) k\left( \xi \cup
\eta\setminus x\right) d{\lambda}(\xi).\nonumber\end{aligned}$$
By Lemma \[Minlos\], for any $G\in{B_{\mathrm{bs}}}({\Gamma}_0)$ we have $$\begin{aligned}
&\int_{{\Gamma}_0}\sum_{\xi \subset
\eta }\int_{{{{\mathbb R}}^d}}e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right)
},\xi \right) e_{{\lambda}}\left( \frac{e^{-{\varepsilon}\phi \left( x-\cdot \right) }-1}{{\varepsilon}},\eta \setminus \xi \right)
\\&\qquad\qquad\times G\left( \xi \cup x\right) dx k(\eta)d{\lambda}(\eta)\\&=
\int_{{\Gamma}_0}\int_{{\Gamma}_0}\int_{{{{\mathbb R}}^d}}e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right)
},\xi \right) e_{{\lambda}}\left( \frac{e^{-{\varepsilon}\phi \left( x-\cdot \right) }-1}{{\varepsilon}},\eta \right)
\\ &\qquad\qquad\times G\left( \xi \cup x\right) dx k(\eta\cup \xi)d{\lambda}(\xi)d{\lambda}(\eta)\\
&=\int_{{\Gamma}_0}\int_{{\Gamma}_0}\sum_{x\in\xi} e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right) },\xi \setminus x\right) e_{{\lambda}}\left(
\frac{e^{-{\varepsilon}\phi \left( x-\cdot \right) }-1}{{\varepsilon}},\eta \right)
\\ &\qquad\qquad\times G\left( \xi \right) dx k(\eta\cup \xi\setminus x)d{\lambda}(\xi)d{\lambda}(\eta),\end{aligned}$$ that implies . The equality may be obtained in the same way or just as a point-wise limit of as ${\varepsilon}{\rightarrow}0$.
The inclusion follows from the estimate ($k\in{{\mathcal K}}_{{{{\alpha}}C}}$) $$\begin{aligned}
&\frac{1}{C^{{|\eta|}}}\sum_{x\in\eta}\int_{{\Gamma}_0} e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right) },\eta\setminus x \right) e_{{\lambda}}\left(
\left\vert\frac{e^{-{\varepsilon}\phi \left( x-\cdot \right) }-1}{{\varepsilon}}\right\vert,\xi \right) \bigl\vert k\left( \xi \cup \eta\setminus
x\right) \bigr\vert d{\lambda}(\xi)\\ \leq &
\frac{\|k\|_{{{\mathcal K}}_{{{{\alpha}}C}}}}{C^{{|\eta|}}}\sum_{x\in\eta}\int_{{\Gamma}_0} e_{{\lambda}}\left( \phi \left( x-\cdot \right),\xi \right) ({{\alpha}}C)^{|\xi \cup
\eta\setminus x|} d{\lambda}(\xi)\\= &\frac{\|k\|_{{{\mathcal K}}_{{{{\alpha}}C}}}\cdot \exp\{{{\alpha}}C\beta\}}{{{\alpha}}C} |\eta| {{\alpha}}^{{{|\eta|}}}\leq \frac{\|k\|_{{{\mathcal K}}_{{{{\alpha}}C}}}\cdot
\exp\{{{\alpha}}C\beta\}}{{{\alpha}}C} \cdot\frac{-1}{e\ln{{\alpha}}},\end{aligned}$$ where we used that $x{{\alpha}}^x \leq -\dfrac{1}{e\ln {{\alpha}}}$ for any ${{\alpha}}\in(0;1)$ and $x\geq0$; and the similar estimates for $$\frac{1}{C^{{|\eta|}}} {{|\eta|}}\left\vert k(\eta) \right\vert, \quad
\frac{1}{C^{{|\eta|}}}\sum_{x\in\eta}\int_{{\Gamma}_0} e_{{\lambda}}\left( \phi \left(
x-\cdot \right) ,\xi \right) \bigl\vert k\left( \xi \cup
\eta\setminus x\right) \bigr\vert d{\lambda}(\xi). \qedhere$$
Let now holds. By Proposition \[descsemigroupexist\], there exist strongly continuous contraction semigroups ${{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ and ${{\hat{T}}}_V(t)$ on ${{\mathcal L}}_C$. Then the corresponding dual semigroups ${{\hat{T}}}'_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ and ${{\hat{T}}}'_V(t)$ act in the space $({{\mathcal L}}_C)'$. Let us denote by ${{\hat{T}}}^\ast_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ and ${{\hat{T}}}^\ast_V(t)$ their corresponding images in ${{\mathcal K}}_C$ under the isometry .
Proposition \[adjgen\] yields that for any ${{\alpha}}\in(0;1)$ the following inclusion holds $$\label{inclusion}
\overline{{{\mathcal K}}_{{{{\alpha}}C}}} \subset \Bigl(\bigcap_{{\varepsilon}>0} \overline{D({{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast)}\,\Bigr)\bigcap \overline{D({{\hat{L}}}_V^\ast)}$$ (all closures are in ${{\mathcal K}}_C$; in particular, $\overline{{{\mathcal K}}_{{{{\alpha}}C}}}$ is a Banach space with norm ).
Moreover, by, e.g., [@vNee1992 Sections 1.2, 1.3] or [@EN2000 Subsection II.2.5], for any ${\varepsilon}>0$ the restrictions ${{\hat{T}}}^\odot_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ and ${{\hat{T}}}^\odot_V(t)$ of ${{\hat{T}}}^\ast_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ and ${{\hat{T}}}^\ast_V(t)$ onto $\overline{D({{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast})}$ and $\overline{D({{\hat{L}}}_V^\ast)}$, correspondingly, are strongly continuous semigroups; their generators ${{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\odot$ and ${{\hat{L}}}_V^\odot$ are the parts of ${{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}$ and ${{\hat{L}}}_V^\ast$, correspondingly. Namely, $$\begin{aligned}
D({{\hat{L}}}^\odot_{{{\varepsilon}, \, \mathrm{ren}}})&=\bigl\{k\in D({{\hat{L}}}^\ast_{{{\varepsilon}, \, \mathrm{ren}}}) \bigm| {{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast k\in \overline{D({{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast})} \bigr\}, \\
D({{\hat{L}}}^\odot_V)&=\bigl\{k\in D({{\hat{L}}}^\ast_V) \bigm| {{\hat{L}}}_V^\ast k\in
\overline{D({{\hat{L}}}_V^\ast)} \bigr\},\end{aligned}$$ and $$\begin{aligned}
{{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}k &={{\hat{L}}}^\odot_{{{\varepsilon}, \, \mathrm{ren}}}k, \qquad k\in D({{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\odot),\\
{{\hat{L}}}^\ast_V k &={{\hat{L}}}^\odot_V k, \qquad k\in D({{\hat{L}}}_V^\odot).\end{aligned}$$
\[sun-inv\] Assume that, as before, $$z\leq \min \left\{ Ce^{-C\beta },2Ce^{-2C\beta }\right\} .
\label{verysmallz-2}$$If $C \beta=\ln2$ we suppose additionally that $z<\frac{C}{2}$. Then, there exists ${{\alpha}}_1={{\alpha}}_1(z,\beta,C)\in(0;1)$ such that for any ${{\alpha}}\in({{\alpha}}_1;1)$ the space ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ will be ${{\hat{T}}}^\odot_V(t)$- and ${{\hat{T}}}^\odot_{{{\varepsilon}, \, \mathrm{ren}}}(t)$-invariant, ${\varepsilon}>0$.
The proof is fully analogous to that of [@FKK2010 Proposition 3.3]. For readers convince we explain it in details.
By , $z\beta\leq\min\{C\beta e^{-C\beta},
2C\beta e^{-2C\beta}\}$. Note that the function $f(x)=x e^{-x}$, $x\geq 0$ is increasing on $(0; 1)$ from $0$ to $e^{-1}$ and it is asymptotically decreasing on $(1;+\infty)$ from $e^{-1}$ to $0$. Therefore, if $C\beta e^{-C\beta}\neq 2C\beta e^{-2C\beta}$ then with necessity implies $z \beta < e^{-1}$. Otherwise, if $C\beta=\ln 2$ then the condition $2z<C$ implies $z\beta<\frac{C\beta}{2}=C\beta e^{-C\beta}=2C\beta e^{-2C\beta}$, and, again, $z\beta<e^{-1}$. As a result, the equation $f(x)=z
\beta$ has exactly two roots, say, $0<x_1<1<x_2<+\infty$. Therefore, $x_1< C \beta < 2 C \beta < x_2$.
If $C\beta>1$ then we set ${{\alpha}}_1:=\max\left\{\frac{1}{2};\frac{1}{C
\beta};\frac{1}{C}\right\}<1$. This yields $2{{\alpha}}C \beta > C
\beta$ and ${{\alpha}}C\beta >1>x_{1}$. If $x_{1}<C\beta \leq 1$ then we set ${{\alpha}}_1:=\max\left\{\frac{1}{2};\frac{x_{1}}{C
\beta};\frac{1}{C}\right\}<1$ that gives $2{{\alpha}}C \beta > C \beta$ and ${{\alpha}}C\beta >x_{1}$. As a result, $$\label{ineq-alpha}
x_1<{{\alpha}}C\beta < C\beta <2 {{\alpha}}C \beta < 2 C \beta < x_2$$ and $1<{{\alpha}}C<C<2{{\alpha}}C<2C$. The last inequality shows that ${{\mathcal L}}_{2C
}\subset{{\mathcal L}}_{2{{\alpha}}C}\subset {{\mathcal L}}_C\subset {{\mathcal L}}_{{{\alpha}}C}$.
By , $z\beta<\min\{f({{{{\alpha}}C}}\beta),f(2{{{{\alpha}}C}}\beta)\}$, hence, $z<\min\{{{{{\alpha}}C}}e^{-{{{{\alpha}}C}}\beta}, 2{{{{\alpha}}C}}e^{-2{{{{\alpha}}C}}\beta}\}$. Then, analogously to Proposition \[descsemigroupexist\], we obtain that the operators $\bigl( {{\hat{L}}}_V ,{{\mathcal L}}_{2{{{{\alpha}}C}}}\bigr) $ and $\bigl( {{\hat{L}}}_{{\varepsilon},\mathrm{ren}},{{\mathcal L}}_{2{{{{\alpha}}C}}}\bigr) $ are closable in ${{\mathcal L}}_{{{\alpha}}C}$ and their closures are generators of contraction semigroups, say, ${{\hat{T}}}_{{{\alpha}},V} (t)$ and ${{\hat{T}}}_{{{\alpha}},{{{\varepsilon}, \, \mathrm{ren}}}} (t)$ on ${{\mathcal L}}_{{{\alpha}}C}$, correspondingly.
It is easy to see, that ${{\hat{T}}}_{{{\alpha}},V} (t) G= {{\hat{T}}}_V (t) G$ and ${{\hat{T}}}_{{{\alpha}},{{{\varepsilon}, \, \mathrm{ren}}}} (t) G= {{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t) G$ for any $G\in{{\mathcal L}}_C$. Indeed, since the contraction mappings ${{\hat{Q}}}_\delta$ and ${{\hat{P}}}_{\delta,{\varepsilon}}$, $\delta,{\varepsilon}>0$ do not depend on ${{\alpha}}$, we obtain, by Proposition \[descsemigroupexist\], that for any $G\in{{\mathcal L}}_{C }\subset{{\mathcal L}}_{{{\alpha}}C}$ we have that ${{\hat{T}}}_V (t)G\in{{\mathcal L}}_{C
}\subset{{\mathcal L}}_{{{\alpha}}C}$ and ${{\hat{T}}}_{{{\alpha}},V} (t) G\in{{\mathcal L}}_{{{\alpha}}C}$ and $$\begin{aligned}
&\| {{\hat{T}}}_V (t)G-{{\hat{T}}}_{{{\alpha}},V} (t) G\|_{{{\alpha}}C}\\
\leq &\Bigl\| {{\hat{T}}}_V (t)G-{{\hat{Q}}}_\delta^{\left[\frac{t}{\delta}\right]}G\Bigr\|_{{{\alpha}}C} + \Bigl\| {{\hat{T}}}_{{{\alpha}},V}
(t) G-{{\hat{Q}}}_\delta^{\left[\frac{t}{\delta}\right]}G\Bigr\|_{{{\alpha}}C}\\ \leq&
\Bigl\| {{\hat{T}}}_V (t)G-{{\hat{Q}}}_\delta^{\left[\frac{t}{\delta}\right]}G\Bigr\|_{ C} +
\Bigl\| {{\hat{T}}}_{{{\alpha}},V} (t)
G-{{\hat{Q}}}_\delta^{\left[\frac{t}{\delta}\right]}G\Bigr\|_{{{\alpha}}C}{\rightarrow}0,\end{aligned}$$ as $\delta{\rightarrow}0$. Therefore, ${{\hat{T}}}_V (t)G={{\hat{T}}}_{{{\alpha}},V} (t) G$ in ${{\mathcal L}}_{{{\alpha}}C}$ (recall that $G\in{{\mathcal L}}_C$) that yields $ {{\hat{T}}}_V (t)G(\eta)={{\hat{T}}}_{{{\alpha}},V} (t)
G(\eta)$ for ${\lambda}$-a.a. $\eta\in{\Gamma}_0$ and, therefore, ${{\hat{T}}}_V
(t)G={{\hat{T}}}_{{{\alpha}},V} (t) G$ in ${{\mathcal L}}_{C}$.
Note that for any $G\in{{\mathcal L}}_C\subset{{\mathcal L}}_{{{\alpha}}C}$ and for any $k\in
{{\mathcal K}}_{{{\alpha}}C}\subset {{\mathcal K}}_C$ we have ${{\hat{T}}}_{{{\alpha}},V} (t) G\in{{\mathcal L}}_{{{\alpha}}C}$ and $${{\left\langle}\!\!{\left\langle}}{{\hat{T}}}_{{{\alpha}},V} (t) G, k{{\right\rangle}\!\!{\right\rangle}}={{\left\langle}\!\!{\left\langle}}G, {{\hat{T}}}^\ast_{{{\alpha}},V} (t) k{{\right\rangle}\!\!{\right\rangle}},$$ where, by construction, ${{\hat{T}}}^\ast_{{{\alpha}},V} (t) k\in{{\mathcal K}}_{{{\alpha}}C}$. But $G\in{{\mathcal L}}_C$, $k\in{{\mathcal K}}_C$ implies $${{\left\langle}\!\!{\left\langle}}{{\hat{T}}}_{{{\alpha}},V} (t) G, k{{\right\rangle}\!\!{\right\rangle}}={{\left\langle}\!\!{\left\langle}}{{\hat{T}}}_V (t) G, k{{\right\rangle}\!\!{\right\rangle}}={{\left\langle}\!\!{\left\langle}}G,
{{\hat{T}}}^\ast(t) k{{\right\rangle}\!\!{\right\rangle}}.$$ Hence, ${{\hat{T}}}^\ast_V(t) k = {{\hat{T}}}^\ast_{{{\alpha}},V} (t) k\in{{\mathcal K}}_{{{\alpha}}C}$ that is what we need.
Since ${{\hat{T}}}^\odot_V(t)$ and ${{\hat{T}}}^\odot_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ are restrictions of ${{\hat{T}}}^\ast_V(t)$ and ${{\hat{T}}}^\ast_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ onto $\overline{D({{\hat{L}}}_V^\ast)}$ and $\overline{D({{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast})}$, correspondingly, one has, by , that the corresponding semigroups coincide on ${{\mathcal K}}_{{{{\alpha}}C}}$. Therefore, ${{\mathcal K}}_{{{{\alpha}}C}}$ is ${{\hat{T}}}^\odot_V(t)$- and ${{\hat{T}}}^\odot_{{{\varepsilon}, \, \mathrm{ren}}}(t)$-invariant, ${\varepsilon}>0$; and the result follows from the continuity of operators which formed semigroups.
Let now ${{\hat{T}}}^{{\odot{{\alpha}}}}_V(t)$ and ${{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ be restrictions of the strongly continuous semigroups ${{\hat{T}}}^\odot_V(t)$ and ${{\hat{T}}}^\odot_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ (which acting on the Banach spaces $\overline{D({{\hat{L}}}_V^\ast)}$ and $\overline{D({{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast})}$, correspondingly) onto the closed linear subspace ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ of all these Banach spaces which are invariant w.r.t. all these $\odot$-semigroups. By the general result (see, e.g., [@EN2000 Subsection II.2.3]), ${{\hat{T}}}^{{\odot{{\alpha}}}}_V(t)$ and ${{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ are strongly continuous semigroups on ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ with generators ${{\hat{L}}}^{{\odot{{\alpha}}}}_V$ and ${{\hat{L}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}$ which are restrictions of the corresponding operators ${{\hat{L}}}_V^\odot$ and ${{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\odot$. Namely, $$\begin{aligned}
D({{\hat{L}}}^{{\odot{{\alpha}}}}_V)&=\bigl\{k\in
{\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}\bigm| {{\hat{L}}}_V^\ast k\in{\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}\bigr\},\label{domVltimes}\\
D({{\hat{L}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}})&=\bigl\{k\in {\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}\bigm| {{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}k\in {\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}\bigr\},\quad {\varepsilon}>0,\label{domrentimes}\end{aligned}$$ and $$\begin{aligned}
{{\hat{L}}}^{{\odot{{\alpha}}}}_V k &= {{\hat{L}}}^\ast_V k, \qquad k\in D({{\hat{L}}}^{{\odot{{\alpha}}}}_V),\\
{{\hat{L}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}k &= {{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}k, \qquad k\in D({{\hat{L}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}).\label{restrren}\end{aligned}$$ By Proposition \[descsemigroupexist\], ${{\hat{T}}}_V(t)$ and ${{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ are contraction semigroups on ${{\mathcal L}}_C$, then, ${{\hat{T}}}'_V(t)$ and ${{\hat{T}}}'_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ are also contraction semigroups on $({{\mathcal L}}_C)'$; but isomorphism is isometrical, therefore, ${{\hat{T}}}_V^\ast(t)$ and ${{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast(t)$ are contraction semigroups on ${{\mathcal K}}_C$. As a result, their restrictions ${{\hat{T}}}^{{\odot{{\alpha}}}}_V(t)$ and ${{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)$ are contraction semigroups on ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$.
To summarize, we have the Banach space ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ and the family of the strongly continuous contraction semigroups ${{\hat{T}}}^{{\odot{{\alpha}}}}_V(t)$ and ${{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)$, ${\varepsilon}>0$ on this space. The generators of these semigroups are satisfied –. Moreover, by construction, ${{\hat{T}}}^{{\odot{{\alpha}}}}_V(t) k = {{\hat{T}}}^\ast_V(t) k$ and ${{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t) k = {{\hat{T}}}^\ast_{{{\varepsilon}, \, \mathrm{ren}}}(t) k$ for any $k\in{\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$.
\[maintheorem\] Let $C, z, \beta, \alpha_1$ be as in Proposition \[sun-inv\]. Suppose additionally that $\bar{\phi}:=\sup_{{{{\mathbb R}}^d}}\phi \left( x\right)
<+\infty $. Then, for any ${{\alpha}}\in({{\alpha}}_1;1)$ and for any $k\in{{\mathcal K}}_{{{{\alpha}}C}}$ $$\bigl\Vert {{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)k - {{\hat{T}}}^{{\odot{{\alpha}}}}_V(t)k\bigr\Vert_{{{\mathcal K}}_C}\leq {\varepsilon}t A \Vert k \Vert_{{{\mathcal K}}_{{{{\alpha}}C}}},
\quad {\varepsilon}>0,$$ where $A$ is depend on ${{\alpha}}$, $C$, $\bar{\phi}$ only.
Let ${{\hat{Q}}}_\delta^\ast$, ${{\hat{P}}}_{\delta,{\varepsilon}}^\ast$, $\delta\in(0;1)$, ${\varepsilon}>0$ be the images of the dual operators ${{\hat{Q}}}_\delta'$, ${{\hat{P}}}_{\delta,{\varepsilon}}'$ under the isometrical isomorphism . Since the norms of dual operators are equal we have that ${{\hat{Q}}}_\delta^\ast$ and ${{\hat{P}}}_{\delta,{\varepsilon}}^\ast$ are linear contractions on ${{\mathcal K}}_C$. Moreover, for any $k\in{{\mathcal K}}_{{{{\alpha}}C}}$ we have $$\begin{aligned}
&\int_{\Gamma_{0}}({{\hat{Q}}}_{\delta }G)\left( \eta \right) k\left( \eta
\right) d\lambda \left( \eta \right) \\
=&\int_{\Gamma_{0}}\sum_{\xi \subset \eta }\left( 1-\delta \right)
^{\left\vert \xi \right\vert }\int_{\Gamma_{0}}\left( z\delta
\right) ^{\left\vert \omega \right\vert }G\left( \xi \cup \omega
\right) \\&\qquad \times e_{\lambda }\left( -E^{\phi }\left( \cdot
,\omega \right) ,\eta \setminus \xi \right) d\lambda \left( \omega
\right) k\left( \eta \right) d\lambda \left( \eta
\right) \\
=&\int_{\Gamma_{0}}\int_{\Gamma_{0}}\int_{\Gamma_{0}}\left( 1-\delta
\right)^{\left\vert \xi \right\vert }\left( z\delta \right)
^{\left\vert \omega \right\vert }G\left( \xi \cup \omega \right) \\
&\qquad \times e_{\lambda }\left( -E^{\phi }\left( \cdot ,\omega
\right) ,\eta \right) d\lambda \left( \omega \right) k\left( \eta
\cup \xi \right) d\lambda \left( \eta \right) d\lambda
\left( \xi \right) \\
=&\int_{\Gamma_{0}}\int_{\Gamma_{0}}\sum_{\omega \subset \xi }\left(
1-\delta \right)^{\left\vert \xi \setminus \omega \right\vert
}\left( z\delta \right)^{\left\vert \omega \right\vert }G\left( \xi
\right)
\\&\qquad \times e_{\lambda }\left( -E^{\phi }\left( \cdot ,\omega \right) ,\eta \right)
k\left( \eta \cup \xi \setminus \omega \right) d\lambda \left( \eta
\right) d\lambda \left( \xi \right)\end{aligned}$$and, therefore, $$\begin{aligned}
({{\hat{Q}}}_{\delta }^{\ast }k)\left( \eta \right) =&\sum_{\omega \subset
\eta }\left( 1-\delta \right)^{\left\vert \eta \setminus \omega
\right\vert }\left( z\delta \right)^{\left\vert \omega \right\vert
}\nonumber\\ &\qquad \times \int_{\Gamma_{0}}e_{\lambda }\left(
-E^{\phi }\left( \cdot ,\omega \right) ,\xi \right) k\left( \xi \cup
\eta \setminus \omega \right) d\lambda \left( \xi \right)
.\label{dualaprrVl}\end{aligned}$$ Then, by , $$\begin{aligned}
&\left( \alpha C\right)^{-\left\vert \eta \right\vert }\left\vert ({{\hat{Q}}}_{\delta }^{\ast }k)\left( \eta \right) \right\vert \\
\leq &\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\left( \alpha
C\right)^{-\left\vert \eta \right\vert }\sum_{\omega \subset \eta
}\left( 1-\delta \right)^{\left\vert \eta \setminus \omega
\right\vert }\left( z\delta \right)^{\left\vert \omega \right\vert
}\\&\qquad \times \int_{\Gamma_{0}}e_{\lambda }\left( E^{\phi
}\left( \cdot ,\omega \right) ,\xi \right) \left( \alpha C\right)
^{\left\vert \xi \cup \eta \setminus \omega
\right\vert }d\lambda \left( \xi \right) \\
=&\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega
\subset \eta }\left( 1-\delta \right)^{\left\vert \eta \setminus
\omega \right\vert }\left( \frac{z\delta }{\alpha C}\right)
^{\left\vert \omega \right\vert }\exp \left\{ \alpha
C\int_{\mathbb{R}^{d}}E^{\phi }\left( x,\omega \right)
dx\right\} \\
=&\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega
\subset \eta }\left( 1-\delta \right)^{\left\vert \eta \setminus
\omega \right\vert }\left( \frac{z\delta }{\alpha C}\right)
^{\left\vert \omega \right\vert
}\exp \left\{ \alpha C\left\vert \omega \right\vert \beta \right\} \\
\leq &\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega
\subset \eta }\left( 1-\delta \right)^{\left\vert \eta \setminus
\omega \right\vert
}\delta^{\left\vert \omega \right\vert }=\left\Vert k\right\Vert_{\mathcal{K}_{{{{{\alpha}}C}}}}.\end{aligned}$$ Therefore, ${{\mathcal K}}_{{{{\alpha}}C}}$ is ${{\hat{Q}}}_V^\ast$-invariant, hence, ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ is also ${{\hat{Q}}}_V^\ast$-invariant due to continuity of ${{\hat{Q}}}_V^\ast$; moreover, ${{\hat{Q}}}_V^\ast$ is a contraction in ${{\mathcal L}}_{{{{\alpha}}C}}$. Absolutely in the same way we may obtain that for any $k\in{{\mathcal K}}_{{{{\alpha}}C}}$ $$\begin{aligned}
({{\hat{P}}}_{\delta ,\varepsilon }^{\ast }k)\left( \eta \right)
=&\sum_{\omega \subset \eta }\left( 1-\delta \right)^{\left\vert
\eta \setminus \omega \right\vert }\left( z\delta \right)
^{\left\vert \omega \right\vert
}e_{\lambda}\left(
e^{-\varepsilon E^{\phi }\left( \cdot ,\omega \right) },\eta
\setminus \omega \right) \nonumber\\&\quad
\times\int_{\Gamma_{0}}e_{\lambda }\left( \frac{e^{-\varepsilon
E^{\phi }\left( \cdot ,\omega \right) }-1}{\varepsilon },\xi \right)
k\left( \xi \cup \eta \setminus \omega \right) d\lambda \left( \xi
\right)\label{dualaprrren}\end{aligned}$$ and that the set ${{\mathcal K}}_{{{{\alpha}}C}}$, and, therefore, the set ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ are ${{\hat{P}}}_{\delta ,\varepsilon }^{\ast }$-invariant; moreover, ${{\hat{P}}}_{\delta ,\varepsilon }^{\ast }$ is a contraction in ${{\mathcal L}}_{{{{\alpha}}C}}$. We preserve the same notations for the restrictions of this contractions onto ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$.
Now, for any fixed ${\varepsilon}>0$ we consider a set $D_{\varepsilon}:=\bigl\{k\in
{{\mathcal K}}_{{{{\alpha}}C}}\bigm| {{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^\ast}k\in {\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}\bigr\}$. By , $D_{\varepsilon}$ is a core for the operator ${{\hat{L}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}$. Next, let us show that for any $k\in D_{\varepsilon}$ $$\label{apprrenest}
\lim_{\delta{\rightarrow}0}\Bigl\Vert \frac{1}{\delta}( {{\hat{P}}}^\ast_{\delta,{\varepsilon}}-{1\!\!1})k - {{\hat{L}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}k\Bigr\Vert_{{{\mathcal K}}_C} =0.$$ Indeed, let $$\begin{aligned}
({{\hat{P}}}_{\delta ,\varepsilon }^{\ast, (0) }k)\left( \eta \right) =&(1-\delta)^{{|\eta|}}k(\eta);\\
({{\hat{P}}}_{\delta ,\varepsilon }^{\ast, (1) }k)\left( \eta \right)
=&\sum_{x\in \eta }\left( 1-\delta \right)^{\left\vert \eta
\right\vert -1} z\delta e_{\lambda}\left( e^{-\varepsilon E^{\phi }\left(
\cdot ,x \right) },\eta \setminus x \right) \nonumber\\&\qquad
\times\int_{\Gamma_{0}}e_{\lambda }\left( \frac{e^{-\varepsilon
E^{\phi }\left( \cdot ,x \right) }-1}{\varepsilon },\xi \right)
k\left( \xi \cup \eta \setminus x \right) d\lambda \left( \xi
\right);\end{aligned}$$ and ${{\hat{P}}}_{\delta ,\varepsilon }^{\ast, (\geq 2) } = {{\hat{P}}}_{\delta
,\varepsilon }^{\ast } - {{\hat{P}}}_{\delta ,\varepsilon }^{\ast, (0) } -
{{\hat{P}}}_{\delta ,\varepsilon }^{\ast, (1) }$. One may improve inequality , namely, for any $n\in{{\mathbb N}}\cup\{0\}$, $\delta\in(0;1)$ $$\begin{aligned}
0 \leq n- \frac{1-(1-\delta)^n}{\delta}\leq\delta \frac{n(n-1)}{2}.\end{aligned}$$ Then, for any $k\in{{\mathcal K}}_{{{{\alpha}}C}}$, $\eta\neq\emptyset$ $$\begin{aligned}
&C^{-{{|\eta|}}}\biggl\vert\frac{1}{\delta}( {{\hat{P}}}^{\ast,(0)}_{\delta,{\varepsilon}}-{1\!\!1})k(\eta) + |\eta|k(\eta)\biggr\vert\\
\leq& \Vert k\Vert_{{{\mathcal K}}_{{{{\alpha}}C}}} {{\alpha}}^{{|\eta|}}\Bigl\vert {{|\eta|}}-
\frac{1-(1-\delta)^{{|\eta|}}}{\delta}\Bigr\vert\leq \frac{\delta}{2} \Vert
k\Vert_{{{\mathcal K}}_{{{{\alpha}}C}}} {{\alpha}}^{{|\eta|}}{{|\eta|}}({{|\eta|}}-1)\nonumber\end{aligned}$$ and the function ${{\alpha}}^x x(x-1)$ is bounded for $x\geq 1$, ${{\alpha}}\in(0;1)$. Next, for any $k\in{{\mathcal K}}_{{{{\alpha}}C}}$, $\eta\neq\emptyset$ $$\begin{aligned}
&C^{-{{|\eta|}}}\Biggl\vert\frac{1}{\delta} {{\hat{P}}}^{\ast,(1)}_{\delta,{\varepsilon}}k(\eta) -z\sum_{x\in\eta}\int_{{\Gamma}_0}
e_{{\lambda}}\left( e^{-{\varepsilon}\phi \left( x-\cdot \right) },\eta\setminus
x \right) \label{genrenadj1}\\&\qquad\times e_{{\lambda}}\left(
\frac{e^{-{\varepsilon}\phi \left( x-\cdot \right) }-1}{{\varepsilon}},\xi \right)
k\left( \xi \cup \eta\setminus x\right) d{\lambda}(\xi)\Biggr\vert\nonumber\\
\leq & \frac{z}{{{{{\alpha}}C}}}{{\alpha}}^{{|\eta|}}\sum_{x\in\eta}\bigl(\left( 1-\delta
\right)^{\left\vert \eta \right\vert -1} -1\bigr)e_{\lambda}\left(
e^{-\varepsilon E^{\phi }\left( \cdot ,x \right) },\eta \setminus x
\right) \nonumber\\&\qquad\times\int_{\Gamma_{0}}e_{\lambda }\left(
{{\alpha}}C \frac{|e^{-\varepsilon
E^{\phi }\left( \cdot ,x \right) }-1|}{\varepsilon },\xi \right) d\lambda \left( \xi \right)\nonumber\\
\leq &\frac{z}{{{{{\alpha}}C}}} {{\alpha}}^{{|\eta|}}\sum_{x\in\eta}\bigl|\left( 1-\delta \right)^{\left\vert \eta
\right\vert -1} -1\bigr| \exp{\{{{\alpha}}C \beta\}}\nonumber\\
\leq& \frac{z}{{{{{\alpha}}C}}} {{\alpha}}^{{|\eta|}}\delta {{|\eta|}}({{|\eta|}}-1) \exp{\{{{\alpha}}C
\beta\}}\nonumber\end{aligned}$$ that is smaller then $\delta$ uniformly in ${{|\eta|}}$. And, finally, $$\begin{aligned}
&\frac{1}{\delta C^{\left\vert \eta \right\vert }} \sum_{\substack{
\omega \subset \eta \crcr \left\vert \omega \right\vert \geq 2}}
\left( 1-\delta \right)^{\left\vert \eta \setminus \omega
\right\vert }\left( z\delta \right)^{\left\vert \omega \right\vert
}e_{\lambda}\left( e^{-\varepsilon E^{\phi }\left( \cdot ,\omega \right)
},\eta \setminus \omega \right)\nonumber\\&\qquad
\times\int_{\Gamma_{0}} e_{\lambda }\left(
\Biggl\vert\frac{e^{-\varepsilon E^{\phi }\left( \cdot ,\omega
\right) }-1}{\varepsilon }\Biggr\vert,\xi \right) |k( \xi
\cup \eta \setminus \omega ) | d\lambda \left( \xi \right) \nonumber \\
\leq &\frac{1}{\delta C^{\left\vert \eta \right\vert }}\sum
_{\substack{ \omega \subset \eta \crcr \left\vert \omega \right\vert \geq 2}}\left( 1-\delta \right)^{\left\vert \eta \setminus \omega
\right\vert }\left( z\delta \right)^{\left\vert \omega \right\vert
}\nonumber\\&\qquad \times \int_{\Gamma_{0}}e_{\lambda }\left(
E^{\phi }\left( \cdot ,\omega \right) ,\xi \right) \left( \alpha
C\right)^{\left\vert \xi \right\vert }\left( \alpha
C\right)^{\left\vert \eta \right\vert -\left\vert \omega \right\vert
}d\lambda
\left( \xi \right) \nonumber\\
=&\alpha^{\left\vert \eta \right\vert }\frac{1}{\delta }
\sum_{\substack{ \omega \subset \eta \crcr \left\vert \omega
\right\vert \geq 2}}\left( 1-\delta \right)^{\left\vert \eta
\setminus \omega \right\vert }\left( \frac{z\delta }{\alpha C}\exp
\left\{ \alpha C\beta \right\} \right) ^{\left\vert \omega
\right\vert } \nonumber\end{aligned}$$ but recall that ${{\alpha}}>{{\alpha}}_1$, therefore, $z\exp\{{{{{\alpha}}C}}\beta\}\leq {{{{\alpha}}C}}$, and one may continue $$\begin{aligned}
\leq &\alpha ^{\left\vert \eta
\right\vert }\frac{1}{\delta }\sum
_{\substack{ \omega \subset \eta \crcr \left\vert \omega \right\vert \geq 2}}\left( 1-\delta \right)^{\left\vert \eta \setminus \omega
\right\vert }\delta^{\left\vert \omega \right\vert }=\delta \alpha
^{\left\vert \eta \right\vert }\sum_{k=2}^{\left\vert \eta
\right\vert }\frac{\left\vert \eta \right\vert !}{k!\left(
\left\vert \eta \right\vert -k\right) !}\left(
1-\delta \right)^{\left\vert \eta \right\vert -k}\delta^{k-2} \nonumber\\
=&\delta \alpha^{\left\vert \eta \right\vert }\sum_{k=0}^{\left\vert
\eta \right\vert -2}\frac{\left\vert \eta \right\vert !}{\left(
k+2\right) !\left( \left\vert \eta \right\vert -k-2\right) !}\left(
1-\delta \right)
^{\left\vert \eta \right\vert -k-2}\delta^{k} \nonumber\\
=&\delta \alpha^{\left\vert \eta \right\vert }\left\vert \eta
\right\vert \left( \left\vert \eta \right\vert -1\right)
\sum_{k=0}^{\left\vert \eta \right\vert -2}\frac{\left( \left\vert
\eta \right\vert -2\right) !}{\left( k+2\right) !\left( \left\vert
\eta \right\vert -k-2\right) !}\left( 1-\delta
\right)^{\left\vert \eta \right\vert -2-k}\delta^{k} \nonumber\\
\leq &\delta \alpha^{\left\vert \eta \right\vert }\left\vert \eta
\right\vert \left( \left\vert \eta \right\vert -1\right)
\sum_{k=0}^{\left\vert \eta \right\vert -2}\frac{\left( \left\vert
\eta
\right\vert -2\right) !}{k!\left( \left\vert \eta \right\vert -k-2\right) !}\left( 1-\delta \right)^{\left\vert \eta \right\vert -2-k}\delta
^{k}\nonumber\\=& \delta \alpha^{\left\vert \eta \right\vert
}\left\vert \eta \right\vert \left( \left\vert \eta \right\vert
-1\right).\nonumber\end{aligned}$$ Combining these inequalities, we obtain .
Analogously, one may obtain that for any $k\in D_V:=\bigl\{k\in
{{\mathcal K}}_{{{{\alpha}}C}}\bigm| {{\hat{L}}}_V^\ast k\in {\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}\bigr\}$ (that is core for ${{\hat{L}}}^{{\odot{{\alpha}}}}_V$) $$\label{apprrenest1}
\lim_{\delta{\rightarrow}0}\Bigl\Vert \frac{1}{\delta}( {{\hat{Q}}}^\ast_{\delta}-{1\!\!1})k - {{\hat{L}}}^{{\odot{{\alpha}}}}_V k\Bigr\Vert_{{{\mathcal K}}_C} =0.$$
By Lemma \[EK\_res-conv\], we obtain that for any $k\in{\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ $$({{\hat{P}}}^\ast_{\delta,{\varepsilon}})^{\bigl[\frac{t}{\delta}\bigr]}k{\rightarrow}{{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)k; \qquad ({{\hat{Q}}}^\ast_{\delta})^{\bigl[\frac{t}{\delta}\bigr]}k{\rightarrow}{{\hat{T}}}^{{\odot{{\alpha}}}}_V(t)k$$ (convergence in ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$, recall that norm in this space is $\|\cdot\|_{{{\mathcal K}}_C}$).
Therefore, to use the same arguments as in the proof of Theorem \[descsemigroupconv\] and to apply Lemma \[powersofcontractions\], we need only to show that for any $k\in{{\mathcal K}}_{{{{\alpha}}C}}$ $$\label{dopex}
\bigl\Vert {{\hat{P}}}^\ast_{\delta,{\varepsilon}}k- {{\hat{Q}}}^\ast_{\delta}k
\bigr\Vert_{{{\mathcal K}}_C}\leq {\varepsilon}\delta A\|k\|_{{{\mathcal K}}_{{{{\alpha}}C}}}.$$
We have the following elementary inequalities. For any $\left\{
a_{k}\right\} _{k=1}^{n}\subset \lbrack 0;1]$, $n\in {{\mathbb N}}$ $$\label{specbern}
1-\prod\limits_{k=1}^{n}a_{k}\leq \sum_{k=1}^{n}\left(
1-a_{k}\right),$$ which can be easily checked by the induction principle. Next, since $$x+e^{-x}-1\leq x^{2},\quad x\geq 0,$$we obtain$$\label{longBern}
E^{\phi }\left( x,\omega \right) \left( 1-\frac{1-e^{-{\varepsilon}E^{\phi
}\left( x,\omega \right) }}{{\varepsilon}E^{\phi }\left( x,\omega \right) }\right) \leq{\varepsilon}\left( E^{\phi }\left( x,\omega \right) \right)
^{2}.$$ Hence, $$\begin{aligned}
& \frac{1}{C^{\left\vert \eta \right\vert }}\sum_{\omega \subset
\eta }\left( 1-\delta \right)^{\left\vert \eta \setminus \omega
\right\vert
}\left( z\delta \right)^{\left\vert \omega \right\vert } \\
& \qquad \times \int_{\Gamma_{0}}\left\vert \left(
e^{-\varepsilon E^{\phi }\left( \cdot ,\omega \right) },\eta
\setminus \omega \right) e_{\lambda }\left( \frac{e^{-\varepsilon
E^{\phi }\left( \cdot ,\omega \right) }-1}{\varepsilon },\xi \right)
-e_{\lambda }\left(
-E^{\phi }\left( \cdot ,\omega \right) ,\xi \right) \right\vert \\
&\qquad \qquad \times k\left( \xi \cup \eta \setminus \omega
\right)
d\lambda \left( \xi \right) \\
\leq &\frac{\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}}{C^{\left\vert \eta \right\vert }}\sum_{\omega \subset \eta }\left(
1-\delta \right)^{\left\vert \eta \setminus \omega \right\vert
}\left( z\delta
\right)^{\left\vert \omega \right\vert } \\
& \qquad \times \int_{\Gamma_{0}}e_{\lambda }\left( E^{\phi
}\left( \cdot ,\omega \right) ,\xi \right) \left\vert \left(
e^{-\varepsilon E^{\phi }\left( \cdot
,\omega \right) },\eta \setminus \omega \right) e_{\lambda }\left( \frac{1-e^{-\varepsilon E^{\phi }\left( \cdot ,\omega \right)
}}{\varepsilon E^{\phi }\left( \cdot ,\omega \right) },\xi \right)
-1\right\vert \\
& \qquad \times\left( \alpha C\right)^{\left\vert \xi \cup \eta
\setminus \omega \right\vert }d\lambda \left( \xi \right)\\
\intertext{and, by \eqref{specbern}, one may continue }
\leq & \alpha^{\left\vert \eta \right\vert }\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega \subset \eta }\left( 1-\delta
\right) ^{\left\vert \eta \setminus \omega \right\vert }\left(
z\delta \right) ^{\left\vert \omega \right\vert }\int_{\Gamma
_{0}}e_{\lambda }\left( E^{\phi }\left( \cdot ,\omega \right) ,\xi
\right) \\ &\qquad \times\sum_{x\in \eta \setminus \omega }\left(
1-e^{-\varepsilon E^{\phi }\left( x,\omega \right) }\right) \left(
\alpha C\right)^{\left\vert \xi \setminus \omega \right\vert
}d\lambda \left( \xi \right) \\ & +\,\alpha^{\left\vert \eta \right\vert }\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega \subset \eta }\left( 1-\delta
\right)^{\left\vert \eta \setminus \omega \right\vert }\left(
z\delta \right)^{\left\vert \omega \right\vert }\int_{\Gamma
_{0}}e_{\lambda }\left( E^{\phi }\left(
\cdot ,\omega \right) ,\xi \right) \\ &\qquad \times \sum_{x\in \xi }\left( 1-\frac{1-e^{-\varepsilon E^{\phi }\left( x,\omega \right) }}{\varepsilon
E^{\phi }\left( x,\omega \right) }\right) \left( \alpha
C\right)^{\left\vert \xi \setminus \omega \right\vert }d\lambda
\left( \xi \right)\\
\intertext{and, by \eqref{longBern},}
\leq & \alpha^{\left\vert \eta \right\vert }\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega \subset \eta }\left( 1-\delta
\right)
^{\left\vert \eta \setminus \omega \right\vert }\left( \frac{z\delta }{\alpha C}\exp \left\{ \alpha C\beta \right\} \right)^{\left\vert
\omega \right\vert }\sum_{x\in \eta \setminus \omega }\varepsilon
E^{\phi }\left( x,\omega \right) \\
& +\,\alpha^{\left\vert \eta \right\vert }\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega \subset \eta }\left( 1-\delta
\right)^{\left\vert \eta \setminus \omega \right\vert }\left(
\frac{z\delta }{\alpha C}\right)
^{\left\vert \omega \right\vert }\\ & \qquad \times \int_{\Gamma_{0}}\int_{\mathbb{R}^{d}}\varepsilon \left( E^{\phi }\left( x,\omega \right) \right)
^{2}e_{\lambda }\left( E^{\phi }\left( \cdot ,\omega \right) ,\xi
\right) \left( \alpha C\right)^{\left\vert \xi \right\vert }\alpha
Cdxd\lambda \left( \xi \right)\\
\intertext{again, $z\exp\{{{{{\alpha}}C}}\beta\}\leq{{{{\alpha}}C}}$ and we continue}
\leq & {\varepsilon}\, \bar{\phi}\,\alpha^{\left\vert \eta \right\vert }\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega \subset \eta }\left( 1-\delta
\right) ^{\left\vert \eta \setminus \omega \right\vert
}\delta^{\left\vert \omega
\right\vert }|\eta \setminus \omega | \cdot |\omega| \\
& + \, {\varepsilon}\alpha C \, \bar{\phi}\alpha^{\left\vert \eta \right\vert }\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\sum_{\omega \subset \eta }\left( 1-\delta
\right)^{\left\vert \eta \setminus \omega \right\vert }\delta
^{\left\vert \omega \right\vert } \left\vert \omega
\right\vert^2=:J.\end{aligned}$$ To complete the proof we need to use the following simple estimates: for any ${\left\vert}\xi {\right\vert}=n\geq 2$ one has $$\begin{aligned}
&\sum_{\omega \subset \xi }{\left\vert}\omega {\right\vert}{\left\vert}\xi \setminus
\omega {\right\vert}\left( 1-\delta \right)^{{\left\vert}\xi
\setminus \omega {\right\vert}}\delta^{{\left\vert}\omega {\right\vert}} \label{est1}\\
=&\sum_{k=1}^{n-1}\frac{n!}{k!\left( n-k\right) !}k\left( n-k\right)
\left(
1-\delta \right)^{n-k}\delta^{k} \nonumber\\
=&\sum_{k=1}^{n-1}\frac{n!}{\left( k-1\right) !\left( n-k-1\right)
!}\left(
1-\delta \right)^{n-k}\delta^{k} \nonumber\\
=&\sum_{k=0}^{n-2}\frac{n!}{k!\left( n-\left( k+1\right) -1\right)
!}\left(
1-\delta \right)^{n-\left( k+1\right) }\delta^{k+1} \nonumber\\
=&\delta \left( 1-\delta \right) n\left( n-2\right) \sum_{k=0}^{n-2}\frac{\left( n-2\right) !}{k!\left( n-2-k\right) !}\left( 1-\delta \right)
^{n-2-k}\delta^{k} \nonumber\\
=&\delta \left( 1-\delta \right) n\left( n-2\right) \left( 1-\delta
+\delta \right)^{n-2}\leq \delta \cdot 2^{n}=\delta \cdot 2^{{\left\vert}\xi
{\right\vert}}\nonumber\end{aligned}$$(and this estimate is trivial for ${\left\vert}\xi {\right\vert}\leq 1$); and, for any $n={\left\vert}\xi {\right\vert}\geq 1$$$\begin{aligned}
&\sum_{\omega \subset \xi }\left( 1-\delta \right)^{{\left\vert}\xi
\setminus \omega {\right\vert}}\delta^{{\left\vert}\omega {\right\vert}}{\left\vert}\omega {\right\vert}^{2} \label{est2}\\
=&\sum_{k=1}^{n}\frac{n!}{k!\left( n-k\right) !}k^{2}\left( 1-\delta
\right)^{n-k}\delta^{k} \nonumber\\
=&\delta \sum_{k=1}^{n}\frac{n!}{\left( k-1\right) !\left(
n-1-\left( k-1\right) \right) !}k\left( 1-\delta \right)^{\left(
n-1\right) -\left(
k-1\right) }\delta^{k-1} \nonumber\\
=&\delta \sum_{k=0}^{n-1}\frac{n!}{k!\left( n-1-k\right) !}k\left(
1-\delta
\right)^{\left( n-1\right) -k}\delta^{k} \nonumber\\
\leq &\delta n\left( n-1\right) \left( 1-\delta +\delta \right)
^{n-1}<\delta \cdot 2^{n}\nonumber\end{aligned}$$(and, again, it is trivial for $\xi =\emptyset $).
Then, by , , we obtain for any $|\eta|\geq
2$ $$J\leq {\varepsilon}\, \bar{\phi}\,\alpha^{\left\vert \eta \right\vert
}\left\Vert k\right\Vert_{
\mathcal{K}_{\alpha C}}\delta {{|\eta|}}({{|\eta|}}-1)+ {\varepsilon}\alpha C \, \bar{\phi}\alpha^{\left\vert \eta \right\vert }\left\Vert k\right\Vert_{\mathcal{K}_{\alpha C}}\delta {{|\eta|}}({{|\eta|}}-1)\leq{\varepsilon}\delta A,$$ where $A$ is independent on $\eta$.
\[maincor\] Let the conditions of Theorem \[maintheorem\] hold. Then for any $\{k^{{({\varepsilon})}}, k\}\subset{{\mathcal K}}_{{{{\alpha}}C}}$, ${\varepsilon}>0$ $$\label{estforeps}
\bigl\Vert {{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}(t)k^{{({\varepsilon})}}- {{\hat{T}}}^{{\odot{{\alpha}}}}_V(t)k\bigr\Vert_{{{\mathcal K}}_C}\leq \bigl\Vert k^{{({\varepsilon})}}-
k\bigr\Vert_{{{\mathcal K}}_C} + {\varepsilon}t A \Vert k \Vert_{{{\mathcal K}}_{{{{\alpha}}C}}}.$$
The proof follows directly from the triangle inequality and the contractive property of the semigroup ${{\hat{T}}}^{{\odot{{\alpha}}}}_{{{\varepsilon}, \, \mathrm{ren}}}$.
And now we will show that our Vlasov limiting dynamics has the properties described in the Subsection \[scalingdescr\].
\[Vlasovscheme\] Let $C, z, \beta, \alpha_1$ be as in Proposition \[sun-inv\], and ${{\alpha}}_2:=\max\bigl\{{{\alpha}}_1,\frac{z}{C}\bigr\}\in(0;1)$. Let $\rho_0$ be a measurable function on ${{{{\mathbb R}}^d}}$ such that there exists ${{\alpha}}\in({{\alpha}}_2;1)$ such that $0\leq\rho_0(x)\leq {{{{\alpha}}C}}$ for a.a. $x\in{{{{\mathbb R}}^d}}$. Then the Cauchy problem $$\label{CauchyVlasov}
\begin{cases}
\dfrac{\partial}{\partial t} k_t = {{\hat{L}}}^\ast_V k_t\\
k_0=e_{\lambda}(\rho_0)
\end{cases}$$ is well-defined on ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ and has a solution $k_t=e_{\lambda}(\rho_t)\in{{\mathcal K}}_{{{{\alpha}}C}}$, where $\rho_t$ is a solution of the Cauchy problem $$\label{CauchyVlasoveqn}
\begin{cases}
\dfrac{\partial}{\partial t} \rho_t(x) = -\rho_t(x) + z \exp\biggl\{\displaystyle-\int_{{{{\mathbb R}}^d}}\rho_t(y) \phi(x-y)dy\biggr\}, \\
\rho_t \bigr|_{t=0}(x)=\rho_0(x),
\end{cases}$$ for a.a. $x\in{{{{\mathbb R}}^d}}$ such that $0\leq\rho_t(x)\leq {{{{\alpha}}C}}$ for a.a. $x\in{{{{\mathbb R}}^d}}$.
First of all, we note that implies $z<C$, therefore, the condition $\frac{z}{C}<1$ holds. Next, if has a solution $\rho_t(x)\geq0$ then $\frac{\partial}{\partial t} \rho_t(x) \leq -\rho_t(x) + z $ and, therefore, $\rho_t(x)\leq r_t(x)$ where $r_t(x)$ is a solution of the Cauchy problem $$\label{CauchyEst}
\begin{cases}
\dfrac{\partial}{\partial t} r_t(x) = -r_t(x) + z , \\
r_t \bigr|_{t=0}(x)=\rho_0(x),
\end{cases}$$ for a.a. $x\in{{{{\mathbb R}}^d}}$, hence, $$r_t(x)=e^{-t}\rho_0(x)+z(1-e^{-t})=z+e^{-t}(\rho_0(x)-z)
\leq \max\{z,\rho_0(x)\}\leq {{{{\alpha}}C}},$$ that yields $0\leq\rho_t(x)\leq {{{{\alpha}}C}}$.
To prove the existence of the solution of let us fix some $T>0$ and define the Banach space $X_T=C([0;T],L^\infty({{{{\mathbb R}}^d}}))$ of all continuous functions on $[0;T]$ with values in $L^\infty({{{{\mathbb R}}^d}})$; the norm on $X_T$ is given by . We denote by $X_T^+$ the cone of the all nonnegative functions from $X_T$.
Let $\Phi$ be a mapping which assign to any $v\in X_T$ the solution $u_t$ of the linear Cauchy problem $$\label{CauchyLin}
\begin{cases}
\dfrac{\partial}{\partial t} u_t(x) = -u_t(x) + z \exp\{\displaystyle-(v_t*\phi)(x)\}, \\
u_t \bigr|_{t=0}(x)=\rho_0(x),
\end{cases}$$ for a.a. $x\in{{{{\mathbb R}}^d}}$, where we use the usual notation for convolution on ${{{{\mathbb R}}^d}}$:$(f*g)(x):=\int_{{{{\mathbb R}}^d}}f(y)g(x-y)dy$. Therefore, $$\label{defPhi}
(\Phi v)_t(x)=e^{-t}\rho_0(x)+z\int_0^te^{-(t-s)}\exp\{-(v_t*\phi)(x)\}ds\geq0.$$ Similarly as before we obtain that $v\in X_T^+$ implies the estimate $|(\Phi v)_t(x)|\leq\max\{z,\rho_0(x)\}$; in particular, $\Phi v\in
X_T^+$. Next, using elementary inequality $|e^{-a}-e^{-b}|\leq|a-b|$ for any $a,b\geq0$, we obtain that for any $v, w\in X_T^+$ $$\begin{aligned}
\bigl| (\Phi v)_t(x)-(\Phi w)_t(x) \bigr|&\leq z\int_0^te^{-(t-s)}\Bigl|\exp\{-(v_t*\phi)(x)-\exp\{-(w_t*\phi)(x)\}\Bigr|ds\\
&\leq z\int_0^te^{-(t-s)}\bigl|(v_t*\phi)(x)-(w_t*\phi)(x)\bigr|ds\\
&\leq z\int_0^te^{-(t-s)}(|v_t-w_t|*\phi)(x) ds\\
&\leq z\beta\|v-w\|_T(1-e^{-t}),\end{aligned}$$ where we used the inequality $|(f*g)(x)|\leq
\|f\|_{L^\infty({{{{\mathbb R}}^d}})}\|g\|_{L^1({{{{\mathbb R}}^d}})}$ and condition . Therefore, $\|\Phi v-\Phi w\|_T\leq
z\beta\|v-w\|_T$. Since implies $z\beta\leq
e^{-1}$ (see the proof of Proposition \[sun-inv\]), hence, $\Phi$ is a contraction mapping on the cone $X_T^+$. Taking, as usual, $v^{(n)}=\Phi^nv^{(0)}$, $n\geq1$ for $v^{(0)}\in X_T^+$ we obtain that $\{v^{(n)}\}\subset X_T^+$ is a fundamental sequence in $X_T$ which has, therefore, a unique limit point $v\in X_T$. Since $X_T^+$ is a closed cone we have that $v\in X_T^+$. Then, identically to the classical Banach fixed point theorem, $v$ will be a fixed point of $\Phi$ on $X_T$ and a unique fixed point on $X_T^+$. Then, this $v$ is the nonnegative solution of on the interval $[0;T]$. By the note above, $v_t(x)\leq {{{{\alpha}}C}}$. Changing initial value in onto $\rho_t
\bigr|_{t=T}(x)=v_T(x)$ we may extend all our considerations on the time-interval $[T;2T]$ with the same estimate $v_t(x)\leq {{{{\alpha}}C}}$; and so on. As a a result, has a global bounded solution $\rho_t(x)$ on ${{\mathbb R}}_+$.
Clearly, $k_0=e_{\lambda}(\rho_0)\in{{\mathcal K}}_{{{{\alpha}}C}}\subset{\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$. Then $k_t={{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0$ will be a strongly differentiable function (in the sense of norm $\|\cdot\|_{{{\mathcal K}}_C}$ in ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$); moreover, $k_t\in{{\mathcal K}}_{{{{\alpha}}C}}$. Next, if we substitute $k_t=e_{\lambda}(\rho_t)$ into , then, by , we obtain $$\begin{aligned}
&\sum_{x\in\eta} \frac{\partial}{\partial t} \rho_t(x)
e_{\lambda}(\rho_t,\eta\setminus x)\\=&-{\left\vert}\eta{\right\vert}e_{\lambda}(\rho_t,\eta)
\\& +z\sum_{x\in\eta}e_{\lambda}(\rho_t,\eta\setminus x)\int_{{\Gamma}_0} e_{{\lambda}}\left(- \phi \left( x-\cdot \right),\xi \right) e_{\lambda}(\rho_t,\xi)
d{\lambda}(\xi)\\=&-\sum_{x\in\eta} \rho_t(x) e_{\lambda}(\rho_t,\eta\setminus
x)\\&+z\sum_{x\in\eta}e_{\lambda}(\rho_t,\eta\setminus
x)\exp\biggl\{-\int_{{{{\mathbb R}}^d}}\phi(x-y)\rho_t(y)dy\biggr\},\end{aligned}$$ that holds since $\rho_t$ is satisfied .
\[RemarkKirkwoodMonroe\] Note that the stationary equation for has the following form $$\label{KirkwoodMonroe}
\rho(x) = z \exp\biggl\{-\int_{{{{\mathbb R}}^d}}\rho(y) \phi(x-y)dy\biggr\}$$ and coincides with the famous Kirkwood–Monroe equation ([@KM1941], see also, e.g., [@GK1976] and references therein, and the recent work [@CP2010]).
Further considerations
----------------------
We have realized the scheme proposed at the end of Subsection \[scalingdescr\]. But let us explain also the rigorous meaning of the equivalence which was background to all our consideration.
Let $C, z, \beta, \alpha_2$ be as in Theorem \[Vlasovscheme\]. Then, for any fixed ${\varepsilon}>0$ we have $1-\exp\{-{\varepsilon}\phi\}\in
L^1({{{{\mathbb R}}^d}})$ and, by [@FKKZ2010 Proposition 3.2], ${{\hat{L}}}_{\varepsilon}$, given by , is a linear operator in ${{\mathcal L}}_{{\varepsilon}^{-1}C}$ with dense domain ${{\mathcal L}}_{2{\varepsilon}^{-1}C}$. Consider the image $\bigl(
{{\hat{L}}}^\ast_{\varepsilon}, D({{\hat{L}}}_{\varepsilon}^\ast)\bigr)$ in ${{\mathcal K}}_{{\varepsilon}^{-1}C}=R_{{\varepsilon}^{-1}}{{\mathcal K}}_C$ under the isometrical isomorphism $R_{{\varepsilon}^{-1}C}$ of the dual operator $\bigl({{\hat{L}}}'_{\varepsilon},
D({{\hat{L}}}'_{\varepsilon})\bigr)$ in $({{\mathcal L}}_{{\varepsilon}^{-1}C})'$.
We are not able to show that ${{\hat{L}}}_{\varepsilon}$ is a generator of a strongly continuous semigroup in ${{\mathcal L}}_{{\varepsilon}^{-1}C}$ since a condition like (with ${\varepsilon}^{-1}C$ instead of $C$) cannot be fulfilled uniformly in ${\varepsilon}>0$. But one can do in the following manner.
Let ${{\alpha}}\in({{\alpha}}_2;1)$ and let us consider the space $\mathbb{K}^{{\alpha}}_{\varepsilon}=\overline{{{\mathcal K}}_{{\varepsilon}^{-1}{{{{\alpha}}C}}}}^{{{\mathcal K}}_{{\varepsilon}^{-1}C}}$. Note that for any $r^{{({\varepsilon})}}\in\mathbb{K}^{{\alpha}}_{\varepsilon}$ there exist $\{r^{{({\varepsilon})}}_n\}\subset{{\mathcal K}}_{{\varepsilon}^{-1}{{{{\alpha}}C}}}$ such that $$0=\lim_{n{\rightarrow}\infty}\|r^{{({\varepsilon})}}_n-r^{{({\varepsilon})}}\|_{{{\mathcal K}}_{{\varepsilon}^{-1}C}}
=\lim_{n{\rightarrow}\infty}\|R_{\varepsilon}r^{{({\varepsilon})}}_n-R_{\varepsilon}r^{{({\varepsilon})}}\|_{{{\mathcal K}}_{C}}$$ and the inclusion $R_{\varepsilon}r^{{({\varepsilon})}}_n \in{{\mathcal K}}_{{{{\alpha}}C}}$, $n\in{{\mathbb N}}$ yields $R_{\varepsilon}r^{{({\varepsilon})}}\in {\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$. Vise versa, for any $k^{{({\varepsilon})}}\in{\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$ we see that $R_{{\varepsilon}^{-1}}k^{{({\varepsilon})}}\in\mathbb{K}^{{\alpha}}_{\varepsilon}$. As a result, $R_{\varepsilon}$ provides an isometrical isomorphism between the Banach spaces $\mathbb{K}^{{\alpha}}_{\varepsilon}$ and ${\overline{{{\mathcal K}}_{{{{\alpha}}C}}}}$. Then, $U_{\varepsilon}^{{\alpha}}(t):=R_{{\varepsilon}^{-1}}{{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}^{{\odot{{\alpha}}}}(t)R_{\varepsilon}$ will be a strongly continuous contraction semigroup on $\mathbb{K}^{{\alpha}}_{\varepsilon}$ with the generator $A_{\varepsilon}^{{\alpha}}=R_{{\varepsilon}^{-1}}{{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^{{\odot{{\alpha}}}}R_{\varepsilon}$ and the domain $D(A_{\varepsilon}^{{\alpha}})=R_{{\varepsilon}^{-1}}D({{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^{{\odot{{\alpha}}}})$. Moreover, since ${{\mathcal K}}_{{{{\alpha}}C}}\cap D({{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^{{\odot{{\alpha}}}})$ is a core for ${{\hat{L}}}_{{{\varepsilon}, \, \mathrm{ren}}}^{{\odot{{\alpha}}}}$, the set $K_{{\varepsilon}^{-1}{{{{\alpha}}C}}}\cap D(A_{\varepsilon}^{{\alpha}})$ is a core for $A_{\varepsilon}^{{\alpha}}$ and on this core the operator $A_{\varepsilon}^{{\alpha}}$ coincides with ${{\hat{L}}}_{\varepsilon}^\ast$. Note that, the semigroup $U_{\varepsilon}^{{\alpha}}(t)$ is the rigorous analog of ${{\hat{T}}}_{\varepsilon}^\ast$ in .
Let now $\{k_0,k_0^{{({\varepsilon})}}|{\varepsilon}>0\}\subset K_{{{{\alpha}}C}}$. Then, by , $$\begin{aligned}
&\bigl\Vert U_{\varepsilon}^{{\alpha}}(t) R_{{\varepsilon}^{-1}} k_{0}^{{({\varepsilon})}}-R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)
k_0\bigr\Vert_{{{\mathcal K}}_{{\varepsilon}^{-1}C}}\label{dop1}\\=&\bigl\Vert R_{\varepsilon}\bigl(U_{\varepsilon}^{{\alpha}}(t) R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}-R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)
k_0\bigr)\bigr\Vert_{{{\mathcal K}}_{C}}\nonumber\\ = &\bigl\Vert{{\hat{T}}}_{{{\varepsilon}, \, \mathrm{ren}}}^{{\odot{{\alpha}}}}(t)k_0^{{({\varepsilon})}}-{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0\bigr\Vert_{{{\mathcal K}}_{C}} \leq A {\varepsilon}t\|k_0\|_{{{\mathcal K}}_{{{{\alpha}}C}}}+\|k_0^{{({\varepsilon})}}-k_0\|_{{{\mathcal K}}_C}.\nonumber\end{aligned}$$ On the other hand, $$\begin{aligned}
&\bigl\Vert U_{\varepsilon}^a(t) R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}-R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0\bigr\Vert_{{{\mathcal K}}_{{\varepsilon}^{-1}C}}\label{dop2}\\=&{\mathop{\mathrm{ess\,sup}}}_{\eta\in{\Gamma}_0}\left\{
({\varepsilon}^{-1}C)^{-{{|\eta|}}} \bigl\vert R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)\bigr\vert \left\vert\frac{U_{\varepsilon}^a(t) R_{{\varepsilon}^{-1}}
k_0^{{({\varepsilon})}}(\eta)}{R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)}-1\right\vert
\right\}\nonumber\\= &{\mathop{\mathrm{ess\,sup}}}_{\eta\in{\Gamma}_0}\left\{ C^{-{{|\eta|}}}
\bigl\vert{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)\bigr\vert
\left\vert\frac{U_{\varepsilon}^a(t) R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}(\eta)}{R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)}-1\right\vert
\right\}.\nonumber\end{aligned}$$
In particular, if $$\label{k-conv}
\lim_{{\varepsilon}{\rightarrow}0}\| k_0^{{({\varepsilon})}}-k_0\|_{{{\mathcal K}}_C}=0$$ then , imply $$\label{equivaa}
\lim_{{\varepsilon}{\rightarrow}0}\frac{U_{\varepsilon}^a(t) R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}(\eta)}{R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)}=1 \quad \mathrm{for}
\ {\lambda}\mathrm{-a.a.} \ \eta\in{\Gamma}_0.$$ The equality is a rigorous realization of the equivalence (with changes $k_0^{({\varepsilon})}$ onto $R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}$).
Moreover, let $T>0$ and suppose that there exists a function $c:{\Gamma}_0\rightarrow(0;+\infty)$ such that $$\label{condveryspec}
q({{\alpha}},T):=\sup_{t\in [0;T]}{\mathop{\mathrm{ess\,sup}}}_{\eta\in{\Gamma}_0}\frac{c(\eta)}{{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)}<+\infty.$$ Then, using the equality $$\begin{aligned}
&c(\eta){C^{-{{|\eta|}}}}
\left\vert\frac{U_{\varepsilon}^a(t) R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}(\eta)}{R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t) k(\eta)}-1\right\vert \\ = &
C^{-{{|\eta|}}} \bigl\vert{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t) k_0(\eta)\bigr\vert
\left\vert\frac{U_{\varepsilon}^a(t) R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}(\eta)}{R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t) k_0(\eta)}-1\right\vert
\frac{c(\eta)}{\bigl\vert{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)\bigr\vert},\end{aligned}$$ we obtain that for such $k_0$ and for any $t\in[0;T]$ $$\left\Vert\frac{U_{\varepsilon}^a(t) R_{{\varepsilon}^{-1}} k_0^{{({\varepsilon})}}(\eta)}{R_{{\varepsilon}^{-1}}{{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)}-1\right\Vert_{C,c} \leq q({{\alpha}},T) A {\varepsilon}t\|k_0\|_{{{\mathcal K}}_{{{{\alpha}}C}}}+\|k_0^{{({\varepsilon})}}-k_0\|_{{{\mathcal K}}_C},$$ where $$\|k\|_{C,c}={\mathop{\mathrm{ess\,sup}}}_{\eta\in{\Gamma}_0} \frac{|k(\eta)|}{C^{{|\eta|}}c^{-1}(\eta)}.$$ This gives that the equivalence may be shown in a proper Banach space which is independent on ${\varepsilon}$.
The condition on $k_0$ is reasonable: for example, for $k_0=e_{\lambda}(\rho_0)$, since, by the Theorem \[Vlasovscheme\], we have ${{\hat{T}}}_V^{{\odot{{\alpha}}}}(t)k_0(\eta)=e_{\lambda}(\rho_t,\eta)$, where $\rho_t$ satisfies ; therefore, holds for any $|\rho_0(x)|\leq {{{{\alpha}}C}}$ such that $$\sup_{t\in[0;T]}\inf_{x\in{{{{\mathbb R}}^d}}}|\rho_t(x)|\geq
\rho_{\min} >0$$ if we set $c(\eta)=e_{\lambda}(\rho_{\min},\eta)=\rho_{\min}^{{|\eta|}}$. Moreover, we obtain that $|\rho_t(x)|\leq {{{{\alpha}}C}}$. The following example shows which function $k_0^{{({\varepsilon})}}$ one can choose in this case.
Let $k_0(\eta)=\rho_0^{{|\eta|}}$, $\rho_0 \in(0;{{{{\alpha}}C}})$. Let us consider the scaled Lebesgue–Poisson exponent $k_0^{{({\varepsilon})}}(\eta)=e_{\lambda}\bigl(\rho_0(1+{\varepsilon}u(\cdot)),\eta\bigr)$, where $\sup_{x\in{{{{\mathbb R}}^d}}}|u(x)|=\bar{u}<\infty$, ${\varepsilon}>0$. Then for any ${\varepsilon}<\frac{{{{{\alpha}}C}}-\rho_0}{\rho_0 \bar{u}}$ we have $|k_0^{{({\varepsilon})}}(\eta)|<({{{{\alpha}}C}})^{{|\eta|}}$. Moreover, $$\begin{aligned}
& C^{-{{|\eta|}}}\bigl\vert k_0^{{({\varepsilon})}}(\eta)-k_0(\eta) \bigr\vert=
\Bigl(\frac{\rho_0}{C}\Bigr)^{{|\eta|}}\bigl\vert e_{\lambda}\bigl(1+{\varepsilon}u(\cdot),\eta\bigr)-1\bigr\vert\\\leq&
\Bigl(\frac{\rho_0}{C}\Bigr)^{{|\eta|}}{\varepsilon}\sup_{s\in(0;{\varepsilon})}\biggl\vert
\frac{d}{ds} e_{\lambda}\bigl(1+s u(\cdot),\eta\bigr)\biggr\vert\\ =&
\Bigl(\frac{\rho_0}{C}\Bigr)^{{|\eta|}}{\varepsilon}\sup_{s\in(0;{\varepsilon})}\biggl\vert
\sum_{x\in\eta}u(x) e_{\lambda}\bigl(1+s u(\cdot),\eta\setminus
x\bigr)\biggr\vert\\ \leq& \Bigl(\frac{\rho_0}{C}\Bigr)^{{|\eta|}}{\varepsilon}\sum_{x\in\eta} \bar{u} e_{\lambda}\bigl(1+{\varepsilon}\bar{u},\eta\setminus
x\bigr)\\ \leq& \Bigl(\frac{\rho_0}{C}\Bigr)^{{|\eta|}}{\varepsilon}{{|\eta|}}\bar{u}
\biggl(1+\frac{{{{{\alpha}}C}}-\rho_0}{\rho_0 \bar{u}} \bar{u}\biggr)^{{{|\eta|}}-1}\\
=&{\varepsilon}\frac{\rho_0}{{{{{\alpha}}C}}}{{|\eta|}}{{\alpha}}^{{|\eta|}}\leq {\varepsilon}\frac{\rho_0}{{{{{\alpha}}C}}}
\frac{-1}{e\ln{{\alpha}}}.\end{aligned}$$ As a result, $\|k_0^{{({\varepsilon})}}-k_0\|_{{{\mathcal K}}_C}{\rightarrow}0$ as ${\varepsilon}{\rightarrow}0$.
Acknowledgement {#acknowledgement .unnumbered}
===============
The financial support of DFG through the SFB 701 (Bielefeld University), German-Ukrainian Projects 436 UKR 113/94, 436 UKR 113/97 and FCT through POCI and PTDC/MAT/67965/2006 is gratefully acknowledged.
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Kozlov, V.V.: The generalized [V]{}lasov kinetic equation. Russian Math. Surveys **63**(4), 691–726 (2008)
Lenard, A.: States of classical statistical mechanical systems of infinitely many particles. [I]{}. Arch. Rational Mech. Anal. **59**(3), 219–239 (1975)
Lenard, A.: States of classical statistical mechanical systems of infinitely many particles. [II]{}. [C]{}haracterization of correlation measures. Arch. Rational Mech. Anal. **59**(3), 241–256 (1975)
van Neerven, J.: The adjoint of a semigroup of linear operators, *Lecture Notes in Mathematics*, vol. 1529. Springer-Verlag, Berlin (1992)
Penrose, M.D.: Existence and spatial limit theorems for lattice and continuum particle systems. Prob. Surveys **5**, 1–36 (2008)
Preston, C.: Spatial birth-and-death processes. Bull. Inst. Internat. Statist. **46**(2), 371–391, 405–408 (1975)
Qi, X.: A functional central limit theorem for spatial birth and death processes. Adv. in Appl. Probab. **40**(3), 759–797 (2008)
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[^1]: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine ([fdl@imath.kiev.ua]{}).
[^2]: Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany ([kondrat@math.uni-bielefeld.de]{})
[^3]: Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany ([kutoviy@math.uni-bielefeld.de]{}).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Non-coherent electronic transport in metallic nanowires exhibits different carrier temperatures for the non-equilibrium forward and backward populations in the presence of electric fields. Depending on the mean free path that characterizes inter-branch carrier backscattering transport regimes vary between the ballistic and diffusive limits. In particular, we show that the simultaneous measurements of the electrical characteristics and the carrier distribution function offer a direct way to extract the carrier mean free path even when it is comparable to the conductor length. Our model is in good agreement with the experimental work on copper nanowires by Pothier [*et al.*]{} \[Phys. Rev. Lett. [**79**]{}, 3490 (1997)\] and provides an elegant interpretation of the inhomogeneous thermal broadening observed in the local carrier distribution function as well as its scaling with external bias.'
address:
- '$^1$ Beckman Institute and Department of Physics, University of Illinois at Urbana-Champaign, IL 61801, USA.'
- '$^2$ Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, IL 61801, USA.'
author:
- 'M. A. Kuroda$^1$, J.-P. Leburton$^{1,2}$'
bibliography:
- 'report.bib'
title: Carrier mean free path and temperature imbalance in mesoscopic wires
---
[*Keywords*]{}: diffusive transport, nanowires, nanotubes, mean free path
Present understanding of transport in mesoscopic systems relies on two different approaches: Landau’s theory of Fermi liquids [@baymlandau] and Tomonaga-Luttinger liquids (TLL) theory [@solyom1979]. The latter describes correlated one dimensional (1D) systems and is characterized by a power law decay at the Fermi level at $T=0$, instead of the discontinuity observed in Fermi liquids. Recent progresses in fabrication technology have made available a variety of material structures to study the electronic properties in quasi-1D systems like quantum wires [@hu1999], nanotubes [@iijima1991] and nanoribbons [@tapaszto2008]. Despite renewed experimental efforts, TLL features of 1D systems have yet to be indisputably proven, as their manifestation is limited fundamentally by their sensitivity to disorder and surface roughness, and the fact that inherent perturbations induced by the measurement should be much smaller than thermal fluctuations. These undesired effects cause loss of coherence amongst particles, and transport becomes diffusive.
In the past, transport experiments on copper nanowires at low temperature [@pothier1997] have shown to exhibit quasi-particle distributions with a two-step profile, a shape expected in a regime with no carrier interactions due to the superposition of the distribution functions in the leads [@nagaev1995]. However, the thermal broadening of the local distribution function (hot carrier effects) suggested significant carrier scattering, thereby invalidating lack of interactions.
In this paper we show that in non-coherent transport and beyond the diffusive limit the non-equilibrium carrier distribution in 1D-systems cannot be described by a single energy distribution but as a superposition of two distinct (forward and backward) carrier populations coupled by mutual scattering. Depending on the strength of the coupling between the two populations, the transport regimes varies between the well-known ballistic and diffusive limits. The model provides a straight correlation between the non-uniform thermal broadening of the carrier distribution and the mean-free path (even when the latter is comparable to the channel length) and presents good agreement with the experimental work in mesoscopic copper wires [@pothier1997], describing both the inhomogeneous local thermal broadening as well as the scaling law of the carrier distribution function observed the high-bias. We also discuss the conditions for the observation of this phenomena in other material systems.
![(a) Band structure of two branch system level. (b) Forward and backward carrier distribution function out of thermal equilibrium in the presence of a field. (c) Effective quasi-particle distribution function.[]{data-label="fig:distrib"}](./distrib.eps){width="4in"}
We assume that close to the Fermi level the energy dispersion in the 1D conductor is well described by linear branches $E_\pm(k)$[^1], as shown in Fig. \[fig:distrib\].a. For simplicity we only consider two branches, but our model and conclusions can be extended to mesoscopic systems with multiple bands provided that, as we show later, in 1D conductors current and heat flow do not depend on the branch Fermi velocity of the system. We assume that each of these two branches exhibits a $2g_c$-degeneracy (where the factor of 2 accounts for the spin). We group the carrier populations according to the sign of the Fermi velocity ($v_F= \hbar^{-1} \partial_k E_\pm$). The effective intra-branch electron-electron (e-e) scattering thermalizes the distribution (i.e. $\tau_{e-e}^{intra}\rightarrow 0$) causing the loss of coherence . Hence, each of these populations is described by a Fermi distribution function $f_\eta(E)$ (with $\eta = +,-$) [@kuroda2008]. In the presence of an electric field a population imbalance between the branches arises because of inefficient inter-branch carrier scattering, which creates a quasi-Fermi level difference ($\mu_+ \neq \mu_-$) and disrupts the thermal equilibrium between the two populations ($T_+\neq T_-$) as depicted in Fig. \[fig:distrib\].b. Under these conditions we have recently shown that the net carrier and heat transport is expressed as [@kuroda2008]: $$\begin{aligned}
I = g_c G_q \frac{\mu_+-\mu_-}{e}\label{eq:current}\\
U = \frac{g_c}{2} \left(G_{th}^+ T_+ -G_{th}^- T_-\right)\label{eq:heatflow}\end{aligned}$$ in terms of the quantum electric ($G_q = e^2/(\pi \hbar) $) [@datta] and thermal ($G_{th}^\pm = \pi k_B^2 T_\pm /(3\hbar)$) [@rego1997] conductance, respectively. Neither the current nor the heat flow depend on the magnitude of the branch Fermi velocity because of the system dimensionality. Because of the constant density of states, the local carrier distribution function measured experimentally [@pothier1997; @pothier1997b] is the average of the branch distribution functions : $$f(E) = \frac{1}{2}\left[f_+(E)+f_-(E)\right] \label{eq:distfunc}$$ as shown in Fig. \[fig:distrib\].c. Two steps in he distribution function are clearly observed when $|\mu_+-\mu_-| \gg k_BT_+,k_BT_-$.
We denote $\lambda$ the mean free path characterizing interactions amongst carriers in [*different*]{} branches, which tends to restore the equilibrium between the two populations. We assume these interactions involve inter-branch e-e, impurity or acoustic phonon (if the sound velocity $v_s\ll v_F$) processes, and only induce quasi-particle backscattering, i.e. no energy is transferred from the carrier populations to the external system. In this case the effective electric field $F$ along the channel has been shown to be [@kuroda2008]: $$F = \frac{I}{g_c G_q \lambda} \label{eq:field}.$$ By integrating this equation along the channel, assuming that the mean free path remains constant and using Eq. \[eq:current\], we find the drain-source bias voltage $V_{ds}$: $$V_{ds} = V_{c}+V_{ch} = \frac{\mu_+-\mu_-}{e}
\left(1+\frac{L}{\lambda}\right) \label{eq:Vds}.$$ The magnitudes $V_c$ and $V_{ch}$ denote voltage drops at the contacts and along channel, respectively, where the former is due to the quantum contact resistance [@datta]. We have also shown that the temperature profiles for forward and backward populations are given by: $$\pm G_{th}^\pm \partial_xT_\pm = \frac{I F}{2} \mp
\frac{U}{\lambda}$$ under the influence of quasi-particle backscattering. In particular, direct integration of this equation with the boundary conditions $T_+(-L/2) = T_{0+}$ and $T_-(L/2)
= T_{0-}$ (perfectly absorbing contacts) yields: $$\begin{aligned}
T_\pm(x) = \sqrt{T_{0\pm}^2\pm\frac{(1/2\pm \tilde{x}) \left(T_{0-}^2 -
T_{0+}^2\right)}{(1 + \tilde{\lambda})}+ \frac{\left(1/2 \pm
\tilde{x}\right) (1/2 \mp \tilde{x} +
\tilde{\lambda})}{(1+\tilde{\lambda})^2 } \frac{V_{ds}^2
}{\mathcal{L}}}\label{eq:tempprof}\end{aligned}$$ where $\mathcal{L}=\frac{\pi^2}{3} \left(\frac{k_B}{e}\right)^2$ is the Lorenz number. The variables $\tilde{\lambda}$ and $\tilde{x}$ stand for the rescaled position ($\tilde{x} = x/L$) and mean free path ($\tilde{\lambda}=\lambda/L$), respectively. The strength of the interaction sets two limiting regimes: (i) ballistic ($\tilde{\lambda} \gg 1$) and (ii) diffusive ($\tilde{\lambda}\ll 1$). In Fig. \[fig:schemes\] we compare the quasi-Fermi levels (top), temperatures profiles (middle) and local energy distribution function different positions along the channel (bottom) for these two limits and the intermediate case with $\tilde{\lambda} = 10$. For illustrative purposes, the bias voltage $V_{ds}=100\times k_BT_{0-}$ and the asymmetric boundary conditions $k_BT_{0+} = 1.1 \times k_BT_{0-}$ are used. On the one hand, in the case of ballistic transport the quasi-Fermi levels for forward and backward populations remain constant along the conductor (Fig. \[fig:schemes\].a). As a result, the electric field vanishes along the channel and the drain-source voltage $V_{ds}$ is reduced to the voltage drop at the contacts: $$V_{ds} = \frac{(\mu_+-\mu_-)}{e} \label{eq:vdsball}$$ In addition, the distribution of forward (backward) carriers exhibits a homogeneous temperature profile along the channel with the value of the left (right) lead temperature. In this regime, the local (average) distribution function $f(E)$ (Eq. \[eq:distfunc\]) exhibits two pronounced steps independently of the position in the wire when $eV_{ds} \gg k_BT_{0\pm}$. On the other hand, if $\tilde{\lambda}\ll 1$ (Fig. \[fig:schemes\].c) a single Fermi-Dirac distribution describes the energy quasi-particle distribution since the quasi-Fermi level difference and thermal imbalance between carrier populations become negligible due to the effective inter-branch backscattering. Indeed, the drain-source voltage is due to the voltage drop along the channel (directly proportional to the channel length) as $V_c$ can be neglected (Eq. \[eq:Vds\]). In this regime, the thermal broadening along the channel (Eq. \[eq:tempprof\]) reads: $$T_\pm(x) = \sqrt{T_{0\pm}^2\pm\left(1/2\pm\tilde{x}\right) \left(T_{0-}^2 - T_{0+}^2\right)+ \left(\frac{1}{4} - \frac{x^2}{L^2}\right) \frac{V_{ds}^2
}{\mathcal{L}}},$$ which presents a maximum a the channel mid-length for large biases. This latter expression reduces to previous results for diffusive limit [@kozub1995] when symmetric boundary conditions are used ($T_{0+}$=$T_{0-}$=$T_0$). When $\tilde{\lambda} = 10$, the voltage drop at the contacts is comparable to that along the channel and the quasi-Fermi level varies along the channel (top of Fig. \[fig:schemes\]b). The forward (backward) temperature $T_+$ ($T_-$) increases monotonously as carriers move away from the source (drain) achieving values that under high biases can be significantly higher than the contact temperature $T_{0+}$ ($T_{0-}$) (middle of Fig. \[fig:schemes\]b). However intra-branch coupling is not strong enough to restore thermal equilibrium between forward and backward populations. In this case, the quasi-Fermi level separation is still significant compared to the branch thermal broadening, so the local carrier distribution function exhibits a two-step shape which, in the presence of backscattering, are more rounded than in the ballistic limit (bottom of Fig. \[fig:schemes\]b).
![Quasi-Fermi level (top) and temperature (middle) profiles for forward and backward populations in different transport regimes: (a) Ballistic $\tilde{\lambda} \gg 1$ (b) $\tilde{\lambda} = 10$ and (c) Diffusive $\tilde{\lambda} \ll 1$. Bottom: quasi-particle distribution function at $\tilde{x}$ = -1/2, -1/4, 0, 1/4 and 1/2 for the same regimes as above.[]{data-label="fig:schemes"}](./profiles.eps){width="4.5in"}
As mentioned above, systems in which the mean free path becomes comparable to the length are of particular interest because they exhibit features of both the ballistic (splitting of quasi-Fermi levels) and diffusive regime (thermal broadening). In Fig. \[fig:inter\_regime\], we show the quasi-particle distribution for the ratios $\tilde{\lambda} = 1$, 3 and 10 at the position $\tilde{x} = 0$ (left column) and -1/4 (right column). We use the symmetric boundary conditions $T_+(-L/2) =T_-( L/2) = T_0$ and bias voltages $eV_{ds}/k_BT_0$ of 0, 50 and 100. As the ratio $\tilde{\lambda}$ increases, the step in the distribution becomes more pronounced as the thermal broadening for forward and backward distribution decreases. At the wire mid-length ($\tilde{x} = 0$), the temperatures of forward and backward distributions have the same value, regardless of the $\tilde{\lambda}$ value. In contrast, the thermal imbalance between forward and backward populations at $\tilde{x} = -1/4$ becomes more prominent as $\tilde{\lambda}$ increases. At this location calculations show that the temperature of the forward population is smaller than that of the backward due to the proximity of the left lead where the carriers moving to the right are injected in the channel. As $-1/2\leq\tilde{x}\leq1/2$, of the forward (backward) carrier temperature (Eq.\[eq:tempprof\]) achieves its maximum value at $\tilde{x}=1/2$ ($\tilde{x}=-1/2$) if $\tilde{\lambda}>1$ or $\tilde{x} = \tilde{\lambda}/2$ ($\tilde{x} = -\tilde{\lambda}/2$) if $\tilde{\lambda}\leq1$. For $T_{0+} = T_{0-}=T_0$, the maximum carrier temperature is: $$T_{max} = \left\{ \begin{array}{ll}
\sqrt{T_0^2+\frac{V_{ds}^2\tilde{\lambda}}{\mathcal{L}(1+\tilde{\lambda} )^2}}& \textrm{if $\tilde{\lambda} > 1$}\\
\sqrt{T_0^2+\frac{V_{ds}^2}{4\mathcal{L}} }& \textrm{if $\tilde{\lambda} \leq 1$}
\end{array} \right. .$$ Hence, $T_{max}\gg T_0$ for large enough $V_{ds}$, even when $\tilde{\lambda} >1$. In particular, when $eV_{ds}\gg k_BT_0$ the carrier temperature profile (Eq.\[eq:tempprof\]) becomes: $$T_\pm(x) = \frac{|V_{ds}|}{1+\tilde{\lambda}} \sqrt{\frac{1}{\mathcal{L}}\left(1/2\pm
\tilde{x}\right)\left(1/2\mp \tilde{x}+\tilde{\lambda}\right)}.\label{eq:scaling_law1}$$ Consequently, as both the branch carrier thermal broadening (Eq. \[eq:scaling\_law1\]) and quasi-Fermi level separation (Eq. \[eq:Vds\]) scale linearly with the applied bias for $eV_{ds} \gg k_BT_0$, the local distribution function (Eq. \[eq:distfunc\]) can be expressed in terms of the reduced parameter $E/eV_{ds}$ as experimentally reported by Pothier et al. [@pothier1997].
We observe that the measurements of the electrical characteristics and the carrier distribution function complement each other in the determination the carrier mean free path in quantum wires. On the one hand, if the system is diffusive ($\tilde{\lambda}\ll 1$), the local thermal imbalance between forward and backward population vanishes (Eq. \[eq:therm\_imb\]), but the mean free path can be estimated by using Eq. \[eq:Vds\] given that most of the voltage drop occurs along the channel. On the other hand, when the mean free path becomes comparable to the length of the system, the determination of $\lambda$ from Eq. \[eq:Vds\], may not be straight forward due to the presence of spurious contact resistances. Nevertheless, the mean free path can be obtained from the forward and backward thermal broadening (Eq. \[eq:tempprof\]). Indeed, combining Eq. \[eq:tempprof\] for both populations with the symmetric boundary conditions ($T_{0+}=T_{0-}$), we obtain: $$\frac{\tilde{\lambda}}{(1+\tilde{\lambda})^2} = \tilde{x} \frac{\mathcal{L} \left[T_+(x)^2-T_-(x)^2\right]}{2 V_{ds}^2}, \label{eq:therm_imb}$$ where the $T_\pm(x)$ values can be easily extracted from the two step distribution function in the high bias regime ($eV_{ds} \gg k_BT_0$), as shown in Fig. \[fig:inter\_regime\].
![Carrier distribution function at the wire $x=0$ (top) and $x=L/4$ (bottom) for $\tilde{\lambda}$ ratios of 1, 3 and 10 and bias voltages $eV_{ds}/k_BT_0$=0, 50 and 100. \[fig:inter\_regime\]](./ctr_sd_distrib.eps){width="4.25in"}
As the carrier mean free path depends on both the quality of the quantum wires and the number of branches (subbands) involved in carrier transport, a reduction in the number of branches crossing the Fermi level causes a decrease in carrier scattering (and a mean free path increase). Therefore, metallic wires with larger aspect ratio are well suited for the observation of quasi-ballistic regime features ($\tilde{\lambda} \gtrsim 1$). However, the larger the aspect ratio, the more significant the effects of surface roughness as a source of scattering. In this regard, pristine carbon nanotubes and graphene nanoribbons are ideal candidates to confirm Eq. \[eq:therm\_imb\] due to their perfect surfaces and weak interaction with substrates. The extremely high Fermi velocity of carbon nanotubes [@saito1992] favors a large mean free path. Unfortunately it simultaneously discretizes the 1D states along the nanotube so that typical micrometer-long nanotubes exhibit 3D confinement as in quantum-dots [@bockrath1999; @chen2009]. Therefore centimeter long nanotubes are necessary to achieve a single degree of freedom as in quantum wires. These phenomena can also be observed in graphene nanoribbons with typical widths of a few tens of nanometers [@han2007] in field-effect-transistor like configuration. In this case, the Fermi level can be tuned by the back gate voltage to populate successive subconducting bands and change the scattering rate. Finally let us point out that in addition to a temperature lower bound imposed by the level spacing, there is an upper bound that limits measurements due to the critical temperature of the superconducting probe. Moreover, high temperatures not only reduce the mean free path of the system, but also favors the occurrence of dissipative collisions, which is beyond the scope of this work.
In conclusion, we have described a model for the non-coherent hot carrier transport in mesoscopic quantum wires in a situation intermediate between the ballistic and diffusive regimes. The model is in good agreement with experimental findings in mesoscopic copper wires; at the sane time, it provides a reinterpretation of the inhomogeneous thermal broadening in the local carrier distribution function and its scaling with external bias. It offers a direct way to extract the inter-branch carrier mean free path from the temperature imbalance in the non-equilibrium quasi-particle distribution functions. The conditions for the observation of this phenomena in 1D conductors such as nanotubes and graphene nanostructures, are also discussed.
Marcelo Kuroda acknowledges the support of the Department of Physics at University of Illinois at Urbana-Champaign. We thank Nadya Mason and Yung-Fu Chen for useful discussion.
References {#references .unnumbered}
==========
[^1]: This is a good approximation in metallic conductors regardless the number of bands as far as no bottom of any subconducting band lies close the Fermi level, while it is the realistic band structure of metallic carbon nanotubes
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We derive the general lagrangian and propagator for a vector-spinor field in $d$-dimensions and show that the physical observables are invariant under the so-called point transformation symmetry. Until now the symmetry has not been exploited in any non-trival way, presumably because it is not an invariance of the classical action nor is it a gauge symmetry. Nevertheless, we develop a technique for exploring the consequences of the symmetry leading to a conserved vector current and charge. The current and charge are identically zero in the free field case and only contribute in a background such as a electromagnetic or gravitational field. The current can couple spin-$\frac{3}{2}$ fields to vector and scalar fields and may have important consequences in intermediate energy hadron physics as well as linearized supergravity. The consistency problem which plagues higher spin field theories is then discussed and and some ideas regarding the possiblity of solutions are presented.'
author:
- |
Terry Pilling[^1]\
Department of Physics,\
North Dakota State University,\
Fargo, ND 58105-5566
title: 'Symmetry of massive Rarita-Schwinger fields'
---
Introduction
============
Theories of interacting high spin fields[^2] have been a subject of considerable interest for many years. This is partly due to the many particles with spin $\geq \frac{3}{2}$ seen in accelerator laboratories and also because there is currently no general field theory description which is relativistic, interacting and also free of inconsistencies[^3]. Over the years one interacting theory after another have been shown to be inconsistent leading many to suggest that all higher spin fields must be composite. On the other hand, higher spin elementary particles, such as the gravitino, play an important role in supersymmetry, which itself represents a fundamental building block of many modern unification schemes. Thus we would like to remain hopeful that a solution to the consistency problems can be found within point particle field theory. Perhaps our interpretation regarding the physical degrees of freedom is misguided or perhaps, as is our present concern, we have neglected symmetries or other aspects of nature that should be included. The hope being that if all of the symmetries are properly included, the result will be a consistent theory. That this hope is reasonable is exemplified by the fact that consistent solutions have already been found in restricted scenarios with curved backgrounds, cosmological constant tuning and Planck scale masses [@madore1975; @deser1977; @rindani1986; @rindani1991; @deser2001; @deser2001-b].
The consistency problems seem to exist for most interacting higher spin field theories and are a main concern of many theorists working in the field and so we will devote the final section of this paper to a discussion of the problem and touch on some possible consequences of the symmetry. Perhaps the ideas that we present will inspire some new angles of attack on the problem.
The main goal of this paper is a review of the $d$-dimensional theory of interacting Rarita-Schwinger fields and an exploration of the symmetries. Our hope is that the general expressions and the new interactions that we present here will be of use in formulating effective theories of interacting hadrons as well as work involving the massive gravitinos of supergravity such as, for example, the AdS/CFT correspondence [@volovich1998; @rashkov1999; @koshelev1998; @matlock1999]. Perhaps the AdS/CFT results can be extended to the case wherein the Rarita-Schwinger fields are not fixed at the start as non-interacting and onshell[^4].
We begin in sections \[spincontent\] and \[conditions\] by (re-)deriving the most general lagrangian and propagator for a Rarita-Schwinger spin-$\frac{3}{2}$ field[^5] using the method of Aurilia and Umezawa [@aurilia1969] extended to $d$-dimensions.
Since we are using the vector-spinor representation of spin-$\frac{3}{2}$, we find the usual result that a lower spin content is retained in the field in order to maintain the desirable properties of the action such as hermiticity, linearity in derivatives and non-singular behavior. However, recently there have been other promising ideas where the lower spin content is given a physical interpretation [@kaloshin2004] or where vector spinor description of spin-$\frac{3}{2}$ is replaced by a pure spin-$\frac{3}{2}$ field [@kirchbach2001; @ahluwalia1992; @ahluwalia1993; @kirchbach2002].
Considering the lower spin components as unphysical, as we do here, leads to a non-unique action depending on an arbitrary complex parameter measuring the lower spin content of the theory. Various choices of the parameter are seen to reduce the general expression to the spin-$\frac{3}{2}$ actions found in the literature. We formulate the equations in $d$ spacetime dimensions in anticipation of diverse applications from effective theories of hadronic interactions involving the spin-$\frac{3}{2}$ baryons to applications in arbitrary dimensional supergravity theories. For example, both the composite $\Delta(1232)$ resonance found in low and intermediate energy nucleon scattering experiments and the gravitino of N-extended supersymmetry after spontaneous symmetry breaking are thought to be described by the massive, spin-$\frac{3}{2}$, Rarita-Schwinger field that we study here.
In section \[group\] we examine the properties of the so-called ‘point’ or ‘contact’ transformations. These form a non-unitary group of transformations of the fields which shifts the parameter, amounting to a sort of rotation among the spin-$\frac{1}{2}$ degrees of freedom. The path integral is seen to be invariant under point transformations which implies that physical correlation functions are invariant under a redefinition of the arbitrary parameter. That the parameter is arbitrary is well known and this has caused many authors to simply fix it to a convenient value. Unfortunately, this has served to hide some of the freedom of the theory. We restore the explicit parameter dependence and, in sections \[interactions\] and \[implications\], we derive and examine new conserved currents resulting from the the symmetry.
Finally, in section \[consistency\] we discuss the consistency problems and point out a few ideas of how the conserved charge found in section \[implications\] might be useful in that context. The analysis we have used should also be generalizable to higher spins as well whenever the theory contains auxiliary fields of lower spin and has a similar symmetry group involving them.
Spin content of the Rarita-Schwinger field {#spincontent}
==========================================
In this section we give a decomposition of the Rarita-Schwinger field into separate spin blocks and derive some general formulas and identities that will be needed later. The result of this and the following section is the expression for the most general free lagrangian. The reader only interested in the result may want to turn immediately to equation (\[action1\]) or (\[Action1\]) below.
A commonly used formulation of the spin-$\frac{3}{2}$ field is the vector-spinor representation[^6] given by Rarita and Schwinger in 1941 [@rarita1941]. The vector-spinor transforms under the Lorentz group as[^7] $$\label{spindecomp}
\left(\frac{1}{2},\frac{1}{2}\right) \otimes \left[ \left(\frac{1}{2}, 0
\right) \oplus \left( 0, \frac{1}{2} \right) \right]
= \left(1, \frac{1}{2}\right) \oplus \left(\frac{1}{2}, 1 \right)
\oplus \left( 0, \frac{1}{2} \right) \oplus \left(\frac{1}{2}, 0 \right)$$ whereas the spin decomposition of the field [*in the rest frame*]{} [@kirchbach2002; @kaloshin2004] is $$\text{spin } \psi_\mu^A = \left( 1 + 0 \right) \otimes \frac{1}{2}
= \frac{3}{2} + \frac{1}{2} + \frac{1}{2}.$$ The vector-spinor field thus contains two spin-$\frac{1}{2}$ components in addition to the physical spin-$\frac{3}{2}$ component. The decomposition of the spin-$\frac{3}{2}$ field that we will use is given by choosing $\left( 0, \frac{1}{2} \right) = p_\mu \psi^\mu$, where $p_\mu = i \partial_\mu$. The complimentary part is $$\left(1, \frac{1}{2}\right) = \left( g_{\mu \nu} - \frac{p_\mu p_\nu}{p^2} \right) \psi^\nu,$$ which can then be written in terms of spin-$\frac{3}{2}$ and spin-$\frac{1}{2}$ projectors as $$g_{\mu \nu} - \frac{p_\mu p_\nu}{p^2}
= \left(P^{\frac{3}{2}}\right)_{\mu \nu} + \left(P^{\frac{1}{2}}_{11}\right)_{\mu \nu}.$$ Defining $\left(P^{\frac{1}{2}}_{22}\right)_{\mu \nu}
= \frac{p_\mu p_\nu}{p^2}$ we have an expansion of the identity $$\label{expanse1}
g_{\mu \nu} = \left(P^{\frac{3}{2}}\right)_{\mu \nu}
+ \left(P^{\frac{1}{2}}_{11}\right)_{\mu \nu}
+ \left(P^{\frac{1}{2}}_{22}\right)_{\mu \nu}.$$ Explicit expressions for the projectors can easily be found by contraction with $\gamma^\mu$ and $p^\mu$, but we will anticipate further applications[^8] and generalize to $d$-dimensions.
We define $d$-dimensional spin projection operators by requiring that they reduce properly to the usual 4 dimensional projection operators [@van1981; @benmerrouche1989; @bernard2003] and yet remain projections in $d$-dimensions satisfying the following orthogonality relations, $$\label{orthog}
\left( P^I_{ij} \right)_{\mu \nu} \left( P^J_{kl} \right)^{\nu
\rho} = \delta^{IJ} \delta_{jk} \left( P^I_{il}
\right)^{\rho}_{\mu}, \quad I, J \in \{ 1/2,3/2 \}, \quad i,j,k,l \in \{1,2\}.$$ The result is $$\label{projectors}
\begin{split}
\left( P^{\frac{3}{2}} \right)_{\mu \nu} &= \frac{1}{p^2 \left(d -
1\right)} \left[ \left(d-1 \right) p^2 g_{\mu \nu} -
\left(d-2\right) p_\mu p_\nu - {\not \negthinspace}{p}
\left( \gamma_\mu p_\nu - p_\mu \gamma_\nu \right) - \gamma_\mu
\gamma_\nu p^2 \right] \\
\left( P^{\frac{1}{2}} \right)_{\mu \nu} &= \frac{1}{p^2 \left(d -
1\right)} \left[ \left(d-2 \right) p_\mu p_\nu + {\not \negthinspace}{p}
\left( \gamma_\mu p_\nu - p_\mu \gamma_\nu \right) + \gamma_\mu
\gamma_\nu p^2 \right] = P^{\frac{1}{2}}_{11} + P^{\frac{1}{2}}_{22} \\
\left( P^{\frac{1}{2}}_{11} \right)_{\mu \nu} &= \frac{1}{p^2 \left(d -
1\right)} \left[ - p_\mu p_\nu + {\not \negthinspace}{p}
\left( \gamma_\mu p_\nu - p_\mu \gamma_\nu \right) + \gamma_\mu
\gamma_\nu p^2 \right] \\
\left( P^{\frac{1}{2}}_{22} \right)_{\mu \nu} &=
\frac{p_\mu p_\nu}{p^2} \\
\left( P^{\frac{1}{2}}_{12} \right)_{\mu \nu} &= \frac{1}{p^2 \sqrt{d -
1}} \left[ p_\mu p_\nu - {\not \negthinspace}{p} \gamma_\mu p_\nu \right] \\
\left( P^{\frac{1}{2}}_{21} \right)_{\mu \nu} &= \frac{1}{p^2 \sqrt{d -
1}} \left[ {\not \negthinspace}{p} p_\mu \gamma_\nu - p_\mu p_\nu \right].
\end{split}$$ From the definitions we see that the total spin-$\frac{1}{2}$ projection operator, $P^{\frac{1}{2}}$, reduces to the sum of the individual projection operators for the two different spin-$\frac{1}{2}$ components of the Rarita-Schwinger field. We also note the following convenient relations, $$\label{relations}
\begin{split}
\left[ {\not \negthinspace}{p}, \left( P^{\frac{1}{2}}_{JJ} \right)_{\mu \nu}
\right] &= \left[ {\not \negthinspace}{p}, \left( P^{\frac{3}{2}} \right)_{\mu \nu}
\right] = \left\{ {\not \negthinspace}{p}, \left( P^{\frac{1}{2}}_{12} \right)_{\mu \nu}
\right\} = \left\{ {\not \negthinspace}{p}, \left( P^{\frac{1}{2}}_{21} \right)_{\mu \nu}
\right\} = 0, \\
\gamma^\mu \left( P^{\frac{3}{2}}\right)_{\mu \nu} &=
\left( P^{\frac{3}{2}}\right)_{\mu \nu} \gamma^\nu =
p^\mu \left( P^{\frac{1}{2}}_{12}\right)_{\mu \nu} =
\left( P^{\frac{1}{2}}_{21}\right)_{\mu \nu} p^\nu = 0, \\
\gamma^\mu \left( P^{\frac{1}{2}}\right)_{\mu \nu} &=
\left( P^{\frac{1}{2}}\right)_{\nu \mu} \gamma^\mu = \gamma_\nu, \\
\gamma^\mu \left( P^{\frac{1}{2}}_{11}\right)_{\mu \nu} &=
\left( P^{\frac{1}{2}}_{11}\right)_{\nu \mu} \gamma^\mu =
\gamma_\nu - \frac{{\not \negthinspace}{p} p_\nu}{p^2}, \\
\gamma^\mu \left( P^{\frac{1}{2}}_{22}\right)_{\mu \nu} &=
\left( P^{\frac{1}{2}}_{22}\right)_{\nu \mu} \gamma^\mu =
\frac{{\not \negthinspace}{p} p_\nu}{p^2}.
\end{split}$$ These projection operators can now be used to derive the most general vector-spinor lagrangian and propagator in a flat[^9] background spacetime.
The free spin-$\frac{3}{2}$ lagrangian {#conditions}
======================================
The free lagrangian for the spin-$\frac{3}{2}$ field can be written as $$\label{freelagrangian}
\mathcal{L} = {{\overline{\psi}}}^\alpha \Lambda_{\alpha \beta} \psi^\beta,$$ where $\Lambda_{\alpha \beta}$ is an operator and $\psi^\beta$ is a vector-spinor field with suppressed spin index[^10]. Using the projectors given in the previous section we will construct the most general operator $\Lambda_{\alpha \beta}$ subject to the following four conditions [@rarita1941; @aurilia1969; @moldauer1956; @nath1971]:
1. [The Euler-Lagrange equations derived from the free action should give the local Rarita-Schwinger equations for a spin-$\frac{3}{2}$ particle. These are a Dirac equation for each of the vector components as well as supplementary conditions to remove the lower spin degrees of freedom: $$\label{raritaschwinger}
\begin{split}
\left( i {\not \negthinspace}{\partial} - m \right) \psi^\mu &= 0, \\
\gamma_\mu \psi^\mu &= 0.
\end{split}$$ ]{}
2. [The lagrangian should be non-singular in the limit $p \rightarrow 0$. In particular, we would like the pole of the propagator to occur at the mass of the particle.]{}
3. [The lagrangian should be linear in derivatives as it describes a fermionic field.]{}
4. [The operator $\gamma^0 \Lambda_{\alpha \beta}$ should be hermitian: $$\label{hermitian}
\gamma^0 \left(\Lambda_{\alpha \beta}\right)^\dagger \gamma^0 = \Lambda_{\beta \alpha}.$$ ]{}
A consequence of equation (\[raritaschwinger\]) in condition 1 is the condition $$\partial_\mu \psi^\mu = 0,$$ as can be seen by multiplying the first equation in (\[raritaschwinger\]) on the left by $\gamma_\mu$ and using the second equation. The condition $\gamma_\mu^{AB} \psi^\mu_{B} = 0$ (where we now explicitly write the spinor indices $A,B$) represents a constraint equation for each value of the spin index $A$ whereas the condition $\partial_\mu \psi^\mu_B = 0$ is an equation of motion for the spinor components $\psi^0_B$. However, the Dirac equation (\[raritaschwinger\]) also gives an equation of motion for the same spinor components and when taken together, these result in another set of constraints. In four spacetime dimensions, these two sets of equations each constitute four constraints[^11] and serve to remove 8 components of the 16 component vector-spinor $\psi^\mu_A$, leaving $2(2 s + 1) = 8$ physical degrees of freedom as required for a massive spin $s = \frac{3}{2}$ particle[^12].
The most general expression for the operator $\Lambda_{\alpha \beta}$ which obeys condition 1 is given by a combination of a spin-$\frac{3}{2}$ part plus an arbitrary amount of spin-$\frac{1}{2}$, $$\label{operator}
\Lambda_{\alpha \beta} = \left( {\not \negthinspace}{p} - m \right)
P^{\frac{3}{2}}_{\alpha \beta}
+ \frac{a_1}{d} m \left( P^{\frac{1}{2}}_{11}\right)_{\alpha \beta}
+ \frac{a_2}{d} m \left( P^{\frac{1}{2}}_{22}\right)_{\alpha \beta},$$ where we have included factors of $m$ and $1/d$ in the two spin-$\frac{1}{2}$ terms for later convenience. The quantities $P^{\frac{3}{2}}_{\alpha \beta}$, $P^{\frac{1}{2}}_{11}$ and $P^{\frac{1}{2}}_{22}$ are the projection operators defined previously which, respectively, project onto the spin-$\frac{3}{2}$ part and the two spin-$\frac{1}{2}$ parts of $\psi^\beta$. The Euler-Lagrange equations $\Lambda_{\alpha \beta} \psi^\beta =
0$ separate because of these projection operators into the Rarita-Schwinger equation plus supplementary conditions as required by condition 1. The equation also contains two complex constants $a_1$ and $a_2$ which are arbitrary at the moment. One can verify, using the projectors (\[projectors\]) that the operator (\[operator\]) satisfies condition 1 for any choice of these constants and so we are free to fix them to whatever values are convenient. We will fix them by requiring the lagrangian to satisfy conditions 2 and 3.
Before proceeding with the other conditions let us take a moment to examine the general action given by (\[operator\]). It contains possible gauge invariances which are already apparent. The field equations are seen to be $$\label{fldeqn}
\left( {\not \negthinspace}{p} - m \right)
P^{\frac{3}{2}}_{\alpha \beta} \psi^\beta
+ \frac{a_1}{d} m \left( P^{\frac{1}{2}}_{11}\right)_{\alpha \beta} \psi^\beta
+ \frac{a_2}{d} m \left( P^{\frac{1}{2}}_{22}\right)_{\alpha \beta} \psi^\beta =
0$$ and using (\[orthog\]) and (\[relations\]) we notice that, if we could set $a_1 = a_2 = 0$, the equation would be invariant under the two variations $\delta \psi^\beta = \left(P^{\frac{1}{2}}_{IJ}\right)^{\beta \lambda}
\chi_\lambda$ for arbitrary spinor $\chi_\lambda$. However, we will see that the remaining conditions require the parameters $a_1$ and $a_2$ to be related to each other in such a way that the vanishing of one implies the other is non-zero and so both parameters cannot be set to zero at the same time. If we set one of them to zero, i.e. $a_I \rightarrow 0$ for $I \in 1,2$, the equations of motion become invariant under $\delta \psi^\beta = \left(P^{\frac{1}{2}}_{II}\right)^{\beta \lambda}
\chi_\lambda$. For $a_I = a_2$ this is the same as $\delta \psi^\beta = \partial^\beta \epsilon$ as can be seen by inserting the explicit form of the projector from (\[projectors\]). We will see that our remaining conditions break this symmetry such that the gauge invariance is lost except in the massless limit (see equation \[gaugeinvariance\]). We will examine the lagrangian written in terms of projection operators again at the end of this section after we have imposed the remaining conditions 2 – 4.
Returning to our conditions, we see that the operator (\[operator\]) as it is written, does not obey condition 2 since it is singular in the limit $p \rightarrow 0$. To remedy this, we use the method of Aurilia and Umezawa [@aurilia1969] and shift the spin-$\frac{1}{2}$ components to form a new operator as $$\label{newoperator}
\tilde{\Lambda}_{\alpha \beta} = \left( \eta_2 \eta_1
\right)_{\alpha}^{\lambda} \Lambda_{\lambda \beta},$$ with $\eta_1$ and $\eta_2$ given by $$\label{transforms}
\begin{split}
\eta_1^{\mu \nu} &= g^{\mu \nu} + \sqrt{d-1}\left[ \frac{g_1}{m} {\not \negthinspace}{p} +
g_2 \right] \left(P^{\frac{1}{2}}_{12}\right)^{\mu \nu}, \\
\eta_2^{\mu \nu} &= g^{\mu \nu} + \sqrt{d-1}\left[ \frac{f_1}{m} {\not \negthinspace}{p} +
f_2 \right] \left(P^{\frac{1}{2}}_{21}\right)^{\mu \nu}.
\end{split}$$ The $f_1, g_1, f_2, g_2$ are new constants that we will fix by requiring the singular terms to vanish. The transformation (\[newoperator\]) is allowed since we are only altering the coefficients of the separate spin-$\frac{1}{2}$ projection operators in (\[operator\]) and thus the equations of motion will still separate properly and condition 1 is still obeyed.
Substituting (\[transforms\]) into (\[newoperator\]) and making use of the relations (\[relations\]) we find that the singular terms will vanish if the constants $f_1, g_1, f_2,
g_2, a_1$ and $a_2$ satisfy the following relations, $$\label{params}
\begin{split}
\frac{a_2}{d} &= - \frac{d}{(d-1)} \left(1 + \frac{1}{a_1}
\right), \\
f_2 &= \frac{a_1 + d}{(d-1) a_1} = -\left(1 +
\frac{a_2}{d}\right), \\
g_2 &= \frac{a_1 + d}{(d-1) a_2} =
- \frac{a_1}{d} \left( 1 + \frac{d}{a_2} \right), \\
a_1 f_1 &= a_2 g_1 + \frac{(d-2)a_1}{(d-1)}.
\end{split}$$ The resulting non-singular operator is then $$\label{nonsingularoperator}
\begin{split}
\tilde{\Lambda}^{\alpha \beta} = \left( {\not \negthinspace}{p} - m \right)
g^{\alpha \beta} &- \left[ \frac{a_1}{d} f_1 + \frac{1}{(d-1)} \right]
\left( \gamma^\alpha p^\beta - p^\alpha \gamma^\beta \right) \\
&+ \frac{(d-2)}{(d-1)} \frac{a_1}{d} \gamma^\alpha p^\beta
- \frac{1}{(d-1)} {\not \negthinspace}{p} \gamma^\alpha \gamma^\beta -
(d-1)\frac{a_2}{d} f_1 g_1
\frac{p^\alpha p^\beta}{m}.
\end{split}$$ This operator now satisfies conditions 1 and 2, and it is written in terms of two parameters, $f_1$ and $a_1$ (since the relations (\[params\]) can be used to write $a_2$ and $g_1$ in terms of $a_1$ and $f_1$).
We would like the field $\psi^\mu$ and its hermitian conjugate to appear symmetrically in the lagrangian, so that the variation of $\psi^\mu$ and that of ${{\overline{\psi}}}^\mu$ both give the field equations. To achieve this, we apply another shift to (\[nonsingularoperator\]) via the transformation[^13] $$\label{pointtrans}
\theta^{\mu \nu}(k) = g^{\mu \nu} + \frac{k}{d} \gamma^\mu \gamma^\nu.$$ Again, we see that the equations of motion remain unaffected (since $\gamma^\nu \psi_\nu = 0$ onshell) and thus the resulting operator will still satisfy conditions 1 and 2. The purpose of this transformation is that now our operator contains a new constant, $k$, which we can use in satisfying the other conditions. Our general operator is then $$\label{blah}
\begin{split}
&\Lambda^{\alpha \beta} = \theta^{\alpha \lambda}(k)
\tilde{\Lambda}_{\lambda}^{\beta} \\
&= \left( {\not \negthinspace}{p} - m \right) g^{\alpha \beta}
- \left[ \frac{a_2}{d} g_1 + \frac{1}{(d-1)} \right]
\left( \gamma^\alpha p^\beta - p^\alpha \gamma^\beta \right)
- \left[ \frac{k a_1 f_1}{d^2} + \frac{1}{(d-1)} \right]
{\not \negthinspace}{p} \gamma^\alpha \gamma^\beta \\
&+ \left[ 1 + \frac{a_1}{d} + k \left(
\frac{1}{d} + \frac{a_1}{d} \right) \right] \frac{m \gamma^\alpha
\gamma^\beta}{(d-1)} - \frac{(d-1) a_2 f_1 g_1}{m d} \left[ \left(
1 + \frac{2 k}{d} \right) p^\alpha p^\beta -
\frac{k}{d} {\not \negthinspace}{p} \gamma^\alpha p^\beta \right] \\
&+ \frac{k}{d} \left[ \frac{(d-2)}{(d-1)} - a_2 g_1 \right]
\gamma^\alpha p^\beta + \left[ \frac{2k}{d^2} a_1 f_1 +
\frac{(d-2)}{d (d-1)} a_1 \right] p^\alpha \gamma^\beta.
\end{split}$$
Condition 3 requires the lagrangian to be linear in derivatives. Inspection of (\[blah\]) reveals that the terms quadratic in $p^\alpha$ can be explicitly cancelled by setting $g_1 = 0$ or $f_1 = 0$. This gives two paths to follow: setting $g_1 = 0$ gives $$\label{op1}
\begin{split}
\Lambda^{\alpha \beta}_1 &= \left( {\not \negthinspace}{p} - m \right) g^{\alpha \beta}
- \frac{1}{(d-1)} \left( \gamma^\alpha p^\beta - p^\alpha \gamma^\beta \right)
- \frac{1}{(d-1)} \left[ \frac{k a_1 (d-2)}{d^2} + 1 \right]
{\not \negthinspace}{p} \gamma^\alpha \gamma^\beta \\
&+ \left[ 1 + \frac{a_1}{d} + \frac{k}{d} \left( 1 + a_1 \right) \right]
\frac{m \gamma^\alpha \gamma^\beta}{(d-1)}
+ \frac{k(d-2)}{d(d-1)} \gamma^\alpha p^\beta
+ \frac{a_1(d-2)}{d(d-1)}\left[ \frac{2k}{d} + 1 \right] p^\alpha \gamma^\beta,
\end{split}$$ whereas setting $f_1 = 0$ gives $$\label{op2}
\begin{split}
\Lambda^{\alpha \beta}_2 = \left( {\not \negthinspace}{p} - m \right) g^{\alpha \beta}
&- \frac{1}{(d-1)} \left( \gamma^\alpha p^\beta - p^\alpha \gamma^\beta \right)
- \frac{1}{(d-1)} {\not \negthinspace}{p} \gamma^\alpha \gamma^\beta \\
&+ \left[ \frac{{{\overline{a}}}_1}{d} + 1 \right] \frac{m \gamma^\alpha \gamma^\beta}{(d-1)}
+ \frac{{{\overline{a}}}_1 (d-2)}{d(d-1)} \gamma^\alpha p^\beta.
\end{split}$$ where we have defined ${{\overline{a}}}_1 = a_1 + k \left( 1 + a_1 \right)$.
We now have two operators, the first (\[op1\]) depending on two parameters, $a_1$ and $k$, and the second (\[op2\]) depending on only one, ${{\overline{a}}}_1$. Both operators satisfy conditions 1 through 3. The last condition to impose on our operators is condition 4, i.e. that the operators be hermitian. Imposing the hermitian requirement (\[hermitian\]) on the first operator (\[op1\]) fixes a relation between $a_1$ and $k$ $$\label{cond1}
a_1^* = k$$ thus reducing the number of parameters to one. The lagrangian (\[op1\]) is then $$\label{lagrangian}
\begin{split}
\Lambda^{\alpha \beta}_1 = \left( {\not \negthinspace}{p} - m \right)& g^{\alpha \beta}
- \frac{1}{(d-1)} \left( \gamma^\alpha p^\beta + p^\alpha \gamma^\beta \right)
+ \frac{(d-2)}{d^2(d-1)} \left[ |a_1|^2 + \frac{d^2}{(d-2)} \right]
\gamma^\alpha {\not \negthinspace}{p} \gamma^\beta \\
&+ \left[ d + a_1 + a_1^* + |a_1|^2 \right]
\frac{m \gamma^\alpha \gamma^\beta}{d(d-1)}
+ \frac{(d-2)}{d(d-1)} \left( a_1^* \gamma^\alpha p^\beta
+ a_1 p^\alpha \gamma^\beta \right).
\end{split}$$ If we were to restrict $a_1$ to be real, and define the number $A$ in terms of $a_1$ as $$\label{defna}
a_1^* = a_1 = \frac{d (d-1)}{d-2}A + \frac{d}{d-2}$$ then $\Lambda^{\alpha \beta}_1$ would become $$\label{benmer}
\begin{split}
\Lambda^{\alpha \beta}_{1A} = \left( {\not \negthinspace}{p} - m \right) g^{\alpha \beta}
&+ A \left( \gamma^\alpha p^\beta + p^\alpha \gamma^\beta \right)
+ \frac{1}{(d-2)} \left[ (d-1)A^2 + 2A + 1\right]
\gamma^\alpha {\not \negthinspace}{p} \gamma^\beta \\
&+ \frac{m \gamma^\alpha \gamma^\beta}{(d-2)^2}
\left[ d(d-1)A^2 + 4(d-1)A + d \right],
\end{split}$$ which is the $d$-dimensional form of a common expression found in the literature [@benmerrouche1989; @moldauer1956; @haberzettl1998; @munczek1967].
Imposing the hermitian requirement on our second operator (\[op2\]), gives the condition $$\label{cond2}
\begin{split}
{{\overline{a}}}_1 &= 0 \\
\Rightarrow a_1 &= \frac{-k}{k+1},
\end{split}$$ and the second lagrangian (\[op2\]) becomes $$\label{rarita}
\Lambda^{\alpha \beta}_2 = \left( {\not \negthinspace}{p} - m \right) g^{\alpha \beta}
- \frac{1}{(d-1)} \left( \gamma^\alpha p^\beta + p^\alpha \gamma^\beta \right)
+ \frac{1}{(d-1)} \gamma^\alpha \left({\not \negthinspace}{p} + m \right) \gamma^\beta
\equiv \Lambda^{\alpha \beta}_{\text{RS}}$$ which is the $d$-dimensional Rarita-Schwinger lagrangian [@rarita1941] and has no arbitrary parameters.
We have thus found the most general set of $d$-dimensional lagrangians for a spin-$\frac{3}{2}$ field which satisfy our four conditions. These are given by (\[lagrangian\]) which depends on a complex parameter, and (\[rarita\]) which is the usual Rarita-Schwinger lagrangian and has no arbitrary parameters. In fact, the second lagrangian (\[rarita\]) corresponds to the case $a_1
\rightarrow 0$ of the first lagrangian (\[lagrangian\]). So the most general expression for the set of operators is given by the single expression (\[lagrangian\]) with the Rarita-Schwinger operator corresponding to the particular value $a_1 = 0$ of the arbitrary parameter. It will be shown in section \[group\] that this general operator can be written in a very simple manner as a transformation of the Rarita-Schwinger lagrangian with $a_1$ being the parameter of the transformation – a fact already noticed by Freedman and van Nieuwenhuizen [@freedman1976] in 1976.
The propagator, $S^{\alpha \beta}$, for the spin-$\frac{3}{2}$ field is the inverse of the quadratic operator in the lagrangian $$\label{invexpr}
S_{\alpha \beta} \Lambda^{\beta \lambda} = \delta_\alpha^\lambda \; .$$ The definition of the projection operators (\[projectors\]) gives the equation $$\left( p^2 - m^2 \right) S_{\alpha \beta} =
\left({\not \negthinspace}{p} + m \right) P^{\frac{3}{2}}_{\alpha \beta}
+ \frac{ \left(p^2 - m^2 \right) d}{m} \left[
\frac{\left(P^{\frac{1}{2}}_{11}\right)_{\alpha \beta}}{a_1}
+ \frac{\left(P^{\frac{1}{2}}_{22}\right)_{\alpha \beta}}{a_2} \right],$$ which can be solved in a similar way as we have done for the lagrangian, but we will skip this lengthy calculation and simply quote the result (\[propagator1\]) below.
To summarize, the most general lagrangian for the spin-$\frac{3}{2}$ field consistent with our four conditions[^14] is $$\label{action1}
\begin{split}
&\mathcal{L}
= {{\overline{\psi}}}^\alpha \Biggl\{
\left( {\not \negthinspace}{p} - m \right) g_{\alpha \beta}
- \frac{1}{(d-1)} \left( \gamma_\alpha p_\beta + p_\alpha \gamma_\beta \right)
+ \frac{1}{(d-1)} \gamma_\alpha \left({\not \negthinspace}{p} + m \right) \gamma_\beta \\
+ &\frac{(d-2)}{d^2(d-1)} |a|^2
\gamma_\alpha {\not \negthinspace}{p} \gamma_\beta + \left[a + a^* + |a|^2 \right]
\frac{m \gamma_\alpha \gamma_\beta}{d(d-1)}
+ \frac{(d-2)}{d(d-1)} \left( a^* \gamma_\alpha p_\beta
+ a p_\alpha \gamma_\beta \right) \Biggr\} \psi^\beta, \\
&\quad = {{\overline{\psi}}}^\alpha \theta_{\alpha \mu}(a^*)
\; \Lambda^{\mu \nu}_{\text{RS}} \; \theta_{\nu \beta}(a) \psi^\beta,
\end{split}$$ where $a$ is a complex parameter with the restriction $a \neq -1$. Notice that we have dropped the index on $a_1$ and will henceforth write it as simply $a$ since only this single parameter remains. The last line in (\[action1\]) is a more compact way of writing the expression as will be shown in the next section. The propagator is $$\label{propagator1}
\begin{split}
S^{\alpha \beta} =
&\frac{1}{\left({\not \negthinspace}{p} - m \right)} \left[ g^{\alpha \beta}
- \frac{\gamma^\alpha \gamma^\beta}{(d-1)}
- \frac{\left(\gamma^\alpha p^\beta - p^\alpha \gamma^\beta \right)}{(d-1)m}
- \frac{(d-2)}{(d-1)} \frac{p^\alpha p^\beta}{m^2} \right] \\
+ &\frac{(d-2)}{m^2(d-1)} \left(
h^* \gamma^\alpha p^\beta + h p^\alpha
\gamma^\beta - |h|^2 \gamma^\alpha {\not \negthinspace}{p}
\gamma^\beta \right)
+ \frac{|h|^2 d - h - h^*}{(d-2)m} \gamma^\alpha \gamma^\beta, \\
= &\; \theta_{\alpha}^{\; \mu}(\tilde{a}) \; S_{\mu \nu}^{RS} \; \theta^{\nu}_{\;
\beta}(\tilde{a}^*),
\end{split}$$ where $h = \frac{d + a^*}{d(1+a^*)}$. Again, the last line in (\[propagator1\]) is a more compact way of writing the expression as will be shown in the next section. The inverse parameter $\tilde{a}^*$ is related to $h$ in our expression as $\tilde{a}^* = \frac{d(1-h)}{1-d}$. With real parameter $A$ defined by (\[defna\]) we get a propagator often used [@benmerrouche1989; @moldauer1956; @nath1971; @haberzettl1998; @amiri1992], $$\begin{split}
S^{\alpha \beta} =
\frac{\left({\not \negthinspace}{p} + m \right)}{\left(p^2 - m^2 \right)} &\left[ g^{\alpha \beta}
- \frac{1}{(d-1)} \gamma^\alpha \gamma^\beta - \frac{1}{(d-1)m}
\left(\gamma^\alpha p^\beta - p^\alpha \gamma^\beta \right)
- \frac{(d-2)}{(d-1)} \frac{p^\alpha p^\beta}{m^2} \right] \\
&+ \frac{A + 1}{m^2(Ad + 2)} \Biggl\{
\left[ \frac{d-4 - d A}{(d-2)(d A + 2)}\right]
m \gamma^\alpha \gamma^\beta \\
&\quad + \frac{(d-2)}{(d-1)} \left(\gamma^\alpha p^\beta + p^\alpha
\gamma^\beta \right)
- \frac{(d-2)(A + 1)}{(d-1) (dA + 2)} \gamma^\alpha {\not \negthinspace}{p} \gamma^\beta
\Biggr\},
\end{split}$$ and setting $A = -1$ gives another common expression [@bernard2003; @dejong1992; @pascalutsa1999; @2pascalutsa2003; @pascalutsa2003]. The Rarita-Schwinger lagrangian and propagator [@rarita1941] correspond to the limits $a \rightarrow 0$ and $h \rightarrow 1$ whereas those of [@van1981; @bernard2003; @dejong1992; @pascalutsa1999; @pascalutsa2003] correspond to $a = -d$ (or $h = 0$). The latter expression is often mistakenly cited as the original Rarita-Schwinger action in the supergravity literature.
Let us now take a moment to discuss the meaning of the arbitrary parameter $a$. The vector-spinor field contains auxiliary lower spin components which are necessary in maintaining Lorentz covariance in our formalism. This is a generic feature of the treatment of higher spin fields in relativistic lagrangian field theory with constraints. In order to get a theory which satisfies our general conditions for the lagrangian, we introduced several arbitrary parameters. These parameters were not fully fixed by our conditions and instead we were left with one parameter, $a$, remaining unfixed. We can see in (\[operator\]) that $a_1$ and $a_2$ measure the relative strengths of the $P_{11}$ and $P_{22}$ parts of the field. Since $a_1$ and $a_2$ are inversely related by (\[params\]) increasing $a$ corresponds to increasing the $P_{11}$ part and decreasing the $P_{22}$ part of the operator. Thus the parameter $a$ is a measure of the proportion of the two different auxiliary spin-$\frac{1}{2}$ components of the theory. The general conditions which we used in formulating the theory has resulted in this proportionality and transforming the parameter is equivalent to changing this proportionality. There is no choice of parameter which will eliminate both spin-$\frac{1}{2}$ components simultaneously and hence our conditions have forced us to retain lower spin components in our theory, the ‘amount’ of each depending on the choice of parameter.
We can now return to the subject of gauge invariances that we touched upon previously and see how the invariances of equation (\[operator\]) have been modified by our conditions. The projection operators can be used to re-write the general lagrangian operator as $$\label{gaugeinvariance}
\begin{split}
\Lambda_{\alpha \beta} &= \left( {\not \negthinspace}{p} - m \right) P^{\frac{3}{2}}_{\alpha \beta} \\
&+ \left[ \frac{(d-2)}{(d-1)} |a|^2 \left( {\not \negthinspace}{p} -
m \right) - m \biggl( a^* b + b^* a \biggr)
\right] \frac{(d-1)}{d^2} \left( P^{\frac{1}{2}}_{11}\right)_{\alpha \beta} \\
&+ \left[ (d-2) a^* b \left( {\not \negthinspace}{p}
- m \right) - m \left( \frac{a^* (b+d)}{d-1} - |b|^2 \right)
\right] \frac{(d-1)^\frac{3}{2}}{d^2} \left( P^{\frac{1}{2}}_{12}\right)_{\alpha \beta} \\
&+ \left[(d-2) |b|^2 \left( {\not \negthinspace}{p} + m
\right) + m \biggl( (b^* + d)b + b^* (b+d) \biggr)
\right] \frac{(d-1)}{d^2} \left( P^{\frac{1}{2}}_{22}\right)_{\alpha \beta} \\
&- \left[ (d-2) a b^* \left(
{\not \negthinspace}{p} + m \right) + m \left( \frac{(b^*+d) a}{d-1} - |b|^2 \right)
\right] \frac{(d-1)^\frac{3}{2}}{d^2} \left( P^{\frac{1}{2}}_{21}\right)_{\alpha \beta},
\end{split}$$ where we have defined $b = \frac{a+d}{(1-d)}$. This is the analogous equation to (\[operator\]) except that the action now satisfies our four conditions and so the symmetry of (\[operator\]) has been reduced.
The free equations of motion $\Lambda_{\alpha \beta} \psi^\beta = 0$ are invariant under $\psi^\beta \rightarrow \psi^\beta + \delta \psi^\beta$ whenever $\delta \psi^\beta$ is annihilated by $\Lambda_{\alpha \beta}$. Using (\[gaugeinvariance\]) along with (\[orthog\]) we see that invariances occur only when the field is massless $m=0$ and also either $a = 0$ or $b = 0$. The case $a=0$ corresponds to the massless Rarita-Schwinger action and gives the invariance $\delta \psi^\beta = \left( p^\beta - \gamma^\beta {\not \negthinspace}{p} \right) \epsilon$ for $\epsilon$ an arbitrary spinor. This can be written as $\delta \psi^\beta = \theta^{\beta \lambda}(-d) \;
\partial_\lambda \epsilon$ using the results of the next section. The case $b = 0$ corresponds to $a = -d$ and is the gravitino part of the linearized supergravity action. In this case, the projectors give the invariance $\delta \psi^\beta = \partial^\beta \epsilon$ which is the linearized version of the gauge invariance in the gravitino action of supergravity[^15].
At first glance it may seem that the gauge invariance in the massless limit only occurs for two values of the parameter. In fact, these two invariances represent merely specific cases of the same general invariance[^16]. We form the general gauge transformation by using the group properties of the point transformations which we discuss in the following sections [@pascalutsa1999], for example, $\delta \psi^\alpha = \theta^{\alpha \beta} ({\tilde{b}}) \; \partial_\beta \epsilon$ is a gauge invariance for arbitrary parameter because the inverse transformation $\theta^{\alpha \beta}({\tilde{b}})$ effectively sets $b=0$, as will soon become clear.
The point transformation group {#group}
==============================
The transformation (\[pointtrans\]) that we have used in the previous section is sometimes referred to as a ‘point’ or ‘contact’ transformation in the literature. The Rarita-Schwinger equations (\[raritaschwinger\]) tell us that the [*onshell*]{} spin-$\frac{3}{2}$ field $\psi^\mu$ satisfies the following equation $$\label{holds}
\theta^\alpha_{\; \mu}(a) \psi^\mu = \psi^\alpha$$ as can be seen by the form of the transformation (\[pointtrans\]): $\theta^{\mu \nu}(a) = g^{\mu \nu} + \frac{a}{d} \gamma^\mu \gamma^\nu$, along with the onshell constraint $\gamma \cdot \psi = 0$. When $a \neq -1$ these transformations form a group with $$\begin{split}
\theta^{\mu \nu}(a) \theta_{\nu \rho}(b)
&= \theta^{\mu}_{\; \rho}(a + b + ab) \equiv \theta^{\mu}_{\; \rho}(a \circ b), \\
\left(\theta^{\mu \nu}\right)^{-1}(a) &=
\theta^{\mu \nu}(\frac{-a}{1+a}) \equiv \theta^{\mu \nu}(\tilde{a}),
\end{split}$$ where we have defined the ‘circle’ operation $a \circ b = a + b + ab$ and the inverse parameter $\tilde{a} = \frac{-a}{1+a}$. The lagrangian is tautologically invariant under the transformation $\psi^\alpha \rightarrow \theta^\alpha_{\; \mu}(k) \psi^\mu$ if, in addition to transforming the fields, we also transform the parameter $a$ as $$\label{halftrans}
a^\prime = \frac{a - k}{1 + k}.$$ Our field transformation together with the transformation (\[halftrans\]) of the parameter $a$ are frequently called a point transformation[^17]
It is worthwhile to notice that separate transformations from the left and from the right act separately on $a$ and $a^*$. If we recall how we formed our lagrangian by applying a transformation from the right and how the requirement that the operator be hermitian fixed $k = a^*$ we can see that $$\begin{split}
\theta_\mu^{\; \alpha}(k^*) \Lambda_{\alpha \beta}(a^*,a) &=
\theta_\mu^{\; \sigma}(k^*) \theta_\sigma^{\; \alpha}(a^*)
\Lambda_{\alpha \beta}(a), \\
&= \theta_\mu^{\; \alpha}(k^* \circ a^*) \Lambda_{\alpha \beta}(a) \\
&= \Lambda_{\alpha \beta}(k^* \circ a^*,a).
\end{split}$$ We find the left hand transformations by taking the hermitian conjugate, $$\begin{split}
\Lambda_{\mu \nu}(a^*,a) \theta^{\nu}_{\; \beta}(k)
&= \Lambda_{\mu \beta}(a^*, a \circ k) \; .
\end{split}$$ Hence transforming $\Lambda_{\mu \nu}(a^*,a)$ from the right affects only $a$, whereas the transformation from the left affects only $a^*$. We can use these left and right transformations, along with the definition of the Rarita-Schwinger lagrangian: $\Lambda^{\alpha \beta}_{\text{RS}}
= \Lambda^{\alpha \beta}(0,0)$, to write our most general lagrangian as $$\label{general}
\Lambda^{\alpha \beta}(a^*,a)
= \Lambda^{\alpha \beta}(a^* \circ 0, 0 \circ a)
=\theta^\alpha_{\; \mu}(a^*) \; \Lambda^{\mu \nu}_{\text{RS}} \;
\theta^{\; \beta}_{\nu}(a) \; .$$ Substituting this expression into the relation (\[invexpr\]) we have $$\begin{split}
S_{\alpha \beta} \; \theta^\beta_{\; \mu}(a^*) \; \Lambda^{\mu \nu}_{\text{RS}} \;
\theta^{\; \lambda}_{\nu}(a) &= \delta_\alpha^\lambda \\
\Rightarrow \left[ \theta^{\delta \alpha}(a) \; S_{\alpha \beta} \;
\theta^{\beta \mu}(a^*) \right] \Lambda_{\mu \nu}^{\text{RS}} &= \delta^{\delta}_{\; \nu}.
\end{split}$$ and the expression in brackets in the last line must therefore equal the Rarita-Schwinger propagator. So the point transformation group properties have given us a convenient way of deriving the general propagator from the parameterless RS propagator which much simpler than the method used in section \[conditions\], namely: $$\label{general2}
\begin{split}
S_{\alpha \beta}^{RS} &=
\theta_{\alpha}^{\; \mu}(a) \; S_{\mu \nu} \; \theta^{\nu}_{\; \beta}(a^*) \\
\Rightarrow S_{\alpha \beta} &=
\theta_{\alpha}^{\; \mu} (\tilde{a}) \; S_{\mu \nu}^{RS} \; \theta^{\nu}_{\; \beta}(\tilde{a}^*).
\end{split}$$
The point transformation becomes singular at the parameter value $a = -1$ as can be seen by the fact that $k \circ -1 = -1$ for any $k$, so that $$\label{singular}
\theta^{\mu \nu}(-1) \theta_{\nu \lambda}(k) = \theta^\mu_{\; \lambda}(-1)
\quad \forall \; k.$$ In the singular case $a = a^* = -1$ the transformations of the fields would no longer change the lower spin content of the operator and this choice may appear quite attractive. The problem with the choice $a = -1$ is that the operator in the lagrangian no longer has an inverse [@benmerrouche1989] as can be seen by the form of $h$ and $h^*$ in the propagator (\[propagator1\]). Interestingly, the singular value, $a = -1$, generates the additive identity element (or ‘zero’) of a ring defined by the following addition rule $$\begin{split}
\theta_{\mu \nu}(a) + \theta_{\mu \nu}(b) &= \theta_{\mu \nu}(a + b + 1), \\
\theta_{\mu \nu}(a) - \theta_{\mu \nu}(b) &= \theta_{\mu \nu}(a - b - 1),
\end{split}$$ where the addition is defined [*modulo zero*]{}, the two-sided ideal generated by $\theta_{\mu \nu}(-1)$. It is easily shown that the multiplication is distributive over addition.
One can redefine the parameter in various ways to make the group more convenient. For example, by shifting the singular point of the parameter space to $- \infty$ by letting $a \rightarrow e^{\alpha} - 1$ the group becomes [@pascalutsa1999] $$\label{lorentz}
\begin{split}
\theta^{\mu \nu}(\alpha) &= g^{\mu \nu} + \frac{e^{\alpha} -
1}{d} \gamma^\mu \gamma^\nu = e^{\frac{\alpha}{d} \gamma^\mu
\gamma^\nu} \\
\theta^{\mu \nu}(\alpha) \theta_{\nu \lambda}(\beta)
&= \theta^\mu_\lambda(\alpha + \beta), \\
\left(\theta^{\mu \nu}\right)^{-1}(\alpha) &= \theta^{\mu \nu}(-\alpha).
\end{split}$$ An even more convenient redefinition is so that the singular point is at 0. One has $a \rightarrow \alpha - 1$ and the ring is then defined by $$\begin{aligned}
\theta^{\mu \nu}(\alpha) =& g^{\mu \nu} + \frac{\alpha - 1}{d} \gamma^\mu \gamma^\nu,
& \text{(definition)} \\
\theta^{\mu \lambda}(\alpha) \theta_{\lambda}^{\; \; \nu}(\beta)
=& \theta^{\mu \nu}(\alpha \beta),
& \text{(multiplication)} \\
\theta^{\mu \nu}(1) =& g^{\mu \nu}, & \text{(multiplicative identity)} \\
\left(\theta^{\mu \nu}\right)^{-1}(\alpha) =& \theta^{\mu \nu}(\frac{1}{\alpha}),
& \text{(multiplicative inverse)} \\
\theta^{\mu \nu}(\alpha) + \theta^{\mu \nu}(\beta) =& \theta^{\mu \nu}(\alpha + \beta),
& \text{(addition)} \\
\theta^{\mu \nu}(0) =& g^{\mu \nu} - \frac{1}{d} \gamma^\mu \gamma^\nu,
& \text{(additive identity)} \\
\theta^{\mu \nu}(\alpha) - \theta^{\mu \nu}(\beta) =& \theta^{\mu \nu}(\alpha - \beta),
& \text{(additive inverse)}\end{aligned}$$ where the addition is again defined modulo the additive identity.
Interactions
============
The path integral is invariant under a global point transformation of the fields since the functional determinant is trivial and factors out of the integral to be cancelled out of the generating functional by the identical factor in the denominator. Hence there are no path integral anomalies and all physical correlation functions are independent of the parameter $a$. This is also true in the interacting theory, as we will discuss shortly, and so all physical Green functions are independent of $a$ and it can be fixed to whatever value is convenient. As was pointed out by Nath, [*et al.*]{} [@nath1971], the meaning of the invariance under point transformations is that the physical content of the theory does not depend on the parameter. However, the classical equations of motion in the presence of interaction are [*not*]{} invariant under shifts of the parameter. This is due to the fact that the transformation is not unitary. This will be discussed further in the next section.
There has been some controversy about to how to include consistent interactions involving spin-$\frac{3}{2}$ fields [@benmerrouche1989; @nath1971; @pascalutsa1999; @hagen1982]. We find the most logical way is to require the interaction terms to transform the same way under point transformations as the free action as well as remaining consistent with the massless gauge invariance of the free action. In other words, we require that a shift of the parameter must leave the form of the entire action unchanged. This ensures that the the path integral remains independent of the parameter and is the reason why many authors require the action to be invariant under a point transformation combined with a compensating parameter shift. This requirement means that we must have a factor of $\theta_{\mu \nu}(a)$ for each $\psi^\nu$ field in the interaction lagrangian (and thus a factor of $\theta_{\mu \nu}(a^*)$ in the hermitian conjugate) so that a transformation $\psi^\alpha \rightarrow \theta^{\alpha}_{\; \beta}(k) \psi^\beta$ will shift $a \rightarrow a \circ k$ everywhere[^18].
Expression (\[general2\]) shows that, for interactions which depend on the parameter in the same way as the lagrangian operator in (\[general\]), the inverse transformations attached to the RS propagator in (\[general2\]) will cancel those of the interaction and will therefore lead to a theory whose Green functions are independent of the parameter. We can see schematically that the correlation functions are invariant under redefinitions of the parameter as follows. In a theory satisfying our requirements the propagator and vertex can be written, respectively, as (suppressing all indices for simplicity) $$\begin{split}
S(a) &= \theta^{-1} S \theta^{-1}, \\
\Lambda(a) &= \theta \Lambda \theta,
\end{split}$$ where the $a$ dependence on the right-hand-side is entirely contained in the $\theta$ factors: $S$ and $\Lambda$ are independent of $a$. This implies that products of these expressions found in Feynman diagrams will always reduce to the form $$\begin{split}
G(a) &\sim \Lambda(a) S(a) \Lambda(a) \cdots \Lambda(a) S(a) \Lambda(a), \\
&= \theta G \theta.
\end{split}$$ Similarly in the case of interactions with scalar fields, spinor fields, etc. For example, consider a diagram which might appear in resonant pion photoproduction at one-loop order shown in figure \[pionfig\].
The parameter dependence of the Green function is contained in the vertices and the spin-$\frac{3}{2}$ resonance propagator. Our conditions require that the vertices and propagators can be written as $$\begin{split}
\Lambda_{\pi N \Delta}(a) &= \Lambda_{\pi N \Delta} \theta, \\
\Lambda_{\gamma N \Delta}(a) &= \Lambda_{\gamma N \Delta} \theta, \\
\Lambda_{\pi \Delta \Delta}(a) &= \theta \Lambda_{\pi \Delta \Delta} \theta, \\
S_{\Delta}(a) &= \theta^{-1} S_{\Delta} \theta^{-1},\\
S_{\pi}(a) &= S_{\pi},
\end{split}$$ where again the $a$ dependence is entirely contained within the $\theta$ factors. The Green functions are found as in the following example $$\begin{split}
G(a) &\sim \Lambda_{\pi N \Delta}(a) S_{\Delta}(a) \Lambda_{\pi \Delta \Delta}(a)
S_{\Delta}(a)
S_{\pi}\Lambda_{\pi \Delta \Delta}(a) S_{\Delta}(a) \Lambda_{\gamma N \Delta}(a), \\
&= \Lambda_{\pi N \Delta} \theta \theta^{-1} S_{\Delta} \theta^{-1} \theta
\Lambda_{\pi \Delta \Delta} \theta \theta^{-1} S_{\Delta} \theta^{-1}
S_{\pi}\theta \Lambda_{\pi \Delta \Delta} \theta \theta^{-1} S_{\Delta} \theta^{-1}
\theta \Lambda_{\gamma N \Delta}, \\
&= \Lambda_{\pi N \Delta} S_{\Delta} \Lambda_{\pi \Delta \Delta} S_{\Delta} S_{\pi}
\Lambda_{\pi \Delta \Delta} S_{\Delta} \Lambda_{\gamma N \Delta}.
\end{split}$$ This example shows the general pattern: all correlation functions for interacting theories are independent of the arbitrary parameter. This is true at every level of perturbation theory and at the non-perturbative level as well since the symmetry under $\theta$ is non-anomalous as we have discussed above.
Conserved vector current {#implications}
========================
We will now examine one of the ways in which the point transformation invariance of the correlation functions can be exploited. We will find that the analysis becomes simpler if we reparametrize our general action in terms of $b = \frac{a+d}{(1-d)}$ so that the general lagrangian (\[action1\]) becomes [@pilling1] $$\label{Action1}
\mathcal{L} = {{\overline{\psi}}}_\alpha
\left( \Gamma^{\alpha \mu \beta} i \partial_\mu - m \Gamma^{\alpha \beta} \right)
\psi_\beta,$$ where $$\begin{split}
\Gamma^{\alpha \mu \beta} &= g^{\alpha \beta} \gamma^\mu - A_1 g^{\mu \beta}
\gamma^\alpha - A_2 g^{\mu \alpha} \gamma^\beta + A_3 \gamma^\alpha \gamma^\mu
\gamma^\beta, \\
\Gamma^{\alpha \beta} &= g^{\alpha \beta} - A_4 \gamma^\alpha \gamma^\beta.
\end{split}$$ The coefficients are defined in terms of $b$ by $$\begin{split}
A_1 &= 1 + \frac{(d-2)}{d}b^*, \quad A_2 = 1 + \frac{(d-2)}{d}b \\
A_3 &= 1 + \frac{(d-2)}{d} \left[\frac{(d-1)}{d} |b|^2 + b^* + b \right] \\
A_4 &= 1 + \frac{(d-1)}{d} \left[|b|^2 + b^* + b\right].
\end{split}$$ The reason for this reparametrization is that we now have $$\label{action2}
\mathcal{L} = {{\overline{\psi}}}^\alpha \theta_{\alpha \mu}(b^*)
\left(\gamma^{\mu \rho \nu} i \partial_\rho + m \gamma^{\mu \nu}
\right) \theta_{\nu \beta}(b) \psi^\beta \quad
\equiv {{\overline{\psi}}}^\alpha \theta_{\alpha \mu}(b^*) \;
\Lambda_{\text{SG}}^{\mu \nu} \; \theta_{\nu \beta}(b) \psi^\beta,$$ where the totally antisymmetric combinations of gamma matrices can be written as $\gamma^{\mu \rho \nu} = \frac{1}{2} \left( \gamma^\mu \gamma^\rho \gamma^\nu
- \gamma^\nu \gamma^\rho \gamma^\mu \right)$ and $\gamma^{\mu \nu} =
\frac{1}{2} \left( \gamma^\mu \gamma^\nu - \gamma^\nu \gamma^\mu \right)$. The parameter choice $b=0$ now corresponds to the expression, ${{\overline{\psi}}}^\mu \; \Lambda^{\text{SG}}_{\mu \nu} \; \psi^\nu$, commonly found [@van1981; @pascalutsa1999; @pascalutsa2003] as the massive gravitino action in linearized supergravity[^19]. The propagator is then $$\label{propagator2}
S^{\alpha \beta} = \theta^{\alpha \mu}(\tilde{b})
\; S^{\text{SG}}_{\mu \nu} \; \theta^{\nu \beta}(\tilde{b}^*),$$ where $S^{\text{SG}}_{\mu \nu}$ is found from (\[propagator1\]) by setting $h = 0$.
Notice that the choice of parameter giving the supergravity action is such that the dimension of spacetime, $d$, does not explicitly appear. This, along with the antisymmetry of $\gamma^{\mu \rho \nu}$ and $\gamma^{\mu \nu}$, make many manipulations simpler and many results more transparent. Hence, we will use this expression for the general action from now on.
As we have seen, the observables of the quantum theory are independent of the parameter choice. We would like to explore the consequences of this invariance. However, the classical action is not invariant under the transformation since the symmetry transformation $\theta(k)$ of the field is in general [*non-unitary*]{}, $\theta(k^*) \neq \theta^{-1}(k)$. Under a global point transformation, $\psi_\mu \rightarrow \theta_{\mu \nu}(k)\psi^\nu$, the lagrangian (\[Action1\]) is not invariant, but transforms as $\mathcal{L}(b) \rightarrow \mathcal{L}(b \circ k)$. In the absence of interactions the equations of motion are invariant since the free field equations of motion imply that $\gamma \cdot \psi = 0$ and this makes the point transformation (\[pointtrans\]) trivial. However, in the interacting theory this is no longer true. In order to explore the consequences of the symmetry we will therefore [*impose it*]{} on the classical action by using the following technique. We simply demand that:
- [*classical actions will be considered equivalent if they lead to the same physical observables*]{}.
Put in another way this says:
- [*classical actions will be considered equivalent if they are related by a circle-shift redefinition of the parameter*]{}.
This makes the point transformations a symmetry of [*equivalence classes*]{} of classical actions and we can examine the consequences.
Let us re-iterate what we mean by this equivalence to avoid any possible confusion. We have shown that $\mathcal{L}(b)$ and $\mathcal{L}(b \circ k)$ lead to exactly the same physical correlation functions. These lagrangians are not the same, nor are they related by any gauge symmetry – point transformations are [*not*]{} gauge transformations. By considering these two different lagrangians as being equivalent, we are saying that any interaction whose only effect is to change $\mathcal{L}(b)$ into $\mathcal{L}(b \circ k)$, for some constant $k$, will have no effect on observable physics. We now want to examine the consequences of this equivalence.
For simplicity, we will re-write our action so that it is symmetric in derivatives[^20] and we will restrict the parameter to be real. Thus $$\label{symaction}
\mathcal{L}(b) = {{\overline{\psi}}}_\alpha \left[\frac{1}{2}
\Gamma^{\alpha \rho \beta}(b)i \overset{\leftrightarrow}{\partial}_\rho
+ m \Gamma^{\alpha \beta}(b) \right] \psi_\beta,$$ where we have written $$\begin{split}
\Gamma^{\alpha \rho \beta}(b) &= \theta^{\alpha}_{\; \mu}(b) \gamma^{\mu \rho \nu}
\theta_{\nu}^{\; \beta}(b), \\
\Gamma^{\alpha \beta}(b) &= \theta^{\alpha}_{\; \mu}(b) \gamma^{\mu \nu}
\theta_{\nu}^{\; \beta}(b), \\
\overset{\leftrightarrow}{\partial}_\rho &=
\overset{\rightarrow}{\partial}_\rho - \overset{\leftarrow}{\partial}_\rho.
\end{split}$$ Under an infinitesimal local point transformation $\theta(k(x))$ the lagrangian varies as $\mathcal{L}(b) \rightarrow \mathcal{L}(b \circ k) + \delta \mathcal{L}$ where $\delta \mathcal{L}$ contains the derivative acting on the parameter and $\mathcal{L}(b \circ k)$ is defined in exactly the same way as $\mathcal{L}(b)$ in (\[symaction\]) with the derivatives acting only on the fields and not on the parameter. Explicit computation gives $$\begin{split}
\delta \mathcal{L} &= \frac{i}{2d} {{\overline{\psi}}}_\alpha \biggl[ \Gamma^{\alpha \rho
\beta} \gamma_\beta \gamma^\nu - \gamma^\alpha \gamma_\beta \Gamma^{\beta
\rho \nu} \biggr] \psi_\nu \left(\partial_\rho k\right).
\end{split}$$ Integrating by parts and discarding the surface term we have $$\label{deltaL}
\mathcal{L}(b) \rightarrow \mathcal{L}(b \circ k)
- \frac{1}{2d} \left(\partial_\rho J^\rho\right) k(x),$$ where $J^\rho$ is given by $$\label{current}
J^\rho = i {{\overline{\psi}}}_\alpha \left[ \Gamma^{\alpha \rho \beta}(b) \gamma_\beta
\gamma^\nu - \gamma^\alpha \gamma_\beta \Gamma^{\beta \rho \nu}(b) \right] \psi_\nu.$$ Our symmetry says that $\mathcal{L}(b) = \mathcal{L}(b \circ k)$ in the limit that $k(x)$ becomes constant. This demands that $\delta \mathcal{L} =0$ in the limit of constant $k(x)$. Hence from (\[deltaL\]) we find a conserved current $J^\rho$ associated to the global symmetry: $\partial_\rho J^\rho = 0$. We see by (\[current\]) that the current changes under point transformations by a circle-shift of the parameter and is therefore invariant according to our symmetry. We can expand the $\Gamma^{\alpha \rho \beta}$ to find a simpler expression of the current as follows $$\begin{split}
J^\rho &= i (1+b) {{\overline{\psi}}}_\alpha \left[ \gamma^\alpha g^{\rho \beta}
- g^{\alpha \rho} \gamma^\beta \right] \psi_\beta, \\
&= i (1+b) \left[{{\overline{\psi}}} \cdot \gamma \psi^\rho
- {{\overline{\psi}}}^\rho \gamma \cdot \psi \right].
\end{split}$$ Under a transformation $\theta(k)$ the only change is the coefficient $(1+b) \rightarrow (1 + b \circ k)$. The conserved charge is given by $$\label{charge}
Q = i (1+b) \int d^{d-1} x
\left[
\left({{\overline{\psi}}} \cdot \gamma \right) \psi^0 - {{\overline{\psi}}}^0 \left(\gamma
\cdot \psi \right) \right].$$ The charge can be put in a more suggestive form by defining $\chi_1 = \gamma \cdot \psi$ and $\chi_2 = \gamma^0 \psi^0$ leaving $$\label{newEMcharge}
Q = i (1+b) \int d^{d-1} x
\left[\chi_1^\dagger \chi_2 - \chi_2^\dagger \chi_1 \right].$$ Since we have a conserved current, $J^\rho$, we can couple a vector field such as the photon to it as follows $$\label{newEM}
\begin{split}
\mathcal{L}_{\gamma} &= g J_\mu A^\mu \\
&= i g (1+b) \left[{{\overline{\psi}}} \cdot \gamma \psi_\mu
- {{\overline{\psi}}}_\mu \gamma \cdot \psi \right] A^\mu,
\end{split}$$ where $g$ is a coupling constant. If this coupling is physically reasonable, then it should, among other things, have a measurable effect on the magnetic moment of the spin-$\frac{3}{2}$ particle. We can also form derivative interactions with scalar fields, such as the pion, as $$\label{newpion}
\mathcal{L}_{\pi} = g_\pi J^\mu \partial_\mu \phi,$$ where $\phi$ is the scalar field. Furthermore, we also have the usual conserved vector current coming from electromagnetic gauge symmetry. This is given by $$j^\mu = {{\overline{\psi}}}_\alpha \Gamma^{\alpha \mu \beta} \psi_\beta,
= {{\overline{\psi}}}^\alpha \theta_{\alpha \mu}(b^*)
\gamma^{\mu \rho \nu} \theta_{\nu \beta}(b) \psi^\beta.$$ We can use (\[Action1\]) to write this as $$\label{oldEM}
\begin{split}
j^\mu &= {{\overline{\psi}}}_\beta \gamma^\mu \psi^\beta
- A_1 \left({{\overline{\psi}}} \cdot \gamma \right) \psi^\mu \\
&- A_1 {{\overline{\psi}}}^\mu \left(\gamma \cdot \psi \right)
+ A_3 \left({{\overline{\psi}}} \cdot \gamma \right) \gamma^\mu
\left(\gamma \cdot \psi \right),
\end{split}$$ where now $A_1 = A_2$ since the parameter is now real. The definitions of $\chi_1$ and $\chi_2$ allow us to write the charge as $$\label{oldEMcharge}
Q_{\text{EM}} = \int d^{d-1} x \left[ \psi_\beta^\dagger \psi^\beta
- A_1 \left( \chi_1^\dagger \chi_2
+ \chi_2^\dagger \chi_1 \right)
+ A_3 \chi_1^\dagger \chi_1 \right],$$ and we see that our new symmetry charge (\[newEMcharge\]) involves only the cross terms contained in the usual electromagnetic charge (\[oldEMcharge\]).
The two currents (\[newEM\]) and (\[oldEM\]) are separately conserved since they come from independent symmetries and hence we can form linear combinations of the two. Both currents couple to the lower spin components of the vector-spinor field and we adjust how much influence these lower spins have. It may be possible, with judicious choices of couplings, to eliminate the contribution of one or the other of the lower spins altogether by eliminating the cross term which contains both spin-$\frac{3}{2}$ and spin-$\frac{1}{2}$ components. Because of this new freedom, it seems that this new current will have some influence on the inconsistency problems that have been found in all interactions involving spin-$\frac{3}{2}$ fields [@johnson1961; @deser2000; @velo1969]. Perhaps the inconsistencies can be made to cancel between the the different conserved currents so that the new symmetry can be used to find solutions to that long-standing problem. We will turn now to a bit of a review of the consistency problem, formulated in $d$-dimensions with arbitrary complex parameter as well as some ideas about the possibility of solutions (or lack thereof).
Consistency problems {#consistency}
====================
As we have mentioned in the introductory section, the problem of finding consistent interactions for the spin-$\frac{3}{2}$ field is an old one. It was first pointed out in paper by Fierz and Pauli in 1939 [@fierz1939] where the two-component spinor formalism was used to derive lagrangians for massive spin-$\frac{3}{2}$ and spin-2 with a minimally coupled electromagnetic field. They pointed out that subsidiary conditions are necessary to reduce the number of independent field components to the physical number. The subject of spin-$\frac{3}{2}$ was revisited by Rarita and Schwinger in 1941 [@rarita1941] who developed the notation and action that are now most often used. They also noticed that the theory was not unique and that a collection of actions all give equivalent theories. This is the same non-uniqueness that we have exploited in the present paper.
Consistency problems with the theory were further discussed in 1961 by Johnson and Sudarshan (JS) [@johnson1961] who showed that in the presence of an electromagnetic field the field anti-commutator becomes indefinite, i.e. a Lorentz frame can always be found in which it is negative. A related discovery by Velo and Zwanziger (VZ) in 1969 [@velo1969] was that there are modes of the field which propagate faster than light. These problems derive from precisely the reasons given by Fierz and Pauli in 1939, namely that the lower spin particles should be removed by subsidiary conditions or they will give rise to negative energy states and indefinite charges[^21] and the number of subsidiary conditions must remain invariant to the presence of interaction.
In higher spin field theories involving auxiliary components and constraints, the time derivative operator of the lagrangian is a singular matrix. That this is necessary can be exemplified with our spin-$\frac{3}{2}$ operator in (\[Action1\]). The dynamical term, which contains the time derivative operator in the equations of motion, is given by $$\label{singop}
\Gamma^{\alpha 0 \beta} i \partial_0 \psi_\beta,$$ where $\Gamma^{\alpha 0 \beta}$ has suppressed spin indices and can be viewed, in 4 dimensions, as a $16 \times 16$ matrix. The equations of motion imply a set of constraints (equations among the field components which do not contain time derivatives) and so not all of the components of $\psi_\beta$ are dynamical and these non-dynamical components will not appear in (\[singop\]). Hence, there must exist vectors, corresponding to the non-dynamical components, which are annihilated by the above matrix. Therefore the matrix is necessarily singular and will have a determinant of zero.
In the case of spin-$\frac{3}{2}$, the constraints implied by the equations of motion are: the primary constraint, found by applying $\theta^{0}_{\; \alpha}(\tilde{b^*})$ to the equations of motion (\[eqmot\]), and the secondary constraint, found by taking the covariant derivative of the equations of motion. In four spacetime dimensions, as we have mentioned, these constitute a total of eight constraints needed to reduce the 16 component vector-spinor field to $2(2s+1)=8$ onshell degrees of freedom. In order to obtain a non-singular operator in (\[singop\]) we must impose the constraint equations and eliminate the non-dynamical components in terms of dynamical ones, thus giving the on-shell equations of motions in terms of a new non-singular matrix.
In the free field case everything works perfectly and the non-dynamical components can be eliminated without trouble. However, a problem occurs with the constraints in the presence of interaction in that the secondary constraint becomes dependent on the external field. This fact can lead to inconsistencies as we will now show for the canonical example of electromagnetic minimal coupling. Our general action in the case of electromagnetic minimal coupling is found by inserting a covariant derivative into (\[action2\]), giving the following equations of motion $$\label{eqmot}
\theta_{\alpha \mu}(b) \left(\gamma^{\mu \rho \nu} i D_\rho + m \gamma^{\mu \nu}
\right) \theta_{\nu \beta}(b) \psi^\beta = 0,$$ where $D_\rho = \partial_\rho + i e A_\rho$ is the covariant derivative. If we apply $\theta^{0 \alpha}(\tilde{b})$ to (\[eqmot\]) we are left with an expression which contains no time derivatives (since $\gamma^{\mu \rho \nu}$ vanishes when two indices are zero) and is the primary constraint. To derive the secondary constraint we apply $\gamma^\alpha$ to (\[eqmot\]) to get the ‘useful relation’ $$\label{rel}
i D \cdot \psi = \left( i \frac{(d-1)b + d}{d} {\not \negthickspace}{D} + \frac{(d-1)(1+b)}{d-2} m
\right) \gamma \cdot \psi.$$ Now take the covariant derivative of (\[eqmot\]), using (\[rel\]) as well as the identities $$\label{ident}
\begin{split}
\gamma_{\mu \nu} D^\mu D^\nu &= \frac{ie}{2} \gamma_{\mu \nu} F^{\mu \nu}, \\
{\not \negthickspace}{D} {\not \negthickspace}{D} &= \frac{ie}{2} \gamma_{\mu \nu} F^{\mu \nu} + \frac{1}{2} D^2,
\end{split}$$ to arrive at the secondary constraint $$\label{const2}
\left( \frac{ie}{2} \frac{(d-2)b + d}{d} \gamma_{\mu \nu} F^{\mu \nu}
+ \frac{(d-1)(1+b)}{d-2} m^2 \right) \gamma \cdot \psi
= ie \gamma_\mu F^{\mu \nu} \psi_\nu.$$ In four[^22] spacetime dimensions we can use the relation[^23] $$\label{ident2}
\frac{1}{2} \gamma_{\mu \nu} F^{\mu \nu} \gamma \cdot \psi
= \gamma_\mu F^{\mu \nu} \psi_\nu - i \gamma^5 \gamma_\mu \star F^{\mu \nu} \psi_\nu,$$ with the dual field strength $\star F^{\mu \nu}
= \frac{1}{2} \epsilon^{\alpha \beta \mu \nu} F_{\alpha \beta}$ to write (\[const2\]) in the simpler form $$\label{const2b}
\gamma \cdot \psi
= - \frac{ie}{3(1+b)m^2} \gamma_\mu \mathcal{F}^{\mu \nu} \psi_\nu,$$ where $$\label{F}
\mathcal{F}^{\mu \nu} = b F^{\mu \nu} + i (b + 2) \gamma^5 \star F^{\mu \nu}.$$ It should now be clear that the free field constraint $\gamma \cdot \psi = 0$ is no longer true unless the interaction vanishes.
Now that we have the constraints in the presence of electromagnetic minimal coupling we can describe the general case of the well known problem [@deser2000]. The secondary constraint (\[const2b\]) can be written as $$\label{const2c}
\left(\gamma^0 + \frac{ie}{3(1+b)m^2} \gamma_\mu \mathcal{F}^{\mu 0} \right) \psi^0
= \left( \gamma^k + \frac{ie}{3(1+b)m^2} \gamma_\mu \mathcal{F}^{\mu k} \right) \psi^k,$$ and one can easily see from the equations of motion (\[eqmot\]) that, in the case $b=0$, the components $\psi^0$ are non-dynamical. In that case they must be fixed by the above condition so that they can be eliminated in terms of the dynamical field components. To do this, one must solve (\[const2c\]) for $\psi^0$. This requires that the matrix $\left(\gamma^0 + \frac{ie}{3(1+b)m^2} \gamma_\mu \mathcal{F}^{\mu 0} \right)$ be non-singular. This implies, since $b=0$, that the matrix $\left( 1 - \frac{2e}{3m^2}\gamma^0 \vec{\gamma} \cdot \vec{B} \gamma^5 \right)$ is non-singular, which in turn requires that $\left( 1 + \frac{2e}{3m^2} \vec{\sigma} \cdot \vec{B} \right)$ is non-singular. Taking the determinant shows that this latter matrix is singular when $|\vec{B}|^2 = \left(\frac{3m^2}{2e}\right)^2$. This is the famous result of [@johnson1961] and [@velo1969] and means that if this equation is used to eliminate $\psi^0$ in the classical equations of motion one would find that, at this value of the external field, not all of the components of $\psi^0$ are determined and these undetermined components lead to space-like characteristic surfaces and the possibility of field modes propagating acausally. These non-physical characteristic surfaces indicate that the number of constraints has changed and that non-dynamical field components are still present in the field equations. For example, the operator coefficient to the time derivative in the equations of motion will still be a singular matrix at this external field value.
Note that we have conducted the analysis for $b = 0$, but we have lost no generality since the point transformation invariance can be used to transform this to the case of arbitrary $b$. To see this, Let us briefly outline the method [@velo1969; @madore1973] that is normally used to analyze the consistency of the classical field equations. The basic idea is to look at the characteristic surfaces given by the classical on-shell differential equation. To do this one plugs the constraints into the equations of motion, thus (hopefully) eliminating all of the non-dynamical degrees of freedom. Then the characteristics are found by replacing the derivative in the equation with a four-vector $n^\mu$. This vector will be a normal to a characteristic surface (a surface along which the maximum velocity solutions to the differential equation are restricted to propagate) if it is the solution to a certain equation (\[characteristics\]). If we want only time-like solutions, so that causality is preserved, we would like the normals to the characteristic surfaces to remain always space-like.
The normals to the characteristic surfaces $n_\mu$ are given by the determinant $$\label{characteristics}
D(n) = \left| \tilde{\Gamma}^{\alpha \mu \beta} n_\mu \right| = 0,$$ where we have replaced the derivatives in the field equations with $n_\mu$ and $\tilde{\Gamma}^{\alpha \mu \beta}$ is the operator in the field equations [*after all of the constraints have been imposed*]{}. If there is a time-like normal we can use Lorentz invariance to write it as $n_\mu = (n,0,0,0)$. Our characteristic equation is then $$\label{det}
D(n) = n^{16} \left| \tilde{\Gamma}^{\alpha 0 \beta} \right| = 0.$$ Notice that if this determinant vanishes for $b=0$, then it also vanishes for arbitrary $b$ by the properties of the point transformations. The vanishing of the determinant means that there are characteristic surfaces which are space-like, indicating the possibility that field components propagate acausally. We say ‘indicating [*the possibility*]{}’ of acausal propagation because if the determinant was zero, so that there are space-like characteristics, one would still have to prove that there were actual physical field modes that propagate along these acausal characteristics. This is usually done with the method of shock discontinuities [@madore1973]. We will not discuss this method further since it has been established many times that there are field modes which propagate acausally in various cases. On the other hand, to prove that a theory is consistent one would only need to show that there are no space-like characteristic surfaces and it would follow that there is no acausal propagation, without need of the method of discontinuities.
Notice that if we didn’t impose constraints, the matrix in the determinant equation (\[det\]) would be the same one that we discussed earlier (\[singop\]) which is necessarily singular. Thus, if the constraints failed to eliminate a non-dynamical field component. This component would remain in the field equations but it would, by definition, not appear in the time derivative part since it is non-dynamical. Hence the matrix coefficient of the time derivative $\partial_0$ would annihilate the 16-component vector representing this component. The matrix then annihilates a non-zero vector and so it must have determinant equal to zero. Conversely, the physical components all have time derivatives which means the matrix coefficient of the time derivative does not annihilate them.
It is interesting that the inconsistency comes from the breakdown of constraints and is caused by non-dynamical, ‘unphysical’ components creeping back into the field equations at certain values of the external field and the presence of these components leads to physical information propagating acausally. Therefore, if we could find a covariant method of preventing these extra components from coming back the problems would be solved. Unfortunately, this seemingly simple task is a subtle one and has not been accomplished in general since the problem was discovered over 40 years ago although there have been many papers written on the subject[^24] [@hagen1971; @singh1973; @hortacsu1974; @prabhakaran1975; @capri1980; @aurilia1980; @sierra1982; @darkhosh1985]. We do not claim to have solved the problem here either, but we will present an idea that may be of some use in the search for a solution or a proof that one can’t exist.
We should mention that, in the context of supergravity, when the mass of the spin-$\frac{3}{2}$ field is tuned in certain ways to the background spacetime, there are theories which seem to be free of inconsistencies [@deser2001]. On the other hand, if we would like to use the theory to effectively model low mass spin-$\frac{3}{2}$ fields (such as the $\Delta(1232)$ nucleon resonance) at low energies one cannot expect the background spacetime or the existence of supersymmetry partners to be of much help.
We have exemplified the problem for the case $b=0$ (i.e. the supergravity action) but, as we have said, the same is true for arbitrary $b$ since the lagrangian operators are related by a non-singular point transformation and so if the determinant (\[characteristics\]) is zero for one value of the parameter, a point transformation will not alter this and it will be zero for any parameter value.
In any case, we can now ask: is it possible that the new symmetry current and charge can help us? The conserved charge arises from a symmetry of the path integral which is not shared by the usual classical theory, so if a solution to this problem could be found from this charge it would have the happy consequence that the inconsistency which is present in the classical theory would not affect the physical correlation functions. Conversely, it is also possible that the extra symmetry of the path integral may make things even worse.
Suppose we begin with no interaction and then slowly turn on an external field. In that case we can argue that the charge should remain zero since it is zero in the free case and is conserved. The charge (\[charge\]) is given in an electromagnetic background by $$\label{interactingcharge}
Q = i (1+b) \int d^3 x \left[
{{\overline{\psi}}}_\nu \left(\mathcal{F}^{\nu \mu}\right)^* \gamma_\mu \psi^0
- {{\overline{\psi}}}^0 \gamma_\mu \mathcal{F}^{\mu \nu} \psi_\nu
\right].$$ The conservation of charge is related to the conservation of constraints since the spin-$\frac{1}{2}$ fields $\gamma \cdot \psi$ and $\psi^0$ in the charge are exactly the ones which are usually removed via the constraint structure of the free field Rarita-Schwinger equations. Hence it seems possible that the loss of constraints, which signals the onset of the inconsistency, will have a direct effect on the charge. This gives us an idea for a possible new direction. When the external field is such that there is a loss of constraints the charge should remain unaffected since it is conserved. It is possible then, that in that case, the equation for the charge would represent an additional constraint which could be used to eliminate the unphysical fields which appear when the usual constraints breakdown. However, we do not want the extra constraint in general, but only in the case when the usual constraints breakdown and then only so as to exactly compensate for the breakdown. This seems like quite a demand. If the conservation of charge implied additional contraints in general we would then have too many constraints as soon as the background is turned on. So physically we want the charge to be identically zero when the constraints are imposed. The only time the situation changes is when the usual constraints break down. In that case we want the charge to no longer vanish identically, but to itself become a constraint compensating for the ones that were lost. The usual constraints are given by $\gamma \cdot \psi \sim \gamma_\mu \mathcal{F}^{\mu \nu} \psi_\nu$ and this combination then also appears in the expression for the charge. So perhaps it is possible that a breakdown of this equation would have a compensating effect in the conserved charge. In any case, we will leave this problem for now with the hope that these thoughts may inspire new angles of attack so that either a new solution, or a new reason for the lack of one, can be found.
Conclusion
==========
We will now summarize the main features of our work. We have derived the general lagrangian and propagator for the Rarita-Schwinger field in $d$-dimensions. These are given by equations (\[Action1\]) and (\[propagator1\]) respectively and should prove useful in calculating higher loop effects in dimensional regularization as would occur in the effective resonance contribution to the imaginary part of pion scattering amplitudes, anomalous magnetic moments, and many other processes for which the $\Delta(1232)$ resonance or any other spin-$\frac{3}{2}$ particles play a significant role.
We studied the point transformation algebra and explored the invariance properties of the general action under rotations of the lower spin, off-shell fields. We found that this invariance implies the existence of a conserved vector current and charge. The conserved current leads to interactions involving spin-$\frac{3}{2}$ fields such as the electromagnetic couplings that we have given in (\[newEM\]) above, possible couplings to other vector fields such as vector mesons, derivative couplings to scalar fields such as the pion, etc. It is important to check the predictions of these interactions.
Finally, we looked at the consistency problems and indicated two possible avenues where progress might be made. The first is a possible cancellation of the problematic terms by tuning interactions based on the conserved current. The second is by using the conserved charge as an additional constraint to compensate for the loss of the usual constraints at the ‘bad’ values of the external magnetic field.
It is important to attempt a generalization of these techniques in the theory of higher spin fields in the same way as done for spin-$\frac{3}{2}$. It seems likely that similar (though more complicated) groups will exist which rotate among the auxiliary lower spin components in those cases as well, leading again to new conserved currents.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Emil Akhmedov and D. V. Ahluwalia-Khalilova for discussion and helpful comments.
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[^1]: terry@member.ams.org
[^2]: By ‘high spin’ we mean particles of spin $\geq \frac{3}{2}$.
[^3]: See section \[consistency\] below.
[^4]: We say non-interacting since onshell fields are used and condition $\Gamma \cdot \psi = 0$ is incompatible with the presence of additional non-gravitational interactions, for example a non-zero background electromagnetic field.
[^5]: We follow the standard procedure of using 4-dimensional terminology for spin even when discussing fields in arbitrary dimensions.
[^6]: However there are other representations which describe spin-$\frac{3}{2}$ also, for example the 3 spinor representation [@aurilia1969] and also the $\left(\frac{3}{2}, 0\right) + \left(0, \frac{3}{2}\right)$ representation of [@ahluwalia1992; @ahluwalia1993].
[^7]: See section 1.2.2 in van Nieuwenhuizen’s supergravity review [@van1981]
[^8]: For example one may want to set $d = 4 - \epsilon$ in dimensional regularization.
[^9]: Curved space results can then be found by the usual technique of introducing vielbeins and a spin connection.
[^10]: See [@wetterich1983] and references therein for properties of spinors in $d$ dimensions, the $d$-dimensional Lorentz group and the group of $d$-dimensional general coordinate transformations.
[^11]: This may not be true in the interacting theory where the interaction may cause a reduction in the number of constraint equations resulting in an increase in the number of degrees of freedom and leading to inconsistencies. This will be discussed further in section \[consistency\].
[^12]: The restriction of condition 2 is perhaps not necessary, both the photon and the gluon propagators also have extra singular terms before gauge fixing and indeed some of the popular spin-$3/2$ actions do not satisfy this condition. However, in our case we have a theory describing a [*massive*]{} field and so extra terms which are singular as $p \rightarrow 0$ give singularities which are not at the physical mass unlike the case of the photon or the gluon.
[^13]: See for example equation (4.2) in Johnson and Sudarshan [@johnson1961] or equation (15) in van Nieuwenhuizen’s supergravity review [@van1981].
[^14]: See also the equivalent, but much simpler, expression given by (\[Action1\]) and (\[action2\]) in section \[implications\] below.
[^15]: In curved space let $\partial_\beta \rightarrow D_\beta = \partial_\beta + \frac{1}{4}
\omega_{\beta}^{\; ab} \gamma_{ab}$ where $a,b$ are flat indices, $\omega_{\beta}^{\; ab}$ is the spin connection and $\gamma_{ab} \equiv \frac{1}{2} [\gamma_a, \gamma_b]$.
[^16]: The author would like to thank V. Pascalutsa for pointing this out.
[^17]: To compare with the notations of other authors, notice that in the case where $k$ and $a$ are real we can use (\[defna\]) to rewrite the transformation of the parameter $a$ in terms of $A$ as $A \rightarrow \frac{A - 2k/d}{1 + k}$. Also note our factor of $\frac{1}{d}$ in the group law.
[^18]: Note that in order to be consistent with the massless gauge invariance of the theory, the interaction must depend only on $d \psi$ rather than simply $\psi$ and also there should be no ‘offshell parameter’ dependence. This is discussed by Pascalutsa and Timmermans in [@pascalutsa1999].
[^19]: Many other expressions are found in the supergravity literature (for examples see Refs. [@madore1975; @deser1977; @cremmer1978]) but they are all found to be equivalent to ours by using some choice of parameter $a$ and/or using the identity $\gamma^{\mu \nu \alpha} = i \epsilon^{\mu \nu \alpha \beta} \gamma_5
\gamma_\beta$.
[^20]: The reason is so that the analysis results in an hermitian current.
[^21]: However, see [@kirchbach2001] for an interesting alternative to this.
[^22]: A similar identity will hold in any even dimension where there exists a $\gamma^5$–like matrix [@wetterich1983].
[^23]: We are using a mostly minus metric, diagonal $\gamma^0$, and $\gamma^5 = -i \gamma^0 \gamma^1 \gamma^2 \gamma^3$.
[^24]: We should mention the recent developments on a new type of spin 1/2 field with mass dimension one (called ‘Elko’) given in reference [@ahluwalia2004]. That work can be extended to spin 3/2 and may lead to new results for the consistency problem.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this are temporal graphs which consist of a set of vertices (entities in the network) and a set of time-stamped binary interactions between the vertices. We focus on enumerating $\Delta$-cliques, an extension of the concept of cliques to temporal graphs: for a given time period $\Delta$, a $\Delta$-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every $\Delta$ time steps within the time interval. Viard, Latapy, and Magnien \[ASONAM 2015, TCS 2016\] proposed a greedy algorithm for enumerating all maximal $\Delta$-cliques in temporal graphs. In contrast to this approach, we adapt the Bron-Kerbosch algorithm—an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs—to the temporal setting. We obtain encouraging results both in theory (concerning worst-case running time analysis based on the parameter “$\Delta$-slice degeneracy” of the underlying graph) as well as in practice[^1] with experiments on real-world data. The latter culminates in an improvement for most interesting $\Delta$-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien.'
author:
- 'Anne-Sophie Himmel'
- Hendrik Molter
- Rolf Niedermeier
- Manuel Sorge
bibliography:
- 'literature.bib'
title: 'Adapting the Bron-Kerbosch Algorithm for Enumerating Maximal Cliques in Temporal Graphs[^2]'
---
Introduction
============
Network analysis is one of the main pillars of data science. Focusing on networks that are modeled by undirected graphs, a fundamental primitive is the identification of complete subgraphs, that is, cliques. This is particularly true in the context of detecting communities in social networks. Finding a maximum-cardinality clique in a graph is a classical NP-hard problem, so super-polynomial worst-case running time seems unavoidable. Moreover, often one wants to solve the more general task of not only finding one maximum-cardinality clique but to list *all maximal* cliques. Their number can be exponential in the graph size. The famous Bron-Kerbosch algorithm (“Algorithm 457” in *Communications of the ACM 1973*, [@bron1973algorithm]) addresses this task and still today forms the basis for the best (practical) algorithms to enumerate all maximal cliques in static graphs [@ELS13]. However, to realistically model many real-world phenomena in social and other network structures, one has to take into account the dynamics of the modeled system of interactions between entities, leading to so-called temporal networks. In a nutshell, compared to the standard static networks, the interactions in temporal networks (edges) appear sporadically over time (while the vertex set remains static). Indeed, as @nicosia2013graph pointed out, in many real-world systems the interactions among entities are rarely persistent over time and the non-temporal interpretation is an “oversimplifying approximation”. In this work, we use the standard model of temporal graphs. A temporal graph consists of a vertex set and a set of edges, each with an integer time-stamp. The generalization of a clique to the temporal setting that we study is called [$\Delta$-clique]{} and was introduced by @Viard2015Dyno [@viard2015computing]. Intuitively, being in a [$\Delta$-clique]{} means to be regularly in contact with all other entities in this [$\Delta$-clique]{}. In slightly more formal terms, each pair of vertices in the [$\Delta$-clique]{} has to be in contact in at least every $\Delta$ time steps. A fully formal definition is given in Section \[sec:preliminaries\]. We present an adaption of the framework of Bron and Kerbosch to temporal graphs. To this end, we overcome several conceptual hurdles and propose a temporal version of the Bron-Kerbosch algorithm as a new standard for efficient enumeration of maximal [$\Delta$-clique]{}s in temporal graphs.
Related Work
------------
Our work relates to two main lines of research. First, enumerating $\Delta$-cliques in temporal graphs generalizes the enumeration of maximal cliques in static graphs, this being subject of many different algorithmic approaches (sometimes also exploiting specific properties such as the “degree of isolation” of the cliques searched for) [@bron1973algorithm; @ELS13; @II09; @HKMN09; @komusiewicz2009isolation; @tomita2006worst]. Indeed, clique finding is a special case of dense subgraph detection. Second, more recently, mining dynamic or temporal networks for periodic interactions [@LB10] or preserving structures [@uno2015mining] (in particular, this may include cliques as a very fundamental pattern) has gained increased attention. Our work is directly motivated by the study of @Viard2015Dyno [@viard2015computing] who introduced the concept of $\Delta$-cliques and provided a corresponding enumeration algorithm for $\Delta$-cliques. In fact, following one of their concluding remarks on future research possibilities, we adapt the Bron-Kerbosch algorithm to the temporal setting, thereby outperforming their greedy-based approach in most cases.
Results and Organization
------------------------
Our main contribution is to adapt the Bron-Kerbosch recursive backtracking algorithm for clique enumeration in static graphs to temporal graphs. In this way, we achieve a significant speedup for most interesting time period values $\Delta$ (typically two orders of magnitude of speedup) when compared to a previous algorithm due to @Viard2015Dyno [@viard2015computing] which is based on a greedy approach. We also provide a theoretical running time analysis of our Bron-Kerbosch adaption employing the framework of parameterized complexity analysis. The analysis is based on the parameter “$\Delta$-slice degeneracy” which we introduce, an adaption of the degeneracy parameter that is frequently used in static graphs as a measure for sparsity. This extends results concerning the static Bron-Kerbosch algorithm [@ELS13]. A particular feature to achieve high efficiency of the standard Bron-Kerbosch algorithm is the use of pivoting, a procedure to reduce the number of recursive calls of the Bron-Kerbosch algorithm. We show how to define this and make it work in the temporal setting, where it becomes a significantly more delicate issue than in the static case. In summary, we propose our temporal version of the Bron-Kerbosch approach as a current standard for enumerating maximal cliques in temporal graphs.
The paper is organized as follows. In Section \[sec:preliminaries\], we introduce all main definitions and notations. In addition, we give a description of the original Bron-Kerbosch algorithm as well as two extensions: pivoting and degeneracy ordering. In Section \[section:bronKerboschDelta\], we propose an adaption of the Bron-Kerbosch algorithm to enumerate all maximal $\Delta$-cliques in a temporal graph, prove the correctness of the algorithm and give a running time upper bound. Furthermore, we adapt the idea of pivoting to the temporal setting. In Section \[sec:degeneracy\] we adapt the concept of degeneracy to the temporal setting and give an improved running time bound for enumerating all maximal $\Delta$-cliques. In Section \[section:implExp\], we present the main results of the experiments on real-world data sets. We measure the $\Delta$-slice degeneracy of real-world temporal graphs, we study the efficiency of our algorithm, and compare its running time to the algorithm of @Viard2015Dyno, showing a significant performance increase due to our Bron-Kerbosch approach. We conclude in Section \[sec:conclusion\], also presenting directions for future research.
Preliminaries {#sec:preliminaries}
=============
In this section we introduce the most important notations and definitions used throughout this article.
Graph-Theoretic Concepts
------------------------
In the following we provide definitions of adaptations to the temporal setting for central graph-theoretic concepts.
### Temporal Graphs
The motivation behind temporal graphs, which are also referred to as temporal networks [@holme2012temporal], time-varying graphs [@nicosia2013graph], or link streams [@Viard2015Dyno], is to capture changes in a graph that occur over time. In this work, we use the well-established model where each edge is given a time stamp [@Viard2015Dyno; @holme2012temporal; @boccaletti2014structure]. Assuming discrete time steps, this is equivalent to a sequence of static graphs over a fixed set of vertices [@michail2015introduction; @Erlebach0K15]. Formally, the model is defined as follows.
A *temporal graph* $\mathbb{G}=(V,E,T)$ is defined as a triple consisting of a set of vertices $V$, a set of *time-edges* $E \subseteq \binom{V}{2} \times T$, and a time interval $T=[\alpha, \omega]$, where $\alpha, \omega \in \mathbb{N}$, $T \subseteq \mathbb{N}$ and $\omega -\alpha$ is the *lifetime* of the temporal graph $\mathbb{G}$.
The notation $\binom{V}{2}$ describes the set of all possible undirected edges $\{v_1,v_2\}$ with $v_1 \not = v_2$ and $v_1,v_2 \in V$. A time-edge $e=(\{v_1,v_2\}, t) \in E$ can be interpreted as an interaction between $v_1$ and $v_2$ at time $t$. Note that we will restrict our attention to discretized time, implying that changes only occur at discrete points in time. This seems close to a natural abstraction of real-world dynamic systems and “gives the problems a purely combinatorial flavor” [@Michail2014].
### $\Delta$-Cliques
A straightforward adaptation of a clique to the temporal setting is to additionally assign a lifetime $I = [a, b]$ to it, that is, the largest time interval such that the clique exists in each time step in said interval. However, this model is often too restrictive for real-world data. For example, if the subject matter of examination is e-mail traffic and the data set includes e-mails with time stamps including seconds, we are not interested in people who sent e-mails to each other every second over a certain time interval, but we would like to know which groups of people were in contact with each other, say, at least every seven days over months. One possible approach would be to generalize the time stamps, taking into account only the week an e-mail was sent, resulting in a loss of accuracy in the data set. The constraint of each pair of vertices being connected in each time step can be relaxed by introducing an additional parameter $\Delta$, quantifying how many time steps may be skipped between two connections of any vertex pair. These so-called $\Delta$-cliques were introduced by @Viard2015Dyno [@viard2015computing] and are formally defined as follows.
Let $\Delta \in \mathbb{N}$. A *$\Delta$-clique* in a temporal graph $\mathbb{G}=(V,E,T)$ is a tuple $C=(X, I=[a,b])$ with $X \subseteq V$, $b-a\geq \Delta$, and $I\subseteq T$ such that for all $\tau \in [a, b-\Delta]$ and for all $v,w\in X$ with $v\not =w$ there exists a $(\{v,w\},t) \in E$ with $t \in [\tau, \tau + \Delta]$.
In other words, for a $\Delta$-clique $C=(X,I)$ all pairs of vertices in $X$ interact with each other at least after every $\Delta$ time steps during the time interval $I$. We implicitly exclude $\Delta$-cliques with time intervals smaller than $\Delta$.
It is evident that the parameter $\Delta$ is a measurement of the intensity of interactions in $\Delta$-cliques. Small $\Delta$-values imply that the interaction between vertices in a $\Delta$-clique has to be more frequent than in the case of large $\Delta$-values. The choice of $\Delta$ depends on the data set and the purpose of the analysis.
We can also consider $\Delta$-cliques from another point of view. For a given temporal graph $\mathbb{G}=(T,V,E)$ and a $\Delta \in \mathbb{N}$, the static graph $G^{\Delta}_{\tau} = (V_{\tau},E_{\tau})$ describes all contacts that appear within the $\Delta$-sized time window $[\tau, \tau+\Delta]$ with $\tau \in [\alpha,\omega-\Delta]$ in the temporal graph $\mathbb{G}$, that is $V_{\tau} = V$ and for every $\{v_1,v_2\} \in E_{\tau}$ there is a time step $t \in [\tau, \tau + \Delta]$ such that $(\{v_1,v_2\},t) \in E$. The existence of a $\Delta$-clique $C=(X, I=[a,b])$ indicates that all vertices in $X$ form a clique in all static graphs $G^{\Delta}_{\tau}$ with $\tau \in [a,b-\Delta]$. This implies that all vertices in $X$ are pairwise connected to each other in the static graphs of all sliding, $\Delta$-sized time windows from time $a$ until $b - \Delta$.
By setting $\Delta$ to the length of the whole lifetime of the temporal graph, every $\Delta$-clique corresponds to a normal clique in the underlying static graph that results from ignoring the time stamps of the time-edges. We are most interested in [$\Delta$-clique]{}s that are not contained in any other [$\Delta$-clique]{}. For this we also need to adapt the notion of maximality to the temporal setting [@Viard2015Dyno; @viard2015computing]. Let [$\mathbb{G}$]{} be a temporal graph. We call a $\Delta$-clique $C=(X, I)$ in [$\mathbb{G}$]{} *vertex-maximal* if we cannot add any vertex to $X$ without having to decrease the clique’s lifetime $I$. That is, there is no [$\Delta$-clique]{} $C' = (X', I')$ in [$\mathbb{G}$]{} with $I \subseteq I'$ and $X \subsetneq X'$. We say that a $\Delta$-clique is *time-maximal* if we cannot increase the lifetime $I$ without having to remove vertices from $X$. That is, there is no [$\Delta$-clique]{} $C' = (X', I')$ in [$\mathbb{G}$]{} with $I \subsetneq I'$ and $X \subseteq X'$. We call a $\Delta$-clique *maximal* if it is both vertex-maximal and time-maximal.
### $\Delta$-Neighborhood, $\Delta$-Cut, and other Temporal Graph Concepts
In this section, we introduce and define further graph theoretical concepts that need to be adapted to the temporal setting.
We refer to a tuple $(v, I=[a, b])$ with $v \in V$ and $I \subseteq T$ as a *vertex-interval pair* of a temporal graph. We call $a$ the *starting point* of interval $I$ and $b$ the *endpoint* of interval $I$. Let $X$ be a set of vertex-interval pairs. The modified element relation $(v,I) { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}X$ (*temporal membership*) expresses that there exists a vertex-interval pair $(v,I') \in X$ with $I \subseteq I'$.
Using these definitions, we can adapt the notion of a neighborhood of a vertex to temporal graphs. Intuitively, we want that two vertex-interval pairs are neighbors if they can be put into a [$\Delta$-clique]{} together.
For a vertex $v \in V$ and a time interval $I \subseteq T$ in a temporal graph, the *$\Delta$-neighborhood* $N^{\Delta}(v,I)$ is the set of all vertex-interval pairs $(w,I'=[a',b'])$ with the property that for every $\tau \in [a',b'-\Delta]$ at least one edge $(\{v,w\},t) \in E$ with $t \in [\tau, \tau + \Delta]$ exists. Furthermore, $b'-a' \geq \Delta$, $I' \subseteq I$, and $I'$ is maximal, that is, there is no time interval $I''\subseteq I$ with $I' \subset I''$ satisfying the properties above.
Notice that being a $\Delta$-neighbor of another vertex is a symmetric relation. If $(w, I') { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v,I)$, then we say that $w$ is a v*$\Delta$-neighbor* of $v$ during the time interval $I'$. In Figure \[figure:TemporalGraph\], we visualize the concepts of $\Delta$-neighborhood and $\Delta$-clique in a temporal graph. See also Example \[example:temporalGraph\] below.
We need to define a suitable way of intersecting of two sets of vertex-interval pairs, so that, as the intuition suggests, a $\Delta$-clique is just the intersection of the “closed” $\Delta$-neighborhoods[^3] of its elements over the lifetime of the clique.
Let $X$ and $Y$ be two sets of vertex-interval pairs. The *$\Delta$-cut* $X\sqcap Y$ contains for each vertex, all intersections of intervals in $X$ and $Y$ that are of size at least $\Delta$. More precisely, $$X \sqcap Y = \{ (v, I \cap I') \mid (v, I)\in X \wedge (v, I')\in Y \wedge |I\cap I'|\ge\Delta\}.$$
In other words, the $\Delta$-cut $X\sqcap Y$ contains all vertex-interval pairs $(v,I)$ such that $(v, I){ \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}X$ and $(v, I) { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}Y$, as well as $|I| \geq \Delta$, and $I$ is maximal under these properties. That is, there is no $J$ with $I \subsetneq J$ and $J \subseteq I'$ and $J \subseteq I''$ such that $(v,I') { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}X$ and $(v,I'') { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}Y$ for some $I'$ and $I''$.
\
\[example:temporalGraph\] In Figure \[figure:TemporalGraph\] we visualize a temporal graph and the concepts of $\Delta$-neighborhood and $\Delta$-clique. We consider a temporal graph $\mathbb{G} = (T, V, E)$ with $T = [0,8]$, $V = \{a,b,c\}$, $E=\{(\{a,b\},2), (\{a,b\},3), (\{a,c\},4), (\{b,c\},5),$ $(\{a,c\},6)\}$, and $\Delta = 2$. The vertices are visualized as horizontal lines. The connections between two vertices at a specific time step represent the time-edges of the temporal graph.
We visualize the $\Delta$-neighborhood of each vertex of the temporal graph over the whole time interval $T$ in Figures \[figure:DeltaNeighborhoodA\]-\[figure:DeltaNeighborhoodC\]:
- In Figure \[figure:DeltaNeighborhoodA\], we consider the $\Delta$-neighborhood $N^{\Delta}(a,T)$ of vertex $a$ during the whole time interval $T$. The yellow shaded bar marks the vertex-interval pair $(b,[0,5]) \in N^{\Delta}(a,T)$. The vertex $b$ is a $\Delta$-neighbor of $a$ during $[0,5]$ because for every $\tau \in [0,5-\Delta =3]$ at least one time-edge $(\{a,b\},t) \in E$ with $t \in [\tau, \tau + \Delta]$ exists since $(\{a,b\},2), (\{a,b\},3) \in E$. The same holds for the vertex-interval pair $(c,[2,8]) \in N^{\Delta}(a,T)$ which is marked in hatched green.
- In Figure \[figure:DeltaNeighborhoodB\], we visualize the $\Delta$-neighborhood $N^{\Delta}(b,T)$ of $b$ over the whole lifetime $T$ of the temporal graph. The vertex-interval pair $(c,[3,7]) \in N^{\Delta}(b,T)$ is marked in hatched green. The vertex-interval pair $(a,[0,5]) \in N^{\Delta}(b,T)$ is shaded in yellow. It becomes evident that being a $\Delta$-neighbor of another vertex is a symmetric relation—if $a$ is a $\Delta$-neighbor of $b$ during $[0,5]$, then $b$ is also a $\Delta$-neighbor of $a$ during $[0,5]$.
- In Figure \[figure:DeltaNeighborhoodC\], we visualize the $\Delta$-neighborhood $N^{\Delta}(c,T)$ of $c$ over the whole lifetime $T$ of the temporal graph. The vertex-interval pair $(b,[3,7]) \in N^{\Delta}(c,T)$ is marked in hatched green. The vertex-interval pair $(a,[2,8]) \in N^{\Delta}(c,T)$ is shaded in yellow.
Figure \[figure:DeltaClique\] shows the maximal $\Delta$-clique $(\{a,b,c\},[3,5])$. During the time interval $[3,5]$, $a$ and $b$ are $\Delta$-neighbors, $b$ and $c$ are $\Delta$-neighbors and $a$ and $c$ are $\Delta$-neighbors, see Figures \[figure:DeltaNeighborhoodA\]-\[figure:DeltaNeighborhoodC\]. We cannot increase the time interval because at time step $2$ the vertices $b$ and $c$ are not yet $\Delta$-neighbors and at time step $6$ the vertices $a$ and $b$ are no longer $\Delta$-neighbors. Further nontrivial maximal [$\Delta$-clique]{}s in this temporal graph are: $(\{a,b\},[0,5])$, $(\{a,c\},[2,8])$, $(\{b,c\},[3,7])$, as well as the trivial $\Delta$-cliques $(\{a\},[0,8])$, $(\{b\},[0,8])$, and $(\{c\},[0,8])$.
Bron-Kerbosch Algorithm {#section:bronKerbosch}
-----------------------
In this section, we explain the basic idea of the (static) Bron-Kerbosch algorithm. We also present two techniques known from the literature which improve the running time of the algorithm.
The Bron-Kerbosch algorithm [@bron1973algorithm] enumerates all maximal cliques in undirected, static graphs. It is a widely used recursive backtracking algorithm which is easy to implement and more efficient than alternative algorithms in many practical applications [@ELS13].
$P \gets P \setminus \{v\}$ $X \gets X \cup \{v\}$
The Bron-Kerbosch algorithm, see Algorithm \[alg:bronker\], receives three disjoint vertex-sets as an input: $P$, $R$, and $X$. The set $R$ induces a clique and $P \cup X$ is the set of all vertices which are adjacent to every vertex in $R$. Each vertex in $P \cup X$ is a witness that the clique $R$ is not maximal yet. The set $P$ contains the vertices that have not been considered yet whereas the set $X$ includes all vertices that have already been considered in earlier steps. In each recursive call, the algorithm checks whether the given clique $R$ is maximal or not. If $P \cup X = \emptyset$, then there are no vertices that can be added to the clique and therefore, the clique is maximal and can be added to the solution. Otherwise, the clique is not maximal because at least one vertex exists that is adjacent to all vertices in $R$ and consequently would form a clique with $R$. For each $v \in P$ the algorithm makes a recursive call for the clique $R \cup \{v\}$ and restricts $P$ and $X$ to the neighborhood of $v$. After the recursive call, vertex $v$ is removed from $P$ and added to $X$. This guarantees that the same maximal cliques are not detected multiple times. For a graph $G=(V,E)$ the algorithm is initially called with $P=V$ and $R=X=\emptyset$.
### Pivoting {#subsection:BKPivoting}
@bron1973algorithm introduced a method to increase the efficiency of the basic algorithm by choosing a pivot element to decrease the number of recursive calls. It is based on the observation that for any vertex $u \in P \cup X$ either $u$ itself or one of its non-neighbors must be contained in any maximal clique containing $R$. This is true since if neither $u$ nor one of the non-neighbors of $u$ are included in a clique containing $R$, then this clique cannot be maximal because $u$ can be added to this clique due to the fact that only neighbors of $u$ were added to $R$. Hence, if we modify the Bron-Kerbosch algorithm (Algorithm \[alg:bronker\]) so that we choose an arbitrary pivot element $u \in P \cup X$ and iterate only over $u$ and all its non-neighbors, then we still enumerate all maximal cliques containing $R$ but decrease the number of recursive calls in the for-loop of Algorithm \[alg:bronker\]. @tomita2006worst have shown that if $u$ is chosen from $P \cup X$ such that $u$ has the most neighbors in $P$, then all maximal cliques of a graph $G=(V,E)$ are enumerated in $O(3^{\mid V\mid /3})$ time, see Algorithm \[alg:bronkerpivot\].
$P \gets P \setminus \{v\}$ $X \gets X \cup \{v\}$
### Degeneracy of a Graph {#subsection:BKDegeneracy}
Degeneracy is a measure of graph sparsity. Real-world instances of static graphs (especially social networks) are often sparse, resulting in a small degeneracy value [@ELS13]. This motivates a modification of the Bron-Kerbosch algorithm which we present in this section and the complexity analysis of this algorithm parameterized by the degeneracy of the input graph. The degeneracy of a graph is defined as follows.
The *degeneracy* of a static graph $G$ is defined as the smallest integer $d \in \mathbb{N}$ such that each subgraph $G'$ of $G$ contains a vertex $v$ with degree at most $d$.
If a graph has degeneracy $d$, we also call it *$d$-degenerated*. It is easy to see that the maximal clique size of a $d$-degenerated graph is at most $d+1$: If there is a clique of size at least $d+2$, then the vertices of this clique would form a subgraph in which every vertex $v$ of the clique has a degree larger than $d$. For each $d$-degenerated graph there is a *degeneracy ordering*, which is a linear ordering of the vertices with the property that for every vertex $v$ we have that at most $d$ of its neighbors occur at a later position in the ordering. The degeneracy $d$ and a corresponding degeneracy ordering for a graph $G=(V,E)$ can be computed in linear time [@ELS13]: For graph $G$, the vertex with the smallest degree is selected in each step and removed from the graph until no vertex is left. The degeneracy of the graph is the highest degree of a vertex at the time the vertex has been removed from the graph and a corresponding degeneracy ordering is the order in which the vertices were removed from the graph.
For a graph $G=(V,E)$ with degeneracy $d$, @ELS13 showed that using the degeneracy ordering of $G$ in the outer-most recursive call and afterwards using pivoting, all maximal cliques can be enumerated in $O(d \cdot | V | \cdot3^{d/3})$ time, see Algorithm \[alg:bronkerdeg\]. In other words, enumerating maximal cliques is fixed-parameter tractable with respect to the parameter degeneracy $d$ of the input graph.
$P \gets N(v_i) \cap \{v_{i+1}, \ldots ,v_{n-1}\}$ $X \gets N(v_i) \cap \{v_{0}, \ldots ,v_{i-1}\}$
Bron-Kerbosch Algorithm for Temporal Graphs {#section:bronKerboschDelta}
===========================================
We adapt the static Bron-Kerbosch algorithm to the temporal setting to enumerate all $\Delta$-cliques, see Algorithm \[alg:bronkerdelta\]. The input of the algorithm consists of two sets $P$ and $X$ of vertex-interval pairs as well as a tuple $R=(C, I)$, where $C$ is a set of vertices and $I$ a time interval. The idea is that in every recursive call of the algorithm, $R$ is a time-maximal $\Delta$-clique, and the sets $P$ and $X$ contain vertex-interval pairs that are in the $\Delta$-neighborhood of every vertex in $C$ during an interval $I' \subseteq I$. In particular, $P \cup X$ includes all vertex-interval pairs $(v,I)$ for which $(C \cup \{v\}, I)$ is a time-maximal $\Delta$-clique. While each vertex-interval pair in $P$ still has to be combined with $R$ to ensure that every maximal $\Delta$-clique will be found, for every vertex-interval pair $(v,I') \in X$ every maximal $\Delta$-clique $(C', I'')$ with $C \cup \{v\} \subseteq C'$ and $I'' \subseteq I'$ has already been detected in earlier steps.
We show below that if $\forall (w, I') \in P \cup X \colon I' \subsetneq I$, then there is no vertex $v$ that forms a $\Delta$-clique together with $C$ over the whole time interval $I$. Consequently, $R=(C, I)$ is a maximal $\Delta$-clique. In one step, for every vertex-interval pair $(v,I') \in P$ a recursive call is initiated for the $\Delta$-clique $R'=(C \cup \{v\}, I')$ with all parameters restricted to the $\Delta$-neighborhood of $v$ in the time interval $I'$, that is, $P \sqcap N^{\Delta}(v,I')$ and $X \sqcap N^{\Delta}(v,I')$. For the set $P'$ for example, we get a set of all time-maximal vertex-interval pairs $(w,I'')$ for which it holds that $(w,I'') { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v,I')$ and $(w,I'') { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}P$. This restriction is made so that for all $(w,I'') \in P'$ of the recursive call the vertex $w$ is not only a $\Delta$-neighbor of all $x \in C$ but also of the vertex $v$ during the time $I'' \subseteq I'$.
After the recursive call for $\Delta$-clique $(C \cup \{v\}, I')$, the tuple $(v,I')$ is removed from the set $P$ and added to the set $X$ to avoid that the same cliques are found multiple times. For a temporal graph $\mathbb{G}=(V,E,T)$ and a given time period $\Delta$, the *initial call* of Algorithm \[alg:bronkerdelta\] to enumerate all maximal $\Delta$-cliques in graph $\mathbb{G}$ is made with $P = \{(v,T) \mid v \in V\}$, $R = (\emptyset, T)$ and $X=\emptyset$. In the remainder of this document we will always assume that [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} is initially called with those inputs.
Analysis
--------
In the following, we prove the correctness of the algorithm and analyze its running time. We start with arguing that the sets $P$ and $X$ behave as claimed.
\[lemma:setPX\] For each recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} with $R=(C,I)$ and $C\neq\emptyset$, it holds that $P \cup X = \bigsqcap_{v \in C} N^{\Delta}(v,I)$.
We prove this by induction on the recursion depth, that is, the number $|C|$ of vertices in the clique in the current recursive call. In the initial call we have that $C = \emptyset$. In each iteration of the first call we have that $P \cup X = \{(v,T) \mid v \in V\}$ since, whenever a vertex-interval pair is removed from $P$, then it is added to $X$, and initially $P = \{(v,T) \mid v \in V\}$. For every recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} with $R'=(C', I')$, $P'$, and $X'$, and $C' = \{v\}$ for some vertex $v$ we have that $P'=P\sqcap N^{\Delta}(v,I')$ and $X' = X \sqcap N^{\Delta}(v,I')$. Hence, we get $$P' \cup X' = \{(v,T) \mid v \in V\} \sqcap N^{\Delta}(v,I') = N^{\Delta}(v,I').$$ Now we assume that we are in a recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} with $R=(C, I)$, $P$, and $X$, where $|C| > 1$. By the induction hypothesis we know that $P \cup X = \bigsqcap_{v \in C} N^{\Delta}(v,I)$. Let $(v,I') \in P$ be the vertex added to the $\Delta$-clique, that is, in the next recursive call we have that $R'=(C', I')$, with $C' = C \cup \{v\}$, and $P' = P \sqcap N^{\Delta}(v,I')$ as well as $X' = X \sqcap N^{\Delta}(v,I')$. Then, $$\begin{aligned}
P' \cup X'
&= (P \sqcap N^{\Delta}(v,I'))\cup (X\sqcap N^{\Delta}(v,I'))\\
&= (P\cup X)\sqcap N^{\Delta}(v,I')\\
&= \underset{w \in C}{\bigsqcap} N^{\Delta}(w,I) \sqcap N^{\Delta}(v,I')\\
&= \underset{w \in C'}{\bigsqcap} N^{\Delta}(w,I').
\end{aligned}$$ This proves the claim.
Next, we show that the set $R$ behaves as claimed, that is, $R$ is indeed a time-maximal $\Delta$-clique in each recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{}.
\[lemma:timeMax\] In each recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{}, $R=(C, I)$ is a time-maximal $\Delta$-clique.
We show by induction on the recursion depth that $R=(C, I)$ is a time-maximal $\Delta$-clique and that all vertex-interval pairs $(v, I')$ in $P$ are $\Delta$-neighbors during $I'$ to all vertices in the $\Delta$-clique $R$ and that $I'$ is maximal under this property. The algorithm is initially called with $R = (\emptyset, T)$, which is a trivial time-maximal $\Delta$-clique, and $P = \{(v,T) \mid v \in V\}$, which fulfills the desired property since the initial $\Delta$-clique is empty and $T$ is the maximum time interval. In each recursive call [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} is called with $(P \sqcap N^{\Delta}(v,I'), (C \cup \{v\}, I'), X \sqcap N^{\Delta}(v,I'))$ for some $(v, I')\in P$. By the induction hypothesis, $v$ is a $\Delta$-neighbor to all vertices in $C$ during time interval $I'$, and $I'$ is maximal. Hence, $(C \cup \{v\}, I')$ is a time-maximal [$\Delta$-clique]{}. Furthermore, each vertex-interval pair $(v', I'')$ in $P \sqcap N^{\Delta}(v,I')$ is in the $\Delta$-neighborhood of each vertex-interval pair $(v'', I')$ with $v'' \in C \cup \{v\}$, since it is both in $P$ and hence in the $\Delta$-neighborhood of each vertex in $C$ and in $N^{\Delta}(v,I')$. The maximality of $I'$ follows from the fact that the $\Delta$-cut and $\Delta$-neighborhood operations preserve maximality of intervals by definition.
Now we can prove the correctness of the algorithm.
\[thm:correctness\] Let $\mathbb{G}=(V,E,T)$ be a temporal graph. If algorithm [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{}$(P, R, X)$ is run on input $(V \times \{T\}, (\emptyset, T), \emptyset)$, then it adds all maximal $\Delta$-cliques of $\mathbb{G}$, and only these, to the solution.
Let $R^*=(C^*, I^*)$ be a maximal $\Delta$-clique with $|C^*|>1$. For a recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} on $(P, R, X)$, say that a vertex is a *candidate*, if there is an interval $I$ with $I^* \subseteq I$ such that $(v, I) \in P$. We show by induction on $|C^*| - \ell$, that for each $\ell = 0, 1, \ldots, |C^*|$ there is a recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} on $(P, R = (C, I), X)$ with $C \subseteq C^*$ and $\ell = |C^* \setminus C|$ candidates.
Clearly, in the initial call, $C = \emptyset \subseteq C^*$ and each vertex in $C^*$ is a candidate. Now assume that there is a recursive call with $(P, R = (C, I), X)$ and $C \subseteq C^*$, and with $\ell - 1 = |C^* \setminus C|$ candidates. Consider the for-loop in that recursive call and consider the first vertex-interval pair $(v, I')$ in that loop in which $v$ is a candidate and $I^* \subseteq I'$. [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} proceeds with a recursive call on $(P \sqcap N^{\Delta}(v, I'), R' = (C \cup \{v\}, I'), X
\sqcap N^{\Delta}(v, I'))$. Observe that each candidate except $v$ remains a candidate also in this recursive call. Furthermore, $|C^* \setminus (C \cup \{v\})| = \ell$. Thus, by induction there is a recursive call with the sets $(P, R, X)$ in which $R^* =
R$. Since $R^*$ is maximal by assumption, for each vertex-interval pair $(w, I'') \in \bigsqcap_{v \in C^*} N^{\Delta}(v, I^*)$ we have $I'' \subsetneq I^*$. By Lemma \[lemma:setPX\] we have $\forall (w, I') \in P \cup X \colon I' \subsetneq I^*$ and hence, $R^*$ is added to the solution.
Now assume that $R=(C,I)$ is added to the solution. By Lemma \[lemma:timeMax\], $R$ is a time-maximal $\Delta$-clique. By Lemma \[lemma:setPX\] and since $P \cup X = \emptyset$, there is no vertex that can be added to $R$. Hence, $R$ is a maximal $\Delta$-clique.
Next, we analyze the running time of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{}. We start with the following observation.
\[lemma:cliquecount\] For every time-maximal $\Delta$-clique $R$ of a temporal graph $\mathbb{G}=(V,E,T)$, there is at most one recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} with $R$ as an input.
Assume that there are two recursive calls $A$ and $B$ of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} with the same $R=(C, I)$. Let $R'=(C', I')$, with $C'\subset C$ and $I \subseteq I'$, occur in the recursive call corresponding to the closest common ancestor of the recursive calls $A$ and $B$ in the recursion tree. Hence, there are two vertex-interval pairs $(v, J), (w, J') \in P$ that lead to the calls $A$ and $B$, respectively, in the for loop.
Consider the case $v = w$. Then, $J$ and $J'$ must overlap in at least $\Delta$ time steps, because $I \subseteq J, J' \subseteq I'$. However, $P$ is contained in the $\Delta$-cut of the $\Delta$-neighborhoods of $C'$ over $I'$ and thus, for each vertex no two vertex-interval pairs in $P$ overlap in $\Delta$ time steps, a contradiction.
Now consider the case $v \neq w$. Without loss of generality due to symmetry assume that $(v, J)$ is processed first in the for loop. Then, when processing $(w, J')$, pair $(v, J)$ has been added to $X$. This is a contradiction to the fact that recursive call $B$ outputs $R$, that is, it outputs a clique with time interval $I \subseteq J$.
Hence, we have that there cannot be two recursive calls of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} with $R=(C, I)$.
Now we upper-bound the running time for computing a $\Delta$-cut.
\[lemma:deltacut\]
Let $X$ and $Y$ be two sets of vertex-interval pairs with the following properties.
For every $(v, I)\in X\cup Y$ we have that $|I|\ge \Delta$,
for every $(v, I)\in X$ and $(v, I')\in X$ we have that $|I\cap I'|<\Delta$,
for every $(v, I)\in Y$ and $(v, I')\in Y$ we have that $|I\cap I'|<\Delta$,
$X$ and $Y$ are sorted lexicographically by first the vertex and then the starting point of the interval.
Then the $\Delta$-cut $X \sqcap Y$ can be computed in $O(|X|+|Y|)$ time such that it is also sorted lexicographically by first the vertex and then the starting point of the interval.
The $\Delta$-cut $X \sqcap Y$ of two sets of vertex-interval pairs $X$ and $Y$ can be computed in the following way.
For every vertex $v$, we do the following:
Select the first vertex-interval pairs $(v, I)$ and $(v, I')$ from $X$ and $Y$, respectively.
If $|I\cap I'|>\Delta$, then add $(v, I\cap I')$ to the output (the $\Delta$-cut). If the endpoint of $I'$ is smaller than the endpoint of $I$, then replace $(v, I')$ with the next vertex-interval pair in $Y$, otherwise replace $(v, I)$ with the next vertex-interval pair in $X$.
Repeat Step 2 until all vertex-interval pairs containing vertex $v$ are processed.
Note that the intervals for each vertex $v$ are added to the output in order of their starting point. Furthermore, by construction of the algorithm we have that for each $(v, I)$ in the output, $(v, I)$ is also in the $\Delta$-cut $X \sqcap Y$. It remains to show that for all $(v, I)\in X$ and $(v, I')\in Y$ with $|I\cap I'|\ge\Delta$ we have that $(v, I\cap I')$ is included in the output. Let $I=[a, b]$ and $I' = [a', b']$. At some point, the procedure processes in Step 2 for the first time one of $(v, I) \in X$ or $(v, I') \in Y$. Without loss of generality, let $(v, I) \in X$ be processed first. If at the same time also $(v, I') \in Y$ is processed, clearly, $(v, I \cap I')$ is added to the output, as required. Now assume that Step 2 processes some other vertex-interval pair $(v, I''=[a'', b''])\in Y$, $a'' < a'$, together with $(v, I) \in X$. Since $|I\cap I'|\ge\Delta$ and $|I'\cap I''|<\Delta$ we have that $b''<b$ and hence, $(v, I)$ is not replaced in this step. Consequently, the procedure eventually adds $(v, I\cap I')$ to the output.
In each step of the procedure at least one new vertex-interval pair is processed and each vertex-interval pair in $X$ and $Y$ is only processed once. Hence, the running time is in $O(|X|+|Y|)$.
Lemmata \[lemma:cliquecount\] and \[lemma:deltacut\] allow us to upper-bound the running time of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} depending on the number of different time-maximal $\Delta$-cliques of the input graph.
\[lemma:runningtime\] Let $\mathbb{G}=(V,E,T)$ be a temporal graph with $x$ distinct time-maximal $\Delta$-cliques. Then [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} enumerates all *maximal* $\Delta$-cliques in $O(x\cdot |E| + |E|\cdot |T|)$ time.
We assume that all edges of the temporal graph are sorted by their time stamp. Note that this can be done in a preprocessing step in $O(|E|\cdot |T|)$ time using Counting Sort. Furthermore, we assume that for each vertex $v$, the $\Delta$-neighborhood $N^{\Delta}(v,T)$ is given. These neighborhoods can be precomputed in $O(|E|)$ time, assuming that the edges are sorted by their time stamps.
By Lemma \[lemma:cliquecount\] we know that for each time-maximal $\Delta$-clique there is at most one recursive call of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{}. By charging the computation of $P'$, $R'$, and $X'$ to the corresponding recursive call, for each recursive call we compute a constant number of $\Delta$-neighborhoods and $\Delta$-cuts. The size of the sets $P$, $X$, and any $\Delta$-neighborhood is upper-bounded by $|E|$ and each of these sets has the property that for every $(v, I)$ and $(v, I')$ out of the same set we have that $|I\cap I'|<\Delta$. Given $N^{\Delta}(v,T)$, $N^{\Delta}(v,I)$ can be computed in $O(|E|)$ time for any $I$ and by Lemma \[lemma:deltacut\], a $\Delta$-cut can be computed in $O(|E|)$ time. Hence, all maximal $\Delta$-cliques can be enumerated in $O(x\cdot|E| + |E|\cdot |T|)$ time.
We now use a general upper bound for the number of time-maximal $\Delta$-cliques in a temporal graph to bound the overall running time of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{}.
Let $\mathbb{G}=(V,E,T)$ be a temporal graph. [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} enumerates all maximal $\Delta$-cliques of $\mathbb{G}$ in $O(2^{|V|} \cdot |T| \cdot |E|)$ time.
Note that the vertex set of each maximal $\Delta$-clique induces a static clique in the static graph $G$ underlying $\mathbb{G}$ that has an edge between two vertices if and only if there is a time-edge in $\mathbb{G}$ between these vertices at some time step. Furthermore, for each clique in $G$, there are at most $|T|$ maximal $\Delta$-cliques because their time intervals are pairwise not contained in one-another. Hence, the number of time-maximal $\Delta$-cliques of any temporal graph is upper-bounded by $2^{|V|} \cdot |T|$. By Theorem \[lemma:runningtime\], we get an overall running time in $O(2^{|V|} \cdot |T| \cdot |E|)$.
Pivoting {#subsection:pivotingBKD}
--------
In this section, we explain how we can decrease the number of recursive calls of [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} by using pivoting. Recall that the idea of pivoting in the classic Bron-Kerbosch algorithm for static graphs is based on the observation that for any vertex $u \in P \cup X$ either $u$ itself or one of its non-neighbors must be contained in any maximal clique containing $R$. Vertex $u$ is also called *pivot*.
A similar observation holds for maximal $\Delta$-cliques in temporal graphs. Instead of vertices, however, we now choose vertex-interval pairs as pivots: For any $(v_p,I_p) \in P \cup X$ and any maximal $\Delta$-clique $R_{\max} = (C_{\max}, I_{\max})$ with $I_{\max} \subseteq I_p$, either vertex $v_p$ or one vertex $w\neq v_p$ which is not a $\Delta$-neighbor of $v_p$ during the time $I_{\max}$, that is, $(w,I_{\max}) \not { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v_p,I_p)$, must be contained in $C_{\max}$.
By choosing a pivot element $(v_p,I_p) \in X \cup P$ we only have to iterate over all elements in $P$ which are not in the $\Delta$-neighborhood of the pivot element, see Algorithm \[alg:bronkerdeltaPivot\]. In other words, we do not have to make a recursive call for any $(w,I') \in P$ which holds $(w,I') { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v_p,I_p)$.
In Figure \[figure:Pivoting\] we give an illustrative example for pivoting. In this example, we assume that the algorithm runs on a temporal graph such that the set $P= \{(a,[0,8 ]),(b,[0,4]),(c,[1,3]),(c,[5,8])\}$ occurs within a recursive call of [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}. For simplicity, we show in Figure \[figure:PElements\] only the subgraph containing the elements of $P$ and the relation between these elements rather than displaying the whole graph. In Figure \[figure:PivotElementWithDeltaNeighborhood\], we choose element $(a,[0,8])$ (hatched) as pivot. It can be seen that the elements $(b,[0,4])$ and $(c,[5,8])$ lie completely in the $\Delta$-neighborhood (dotted) of the pivot, that is, $(b,[0,4]),(c,[5,8]) { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(a, [0,8])$. These two elements can therefore be left out in the iteration over the elements in $P$ of the [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{}. We only have to iterate over the pivot $(a,[0,8])$ and the element $(c,[1,3])$ which is not completely in the $\Delta$-neighborhood of our chosen pivot. In Figure \[figure:DeltaCliques\], we can see that for every maximal [$\Delta$-clique]{} $(C,I)$ with respect to $P$ either $a \in C$, $I \subseteq [0,8]$ or $c \in C$, $I \subseteq [1,3]$. The figure hence shows that iterating over the elements $(b,[0,4])$ and $(c,[5,8])$ will not find any maximal [$\Delta$-clique]{} that we do not find via one of the elements $(a,[0,8])$ and $(c,[1,3])$.
Next, we formally prove the correctness of this procedure.
\[lemma:pivoting\] For each $\Delta$-clique $R=(C,I)$ and a pivot element $(v_p,I_p) \in P \cup X$, the following holds: for every $R_{\max} = (C_{\max}, I_{\max})$ with $C \subset C_{\max}$ and $I_{\max} \subseteq I_p \subseteq I$ it either holds that $v_p \in C_{\max}$ or otherwise there is a vertex $w \in C_{\max}$ that satisfies $(w,I') \in P \cup X $, $I_{\max} \subseteq I'$, and $(w,I_{\max}) \not { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v_p,I_p)$, and consequently $(w,I') \not { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v_p,I_p)$.
Let $R_{\max} = (C_{\max}, I_{\max})$ be a maximal $\Delta$-clique with $C \subset C_{\max}$ and $I_{\max} \subseteq I_p \subseteq I$. Assume that $v_p \notin C_{\max}$ and for each $w \in C_{\max}$ it holds that $(w ,I_{\max}) { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v_p,I_p)$. Consequently, for each $w \in C_{\max} \setminus C$ there exists a $(w,I') \in P \cup X$ with $I_{\max} \subseteq I'$. Because $v_p$ is a $\Delta$-neighbor of all vertices in $C_{\max} \setminus C$ at least during $I_{\max}$ and a $\Delta$-neighbor of all vertices in $C$ during $I_p$, the vertex $v_p$ can be added to the $\Delta$-clique $R_{\max}$, yielding another [$\Delta$-clique]{} with at least the same lifetime. This is a contradiction to the assumption that $R_{\max}$ is maximal.
choose pivot element $(v_p,I_p) \in P \cup X$
An optimal pivot element is chosen in such a way that it minimizes the number of recursive calls. It is the element in the set $P \cup X$ having the largest number of elements in $P$ in its $\Delta$-neighborhood. We have seen that the whole procedure is quite similar to pivoting in the basic Bron-Kerbosch algorithm but with one difference: we are able to choose more than one pivot element. The only condition that has to be satisfied is that the time intervals of the pivot elements cannot overlap: For each $\Delta$-clique $R=(C,I)$ in a recursive call of the algorithm, choosing a pivot element $(v_p,I_p) \in P \cup X$ only affects maximal $\Delta$-cliques $R_{\max} = (C_{\max}, I_{\max})$ fulfilling $I_{\max} \subseteq I_p$. Moreover, for all elements $(w,I') \in P$ satisfying $(w,I') { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v_p,I_p)$ it holds $I' \subseteq I_p$. Consequently, a further pivot element $(v_p',I_p') \in P\cup X$ fulfilling that $I_p'$ does not overlap with $I_p$ neither interferes with the considered maximal $\Delta$-cliques nor with the vertex-interval pairs in $P$ that are in the $\Delta$-neighborhood of the pivot element $(v_p,I_p)$. The problem of finding the optimal set of pivot elements in $P \cup X$ can be formulated as a weighted interval scheduling maximization problem:
A set $J$ of jobs $j$ with a time interval $I_j$ and a weight $w_j$ each.
Find a subset of jobs $J' \subseteq J$ that maximizes $\sum_{j \in J'}w_j$ such that for all $i, j \in J'$ with $i \not = j$, the time intervals $I_i$ and $I_j$ do not overlap.
In our problem, the jobs are the elements of $P \cup X$ and the weight of an element is thereby the number of all elements that are in $P$ and lie in the $\Delta$-neighborhood of this element. Formally, the jobs are the elements $(v,I') \in P \cup X$, the corresponding time interval is $I'$ of the element $(v,I')$ and the corresponding weight $w_{(v,I')}= | \{(v,I) \mid (v,I) \in P \wedge (v,I) { \mathrel{\vphantom{\sqsubset}\text{ \mathsurround=0pt
\ooalign{$\sqsubset$\cr$-$\cr} }}}N^{\Delta}(v,I')\} |$. This problem can be solved efficiently in $O(\min(|E|, |V|\cdot |T|) \cdot \log (\min(|E|, |V|\cdot |T|)))$ time by using dynamic programming [@kleinberg2006algorithm Chapter 6.1] under the assumption that the weights of the potential pivot elements are known.
Degeneracy of Temporal Graphs {#sec:degeneracy}
=============================
Recall from Section \[subsection:BKDegeneracy\] that one can upper-bound the running time of the static Bron-Kerbosch algorithm using the degeneracy of the input graph. The degeneracy of a graph $G$ is the smallest integer $d$ such that every non-empty subgraph of $G$ contains a vertex of degree at most $d$. We now give an analogue for the temporal setting, motivated by the fact that static graphs are often sparse in practice as measured by small degeneracy [@ELS13]. Intuitively, we want to capture the fact that a temporal graph keeps its degeneracy value during its whole lifetime.
A temporal graph $\mathbb{G}=(V,E,T)$ has $\Delta$-slice degeneracy $d$ if for all $t \in T$ we have that the graph $G_t=(V, E_t)$, where $E_t = \{\{v, w\} \ | \ (\{u, w\}, t') \in E \text{ for some } t' \in [t, t+\Delta]\}$, has degeneracy at most $d$.
Using the parameter $\Delta$-slice degeneracy, we can upper-bound the number of time-maximal $\Delta$-cliques of a temporal graph.
\[lemma:cliquecountdeg\] Let $\mathbb{G}=(V,E,T)$ be a temporal graph with $\Delta$-slice degeneracy $d$. Then, the number of time-maximal $\Delta$-cliques in $\mathbb{G}$ is at most $3^{d/3}\cdot 2^{d+1}\cdot |V|\cdot |T|$.
Let $\mathbb{G}=(V,E,T)$ be a temporal graph with $\Delta$-slice degeneracy $d$. Then we call the graph $G_t=(V, E_t)$, where $E_t = \{\{v, w\} \ | \ (\{u, w\}, t') \in E \text{ for some } t' \in [t, t+\Delta]\}$ a *$\Delta$-slice* of $\mathbb{G}$ at time $t$. The vertex set of each time-maximal $\Delta$-clique which starts at time $t$ is also a clique in $G_t$, otherwise there would be two vertices which are disconnected for more than $\Delta$ time-steps. Since $G_t$ has degeneracy at most $d$, the number of maximal cliques of $G_t$ is upper-bounded by $3^{d/3}\cdot |V|$ [@ELS13]. Furthermore, the maximum size of a clique is upper-bounded by $d+1$. Hence, the total number of cliques is upper-bounded by $3^{d/3}\cdot 2^{d+1}\cdot |V|$. Note that for each of those cliques we have at most one time-maximal $\Delta$-clique starting at time $t$. Hence, the total number of $\Delta$-cliques is at most $3^{d/3}\cdot 2^{d+1}\cdot |V|\cdot|T|$.
Lemma \[lemma:cliquecountdeg\] now allows us to bound the running time of Algorithm \[alg:bronkerdelta\] using the $\Delta$-slice degeneracy $d$ of the input graph $\mathbb{G}$.
\[thm:fpt\] Let $\mathbb{G}=(V,E,T)$ be a temporal graph with $\Delta$-slice degeneracy $d$. Then, [<span style="font-variant:small-caps;">BronKerboschDelta</span>]{} enumerates all $\Delta$-cliques of $\mathbb{G}$ in $O(3^{d/3}\cdot 2^d\cdot |V|\cdot|T|\cdot |E|)$ time.
By Lemma \[lemma:cliquecountdeg\] we know that the number of time-maximal $\Delta$-cliques in a temporal graph with $\Delta$-slice degeneracy $d$ is at most $3^{d/3}\cdot 2^{d+1}\cdot |V|\cdot |T|$. Hence, by Theorem \[lemma:runningtime\], we get an overall running time in $O(3^{d/3}\cdot 2^d\cdot |V|\cdot|T|\cdot |E|)$.
Note that Theorem \[thm:fpt\] implies that enumerating all maximal $\Delta$-cliques is fixed-parameter tractable with respect to the parameter $\Delta$-slice degeneracy. Hence, while NP-hard in general, the problem can be solved efficiently if the $\Delta$-slice degeneracy of the input graph is small.
Experimental Results {#section:implExp}
====================
In this section we present our experimental results. We give the [$\Delta$-slice degeneracy]{} of several real-world temporal graphs for several values for $\Delta$. Then we show the behavior of our implementation of [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{} (Algorithm \[alg:bronkerdeltaPivot\]) applied to these real-world temporal graphs and compare it to the algorithm implemented by [@viard2015computing]{}.
Setup and Statistics
--------------------
We now give details of the implementation and the used reference algorithm, and introduce the data sets we used in the experiments. Furthermore, we explain how the values of $\Delta$ were chosen, give some statistics for the data set, and calculate the [$\Delta$-slice degeneracy]{} of the data sets for the chosen values of $\Delta$.
#### Implementation.
We implemented[^4] [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{} with slight modifications that allow the algorithm to use multiple pivot elements (we refer to this version as [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\*). Furthermore, we implemented a simple algorithm to compute the [$\Delta$-slice degeneracy]{}. Both implementations are in Python 2.7.12 and all experiments were carried out on an Intel Xeon E5-1620 computer with four cores clocked at 3.6GHz and 64GB RAM. We did not utilize the parallel-processing capabilities although it should be easy to achieve almost linear speed-up with growing number of cores due to the simple nature of [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}. The operating system was Ubuntu 16.04.4 with Linux kernel version 4.4.0-57. We compared [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\* with the algorithm by [@viard2015computing]{} which was also implemented in Python. We modified their source code[^5] by removing the text output in their implementation in order to avoid speed differences. We call their algorithm Algorithm VLM below.
#### Data Sets.
We chose several freely available real-world temporal graphs aiming for an overview over the different kinds of contexts in which such graphs arise, that is, an overview over different modes of communication and different kinds of entities and environments in which this communication takes place. However, a focus is on temporal graphs based on physical proximity of individuals, since previous work on [$\Delta$-clique]{}s also focused on these [@viard2015computing; @Viard2015Dyno]. The contexts and sources of our test set of temporal graphs are as follows:
- internet-router communication: [@leskovec2005graphs],
- email communication: [@EMT2011],
- social-network communication: [@opsahl2009clustering], and
- physical-proximity[^6] between
- high school students: , , [@gemmetto2014mitigation; @stehle2011high; @fournet2014contact],
- patients and health-care workers: [@vanhems2013estimating],
- attendees of the ACM Hypertext 2009 conference: [@isella2011s],
- attendees of the Infectious SocioPatterns event: [@isella2011s], and
- children and teachers in a primary school: [@stehle2011high].
Table \[tab:stats\] contains the number of vertices, edges, temporal resolution, and lifetime of the corresponding temporal graphs. As a time step we fixed one second for each of the data sets. [@viard2015computing]{}, as the first work on enumerating [$\Delta$-clique]{}s, used the data set in their experiments.
#### Chosen values of $\Delta$.
In order to limit the influence of time scales in the data and to make running times comparable between instances, as well as to be able to present the results in a unified way, we chose the $\Delta$-values as follows. We decided on a reference point of the *edge appearance rate* that is, of the average number of edges per time step and we fixed a set of $\Delta$-values for this reference point. For each considered instance we then scaled the reference $\Delta$-values proportionally to the quotient of the reference edge appearance rate and the edge appearance rate in the instance.
As the reference point we chose the edge appearance rate of $1/5$ edges per time step; this value was chosen for convenience within the interval of edge-appearance rates in the studied data sets (see Table \[tab:stats\]). Since, intuitively, the $\Delta$-values of interest in practice increase exponentially, we chose as reference $\Delta$-values the numbers $0$ and $5^i$ for $i = 1, 2, \ldots$. As mentioned, for each instance, these values are then multiplied by the quotient of edge appearance rates. That is, if the instance has $m$ edges and lifetime $\ell$, then we scaled the reference $\Delta$-values by the factor $(1/5)/(m/\ell) = \ell/(5m)$. For example, for we obtain the $\Delta$-values $\{0, 80, 404, 2024, 10121, 50606,$ $253034, \ldots\}$. For reference, recall that each time step in corresponds to one second (a day has 86,000 seconds and a week has 604,800 seconds). In figures, we refer to each scaled value of $\Delta$ by $\Delta \sim 5^i$ for some concrete $i$. @Viard2015Dyno used $\Delta$-values according to $60$ seconds, $15$ minutes, $1$ hour and $3$ hours.
\[tab:stats\]
\[tab:dsd\]
#### [$\Delta$-Slice Degeneracy]{}.
The [$\Delta$-slice degeneracies]{} for our set of instances are shown in Table \[tab:dsd\] together with the static degeneracy of the underlying static graph which has an edge whenever there is an edge at some time step in the temporal graph. Clearly, as the value of $\Delta$ increases, the [$\Delta$-slice degeneracy]{} approaches—and is upper-bounded by—the static degeneracy. The static degeneracy of our instances is small in comparison with the size of the graph. This falls in line with the analysis by @ELS13 for many real-world graphs. Moreover, for many practically relevant values of $\Delta$ the [$\Delta$-slice degeneracy]{} is still significantly smaller. For example, in the instance , the scaled value of $\Delta$ corresponding to $5^3$ equals 2204 time steps (seconds) and the corresponding [$\Delta$-slice degeneracy]{} is 5. This indicates that [$\Delta$-slice degeneracy]{} can be a very promising (that is, also small) parameter when designing and analyzing algorithms for temporal graphs.
We computed the [$\Delta$-slice degeneracies]{} using a straightforward approach. We iteratively computed for each $\Delta$-long time interval the graph induced by the edges in that time interval. For each of these graphs we computed the static degeneracy using an implementation from the NetworkX python library [@HSS08]. This approach is rather inefficient. For example, it took about seven hours to compute the [$\Delta$-slice degeneracy]{} for with $\Delta \sim 5^5$ (equating to a $\Delta$ value of about two hours).
Results and Running Times
-------------------------
We now study the efficiency of [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\*, evaluate pivoting strategies, and compare the result to Algorithm VLM.
#### Pivoting.
Generally we observed that pivoting plays a negligible role when $\Delta$ is small compared to the overall lifetime of the graph, that is, when $\Delta$ is less than roughly one third of the lifetime. In this case, pivoting has almost no effect on the running time and the number of recursive calls. For larger values of $\Delta$, however, pivoting can make a clear difference depending on the type of temporal graph.
We tested five strategies for selecting a set of pivots from $P$ in [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\*. Call a set of pivots is *maximal* if the interval of each element from $P$ overlaps with at least one pivot. We tested the following variants of pivot sets:
1. a single arbitrary pivot,
2. a single pivot maximizing the number of elements removed from $P$,
3. an arbitrary maximal set of pivots (pivots picked one-by-one arbitrarily),
4. a maximal set of pivots (pivots picked one by one according to the maximum number of further elements removed from $P$), and
5. a set of pivots which maximizes the number of elements removed from $P$.
Clearly, each strategy has its own trade-off between the time needed to compute the pivots and the possible reduction in recursive calls.
![Running time for different pivoting strategies on .[]{data-label="fig:pivot-run-hsb"}](pivot-run-hsb)
![Running time for different pivoting strategies on .[]{data-label="fig:pivot-run-soc"}](pivot-run-soc)
Running times are given for in Figure \[fig:pivot-run-hsb\] with $\Delta$ between 15,000 and 725,000. We note that running times for some very small values of $\Delta$ below 15,000 are larger than 30s and hence do not fit in the chart. We consider this phenomenon more closely below. For $\Delta \leq {}$15,000 there is no appreciable difference between the pivoting strategies. In terms of relative difference between pivoting strategies, seems to be a representative example. Strategies 1G and MG seem to be the best options: they do not incur much overhead compared to no pivoting for small $\Delta$ and yield strong running time improvements for larger $\Delta$. In comparison to no pivoting, strategies 1G and MG achieve a 60% reduction in recursive calls for $\Delta$-values of around $7 \cdot 10^6$ in . Since the running times of strategy 1G and MG are so close to each other we conclude that in most cases there is only one important pivot that should be selected. We were surprised to see that maximizing the overall number of elements removed from $P$ via the pivot set (strategy MM) results in slightly worse running times and slightly larger numbers of recursive calls. The number of elements that are removed by a pivot in one recursive call of the algorithm ranges between one and 14 while many of the calls remove two to four elements. Notice that occasional reduction by ten or more elements can substantially decrease the search space, because in general its size depends exponentially on the size of $P$.
Figure \[fig:pivot-run-soc\] shows running times for . On this graph, pivoting seldom removes more than one element from the candidate set $P$ in one call of the recursive procedure. Hence, for this instance, pivoting mainly incurs overhead for computing the pivots, but do not substantially decrease the search space. We consequently observe about 10% slower running times, regardless of the pivoting strategy.
In conclusion, strategy 1G offers the best trade-off between additional running time spent with computing the pivot(s) and running time saved due to decreased number of recursive calls. Overall, the the possible benefits seem to outweigh the overhead incurred by pivoting on some instances. All remaining experiments were thus carried out with strategy 1G.
#### Running Times and Comparison with Algorithm VLM.
\
\
\[tab:holist\]
We experimented with [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\* (Algorithm \[alg:bronkerdeltaPivot\]) using pivoting strategy 1G and with Algorithm VLM for $\Delta = 0$ and $\Delta \sim 5^i$ with $i = 1, 3, 5, 7, 9$ (where the lifetime allowed such values of $\Delta$). An excerpt of the results is given in Table \[tab:holist\]. Clearly, larger instances with more vertices or edges demand a longer running time. However, even large instances like can still be solved within one hour.
From our theoretical results in Section \[section:bronKerboschDelta\] we expected that the running time of [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\* increases exponentially with growing [$\Delta$-slice degeneracy]{}. As the [$\Delta$-slice degeneracy]{} grows very slowly with increasing $\Delta$ (see Table \[tab:dsd\]), we expected a corresponding moderate growth in running time with respect to $\Delta$. For larger $\Delta$, this is consistent with the experimental results, as shown in Figures \[fig:pivot-run-hsb\], \[fig:pivot-run-soc\] and Table \[tab:holist\]. However, for (very) small $\Delta$ we often observe an initial spike in the running time (and number of $\Delta$-cliques) which then subsides. This is also shown in Figure \[fig:smalld-hsb\]. A possible explanation for this spike is that, for small $\Delta$, the $\Delta$-neighborhood of many vertices becomes very fragmented, leading to large candidate sets $P$ in the algorithm (although the size of $P$ is still linear in the input size for constant [$\Delta$-slice degeneracy]{}). Furthermore, if $\Delta$ is small, then many singleton edges may form maximal $\Delta$-cliques themselves. These $\Delta$-cliques then get taken up into larger maximal $\Delta$-cliques when $\Delta$ increases, which decreases the number of $\Delta$-cliques and running times for [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\*.
On our algorithm notably is comparably efficient given the relatively large size (see Figures \[fig:pivot-run-soc\] and \[fig:smalld-soc\]). Furthermore, the number of [$\Delta$-clique]{}s does not seem to vary strongly with changing values of $\Delta$. These two facts may hint at some special structure that is present in temporal graphs based on online social networks, in addition to small [$\Delta$-slice degeneracy]{}.
Algorithm VLM is usually faster than [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\* for small values of $\Delta$ below the $\Delta \sim 5^3$ threshold. Starting from there, however, [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\* outperforms Algorithm VLM with running times smaller by at least one order of magnitude and up to three orders of magnitude (see Table \[tab:holist\]). In terms of main memory, 385MB is the maximum used by [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\* over all solved instances, attained on for $\Delta = 0$. On this instance, Algorithm VLM uses 494MB and often more than 1GB.
![Running time vs. $\Delta$ on .[]{data-label="fig:smalld-hsb"}](smalld-hsb)
![Running time vs. $\Delta$ on .[]{data-label="fig:smalld-soc"}](smalld-soc)
Finally we mention that, when increasing the time limit to six hours, [<span style="font-variant:small-caps;">BronKerboschDeltaPivot</span>]{}\* can solve all instances of for $\Delta = 0$ and $\Delta \sim 5^i$ for $i = 1, 3, 5, 7, 9$ wherein the last value of $\Delta$ involves enumerating $43 \cdot 10^6$ maximal $\Delta$-cliques.
Conclusion and Outlook {#sec:conclusion}
======================
We studied the algorithmic complexity of enumerating $\Delta$-cliques in temporal graphs. We adapted the Bron-Kerbosch algorithm ([@bron1973algorithm]), including the procedure of pivoting to reduce the number of recursion calls, to the temporal setting and provided a theoretical analysis. For the theoretical analysis, we formalized and employed the concept of $\Delta$-slice degeneracy which may be a useful parameter when analyzing problems in sparse temporal graphs. In experiments on real-world data sets, we showed that our algorithm is notably faster than the first approach for enumerating all maximal $\Delta$-cliques in temporal graphs due to @Viard2015Dyno [@viard2015computing]. Our experimental results further reveal that pivoting can notably decrease the running time for large values of $\Delta$. Furthermore, we measured the $\Delta$-slice degeneracy for different $\Delta$-values and showed that it is reasonably small in many real-world data sets.
As to future research, an algorithmic challenge is to find a more efficient way to compute the [$\Delta$-slice degeneracy]{} of a given temporal graph, perhaps via different characterizations as in the case of static graphs. See [@ELS13] for an account of several equivalent definitions of the degeneracy of a static graph. Regarding the adapted version of the Bron-Kerbosch algorithm, our theoretical analysis (based on the $\Delta$-slice degeneracy parameter) of the running time still leaves room for improvement. In particular, we leave the impact of pivoting on the running time upper bound as an open question for future research. It furthermore makes sense to try and implement further improved branching rules on top of pivoting. This was also successful for the static Bron-Kerbosch algorithm [@Nau16]. Another interesting question is whether an analogue to the degeneracy ordering can be defined in the temporal setting and, if so, whether it can be used to further improve the algorithm.
#### Acknowledgements. {#acknowledgements. .unnumbered}
Anne-Sophie Himmel, Hendrik Molter and Manuel Sorge were partially supported by DFG, project DAPA (NI 369/12). Manuel Sorge gratefully acknowledges support by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 631163.11 and by the Israel Science Foundation (grant no. 551145/14).
We are grateful to two anonymous SNAM reviewers whose feedback helped to significantly improve the presentation and to eliminate some bugs and inconsistencies.
[^1]: Code freely available at <http://fpt.akt.tu-berlin.de/temporalcliques/> (GNU General Public License).
[^2]: A preliminary version of this article appeared in the Proceedings of the 2016 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining [@HMNS16]. Parts of this work are based on the first author’s Bachelor thesis at TU Berlin [@Him16].
[^3]: In static graphs, the closed neighborhood of a vertex includes the vertex itself.
[^4]: Code freely available at <http://fpt.akt.tu-berlin.de/temporalcliques/> (GNU General Public License).
[^5]: Code freely available at <https://github.com/TiphaineV/delta-cliques> .
[^6]: Available at <http://www.sociopatterns.org/datasets> .
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The location of the images in a multiple-image gravitational lens system are strongly dependent on the orientation angle of the mass distribution. As such, we can use the location of the images and the photometric properties of the visible matter to constrain the properties of the dark halo. We apply this to the optical Einstein Ring system 0047-2808 and find that the dark halo is almost spherical and is aligned in the same direction as the stars to within a few degrees.'
author:
- 'Randall B. Wayth'
- 'Rachel L. Webster'
title: 'The dark matter halo of the gravitational lens galaxy 0047-2808'
---
Introduction
============
Numerical simulations of Cold Dark Matter (CDM) have been very successful in reproducing the observed large scale structure of the universe. The CDM model predicts that the dark matter (DM) haloes of today’s galaxies are assembled through successive mergers of smaller haloes. Simulations using only dark matter predict that the haloes should be quite prolate, however it is not clear how gas and/or stars interacting with the dark matter will change the shape of the halo. Studies have suggested that the DM halo can become more or less cuspy ([El-Zant]{}, [Shlosman]{}, & [Hoffman]{}, 2001; [Tissera]{} & [Dominguez-Tenreiro]{}, 1998) and rounder ([Evrard]{}, [Summers]{}, & [Davis]{}, 1994; [Dubinski]{}, 1994) after the interaction with stars and gas. An important test of galaxy formation and evolution models will be to compare the shape and profile of galaxy haloes with observed haloes. Thus, simple questions such as: “Do we expect the visible and dark matter to be aligned in elliptical galaxies?” and “Is the dark matter density in the central regions changed by the gravitational dominance of the stars?” must be answered with observations. For instance: the Milky Way, despite being a spiral galaxy, appears to have an almost spherical halo ([Ibata]{} [et al.]{}, 2001).
Gravitational lensing offers a method to tightly constrain the shape of DM haloes in the population of medium redshift ($0.1 < z < 1.0$) lens galaxies. The image positions in a lens system are highly sensitive to the orientation of the overall mass profile. [Keeton]{}, [Kochanek]{}, & [Falco]{} (1998) showed that the *overall* mass distribution is typically aligned with the visible matter using a sample of lens galaxies and a simple SIE mass model. However, depending on the lens galaxy, the stellar mass can contribute a substantial fraction of the total mass inside the image. The extreme case is the lensed QSO 2237+0305 where the dark matter constitutes only 4% of the projected mass inside the images ([Trott]{} & [Webster]{}, 2002). In this case we expect the visible matter orientation and the total matter orientation derived from a lensing analysis to be very similar. The logical next step is to use a more complicated (stars + halo) model for the lens galaxy to determine the properties of the DM halo alone.
In this paper we use an implementation of the LensMEM algorithm ([Wallington]{}, [Kochanek]{}, & [Narayan]{}, 1996) and a stars+halo lens model to study the optical Einstein Ring 0047-2808 ([Warren]{} [et al.]{}, 1999, 1996) using data from the HST. This system is well suited for the study because it is an isolated lens galaxy so we expect any external shear contributions to be small. The system is a $z=0.485$ elliptical which is lensing a background starbursting galaxy at $z=3.6$.
The algorithm we employ performs a non-parametric source reconstruction to match the observed data for a given lens model. The goodness-of-fit of the model is calculated using a $\chi^2$ taking into account the degrees of freedom used in the source. In this paper we assume $H_0=70$ kms$^{-1}$Mpc$^{-1}$ and $(\Omega_m,\Omega_{\lambda})=(0.3,0.7)$.
Method
======
The data was reduced as described in [Wayth]{} [et al.]{} (2002). The final image of the “ring” is 133x133 with $0.05\arcsec$ pixels as shown in Figure \[fig:img\_and\_model\]. The lens galaxy was best fit with a Sersic profile, where the surface brightness as a function of radius $r$ is $\Sigma=\Sigma_{1/2} \exp\{-B(n)\lbrack(r/r_{1/2})^{1/n}-1\rbrack\}$. The parameter $n$ quantifies the shape of the profile: the values $n=0.5$, $n=1$, and $n=4$ correspond to the Gaussian, exponential, and de Vaucouleurs profiles. Profiles with larger $n$ are more cuspy. $B(n)$ is a constant for a particular $n$ and we used the series asymptotic solution for $B(n)$ provided by [Ciotti]{} & [Bertin]{} (1999). Additional parameters used for the light profile are the axis ratio ($q$) and orientation angle ($\theta_s$). The fitted parameters are shown in Table \[tab:phot\_fits\].
------------------------- ---------
$R_{1/2}$ (pixels) $21.69$
$\Sigma_{1/2}$ (counts) $0.7$
$q$ $0.693$
$\theta_s$ ($\deg$) $125$
n $3.115$
------------------------- ---------
: Photometric parameters for the lens galaxy.[]{data-label="tab:phot_fits"}
We model the galaxy stellar component with fixed parameters from the photometry and allow only the M/L to vary. The halo is modelled as a Pseudo-Isothermal Elliptic Potential (PIEP) with a finite core. The PIEP model is defined by the lensing potential $\psi = b[r_c^2 + (1-\epsilon)x^2 + (1+\epsilon)y^2]^{1/2}$ where $r_c$ is the core radius, $b$ is the mass scale (Einstein radius) and $\epsilon$ is the ellipticity. An additional parameter is used for the orientation angle ($\theta_h$, measured anti-clockwise from horizontal). It is worth noting that the lens can be fit with the PIEP model alone (without a core) with the parameters $b=1.165, \epsilon=0.08$ and $\theta_h=129$. We use this mass scale for the halo model. The source in this system actually has two distinct components. The two-component model explains the location and brightness of all features in the image with a standard lens model. Figure \[fig:img\_and\_model\] shows the model source and corresponding image for the plain PIEP model.
The mass enclosed inside the image is tightly constrained by the Einstein radius. We use this constraint to normalise the stellar M/L for a halo of a given core radius. A large core is equivalent to a constant M/L mass model, whereas a small core will generate an unrealistically low M/L for the observed stellar component of the lens.
In preliminary tests, we found that we cannot fit the data for $r_c \ga 7\arcsec$ (42kpc physical scale length) i.e. constant M/L models cannot fit the data. Therefore we have restricted our analysis to halo core radii $< 7\arcsec$. For the range of allowed core radius values, we have calculated the range of halo ellipticity and orientation angle which produce acceptable fits to the data.
Results
=======
Figure \[fig:halores\] plots the acceptable range of halo ellipticity and orientation angle as a function of core radius. On the left, we see that the halo ellipticity is consistently less than the stellar ellipticity. For $1.5\arcsec < r_c < 2.5\arcsec$, the data permit a halo with projected mass density which is circular, although in all cases the best solution has a halo with non-zero ellipticity.
On the right of Figure \[fig:halores\], the plot shows that the halo orientation angle is independent of the core radius and is in the same direction as the projected stellar major axis (within errors). The acceptable range of orientation angles for $1.5\arcsec < r_c < 2.5\arcsec$ are for non-zero ellipticity.
Conclusion
==========
By using a lens model which separates the stars from the halo, we have been able to determine some of the basic properties of the dark matter halo in the lens system 0047-2808. We find that the projected ellipticity of the halo is not circular, but is substantially rounder than the observed stellar ellipticity. A small range of halo core radii values ($1.5\arcsec < r_c < 2.5\arcsec$) allow the projected halo mass to be circular.
The halo’s core, modelled as a constant density region, must be $< 7\arcsec$ to fit the observation. The core size could be further constrained by applying realistic limits to the stellar M/L which we intend to do in further work.
Finally, we find that although the halo is less elliptical than the stars, the orientation angle of the star’s and halo’s major axis are the same within errors.
, L. & [Bertin]{}, G. 1999, , 352, 447
, J. 1994, , 431, 617
, A., [Shlosman]{}, I., & [Hoffman]{}, Y. 2001, , 560, 636
, A. E., [Summers]{}, F. J., & [Davis]{}, M. 1994, , 422, 11
, R., [Lewis]{}, G. F., [Irwin]{}, M., [Totten]{}, E., & [Quinn]{}, T. 2001, , 551, 294
, C. R., [Kochanek]{}, C. S., & [Falco]{}, E. E. 1998, , 509, 561
, P. B. & [Dominguez-Tenreiro]{}, R. 1998, , 297, 177
, C. M. & [Webster]{}, R. L. 2002, , 334, 621
, S., [Kochanek]{}, C. S., & [Narayan]{}, R. 1996, , 465, 64+
, S. J., [Hewett]{}, P. C., [Lewis]{}, G. F., [Moller]{}, P., [Iovino]{}, A., & [Shaver]{}, P. A. 1996, , 278, 139
, S. J., [Lewis]{}, G. F., [Hewett]{}, P. C., [M[ø]{}ller]{}, P., [Shaver]{}, P., & [Iovino]{}, A. 1999, , 343, L35
, R. B., [Lewis]{}, G. F., [Warren]{}, S. J., & [Hewett]{}, P. C. 2002, in prep.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well- posed. This often requires setting the value of the so-called regularisation parameters that control the amount of regularisation enforced. These parameters are notoriously difficult to set a priori, and can have a dramatic impact on the recovered estimates. In this paper, we propose a general empirical Bayesian method for setting regularisation parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularisation parameters directly from the observed data by maximum marginal likelihood estimation, and can simultaneously estimate multiple regularisation parameters. A main novelty is that this maximum marginal likelihood estimation problem is efficiently solved by using a stochastic proximal gradient algorithm that is driven by two proximal Markov chain Monte Carlo samplers, thus intimately combining modern high-dimensional optimisation and stochastic sampling techniques. Furthermore, the proposed algorithm uses the same basic operators as proximal optimisation algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimisation techniques. We also present a detailed theoretical analysis of the proposed methodology, including asymptotic and non-asymptotic convergence results with easily verifiable conditions, and explicit bounds on the convergence rates. The proposed methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, non-blind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the $\ell_1$, total-variation, total-variation and $\ell_1$, and total-generalised-variation pseudo-norms.'
author:
- 'Ana F. Vidal [^1]'
- 'Valentin De Bortoli [^2]'
- 'Marcelo Pereyra [^3]'
- 'Alain Durmus [^4]'
bibliography:
- 'refs.bib'
title: 'Maximum likelihood estimation of regularisation parameters in high-dimensional inverse problems: an empirical Bayesian approach'
---
Acknowledgements
================
We are grateful to Dr. Charles Deledalle for providing us with a SUGAR implementation for an ADMM solver available at <https://github.com/deledalle/sugar/blob/master/solvers/admm.m>. AD acknowledges financial support from Polish National Science Center grant: NCN UMO-2018/31/B/ST1/00253.
[^1]: Email: af69@hw.ac.uk
[^2]: Email: valentin.debortoli@cmla.ens-cachan.fr
[^3]: Email: m.pereyra@hw.ac.uk
[^4]: Email: alain.durmus@cmla.ens-cachan.fr Part of this work has been presented at the 25th IEEE International Conference on Image Processing (ICIP) [@vidal2018maximum]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we investigate how semantic relations between concepts extracted from medical documents can be employed to improve the retrieval of medical literature. Semantic relations explicitly represent relatedness between concepts and carry high informative power that can be leveraged to improve the effectiveness of retrieval functionalities of clinical decision support systems. We present preliminary results and show how relations are able to provide a sizable increase of the precision for several topics, albeit having no impact on others. We then discuss some future directions to minimize the impact of negative results while maximizing the impact of good results.'
author:
- 'Maristella Agosti, Giorgio Maria Di Nunzio, Stefano Marchesin, Gianmaria Silvello'
bibliography:
- 'Marchesin.bib'
title: A Relation Extraction Approach for Clinical Decision Support
---
<ccs2012> <concept> <concept\_id>10002951.10003317.10003318</concept\_id> <concept\_desc>Information systems Document representation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003318.10011147</concept\_id> <concept\_desc>Information systems Ontologies</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003325.10003330</concept\_id> <concept\_desc>Information systems Query reformulation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003347.10003352</concept\_id> <concept\_desc>Information systems Information extraction</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003371</concept\_id> <concept\_desc>Information systems Specialized information retrieval</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Katarina Uzelac
- Zvonko Glumac
- 'Osor S. Barišić'
date: 'Received: date / Revised version: date'
title:
- 'Short-time dynamics in the 1D long-range Potts model'
- 'Short-time dynamics in the 1D long-range Potts model'
---
Introduction
============
Short-time dynamics (STD) in systems quenched to criticality has attracted considerable attention in the last decade due to the appealing fact that systems even in the early period of relaxation to equilibrium exhibit universal scaling properties which involve both static and dynamic critical exponents [@JSS89; @Huse]. The interest in this phenomenon exists at different levels. From a practical point of view, it offers a useful numerical tool for calculating both dynamic and static critical properties where the critical slowing down is turned into advantage. From a fundamental point of view, it opened a series of questions of current interest from the universal amplitudes to the universality of the fluctuation-dissipation ratio [@Cugliandolo93] in a wider context of ageing phenomena in pure systems [@CG05]. One of the first points of conceptual interest was the emergence of a new independent universal dynamical exponent describing the initial increase of the magnetization in this regime [@JSS89], but related also to the persistence probability of the global order parameter [@Majumdar96]. Since the STD was formulated in the context of the dynamical renormalization group (RG) and the new exponent evaluated within the $\epsilon$-expansion [@JSS89] it has been further investigated, mostly numerically, in a variety of models in two and three dimensions for equilibrium phase transitions [@SZ95; @OSchYZ97; @daSilva02; @Zheng98] and also for out-of-equilibrium ones [@TO98].
Quite a few studies were carried out on models with long-range (LR) interactions. The RG approach of Janssen et al. [@JSS89] was extended to the case of power-law decaying interactions of the form $r^{-d-\sigma}$ in the same continuous n-vector model [@CGLMS00], in the random Ising model [@Chen02], and in the kinetic spherical model [@CGLY00; @BDH07]. Studies of STD at criticality in discrete models with LR interactions, where such an approach does not apply, are still absent. Numerical “advantage” is there rather reduced due to the fast relaxation in the presence of LR interactions.
In this paper we present the first and preliminary numerical study of the 1D LR Potts model, useful as a paradigm that comprises different universality classes obtained by variation of the number of states $q$. We show that, in spite of the difficulties of the numerical approach in the LR case, the scaling properties characteristic for the STD may be well reproduced with a reasonable numerical effort and derive the two dynamical critical exponents in the wide extent of the range-parameter $\sigma$ for two different universality classes.
The outline of the paper is as follows. In Section \[sec:Model\] we give an overview of the model and basic STD properties considered in the paper, followed by the details of our numerical approach. The Section \[sec:Results\] contains the results for two special cases of the Potts model: $q=2$, corresponding to the Ising model, which is compared to the previous RG results, and $q=3$, where the new results are derived in the regime where the transition is of the second order. The conclusion is given in Section \[sec:Conclusion\].
Model and short-time dynamics approach {#sec:Model}
======================================
We consider the 1D Potts model defined by the Hamiltonian $$H = - \sum_{i < j} \; \frac{J}{|i - j|^{1+\sigma}} \; \delta _{s_i, s_j} \; ,
\label{hamilt}$$ where $J>0$, $s_i$ denotes a $q$-state Potts spin at the site $i$, $\delta$ is the Kronecker symbol and the summation is over all the pairs of the system. Hereafter $J=k_B=1$ is used. As is well known [@ACCN88], for $0 <\sigma \leq 1$ the model (\[hamilt\]) has a phase transition at nonzero temperature for all $q$. Only a few exact results are available for its equilibrium critical behavior, but the model was studied in detail by several approximate methods [@GU93; @LB97; @BDD99]. It has a rather complicated phase diagram in the $(q,\sigma)$ plane, involving similar variety of critical regimes, that is encountered in the $(q,D)$ plane of the same model with short-range (SR) interactions. This gives the additional motivation to examine also the dynamical scaling properties in the STD regime depending on $q$ and $\sigma$.
In the present work we are interested in two special cases, $q=2$ and $q=3$ in the range of parameter $\sigma$ corresponding to the nontrivial (non mean-field (MF)) critical regime, where the initial slip of the magnetization can be observed. For $q=2$ this is accomplished for $0.5 <\sigma <1$ [@FMN72]. In the latter case, $q=3$, which belongs to a different universality class, this region is restrained to $\sigma_{c}(q=3) < \sigma < 1$, where $\sigma_{c}(q) > 0.5$ denotes the point of the onset of the first-order phase transition, occurring for $q>2$ and known only approximately [@UG97GU98; @ReynalDiep04]. For these two cases we shall study the nonequilibrium evolution to criticality in early times of several quantities, magnetization, autocorrelation function and time correlations of the magnetization. Let us first briefly remind their scaling properties in the STD regime and explain their implementation to the model (\[hamilt\]).
STD approach
------------
As shown by Janssen [*et al*]{} [@JSS89], if the system is brought out of equilibrium by a quench from high temperature to criticality, and left to evolve following the nonconservative dynamics of Model A (in the sense of reference [@HH77]), then, during the early stage of relaxation it will display universal scaling properties characterized by the static exponents and the new universal dynamic exponent. Consequently, in the system of size $L$ after a quench from high temperature to the critical region in the presence of small initial magnetization $m_0$, the magnetization will obey the scaling relation $$M(t, \tau, L, m_0) = b^{-\beta/\nu} M(t/b^z, b^{1/\nu} \tau, L/b, b^{x_0} m_0),
\label{scgen}$$ where $\tau = (T-T_c)/T_c$, $b$ is a scaling factor and $\beta, \nu$ are the static critical exponents. Besides the dynamical exponent $z$, the scaling involves a new exponent $x_0$ as the anomalous dimension of the initial magnetization $m_0$.
At criticality ($\tau = 0$), and for $L\gg \xi$, equation (\[scgen\]) may be reduced to $$M(t, m_0) = t^{-\beta/(\nu z)} M(1, t^{x_0/z} m_0).
\label{scTc}$$ For early times satisfying $t \ll t_x \approx m_0^{-z/x_0}$, but larger than the microscopic time $t_{micro}$, the r. h. s. can be expanded giving the power-law increase of the magnetization known as the initial slip, $$M(t) \sim m_0 t^{\theta'},
\label{deftheta1}$$ with $\theta' = x_0/z - \beta/(\nu z)$. The magnetization in the model (\[hamilt\]) is defined in a standard way $$\label{eq:m1}
M(t) = \langle M_1(t) \rangle\; = \frac{q}{(q-1)\;L}\;\left< \sum_{i}\; \left( \delta_{s_i(t),1} - \frac{1}{q} \right) \right>,$$ where $1$ denotes the preferential direction among $q$ possible Potts states ${\alpha}$. The brackets $\langle...\rangle$ denote the average over initial conditions and random force.
During the short time after the quench, the correlation length is small compared to the system size, and the exponent $\theta'$ can be derived directly from the power law (\[deftheta1\]) by performing simulations on the chain of a single large size and averaging over a great number of independent runs.
In the absence of the initial magnetization ($m_0 = 0$), equation (\[scgen\]) gives the scaling relation for the $k$-th moment of the magnetization, $$M^{(k)}(t,L) = b^{-k\beta/\nu} M^{(k)}(t/b^{z},L/b).
\label{eq:scMk}$$ In early times, when $\xi(t) \ll L$, the second moment also displays a power-law behavior, $$M^{(2)}(t,L) \sim t^{(d-2\beta/\nu)/z},
\label{eq:Mk}$$ which can be used to derive the anomalous dimension of the order parameter $\beta/\nu$, or the dynamical exponent $z$ directly from the single large chain. To this purpose we use the alternative definition of the order parameter $$\label{eq:mx}
M_x(t) = \frac{q}{(q-1)\;L}\; max_{\alpha} \left[ \sum_{i}\; \left( \delta_{s_i(t),\alpha} - \frac{1}{q} \right) \right]
% M_x(t) = \frac{max_\alpha[\frac{1}{L}\sum_{i}\; (q \; \delta_{s_i(t),\alpha} - 1)]}{q - 1}$$ and the moments of magnetization are obtained as the average $$M^{(k)}(t) = \langle M^{k}_x(t) \rangle.
\label{mxk}$$ Equation (\[eq:mx\]) describes the absolute value of the magnetization and allows us to apply the scaling relation (\[eq:scMk\]) already to the first momentum, and obtain $$\langle M_{x}(t) \rangle\; \sim \; t^{(d/2-\beta/\nu)/z}.
\label{eq:mxpl}$$
The autocorrelation function of the local order parameter is defined in a standard way and also obeys the power-law form $$A(t) = \frac{q}{(q-1)\;L}\; \left< \sum_{i}\; \left(\delta_{s_i(0),s_i(t)} - \frac{1}{q}\right) \right> \; \sim \; t^{-\lambda/z},
\label{autocorr}$$ depending on the combination of both dynamical exponents $\lambda/z=d/z - \theta'$.
For the calculation of the exponent $\theta'$ we shall use another quantity which represents the autocorrelation of the global order parameter. It was shown by Tomé and de Oliveira [@TO98] that the time correlation of magnetization defined as $$Q(t) = \langle M_1(0)M_1(t) \rangle
\label{eq:defQ}$$ also exhibits the initial increase of the power-law form $$Q(t) \sim t^{\theta'}
\label{gcorr}$$ even in absence of the imposed initial magnetization. (Notice that in Equation (\[eq:defQ\]) the definition (\[eq:m1\]) of magnetization should be used and not its absolute value.) For numerical calculations of the exponent $\theta'$, Equation (\[gcorr\]) has a technical advantage compared to the expression (\[deftheta1\]), where the runs should be performed first for several values of the initial magnetization $m_0$ and than the extrapolation to the limit $m_0 \rightarrow 0$ taken in order to obtain the exponent $\theta'$. In return, however, the fluctuations are more pronounced for $Q(t)$ and its calculation requires better statistics.
Numerical calculations
----------------------
Monte Carlo simulations were done on finite chains with periodic boundary conditions by using simple Metropolis dynamics. The system was quenched from a random configuration (high-temperature state) to criticality.
Unlike the earlier studies for the short-range Potts model in 2D, where the critical temperatures are known exactly, in the LR case only the approximate results are available. Satisfactory results for series of different values of $\sigma$ were obtained by the finite-range scaling (FRS) approach [@UG88], cluster mean-field approach [@Monroe99], or Monte Carlo calculations [@ReynalDiep04]. In the present study we use the values for $T_c$ obtained by the FRS [@GU93].
Two approaches were examined - a direct derivation of exponents and a derivation from the finite-size scaling (FSS). In the former approach where the exponents are calculated using a single large system, the correlation length has to be small compared to the system size during times which are taken into account in the evaluation of the power laws. Due to LR interactions, the correlation length increases much faster than in systems with short-range interactions. For illustration we supply here a rough estimate of the increase of the correlation length $\xi(t)$ calculated from the second moment of the spin-spin correlation function [@Brezin82] at the instant t, $$\xi^{2}(t) = \frac{ \sum_{l=1}^{L/2} \; l^2 \; C(l,t)}{\sum_l \; C(l,t)},
\label{eq:defksi}$$ where the correlation function $C(l,t)$ is given by $$C(l,t) = \frac{q}{(q-1)\;L}\; \left< \sum_i \; \left( \;\delta_{s_i(t)s_{i+l}(t)} - \frac{1}{q}\right) \right>.
\label{defcorr}$$ Summation in equation (\[eq:defksi\]) runs only up to $L/2$ because of the periodic boundary conditions.
[crrrrr]{} $\sigma \backslash L$ &100&400 & $1\,000$ & $3\,000$\
0.9 & 2 & 13 & 40 & 168\
0.8 & 1 & 7 & 20 & 54\
0.7 & 1 & 4 & 10 & 30\
0.6 & 0 & 3 & 6 & 15\
As shown in Table \[tb:Table1\], the correlation length increases very rapidly indeed, especially for lower values of $\sigma$. Values for $q=3$ are similar. Consequently, in order to reach sufficiently long time intervals in the power-law regime, all the direct calculations were performed with chains of $3\,000$ sites. All the quantities were averaged over $200\,000$ to $350\;000$ independent runs. Larger numbers of independent runs were used for smaller values of $\sigma$, where the fluctuations are more pronounced. Finally, in the FSS approach small sizes ranging from $L=100$ to $L=400$ were compared.
Results {#sec:Results}
=======
Systematic calculations in cases $q=2$ and $q=3$ were performed for four characteristic values of parameter of range $\sigma =~0.6,~0.7,~0.8,~0.9$.
Increasing of $\sigma$ by moving away from the MF regime up to the limits of relevance of long-range interactions has similar effect in this 1D model as leaving the MF regime by lowering dimensionality down to the lower critical dimensionality in its SR analogue, and we expect to observe similar features. One of them is dependence of dynamical exponents $\theta'$ and $z$ on $\sigma$.
The above choice of $\sigma$ allows to cover evenly the nontrivial critical regime $0.5 < \sigma < 1$ for $q=2$. In the case $q=3$ it covers both first- and second-order transition regimes, but the detailed analysis is focused on the region where the second-order phase transition is expected.
Case $q=2$
----------
### Time correlations of the magnetization and the exponent $\theta'$
The dynamical exponent $\theta'$, which in the SR analogue increases with decreasing of dimensionality [@JSS89], in the present LR case should increase with $\sigma$, which is also in agreement with the RG results [@CGLMS00].
The principal quantity that we used to derive the exponent $\theta'$ is the function $Q(t)$ (\[eq:defQ\]). A summary graph of our numerical simulations for the selected values of $\sigma$ is presented in Figure \[fg:fig1\].
One observes that the microscopic time $t_{micro}$ is short and the linear behavior on the logarithmic scale is established immediately after the first 2-3 steps. The linear regime in the log-log scale becomes shorter as $\sigma$ decreases, but the size $L=3\,000$ is sufficiently long for an accurate evaluation of the exponent $\theta'$ by a fit to equation (\[gcorr\]), which deteriorates only for the lowest $\sigma$ considered.
The errors in present results are not easy to estimate, because they may be introduced by several sources: insufficient statistics, arbitrariness in the selection of the linear segment of the plot, or using the approximate values for $T_c$. The error bars given in tables cover the first two sources and could be systematically reduced by increasing the number of independent runs and the size of the chains considered. Yet, we stress that the third one cannot be estimated directly.
The obtained values for $\theta'$ are presented in Table \[tb:Table2\] compared to the RG results by the two-loop $\epsilon$ expansion of reference [@CGLMS00].
[8.2cm]{}[@c@c@c@c@c@c@]{} $~\sigma$ & $\theta'$ & $\theta'_{RG}$ & $z$ & $z_{RG}$ & $\lambda/z$\
0.9 & $0.212\pm .005$ & 0.3346 & $1.18\pm.04$ & 0.9532 & $0.635\pm .004$\
0.8 & $0.188\pm .004$ & 0.2587 & $0.96\pm.04$ & 0.8340 & $0.85~\pm .01~$\
0.7 & $0.137\pm .006$ & 0.1733 & $0.81\pm.01$ & 0.7174 & $1.136\pm .02~$\
0.6 & $0.07~\pm .01~$ & 0.0821 & $0.70\pm.01$ & 0.6052 & $1.47~\pm .02~$\
Since the accuracy of our results improves with increasing $\sigma$, one may conclude that the RG results are overestimated due to the insufficiency of the two-loop expansion in that regime. Similar overestimation of $\theta'$ was observed in the SR case, where the same RG $\epsilon$-expansion in the SR limit [@CGLMS00] gives e.g. $\theta'= 0.131$ and $\theta'= 0.356$ for $d=3$ and $d=2$ respectively, while the MC simulations give respectively $\theta'=0.104$ [@Grassberger] and $\theta'=0.191$ [@Grassberger; @OSchYZ97].
A more standard way to calculate the exponent $\theta'$ is from the initial slip of the magnetization given by equation (\[deftheta1\]). In the present problem we find it less advantageous both for precision and for the numerical effort needed. For this reason we do not proceed with the systematic analysis using this approach. Just for illustration, we present the $\sigma = 0.9$ data in Table \[tb:Table3\], limiting ourselves to a very rough estimation.
[8.2cm]{}[@c@c@c@c@c@]{} $~m_0~~$ & 0.1 & 0.05 & 0.01 & $m_0 \rightarrow 0$\
$\theta'(m_0)$ & $.187\pm .005$ & $.196\pm .006$ & $.202\pm .008$ & $.204\pm .009$\
The result is consistent with the one cited in Table \[tb:Table2\]. Improving the accuracy would imply performing the calculations on several smaller initial values $m_0$, each of them requiring the same amount of numerical effort spent for the calculation of $Q(t)$.
### Magnetization and exponent $z$
As discussed earlier in Section 2.2, the dynamical exponent $z$ is expected to increase with $\sigma$, since the relaxation becomes slower with decreasing range of interactions. The exponent $z$ was calculated from the magnetization using equation (\[eq:mxpl\]). The log-log plot of the simulation data is illustrated in Figure \[fg:fig2\] for
the case $\sigma=0.8$. The values of $z$ presented in Table \[tb:Table2\] were obtained by substituting into equation (\[eq:mxpl\]) the exact value for the anomalous dimension of the order parameter $\beta/\nu$, which is equal to $(1-\sigma)/2$ [@Brezin76]. As $\sigma$ increases our results become significantly larger than those obtained by the $\epsilon$ expansion [@CGLMS00]. Again, we may attribute this discrepancy to an underestimation of the RG results by the two loop expansion and may observe similar behavior in the SR case, where, for the Ising model, the $(4-d)$-expansion to the second order [@HHM72] gives $z=2.013$ and $z=2.052$ for $d=3$ and $d=2$ respectively, while the best MC calculations give z close to 2.04 [@CG05] for d=3, and z=2.1667 [@NB00] for $d=2$.
### Autocorrelation function
The example of simulations of the autocorrelation function for $\sigma = 0.8$ is illustrated by the log-log plots in Figure \[fig3\].
The power-law fit to equation (\[autocorr\]) gives the exponent $\lambda/z$ presented in Table \[tb:Table2\]. Although not suitable for the calculation of the exponent $\theta'$ when the exponent $z$ is not known with sufficient precision, the values for $\lambda/z$ were used for a check of independent calculations of $\theta'$ and $z$. Within given error bars, the agreement is obtained.
### Finite-size scaling
An alternative way of evaluating the exponent $z$ is to perform the simulations on several small systems of different sizes and apply FSS by using the overlapping fits [@LSchZ96]. To this purpose one may consider the magnetization, Binder’s fourth-order cumulant, but also the correlation length defined by equation (\[eq:defksi\]).
We illustrate two such fits, involving sizes $L=100, 200$ and 400, for the magnetization and the correlation length in Figures \[fg:fig4\] and \[fg:fig5\].
The fit for the magnetization was performed by applying the scaling relation (\[eq:scMk\]) to the magnetization defined by (\[eq:mx\]). An example for $q=2$ and $\sigma=0.8$ is given in Figure \[fg:fig4\]. The magnetization is rescaled by using the exact value for $\beta/\nu$. The time axis is rescaled by using the earlier calculated value of the exponent $z$ cited in Table \[tb:Table2\].
The scaling fit may also be applied directly to the correlation length defined by equation (\[eq:defksi\]), since at the criticality it should scale as $$\xi_L(t) = L \; f(t/L^z).
\label{scksi}$$ Figure \[fg:fig5\] gives the scaling fit for the case $q=2$ and $\sigma=0.9$. By $\xi_{Lmax}$ we denote the saturation value that $\xi_L$ attains according to the expression (\[eq:defksi\]). It is proportional to the size $L$ in the limit of large $L$.
As in the previous example, the scaling is performed with the same value of $z$ as given in Table \[tb:Table2\].
The agreement in both cases is very good. Nevertheless, these fits are generally less accurate than direct calculations from systems of large sizes.
Case $q=3$
----------
For the three-state Potts model we expect to obtain different dynamical exponents. Also, we should be able to distinguish, depending of $\sigma$, two regimes, corresponding to the first- and second-order phase transition.
The calculations were performed along the same lines and with similar parameters as for the preceding case, since the increase of the correlation length with time is very similar to that for $q=2$.
Yet, the microscopic time period $t_{micro}$ was found to be larger by several steps than the one for $q=2$. This property is clearly seen in Figure \[fg:fig6\].
![ Comparison of $t_{micro}$ for the cases $q=2$ and $q=3$ on the example of the function $Q(t)$ (equation (\[eq:defQ\])) for $\sigma = 0.9$, $L=3\,000$. The straight lines are the linear extrapolations performed in the scaling regime. The plot for $q=3$ was shifted by 0.5 in the y direction in order to display the two plots on the same graph.[]{data-label="fg:fig6"}](fig6.eps)
In spite of such behavior, owing to the fact that the analysis for $q=3$ is limited to larger values of $\sigma$ (as explained later), it was sufficient to use the same size $L=3\,000$ as in the $q=2$ case, but the statistics had to be increased systematically up to the 350000 independent runs.
In Figure \[fig7\] we present the results for the time correlation function of the magnetization, $Q(t)$, for the same values of $\sigma$ as in the previous case $q=2$.
The behavior is qualitatively different from the one in Figure \[fg:fig1\]. An initial increase is observed only for $\sigma=0.8$ and $\sigma=0.9$, while the change of behavior for lower values of $\sigma$ announces the expected onset of the first-order transition regime, which lies between $\sigma=0.7$ and $\sigma = 0.8$. The onset of the first-order phase transition in the present model is a challenging question in itself, since the position of the tricritical point $\sigma_c(q)$ is still not known with precision [@BDD99; @UG00; @ReynalDiep04]. In this connection, it is important to notice, that STD has proven as an efficient approach in cases involving short-range interactions for studying both first-order phase transitions [@SchZ00; @YZT04; @YZPT06] and a the tricritical point [@JanssenOerding94]. A more detailed study of these issues in the present LR model requires a separate study [@GUprep]. We mention here only, that our preliminary results for $Q(t)$ on a finer scale of $\sigma$ locate the change of regime between 0.72 and 0.74, which is in agreement with most recent estimates [@ReynalDiep04] that give $\sigma_c(q=3)=0.72(1)$. Here we shall limit the scope to the second-order transition regime analyzing further only the behavior for $\sigma=0.8$ and $\sigma=0.9$.
The dynamical exponent $\theta'$ derived from the log-log plot of the function $Q(t)$ is given in Table \[tb:Table4\].
[8.0cm]{}[@c@c@c@c@c@]{} $~\sigma~~$ & $\theta'$ & z & $z_{FSS}$ & $\lambda/z$\
$0.9$ & $0.120\pm .004$ & $1.21\pm .01 $ & $1.26\pm.04$ & $0.704\pm .008$\
$0.8$ & $0.058\pm .004$ & $1.01\pm .004$ & $1.02\pm.04$ & $0.935\pm .006$\
It strongly decreases as the first-order regime approaches. Compared to the Ising case, the exponents $\theta'$ for $q=3$ turn out to be significantly lower. This is similar to what was observed for the $2D$ SR Potts model, where $\theta'=0.191$ for $q=2$, while $\theta'=0.075$ for $q=3$ [@OSchYZ97].
In Table \[tb:Table5\] we also present the alternative derivation of the exponent $\theta'$ by investigating the initial slip of the magnetization for the case $\sigma=0.9$.
[8.0cm]{}[@c@c@c@c@c@]{} $~m_0~$ & 0.1 & 0.05 & 0.02 & $m_0 \rightarrow 0$\
$\theta'(m_0)$ & $.133\pm.003$ & $.118\pm.004$ & $.112\pm .006$ & $.106\pm .09$\
A rough linear extrapolation to $m_0=0$ gives a slightly smaller value for $\theta'$ than the one cited in the Table \[tb:Table4\]. In the same time the precision of the calculations from the magnetization was considerably lower.
Within this approach, one can observe one feature common to earlier numerical calculations for the 2D short-range Potts model [@OSchYZ97], that $\theta'(m_0)$ converge to the limit $m_0 \rightarrow 0$ from different sides for $q=2$ and $q=3$, which was there attributed to the opposite positions of the related fixed points.
The results for the exponent $z$ obtained from equation (\[eq:mxpl\]) using the same procedure as in the case $q=2$ are presented in the third column of Table \[tb:Table4\] and illustrated in Figure \[fg:fig8\] for $\sigma=0.8$. The values of $z$ are slightly larger than for the Ising case, similar as it was obtained for the $2D$ SR Potts model [@OSchYZ97].
For want of prior results for dynamical exponents in the case $q=3$, we also applied the FSS to the magnetization (\[eq:mx\]) and the correlation length (\[eq:defksi\]) by performing independent evaluations of the exponent $z$ by using the collapsing fits. The results are included in Table \[tb:Table4\] for comparison.
As in previous case, the results for $\lambda/z$ obtained from the autocorrelation function (\[autocorr\]) (cited in the last column of Table \[tb:Table4\]) agree with the independently calculated values of $\theta'$ and $z$ within the accuracy limits.
Conclusion {#sec:Conclusion}
==========
We presented a numerical study of scaling properties related to the short-time dynamics at criticality in the 1D LR Potts model. Based on the analysis of several physical quantities, we showed that in spite of the fast relaxation in presence of the LR interactions, the STD scaling regime can be observed numerically and dynamic critical exponents evaluated with satisfactory accuracy. We focused here on studying the problem in larger range of $\sigma$, but the accuracy of each individual result may still be improved with reasonable numerical effort.
The dynamical exponents $\theta'$ and $z$ were evaluated in the cases $q=2$ and $q=3$, for several values of $\sigma$ belonging to the nontrivial critical regime. The exponents are found to differ for the two cases and depend on $\sigma$, in similar way they depend on dimensionality in the SR analogue of this model.
For the Ising case, the comparison could be made with the existing RG results. A fair agreement for values of $\sigma$ close to the MF border ($\epsilon=2\sigma-1 \ll 1$) is obtained, but the discrepancy reaches far beyond the estimated error bars as $\sigma$ increases, which could be attributed to the shortcomings of the $\epsilon$-expansion. Our results are in favor of significantly smaller increase of $\theta'$ and larger increase of $z$ with decreasing range of interactions.
For $q=3$, new values for the exponents $\theta'$ and $z$ were obtained in, more restrained, second-order phase transition regime. The value of the exponent $z$ is found to be slightly larger than the one for $q=2$, while increasing number of Potts states had larger impact on the critical exponent $\theta'$ which is appreciably smaller and tends to vanish as the first-order transition regime approaches. We also found the change in the behavior of the time correlations of the magnetization as the first-order transition sets in with lowering of the parameter $\sigma$.
Besides the onset of the first-order transition regime in this model which is already a subject of a separate study, a number of issues remain to be examined further, such as the possible effects of different dynamics, or a complementary analysis of the exponent describing the persistence probability of the global order-parameter at criticality.
This work was supported by the Croatian Ministry of Science, Education and Sports through grant No. 035-0000000-3187.
[99]{}
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Bahman Dehnadi ${}^a$\
E-mail:
- |
Andre H. Hoang ${}^{b,c}$\
E-mail:
- |
Vicent Mateu ${}^{c,d,e}$\
E-mail:
- |
S. Mohammad Zebarjad ${}^a$\
E-mail:\
${}^a$ Shiraz University, Physics Department, Shiraz 71454, Iran.\
${}^b$ University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Vienna, Austria.\
${}^c$ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München, Germany.\
${}^d$ Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139.\
${}^e$ Instituto de Física Corpuscular, UVEG - Consejo Superior de Investigaciones Científicas, Apartado de Correos 22085, E-46071, Valencia, Spain.
bibliography:
- 'charm2.bib'
title: 'Charm Mass Determination from QCD Charmonium Sum Rules at Order $\alpha_s^3$'
---
Introduction {#sectionintroduction}
============
Accurate determinations of the charm quark mass are an important ingredient in the prediction of inclusive and radiative $B$ decays or exclusive kaon decays such as $K\to\pi\nu\bar{\nu}$. Since these decays are instruments to either measure CKM matrix elements or to search for new physics effects, appropriate and realistic estimates of the uncertainties are also an important element of these analyses [@Antonelli:2009ws].
One of the most powerful methods to determine the charm quark mass is based on sum rules for the charm-anticharm production rate in $e^{+}e^{-}$ annihilation [@Novikov:1977dq]. Here, moments of the correlation function of two charm vector currents at zero momentum transfer $$\begin{aligned}
\label{momentdef1}
M_{n}^{{\rm th}} & = & \dfrac{12\pi^2 Q_c^2}{n!}\,\dfrac{{\rm d}}{{\rm
d}q^{2n}}\left.\Pi(q^{2})\right|_{q^{2}=0}\,,\\
\left(g_{\mu\nu}q^{2}-q_{\mu}q_{\nu}\right)\,\Pi(q^{2})
& = &
-\, i\int\mathrm{d}x\, e^{iqx}\left\langle \,0\left|T\,
j_{\mu}(x)j_{\nu}(0)\right|0\,\right\rangle
\,,\nonumber \\[2mm]
j^{\mu}(x)
& = &
\bar{\psi}(x)\gamma^{\mu}\psi(x)
\,,\nonumber \end{aligned}$$ $Q_c$ being the charm quark electric charge, can be related to weighted integrals of the normalized charm cross section $$\begin{aligned}
\label{momentdef2}
M_{n} & = &
\int\dfrac{{\rm d}s}{s^{n+1}}R_{e^{+}e^{-}\to\, c\bar{c}\,+X}(s)\,,\\
R_{e^{+}e^{-}\to\, c\bar{c}\,+X}(s) & = & \dfrac{\sigma_{e^{+}e^{-}\to\, c\bar{c}\,+X}(s)}{\sigma_{e^{+}e^{-}\to\,\mu^{+}\mu^{-}}(s)}\,,\nonumber \end{aligned}$$ which can be obtained from experiments. For small values of $n$ such that $m_{c}/n\gtrsim\Lambda_{{\rm QCD}}$ the theoretical moments $M_{n}^{{\rm th}}$ can be computed in an operator product expansion (OPE) where the dominant part is provided by perturbative QCD supplemented by small vacuum condensates that parametrize nonperturbative effects [@Shifman:1978bx; @Shifman:1978by]. The leading gluon condensate power correction term has a surprisingly small numerical effect and is essentially negligible for the numerical analysis as long as $n$ is small.
This allows to determine the charm mass in a short distance scheme such as $\overline{{\rm MS}}$ to high precision. This method to determine the $\overline{{\rm MS}}$ charm mass is frequently called charmonium sum rules. For the theoretical moments the perturbative part of the OPE is known at ${\mathcal O}(\alpha_{s}^{0})$ and ${\mathcal O}(\alpha_{s})$ for any value of $n$ [@Kallen:1955fb]. At ${\mathcal O}(\alpha_{s}^{2})$ the first 30 moments are known [@Boughezal:2006uu; @Maier:2007yn], and to ${\mathcal O}(\alpha_{s}^{3})$ for $n=1$ [@Chetyrkin:2006xg; @Boughezal:2006px], $n=2$ [@Maier:2008he], and $n=3$ [@Maier:2009fz]. Higher moments at ${\cal O}(\alpha_s^3)$ have been determined by a semianalytical procedure [@Hoang:2008qy; @Kiyo:2009gb] (see also [@Greynat:2010kx]). The Wilson coefficient of the gluon condensate contribution is known to ${\mathcal O}(\alpha_{s})$ [@Broadhurst:1994qj]. On the experimental side the total hadronic cross section in $e^{+}e^{-}$ annihilation is known from various experimental measurements for c.m. energies up to $10.538\,$GeV. None of the experimental analyses actually ranges over the entire energy region between the charmonium region and $10.538$ GeV, but different analyses overlapping in energy exist such that energies up to $10.538\,$GeV are completely covered [@Bai:1999pk; @Bai:2001ct; @Ablikim:2004ck; @Ablikim:2006aj; @Ablikim:2006mb; @:2009jsa; @Osterheld:1986hw; @Edwards:1990pc; @Ammar:1997sk; @Besson:1984bd; @:2007qwa; @CroninHennessy:2008yi; @Blinov:1993fw; @Criegee:1981qx; @Siegrist:1976br; @Rapidis:1977cv; @Abrams:1979cx; @Siegrist:1981zp].[^1] Interestingly, to the best of our knowledge, the complete set of all available experimental data on the hadronic cross section has never been used in previous charmonium sum rule analyses to determine the experimental moments. Rather, sum rule analyses have relied heavily on theoretical input using different approaches to determine the corresponding “experimental error” and intrinsically leading to a sizable modeling uncertainty for energy regions below $10.538\,$GeV for low values of $n$ [@Hoang:2004xm].
The most recent charmonium sum rule analysis based on Eqs. (\[momentdef1\]) and (\[momentdef2\]), carried out by Kühn et al. [@Chetyrkin:2009fv; @Kuhn:2007vp] using input from perturbative QCD (pQCD) at ${\mathcal O}(\alpha_{s}^{3})$ for the perturbative contribution, obtained $\overline{m}_{c}(\overline{m}_{c})=
1279\pm(2)_{{\rm pert}}\pm(9)_{{\rm exp}}
\pm(9)_{\alpha_{s}}\pm(1)_{{\rm \left\langle GG\right\rangle}}\,$ MeV where the first error is the perturbative uncertainty and the second is the experimental one. The third and the fourth uncertainties come from $\alpha_s$ and the gluon condensate correction, respectively. To our knowledge this result, the outcome of similar analyses in Ref. [@Chetyrkin:2006xg] and by Boughezal, Czakon and Schutzmeier [@Boughezal:2006px][^2], and a closely related analysis based on lattice results instead of data for pseudoscalar moments [@Allison:2008xk; @McNeile:2010ji] represent the analyses with the highest precision achieved so far in the literature. If confirmed, any further investigations and attempts concerning a more precise charm quark $\overline{{\rm MS}}$ mass would likely be irrelevant for any foreseeable future.
We therefore find it warranted to reexamine the charmonium sum rule analysis with special attention on the way how perturbative and experimental uncertainties have been treated in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp]. A closer look into their analysis reveals that the quoted perturbative uncertainty results from a specific way to arrange the $\alpha_s$ expansion for the charm mass extractions and, in addition, by setting the $\overline{{\rm MS}}$ renormalization scales in $\alpha_{s}$ and in the charm mass (which we call $\mu_{\alpha}$ and $\mu_{m}$, respectively) equal to each other (i.e., they use $\mu_{\alpha}=\mu_{m}$). Moreover, concerning the experimental moments, only data up to $\sqrt{s}=4.8\,$GeV from the BES experiments [@Bai:2001ct; @Ablikim:2006mb] were used, while for $\sqrt{s}>4.8$ GeV perturbative QCD predictions were employed. Conceptually this approach is somewhat related to the method of finite energy sum rules (see e.g. Ref. [@Penarrocha:2001ig]), which we, however, do not discuss in this work. While this approach might be justified to estimate the overall nominal contribution for the experimental moments from $\sqrt{s}>4.8$ GeV, since perturbative QCD predictions describe quite well the measured total hadronic cross section outside the resonance regions, it is certainly not a suitable method to determine the experimental uncertainty. Since the region $\sqrt{s}>4.8$ GeV constitutes about $30\%$ of the first moment $M_{1}$, which is theoretically most reliable, this approach contains a significant intrinsic model dependence that cannot be quantified unambiguously.
In this work we reexamine the charmonium sum rules analysis for low values of $n$ using the latest ${\mathcal O}(\alpha_{s}^{3})$ perturbative results, and we implement improvements which concern the two issues just mentioned:
1. We analyze several different types of perturbative expansions and examine in detail how the result for the $\overline{{\rm MS}}$ charm mass depends on independent choices of $\mu_{\alpha}$ and $\mu_{m}$. We show in particular that the interplay of the perturbative expansion and the scale setting $\mu_{\alpha}=\mu_{m}$ used in previous ${\mathcal O}(\alpha_s^3)$ analyses leads to sizable cancellations of the dependence on $\mu_{\alpha}$ and $\mu_{m}$ that in the light of our new analysis has to be considered as accidental. As the outcome of our analysis we estimate the current ${\mathcal O}(\alpha_{s}^{3})$ perturbative error as around $20$ MeV, which is an order of magnitude larger than that of Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp; @Chetyrkin:2006xg; @Boughezal:2006px].
2. Using a clustering method [@D'Agostini:1993uj; @Takeuchi:1995xe; @Hagiwara:2003da] to combine correlated data from many different experimental measurements we show that the $e^{+}e^{-}$ total hadronic cross section relevant for the charmonium sum rules can be determined with a complete coverage of center of mass energies above the $J/\psi$ and $\psi'$ resonances up to $10.538\,$GeV. Conservatively estimated modeling uncertainties coming from the energy range above $10.538\,$GeV then only lead to an insignificant contribution to the total uncertainty of the experimental moments.
This paper is organized as follows: In Sec. \[sectiontheory\] we introduce the theoretical framework and review the current status of perturbative computations. We also show various equivalent ways of arranging the perturbative series in $\alpha_s$ for the charm mass. Finally we discuss how to properly estimate theoretical uncertainties due to the truncation of the perturbative series. In Sec. \[sectiondata\] we present all the experimental information that goes into our analysis. We discuss a clustering fit procedure that allows to combine data from different experiments accounting for their correlation and show the results. In Sec. \[sectionanalysis\] we carry out the numerical charm mass analysis concentrating on the first moment using arbitrary values of $\alpha_s$, and we present our final charm mass result. In Appendix A we present more details on the outcome of our clustering fit procedure for the charm , and in Appendix B we prove the equivalence of different versions of $\chi^2$ functions when auxiliary fit parameters are employed.
Theoretical Input {#sectiontheory}
=================
Perturbative Contribution {#subsectionperturbative}
-------------------------
The moments of the vector current correlator are defined in Eq. (\[momentdef1\]). Their perturbative contribution in the framework of the OPE has a non-linear dependence on the charm quark mass. Thus in principle no conceptual preference can be imposed on any of the possible perturbative series that arises when solving for the charm mass. As a consequence, different versions of the expansion should be considered to obtain reliable estimates of the perturbative uncertainty. As indicated in Sec. \[sectionintroduction\] we use in the following $\mu_\alpha$ as the renormalization scale in $\alpha_s$ and $\mu_m$ as the renormalization scale in the $\overline{\rm MS}$ charm quark mass $\overline m_c$.
[**(a) Standard fixed-order expansion**]{}\
Writing the perturbative vacuum polarization function as $$\begin{aligned}
\label{Mnpertfixedorder1}
\Pi^{\rm pert}(q^2,\alpha_s(\mu_\alpha), \overline m_c(\mu_m),
\mu_\alpha, \mu_m)
\, = \, \dfrac{1}{12\pi^2 Q_c^2}\sum_{n=0}^\infty q^{2n} M_n^{\rm pert}
\,,\end{aligned}$$ we have for the perturbative moments $M_n^{\rm pert}$ $$\begin{aligned}
\label{Mnpertfixedorder2}
M_n^{\rm pert} & = &
\frac{1}{(4\overline m^2_c(\mu_m))^{n}}
\sum_{i,a,b} \left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i
C^{a,b}_{n,i}\,\ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right)
\ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right).\end{aligned}$$ This is the standard fixed-order expression for the perturbative moments. At ${\cal O}(\alpha_s^3)$ the coefficients $C^{0,0}_{n,3}$ were recently determined for $n=1$ [@Chetyrkin:2006xg; @Boughezal:2006px], $n=2$ [@Maier:2008he], $n=3$ [@Maier:2009fz] and higher [@Hoang:2008qy; @Kiyo:2009gb; @Greynat:2010kx]. We refer to Ref. [@Boughezal:2006uu; @Maier:2007yn] for the coefficients at ${\cal O}(\alpha_s^2)$. We have summarized the numerical expressions for the $C^{a,b}_{n,i}$ coefficients of the first four moments in Tab. \[tabcfixedorder\].
The standard fixed-order expansion in Eq. (\[Mnpertfixedorder2\]) is the common way to represent the perturbative moments. However, written in this form the non-linear dependence on $\overline m_c$ does for some values of the experimental moments and the renormalization scales not yield numerical solutions for $\overline m_c$.[^3]
[**(b) Linearized expansion**]{}\
Concerning the charm mass dependence, a more linear way to organize the perturbative expansion is to take the root of Eq. (\[Mnpertfixedorder2\]): $$\begin{aligned}
\label{Mnpertlinearized1}
\Big(M_n^{\rm th, pert}\Big)^{1/2n} & = &
\frac{1}{2\overline{m}_c(\mu_m)} \,\sum_{i,a,b}\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i \tilde C_{n,i}^{a,b}
\ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right)
,\end{aligned}$$ or equivalently $$\begin{aligned}
\label{Mnpertlinearized2}
\overline m_c(\mu_m) & = & \frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}} \sum_{i,a,b}\,\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i\tilde C_{n,i}^{a,b}
\ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right)
.\end{aligned}$$ The coefficients $\tilde C_{n,i}^{a,b}$ using again $\mu_\alpha$ for the renormalization scale in $\alpha_s$ and $\mu_m$ for the renormalization scale in the $\overline{\rm MS}$ charm mass are given in Tab. \[tabctildefixedorder\]. Although relation (\[Mnpertlinearized2\]) involves a non-linear dependence on ${\overline m}_c$, we find that it always has a numerical solution.
[**(c) Iterative linearized expansion**]{}\
For the standard and the linearized expansions in Eqs. (\[Mnpertfixedorder2\]) and (\[Mnpertlinearized1\]) one searches for numerical solutions of the charm mass $\overline m_c(\mu_m)$ keeping the exact mass dependence on the respective RHS at each order in $\alpha_s$. An alternative way that is consistent within perturbation theory is to solve for $\overline
m_c(\mu_m)$ iteratively order-by-order supplementing appropriate lower order values for $\overline m_c(\mu_m)$ in higher order perturbative coefficients. To be more explicit, we describe the method in the following. As the basis for the iterative expansion carried out in our analysis we use the linearized expansion of Eq. (\[Mnpertlinearized2\]).
In the first step we determine $\overline m_c(\mu_m)$ employing the tree-level relation $$\begin{aligned}
\label{Mnpertiterative1}
\overline m_c^{(0)}(\mu_m) =
\frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}} \,\tilde C_{n,0}^{0,0}
\,,\end{aligned}$$ giving the tree-level charm mass $\overline m_c^{(0)}(\mu_m)$. In the next step one employs the relation $$\begin{aligned}
\label{Mnpertiterative2}
\overline m_c^{(1)}(\mu_m) &=&
\frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}}\,\Biggr\{\,
\tilde C_{n,0}^{0,0} \, + \,
\frac{\alpha_s(\mu_\alpha)}{\pi}\bigg[\tilde C_{n,1}^{0,0}+\tilde C_{n,1}^{1,0}
\ln\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right)\Bigg]\Biggr\}
\,,\quad\end{aligned}$$ to determine the ${\cal O}(\alpha_s)$ charm mass $\overline m_c^{(1)}(\mu_m)$. In the ${\cal O}(\alpha_s)$ terms on the RHS of Eq. (\[Mnpertiterative2\]) the tree-level charm mass $\overline m_c^{(0)}$ is used, which is consistent to ${\cal O}(\alpha_s)$. At ${\cal O}(\alpha_s^2)$ for the determination of $\overline m_c^{(2)}(\mu_m)$ one uses $\overline m_c^{(0)}$ for the ${\cal
O}(\alpha_s^2)$ coefficient and $\overline m_c^{(1)}(\mu_m)$ for the ${\cal
O}(\alpha_s)$ correction, which in the strict $\alpha_s$ expansion yields $$\begin{aligned}
\label{Mnpertiterative3}
\overline m_c^{(2)} & = &
\frac{1}{2\Big(M_n^{\rm th, pert}\Big)^{1/2n}}\,\Biggr\{\,
\tilde C_{n,0}^{0,0} \, +\,
\frac{\alpha_s(\mu_\alpha)}{\pi}\Bigg[\tilde C_{n,1}^{0,0}+\tilde C_{n,1}^{1,0}
\ln\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right)\Bigg]+\nonumber\\
&&\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^2\Bigg[2\,\dfrac{\tilde C_{n,1}^{1,0}\,\tilde C_{n,1}^{0,0}}{\tilde C_{n,0}^{0,0}}+
2\,\dfrac{(\tilde C_{n,1}^{1,0})^2}{\tilde C_{n,0}^{0,0}}\ln\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right)+\nonumber\\
&&\sum_{a,b}\,\tilde C_{n,2}^{a,b}
\ln^a\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}_c^{(0)\,2}(\mu_m)}{\mu^2_\alpha}\right)\Bigg]\Biggr\}
\,.\end{aligned}$$ Here the second line contains the derivative of the ${\mathcal O}(\alpha_s)$ terms with respect to the charm mass. The determination of the ${\cal O}(\alpha_s^3)$ charm mass $\overline
m_c^{(3)}(\mu_m)$ is then carried out in an analogous way involving the second (first) derivative with respect to the mass in the ${\cal O}(\alpha_s)$ (${\cal
O}(\alpha_s^2)$) correction and using again $\overline m_c^{(0)}(\mu_m)$ for the ${\cal O}(\alpha_s^3)$ coefficient.
In general we can write the iterative expansion as follows: $${\overline m}_c(\mu_m)\,=\,{\overline m}_c^{(0)}
\sum_{i,a,b}\left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i \hat C_{n,i}^{a,b}\,
\ln^a\left(\frac{{\overline m}_c^{(0)\,2}}{\mu^2_m}\right)
\ln^b\left(\frac{{\overline m}_c^{(0)\,2}}{\mu^2_\alpha}\right),
\label{eq:iterative-general}$$ where the numerical value of the coefficients $\hat C_{n,i}^{a,b}$ are collected in Tab. \[tab:chat\].
The iterative way to treat the perturbative series for the charm mass has the advantage that solving for the charm mass involves equations that are strictly linear in the charm mass at any order of the $\alpha_s$ expansion and thus always have solutions. In this way any possible influence on the analysis arising from a non-linear dependence is eliminated.
[**(d) Contour improved expansion**]{}\
For the expansion methods (a)-(c) the moments and the charm quark mass are computed for a fixed choice of the renormalization scale $\mu_\alpha$ in the strong coupling $\alpha_s$. In analogy to the contour improved methods used for (see e.g. Refs. [@LeDiberder:1992te; @Pivovarov:1991rh; @Braaten:1991qm; @Narison:1988ni; @Braaten:1988ea; @Braaten:1988hc]) one can employ a path-dependent $\mu_\alpha$ in the contour integration that defines the perturbative moments [@Hoang:2004xm], see Fig. \[figcontour\], $$\begin{aligned}
\label{Mnpertcontour1}
M_n^{\rm c, pert} & = &
\frac{6\pi Q_c^2}{i}\,\int_c\,\frac{{\rm d}s}{s^{n+1}}
\Pi(q^2, \alpha_s(\mu_\alpha^c(s,\overline m_c^2)), \overline{m}_c(\mu_m),
\mu_\alpha^c(s,\overline m_c^2), \mu_m)
\,.\end{aligned}$$
![ Path of integration in the complex for the computation of the moments. \[figcontour\] ](contour.eps){width="30.00000%"}
Due to the independence of the moments on $\mu_\alpha$ and since no large logarithms are being generated anywhere for a path with distance of order $\overline m_c$ from the cut on the real axis, this method is a viable alternative to carry out the perturbative expansion. The different orders in the expansion of the contour improved moments $M_n^{\rm c, pert}$ are generated from the fixed-order $\alpha_s$ expansion of the vacuum polarization function $\Pi$ in Eq. (\[Mnpertcontour1\]).
A useful path-dependent choice for $\mu_\alpha^c$ is given by [@Hoang:2004xm] $$\begin{aligned}
\label{mualphacontour}
(\mu_\alpha^c)^2(s,\overline m_c^2) & = &
\mu_\alpha^2\,\bigg(\,1-\frac{s}{4\overline m_c^2(\mu_m)}\,\bigg)
\,,\end{aligned}$$ which implements a modified weighting of threshold versus high energy contributions. It is straightforward to prove that the resulting moments $M_n^{\rm c,
pert}$ can be obtained from the small-$q^2$ expansion of the perturbative vacuum polarization function using $\mu_\alpha^c$ as the renormalization scale of $\alpha_s$, $$\begin{aligned}
\label{Mnpertcontour2}
\Pi^{\overline {\rm MS}}\Big(q^2,
\alpha_s(\mu_\alpha^c(q^2,\overline m_c^2)),
\overline m_c(\mu_m),
\mu_\alpha^c(q^2,\overline m_c^2),
\mu_m\Big) & = &
\sum\limits_{n=0}^\infty \, q^{2n}\,M_n^{\rm c, pert}
\,.\end{aligned}$$ From Eq. (\[Mnpertcontour2\]) we can see that the $M_n^{\rm c, pert}$ can be derived from the expressions for the fixed-order moments $M_m^{\rm pert}$ with $m\le n$ given in Eq. (\[Mnpertfixedorder2\]).[^4] They also depend on the QCD and its derivatives, which arise in the small-$q^2$ expansion of $\alpha_s(\mu^c_\alpha(q^2,\overline m_c^2))$. Note that the that has to be employed must be exactly the same that is used for the contour integration in Eq. (\[Mnpertcontour1\]). Expanding the dependence of the on $\alpha_s$ strictly in fixed-order one recovers the fixed-order moments $M_n^{\rm pert}$. So the dependence of the contour improved moments $M_n^{\rm c, pert}$ on the fixed-order moments $M_m^{\rm pert}$ with $m<n$ is only residual due to the truncation of the $\alpha_s$ series, representing yet another alternative parametrization of higher order perturbative corrections. The contour improved moments do in particular have a residual dependence on the UV-subtraction scheme for the vacuum polarization function, i.e. on $\Pi(0)=M_0^{\rm pert}$. Using the “on-shell” scheme with $\Pi(0)=0$ one finds that $M_1^{\rm c,pert} = M_1^{\rm pert}$. For our analysis we employ the $\overline{\rm MS}$ scheme for $\Pi(0)$ defined for $\mu={\overline m}_c({\overline m}_c)$. Expressed in terms of $\alpha_s(\mu_\alpha)$ and ${\overline m}_c(\mu_m)$ it has the form [@Chetyrkin:2006xg] $$\begin{aligned}
\label{Pi0msbar}
\Pi^{\overline {\rm MS}}(0) & = &
\sum_{i,a,b} \left(\frac{\alpha_s(\mu_\alpha)}{\pi}\right)^i
C^{a,b}_{0,i}\,\ln^a\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_m}\right) \ln^b\left(\frac{\overline{m}^2_c(\mu_m)}{\mu^2_\alpha}\right)
\,.\end{aligned}$$ The numerical values for the coefficients $C^{a,b}_{0,i}$ can be found in Tab. \[tabcPi0\]. Using contour improved moments to determine $\overline m_c$ also involves non-linear relations, which implies that in some cases there is no solution. Again this can happen for $n=>1$ and for $\mu_m\sim 3$ GeV at any order.
Gluon Condensate Contribution {#subsectioncondensate}
-----------------------------
The dominant subleading contribution in the OPE for the theory moments $M_n^{\rm
th}$ is arising from the gluon condensate. Writing $$\begin{aligned}
\label{MnOPE1}
M_n^{\rm th} & = & M_n^{\rm pert} + \Delta M_n^{\langle G^2\rangle}\,+\,\ldots
\,,\end{aligned}$$ and neglecting any higher order power-suppressed condensate contributions of the OPE we have [@Novikov:1977dq; @Baikov:1993kc; @Broadhurst:1994qj] $$\label{DeltaMncondensate1}
\Delta M_n^{\langle G^2\rangle}
= \dfrac{1}{(4\overline{m}_c^2(\mu_m))^{n+2}}\Big\langle\frac{\alpha_s}{\pi} G^2\Big\rangle_{\rm RGI} \left\{ a^{0,0}_{n}+\dfrac{\alpha_{s}(\mu_\alpha)}{\pi}\left[a^{1,0}_{n}+a^{1,1}_n \log\dfrac{\overline{m}_c^2(\mu_m)}{\mu_m^{2}}\right]\right\}\,.$$ We have written Eq. (\[DeltaMncondensate1\]) using the renormalization group invariant (RGI) scheme for the gluon condensate [@Narison:1983kn] and the $\overline{\rm
MS}$ charm quark mass. In Ref. [@Chetyrkin:2010ic] it is suggested that the Wilson coefficient of the gluon condensate should be expressed in terms of the pole rather than the $\overline{\mbox{MS}}$ mass based on the observation that the pole mass leads to a condensate correction that is numerically quite stable for higher moments. We disagree with this prescription due to the infrared renormalon issue in the pole mass and since it is expected that the condensate corrections for the moment increases with $n$. On the other hand, for a sum rule analysis of the first moment with $n=1$ this issue is absolutely irrelevant due to the small size of the gluon condensate contribution compared to the level of the remaining perturbative uncertainties. Numerical values for the coefficients $a_n^{i,j}$ are given in Tab. \[tabgluoncondensate\] for $n=1,2,3,4$. For the RGI gluon condensate we adopt [@Ioffe:2005ym] $$\begin{aligned}
\label{condensatevalue1}
\Big\langle\frac{\alpha_s}{\pi} G^2\Big\rangle_{\rm RGI}
& = & 0.006\pm0.012\;\mathrm{GeV}^4\,.\end{aligned}$$ The overall contribution of the gluon condensate correction in Eq. (\[DeltaMncondensate1\]) in the charm quark mass analysis is quite small. Its contribution to the moments amounts to around $0.2\%$, $0.6\%$, $2\%$, and $3\%$ for the first four moments, respectively. For $n=1$ it leads to a correction in the $\overline{\rm MS}$ charm quark mass at the level of $1$ MeV and is an order of magnitude smaller than our perturbative uncertainty. We therefore ignore the condensates correction for the discussion of the perturbative uncertainties in Sec. \[subsectionmcerror\]. Its contribution is, however, included in the final charm mass results presented in Sec. \[sectionanalysis\].
Running Coupling and Mass {#subsectionrunning}
-------------------------
The analysis of the charmonium sum rules naturally involves renormalization scales around the charm mass, $\mu\sim {\overline m}_c\sim 1.3$ GeV, which are close to the limits of a perturbative treatment. In fact, parametrically, the typical scale relevant for the perturbative computation of the moment $M_n^{\rm th}$ is of order $\mu\sim {\overline m}_c/n$ (see e.g. Ref. [@Hoang:1998uv]) because the energy range of the smearing associated to the weight function $1/s^{n+1}$ in Eq. (\[momentdef2\]) decreases with $n$. We will therefore use $n=1$ for our final numerical analysis. Moreover, it is common practice to quote the $\overline{\rm MS}$ charm mass $\overline m_c(\overline m_c)$, i.e. for the scale choice $\mu_m=\overline m_c$. It is therefore useful to have a look at the quality of the perturbative behavior of the renormalization group evolution of the strong $\overline{\rm MS}$ coupling $\alpha_s$ and the $\overline{\rm MS}$ charm quark mass.
In Fig. \[fig:alphasevolutionexact\][^5] we have displayed $\alpha_s^{\rm N^3LL}(\mu)/\alpha_s^{\rm N^kLL}(\mu)$ using for $\alpha_s^{\rm N^kLL}$ the QCD and the respective exact numerical solution for $\alpha_s(3~\mbox{GeV})= 0.2535$ as the common reference point. We see that the convergence of the lower order results towards the 4-loop evolution is very good even down to scales of around $1$ GeV. The curves indicate that the remaining relative perturbative uncertainty in the 4-loop evolution might be substantially smaller than $1\%$ for scales down to $\overline m_c\sim 1.3$ GeV. It is also instructive to examine the evolution using a fixed-order expansion. In Fig. \[fig:alphasevolutionexpanded\] we display $\alpha_s^{\rm N^3LL}(\mu)/\alpha_s^{(m)}(\mu)$ where $\alpha_s^{(m)}(\mu)$ is the ${\cal O}(\alpha_s^{m+1})$ fixed-order expression for $\alpha_s(\mu)$ using the reference value $\alpha_s(3~\mbox{GeV})=0.2535$ as the expansion parameter. The convergence of the fixed-order expansion for $\alpha_s(\mu)$ towards the exact N${}^3$LL numerical solution $\alpha_s^{\rm N^3LL}(\mu)$ is somewhat worse compared to the renormalization group resummed results since the deviation of the ratio from one is in general larger. However, convergence is clearly visible. In particular there are not any signs of instabilities. It therefore seems to be safe to use renormalization scales down to the charm mass and associated renormalization scale variations as an instrument to estimate the perturbative uncertainties.
In Figs. \[fig:mcevolutionexact\] and \[fig:mcevolutionexpanded\] an analogous analysis has been carried out for the $\overline{\rm MS}$ charm quark mass. In Fig. \[fig:mcevolutionexact\] $\overline m_c^{\rm N^3LL}(\mu)/\overline m_c^{\rm N^kLL}(\mu)$ is plotted for $k=0,1,2,3$ using the exact numerical solutions of the renormalization group equations and $\overline m_c(\mu=3~\mbox{GeV})$ as the respective reference value.[^6] Compared to the Fig. \[fig:alphasevolution\] we observe a very similar convergence. In Fig. \[fig:mcevolutionexpanded\], finally, we show $\overline m_c^{\rm N^3LL}(\mu)/\overline m_c^{(m)}(\mu)$, where $\overline m_c^{(m)}(\mu)$ is the ${\cal O}(\alpha_s^m)$ fixed-order expression for $\overline m_c(\mu)$ using $\alpha_s(3~\mbox{GeV})=0.2535$ as the expansion parameter. Again, the convergence towards the exact N${}^3$LL evolved result is very similar to the corresponding results for the strong coupling, and we find again no evidence for perturbative instabilities. Of course the corrections are somewhat larger when the fixed-order expansion is employed. We therefore conclude that perturbative evolution and renormalization scale variations for the $\overline{\mbox{MS}}$ charm quark mass can be safely used down to scales above $\overline m_c\sim 1.3$ GeV. One should of course mention that the lines in Fig. \[fig:mcevolutionexpanded\] also give an indication about the expected size of scale variations depending on the range of the variations. Scales above $2$ GeV can lead to sub-MeV variations, while scales down to the charm mass will result at best in percent precision (i.e. ${\cal O}(10~\mbox{MeV})$).
Perturbative Uncertainties in the $\overline{\rm MS}$ Charm Mass {#subsectionmcerror}
----------------------------------------------------------------
In this section we discuss in detail the perturbative series for the determination of the $\overline{\rm MS}$ charm mass $\overline m_c$ and how to set up an adequate scale variation to estimate the perturbative uncertainty. In the previous subsections we have discussed four different ways to carry out the perturbative expansion and we presented the corresponding order-by-order analytic expressions. As described there, we can determine at each order of the perturbative expansion for the moments $M_n^{\rm th}$ a value for $\overline
m_c(\mu_m)$ which also has a residual dependence on $\mu_\alpha$, the renormalization scale used for $\alpha_s$. To compare the different mass determinations we then evolve $\overline m_c(\mu_m)$ to obtain $\overline
m_c(\overline m_c)$ using the 4-loop renormalization group equations for the mass and the strong coupling [@Tarasov:1980au; @Larin:1993tp; @vanRitbergen:1997va; @Chetyrkin:1997dh; @Vermaseren:1997fq].[^7] The obtained value of $\overline m_c(\overline m_c)$ thus has a residual dependence on the scales $\mu_m$ and $\mu_\alpha$, on the order of perturbation theory and on the expansion method.[^8] For the results we can therefore use the notation $$\begin{aligned}
\label{mcnotation1} &&
\overline m_c(\overline m_c)[\mu_m,\mu_\alpha]^{i,n}
\,,\end{aligned}$$ where $n=0,1,2,3$ indicates perturbation theory at ${\cal O}(\alpha_s^n)$ and $$\begin{aligned}
\label{methodnotation1}
i & = & \left\{
\begin{array}[c]{ll}
a \qquad & \mbox{(fixed-order expansion),} \\
b & \mbox{(linearized expansion),} \\
c & \mbox{(iterative expansion),} \\
d & \mbox{(contour improved expansion).}
\end{array}\right.\end{aligned}$$
To initiate the discussion, we show in Fig. \[fig:trumpet1\] results for $\overline m_c(\overline m_c)$ at ${\cal O}(\alpha_s^n)$ for expansions a–d using $\mu_m=\mu_\alpha$ (upper four graphs) and using $\mu_m=\overline
m_c(\overline m_c)$ (lower four graphs). For each method and order we have displayed the range of $\overline m_c(\overline m_c)$ values for a variation of $2~\mbox{GeV}\le \mu_\alpha \le 4~\mbox{GeV}$, which corresponds to the scale variation employed in Refs. [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp]. Their analysis used the fixed-order expansion with the setting $\mu_m=\mu_\alpha$ and is represented by graph 1. We make several observations:
- Choosing $\mu_m$ and $\mu_\alpha$ both larger than $2$ GeV makes the $\overline m_c(\overline m_c)$ value decrease with the order of perturbation theory.
- Choosing $\mu_m$ smaller than $1.5$ GeV and $\mu_\alpha$ larger than $2$ GeV makes the $\overline m_c(\overline m_c)$ value increase with the order of perturbation theory.
- For most choices of the scale setting and the expansion method the spread of the $\overline m_c(\overline m_c)$ values from the variation of $\mu_\alpha$ does not decrease in any substantial way with the order. However, viewing all methods and scale setting choices collectively a very good convergence is observed.
We have checked that these statements apply also in general beyond the specific cases displayed in Fig. \[fig:trumpet1\].
Quite conspicuous results are obtained for the scale choice $\mu_m=\mu_\alpha$ for the fixed-order (graph 1) and linearized expansions (graph 2). Here, extremely small variations in $\overline m_c(\overline m_c)$ are obtained. They amount to $1.8$ MeV ($4$ MeV) and $0.6$ MeV ($1.4$ MeV) at order $\alpha_s^2$ and $\alpha_s^3$, respectively, for the fixed-order expansions (linearized expansions). We note that our scale variation for the fixed-order expansion at ${\cal O}(\alpha_s^3)$ is consistent with the corresponding numbers quoted in Ref. [@Chetyrkin:2006xg; @Boughezal:2006px], where the ${\cal O}(\alpha_s^3)$ corrections to the first moment were computed,[^9] but differs from the scale variations given in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] which also quoted numerical results from the fixed-order expansion. Interestingly the ${\cal O}(\alpha_s^2)$ and ${\cal O}(\alpha_s^3)$ variations we find do not overlap and the ${\cal O}(\alpha_s^3)$ ranges appear to be highly inconsistent with the ${\cal O}(\alpha_s^3)$ results from the iterative method (graph 3). A visual display of scale variations obtained from the four expansion methods at ${\cal O}(\alpha_s^3)$ with different types of variation methods is given in Fig. \[fig:trumpet2\].
An illustrative way to demonstrate how a small scale variation can arise is given in Fig. \[fig:mccontour1fixed\] - \[fig:mccontour1improved\]. For all four expansion methods contour curves of constant $\overline m_c(\overline m_c)$ are displayed as a function of (the residual dependence on) $\mu_m$ and $\mu_\alpha$. For the fixed-order (a) and the linearized expansions (b) we see that there are contour lines closely along the diagonal $\mu_m=\mu_\alpha$. For the fixed-order expansion (a) this feature is almost exact and thus explains the extremely small scale-dependence seen in graph 1 of Fig. \[fig:trumpet1\]. For the linearized expansion (b) this feature is somewhat less exact and reflected in the slightly larger scale variation seen in graph 2 of Fig. \[fig:trumpet1\]. On the other hand, the contour lines of the iterative (c) and contour improved (d) expansions have a large angle with respect to the diagonal $\mu_m=\mu_\alpha$ leading to the much larger scale variations of $9$ and $5$ MeV, respectively, at ${\cal O}(\alpha_s^3)$ visible in graphs 3 and 4 of Fig. \[fig:trumpet1\]. The contour plots shown in Fig. \[fig:mccontour1\] also show the variation along the line $\mu_m$ or $\mu_\alpha=\overline
m_c(\overline m_c)$ (border of gray shaded areas). Here, the contour lines are relatively dense leading to scale variation of around $15$ MeV at ${\cal O}(\alpha_s^3)$.
Overall, we draw the following conclusions:
- The small scale variations observed for the fixed-order and linearized expansions for $\mu_m=\mu_\alpha$ result from strong cancellations of the individual $\mu_m$ and $\mu_\alpha$ dependences that arise for this correlation.
- Other correlations between $\mu_m$ and $\mu_\alpha$ that do not generate large logarithms do not lead to such cancellations. One therefore has to consider the small scale variations observed for $\mu_m=\mu_\alpha$ in the fixed-order and linearized expansions as accidental.
- For an adequate estimate of perturbative uncertainties specific correlations between $\mu_m$ and $\mu_\alpha$ that are along contour lines of constant $\overline m_c(\overline m_c)$ have to be avoided. Moreover, adequate independent variations of $\mu_m$ and $\mu_\alpha$ should not induce large logarithms.
As the outcome of this discussion we adopt for our charm mass analysis an independent and uncorrelated variation of $\mu_m$ and $\mu_\alpha$ in the range $$\begin{aligned}
\label{muammvariation}
\overline m_c(\overline m_c) & \le & \mu_m,\mu_\alpha \, \le \, 4~\mbox{GeV}\,,\end{aligned}$$ in order to estimate perturbative uncertainties. The excluded region $\mu_m,\mu_\alpha<\overline{m}_c(\overline{m}_c)$ in the plane in the contour plots of Fig. \[fig:mccontour1\] is indicated by the gray shaded areas. This two-dimensional variation avoids accidental cancellations from correlated variations, large logarithms involving ratios of the scales $\overline m_c(\mu_m)$, $\mu_m$, $\mu_\alpha$ and remains well in the validity ranges for the perturbative renormalization group evolution of the $\overline{\rm MS}$ charm quark mass and $\alpha_s$ (see Sec. \[subsectionrunning\]). The range of Eq. (\[muammvariation\]) is also consistent with a scale variation $\mu\sim 2\,\overline m_c\times (1/2,1,2)$ one might consider as the standard choice with respect to the to particle threshold located at $\sqrt{s}=2\,{\overline m}_c$. As a comparison the range $\mu=(3\pm 1)$ GeV corresponds to $\mu=2\,{\overline m}_c(0.8,1.2,1.6)$. We emphasize that the lower boundary $\overline m_c(\overline m_c)\sim
1.25$ GeV is also reasonable as it represents the common flavor matching scale where gauge coupling evolution remains smooth up to NLL order. We do not see any evidence for perturbation theory for the first moment $M_1$ being unstable at the charm mass scale, in contrast to claims made in Ref. [@Kuhn:2007vp].
In Fig. \[fig:mmcerror1\] we show the ranges of $\overline m_c(\overline m_c)$ at ${\cal O}(\alpha_s^{1,2,3})$ for the four expansion methods employing the scale variations of Eq. (\[muammvariation\]). We see that all four expansion methods now lead to equivalent results. The variations are also compatible with the overall variations shown in Fig. \[fig:trumpet1\]. At ${\cal
O}(\alpha_s^3)$ we obtain a scale variation for $\overline m_c(\overline m_c)$ of around $20$ MeV. This is an order of magnitude larger than the perturbative uncertainties quoted in Refs. [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp]. In Fig. \[fig:m3cerror1\] we have displayed the corresponding results for $\overline m_c(3~\mbox{GeV})$. At ${\cal
O}(\alpha_s^3)$ they also exhibit a scale variation of around $20$ MeV.
Experimental Data {#sectiondata}
=================
Data Collections {#subsectioncollections}
----------------
[**Narrow resonances**]{}\
Below the open charm threshold there are the $J/\psi$ and $\psi^\prime$ narrow charmonium resonances. Their masses, and electronic widths are taken from the PDG [@Nakamura:2010zzi] and are collected in Tab. \[tabpsidata\] together with the value of the $\overline{\rm MS}$ QED coupling at their masses. The total widths are not relevant since we use the narrow width approximation for their contributions to the moments. The uncertainty for the contribution to the moments coming from the masses can be neglected.
$J/\psi$ $\psi^\prime$
-------------------------- ---------------- ----------------
$M$ (GeV) $3.096916(11)$ $3.686093(34)$
$\Gamma_{ee}$ (keV) $5.55(14)$ $2.48(6)$
$(\alpha/\alpha(M))^{2}$ $0.957785$ $0.95554$
: Masses and electronic widths [@Nakamura:2010zzi] of the narrow charmonium resonances and effective electromagnetic coupling [@Kuhn:2007vp]. $\alpha=1/137.035999084(51)$ stands for the fine structure constant, and $\alpha(M)$ stands for the $\overline{\mbox{MS}}$ electromagnetic coupling at the scale $M$.\[tabpsidata\]
[**Threshold and data continuum region**]{}\
The open charm threshold is located at $\sqrt{s}=3.73$ GeV. We call the energies from just below the threshold and up to $5$ GeV the threshold region, and the region between $5$ GeV and $10.538$ GeV, where the production rate is dominated by multiparticle final states the data continuum region. In these regions quite a variety of measurements of the total hadronic cross section exist from BES [@Bai:1999pk; @Bai:2001ct; @Ablikim:2004ck; @Ablikim:2006aj; @Ablikim:2006mb; @:2009jsa], CrystalBall [@Osterheld:1986hw; @Edwards:1990pc], CLEO [@Ammar:1997sk; @Besson:1984bd; @:2007qwa; @CroninHennessy:2008yi], MD1 [@Blinov:1993fw], PLUTO [@Criegee:1981qx], and MARKI and II [@Siegrist:1976br; @Rapidis:1977cv; @Abrams:1979cx; @Siegrist:1981zp]. Taken together, the entire energy region up to $10.538$ GeV is densely covered with total cross section measurements from these 19 data sets.[^10] The measurements from BES and CLEO have the smallest uncertainties. They do, however, not cover the region between $5$ and $7$ GeV. Here CrystalBall and MARKI and II have contributed measurements albeit with somewhat larger uncertainties. The statistical and total systematical uncertainties of the measurements can be extracted from the respective publications. For some data sets the amount of uncorrelated and correlated systematical uncertainties is given separately (BES [@Bai:2001ct; @Ablikim:2006aj; @Ablikim:2006mb], CrystalBall [@Osterheld:1986hw; @Edwards:1990pc], CLEO [@:2007qwa], MARKI and II [@Siegrist:1981zp; @Abrams:1979cx], MD1 [@Blinov:1993fw]) while for all the other data sets only combined systematical uncertainties are quoted. All these data sets are shown in Figs. \[figdatacompilation\], where the displayed error bars represent the (quadratically) combined statistical and systematical uncertainties.
Interestingly, none of the previous charm mass analyses, to the best of our knowledge, ever used the complete set of available data. As examples, Bodenstein et al. [@Bodenstein:2010qx] used data sets [@Bai:1999pk; @Bai:2001ct; @Ablikim:2006mb; @CroninHennessy:2008yi] from BES and CLEO. Jamin and Hoang [@Hoang:2004xm] used the data sets of Refs. [@Bai:2001ct], [@Blinov:1993fw] and [@Ammar:1997sk] from BES, MD1 and CLEO, covering the regions $2~\mbox{GeV} \leq E \leq 4.8~\mbox{GeV}$ and $6.964~\mbox{GeV} \leq E \leq 10.538~\mbox{GeV}$. Boughezal et al. [@Boughezal:2006px], Kuhn et al. [@Kuhn:2001dm], and Narison [@Narison:2010cg] use only one data set from BES [@Bai:2001ct]. Kuhn et al. [@Chetyrkin:2009fv; @Kuhn:2007vp] used the data sets of Refs. [@Bai:2001ct; @Ablikim:2006mb] from BES covering the energy region $2.6~\mbox{GeV} \leq E \leq 4.8~\mbox{GeV}$.
We consider three different selections of data sets to study the dependence of the experimental moments on this choice:
- The [*minimal selection*]{} contains all data sets necessary to cover the whole energy region between $2$ and $10.538$ GeV without any gaps and keeping only the most accurate ones. These 8 data sets are from BES [@Bai:1999pk; @Bai:2001ct; @Ablikim:2006aj; @:2009jsa], CrystalBall [@Edwards:1990pc], CLEO [@CroninHennessy:2008yi; @:2007qwa] and MD1 [@Blinov:1993fw] corresponding to the data sets 1, 2, 5, 6, 9, 12, 13, and 14 (see Tab. \[tab:datasets\] for references).
- The [*default selection*]{} contains all data sets except for the three ones with the largest uncertainties. It contains 16 data sets and fully includes the minimal selection. It contains all data sets except for Mark I and II data sets 16, 17 and 19 from Refs. [@Siegrist:1981zp; @Rapidis:1977cv; @Siegrist:1976br].
- The [*maximal selection*]{} contains all 19 data sets.
We use the default selection as our standard choice for the charm mass analysis, but we will also quote results for the other data selections.
[**Perturbative QCD region**]{}\
Above $10.538$ GeV there are no experimental measurements of the total hadronic that might be useful for the experimental moments. In this energy region we will therefore use perturbative QCD to provide estimates for the charm production . As a penalty for not using experimental data we assign a $10$% total relative uncertainty to the contribution of the experimental moments coming from this region, which we then treat like an uncorrelated experimental uncertainty for the combination with the moment contribution from lower energies. As we see in Sec. \[subsectionmoments\] the energy region above $10.538$ GeV contributes only $(6, 0.4, 0.03, 0.002)\%$ to $M^{\rm
exp}_{1,2,3,4}$. In the first moment $M_1^{\rm exp}$ the total contribution is about three times larger than the combined statistical and systematical (true) experimental uncertainties from the other energy regions. So the 10% penalty we assign to this approach represents a subleading component of the final quoted uncertainty.[^11] In the second and higher moments $M_{n\ge 2}^{\rm exp}$ the contributions from above $10.538$ GeV and the corresponding uncertainty are negligible compared to the uncertainties from the lower energy regions.
As the theoretical formula to determine the moment contribution from the perturbative QCD region we use the ${\cal O}(\alpha_s^3)$ non-singlet massless quark cross section including charm mass corrections up to ${\cal O}(\overline m_c^4/s^2)$: $$\begin{aligned}
R_{cc}^{\rm th}(s) & = &
N_c \,Q_c^2\,R^{\rm ns}(s,\overline m_c^2(\sqrt{s}), n_f=4,\alpha_s^{n_f=4}(\sqrt{s}))\,,
\label{Rcchighdef1}\end{aligned}$$ where $$\begin{aligned}
\label{Rnsdef1}
\lefteqn{R^{\rm ns}(s,\overline m_c^2(\mu), n_f=4,\alpha_s^{n_f=4}(\mu),\mu)}\\
& = &1+\dfrac{\alpha_{s}}{\pi}
+\left(\dfrac{\alpha_{s}}{\pi}\right)^{2}\left[
1.52453-2.08333 L_s\right]
+\left(\dfrac{\alpha_{s}}{\pi}\right)^{3}
\left[-11.52034+34028 L_s^2 -9.56052 L_s\right]\nonumber\\
& + & \dfrac{\overline m_c^{2}(\mu)}{s}
\Big\{ 12\,\dfrac{\alpha_{s}}{\pi}+
\left(\dfrac{\alpha_{s}}{\pi}\right)^{2}\left[109.167-49 L_s\right]
+\left(\dfrac{\alpha_{s}}{\pi}\right)^{3}\left[634.957-799.361 L_s+
151.083 L_s^2\right]\!\!\Big\} \nonumber\\
& + & \dfrac{\overline m_c^4(\mu)}{s^2}\Big\{ \!\!
-6+\dfrac{\alpha_{s}}{\pi}\left[-22+24 L_s\right]
+ \left(\dfrac{\alpha_{s}}{\pi}\right)^{2}
\left[140.855+221.5 L_s-5.16667 L_m-73 L_s^2\right]\nonumber\\
&& \qquad + \left(\dfrac{\alpha_{s}}{\pi}\right)^{3}\left[
3776.94-509.226 L_s -174.831 L_m-1118.29 L_s^2\right.\nonumber\\
&& \qquad\qquad + 10.3333 L_m^2+\left.42.1944 L_m L_s
+198.722 L_s^3\right]\Big\}
\,,\nonumber\end{aligned}$$ with $$L_s \, \equiv \, \ln\Big(\frac{s}{\mu^{2}}\Big)\,,\qquad
L_m \, \equiv \, \ln\Big(\dfrac{\overline m_c^{2}(\mu)}{s}\Big)\,.$$ For the computation of the contribution to the experimental moments we determine $\overline m_c(\sqrt{s})$ and $\alpha_s(\sqrt{s})$ appearing in Eq. (\[Rcchighdef1\]) using $\overline m_c(\overline m_c)=1.3$ GeV and $\alpha_s(m_Z)=0.118$ as initial conditions.
It is instructive to examine for the moment contributions from $\sqrt{s}>10.538$ GeV terms related to charm production that we do not account for in Eq. (\[Rnsdef1\]). In Tab. \[tabneglected\] the relative size with respect to the full first four moments (in percent) of the most important neglected contributions are given. In the second column the size of the mass corrections up to order $\overline m_c^4$, which we have included in $R^{\rm th}_{cc}$, are shown as a reference. The third column shows the contributions coming from secondary $c\bar c$ radiation through gluon splitting. The fourth column depicts the contributions from the ${\cal O}(\alpha_s^3)$ singlet corrections (including the mass corrections up to order $\overline m_c^4$), which one can take as an rough estimate for the actual contributions from the charm cut. Finally in the last column we show the size of the Z-boson exchange terms integrated from threshold to $10.538$ GeV. This contribution represents the Z-exchange contribution that is contained in the data, but - by definition - not accounted for in the theory moments. We see that at least for the first two moments, the contributions neglected are much smaller than the charm mass corrections we have accounted for in the nonsinglet production rate, which are already constituting a very small effect. Overall the numerical effect on the charm mass of all these contributions is tiny considering the scaling $\overline m_c\sim M_n^{1/2n}$. Since we assign a $10$% error on the moments’ contribution from the energy region $\sqrt{s}>10.538$ GeV where we use theory input, our approach to neglect subleading effects is justified.
[**Non-charm background**]{}\
Experimentally only the total hadronic cross section is available. Although charm-tagged rate measurements are in principle possible [@CroninHennessy:2008yi] they have not been provided in publications. On the other hand, they would also exhibit sizable additional uncertainties related to the dependence on simulations of the decay of charmed mesons into light quark final states. So to obtain the charm production cross section from the data we have to subtract the non-charm background using a model based on perturbative QCD related to the production of $u$, $d$ and $s$ quarks. A subtle point is related to the secondary radiation of $c\bar c$ pairs off the $u$, $d$ and $s$ quarks from gluon splitting and to which extent one has to account theoretically for the interplay between real and virtual secondary $c\bar c$ radiation which involves infrared sensitive terms [@Hoang:1997ca]. Since in this work we define the moments from primary $c\bar c$ production (see Eq. (\[momentdef1\])), secondary $c\bar c$ production is formally counted as non-charm background. Thus for the model for the non-charm background for $\sqrt{s}>2\, \overline m_c$ we employ the expression $$\begin{aligned}
\label{Rudsdef}
R_{uds}(s) & = &
N_c(Q_u^2 + 2 Q_d^2)\,R^{\rm ns}(s,0,n_f=3,\alpha_s^{n_f=4}(s))\\
&&+\,
N_c\, Q_c^2\left(\dfrac{\alpha_s^{n_f=4}(s)}{\pi}\right)^{2}\left(\rho^{V}+\rho^{R}+\dfrac{1}{4}\,\log\dfrac{\overline m_{c}^{2}(\overline m_{c})}{s}\right)
\,.\nonumber\end{aligned}$$ The second term on the RHS describes the contributions from real and virtual secondary $c\bar c$ radiation. The analytic expressions for $\rho^R$ and $\rho^V$ can be found in Eqs. (2) and (6) of Ref. [@Hoang:1994it]. We have checked that the numerical impact of real ($\rho^R$) and virtual ($\rho^V$) secondary radiation individually as well as the complete second term on the RHS of Eq. (\[Rudsdef\]) on the moments is negligible, see Tab. \[tabneglected\]. We use Eq. (\[Rudsdef\]) and fit the non-charm background including also data in the region $2~\mbox{GeV}\le E\le 3.73~\mbox{GeV}$ via the ansatz $R_{{\rm non}-c\bar c}(s) = n_{\rm ns}\,R_{uds}(s)$, where the constant $n_{\rm
ns}$ represents an additional fit parameter.
Data Combination {#subsectioncombination}
----------------
Combining different experimental measurements of the hadronic cross section one has to face several issues: (a) the measurements are given at individual separated energy points, (b) the set of measurements from different publications are not equally spaced, cover different, partly overlapping energy regions and have different statistical and systematical uncertainties, (c) the correlations of systematical errors are only known (or provided) for the data sets within each publication, (d) there are a number of very precise measurements at widely separated energies.
In this section we discuss the combination of the experimental data from the threshold and the data continuum regions between $2$ and $10.538$ GeV using a method based on a fitting procedure used before for determining the hadronic vacuum polarization effects for $g-2$ [@Hagiwara:2003da]. In this work we extend this approach and also account for the subtraction of the non-charm background.
[**Combination method**]{}\
The method uses the combination of data in energy bins (clusters) assuming that the within each cluster changes only very little and can thus be well approximated by a constant. Thus clusters for energies where $R$ varies rapidly need to be small (in this case the experimental measurements are also denser). The in each cluster is then obtained by a $\chi^2$ fitting procedure. Since each experimental data set from any publication covers an energy range overlapping with at least one other data set, the clusters are chosen such that clusters in overlapping regions contain measurements from different data sets. Through the fitting procedure correlations are then being communicated among different data sets and very accurate individual measurements can inherit their precision into neighbouring clusters. Both issues are desirable since the hadronic is a smooth function with respect to the sequence of clusters.
To describe the method we have to set up some notation:
- All measurements $R(E)$ are distinguished according to the energy $E$ at which they have been carried out.
- Each such energy point having a measurement is written as $E_i^{k,m}$, where $k=1,\ldots,N_{\rm exp}$ refers to the $N_{\rm exp}$ data sets, $m=1,\ldots,N_{\rm cluster}$ runs over the $N_{\rm cluster}$ clusters and $i=1,\ldots,N^{k,m}$ assigns the of the $N^{k,m}$ measurements.
- Each individual measurement of the is then written as $$\begin{aligned}
\label{Rexp}
R(E^{k,m}_i) & = & R_i^{k,m} \, \pm \, \sigma_i^{k,m} \,
\pm \, \Delta_i^{k,m}
\,,\end{aligned}$$ where $R_i^{k,m}$ is the central value, $\sigma_i^{k,m}$ the combined statistical and uncorrelated systematical uncertainty and $\Delta_i^{k,m}$ the correlated systematical experimental uncertainty.
- For convenience we define $\Delta
f_i^{k,m}=\Delta_i^{k,m}/R_i^{k,m}$ to be the relative systematical correlated uncertainty.
As our standard choice concerning the clusters we use 5 different regions each having equidistant cluster sizes $\Delta E$. The regions are as follows:
- [**non-charm region**]{}: has 1 cluster for $2~\mbox{GeV}\leq E \leq
3.73~\mbox{GeV}$ ($\Delta E=1.73$ GeV).
- [**low charm region**]{}: has 2 clusters for $3.73~\mbox{GeV}< E \leq
3.75~\mbox{GeV}$ ($\Delta E=10$ MeV).
- [**$\psi(3S)$ region/threshold region 1**]{}: has 20 cluster for $3.75~\mbox{GeV}< E \leq
3.79~\mbox{GeV}$ ($\Delta E=2$ MeV).
- [**resonance region 2**]{}: has 20 cluster for $3.79~\mbox{GeV}< E \leq
4.55~\mbox{GeV}$ ($\Delta E=38$ MeV).
- [**continuum region**]{}: has 10 cluster for $4.55~\mbox{GeV}< E \leq
10.538~\mbox{GeV}$ ($\Delta E=598.8$ MeV).
We assign to this choice of $52+1$ clusters the notation (2,20,20,10) and later also examine alternative cluster choices demonstrating that the outcome for the moments does within errors not depend on them. The cluster in the non-charm region is used to fit for the normalization constant $n_{\rm ns}$ of the non-charm background contribution, see Eq. (\[Rudsdef\]).
Our standard procedure to determine the central energy $E_m$ associated to each cluster is just the weighted average of the energies of all measurements falling into cluster $m$, $$\begin{aligned}
\label{Eclusterdef}
E_m & = &
\dfrac{\sum_{k,i}\dfrac{E_{i}^{k,m}}
{(\sigma_{i}^{k,m})^2+(\Delta_{i}^{k,m})^2}}
{\sum_{k,i}\dfrac{1}{(\sigma_{i}^{k,m})^2+(\Delta_{i}^{k,m})^2}}
\,.\end{aligned}$$ The corresponding for the charm cross section that we determine through the fit procedure described below is called $$\begin{aligned}
\label{Rclusterdef}
R_m & \equiv & R_{c\bar c}(E_m)
\,. \end{aligned}$$ We note that using instead the unweighted average or simply the center of the cluster has a negligible effect on the outcome for the moments since the clusters we are employing are sufficiently small.
[**Fit procedure and** ]{}\
We determine the charm cross section $R_{c\bar c}$ from a that accounts for the positive correlation among the systematical uncertainties $\Delta_i^{k,m}$ within each experiment $k$ and, at the same time, also for the non-charm background. To implement the correlation we introduce the auxiliary parameters $d_k$ ($k=1,\ldots, N_{\rm exp}$) that parametrize the correlated deviation from the experimental central values $R_i^{k,m}$ in units of the correlated systematical uncertainty $\Delta_i^{k,m}$, see Eq. (\[Rexp\]). In this way we carry out fits to $R_i^{k,m}+d_k \,\Delta_i^{k,m}$ and treat the $d_k$ as additional auxiliary fit parameters that are constraint to be of order one (one standard deviation) by adding the term $$\begin{aligned}
\label{chi2totald}
\chi^2_{\rm corr}(\{d_k\}) & = &
\sum_{k=1}^{N_{\rm exp}} \, d_k^2\,,\end{aligned}$$ to the . To implement the non-charm background we assume that the relative energy dependence of the non-charm cross section related to primary production of $u$, $d$ and $s$ quarks is described properly by $R_{uds}$ given in Eq. (\[Rudsdef\]). We then parametrize the non-charm background cross section by the relation $$\begin{aligned}
\label{Rncdef}
R_{{\rm non}-c\bar c}(E) &= & n_{\rm ns}\,R_{uds}(E)
\,,\end{aligned}$$ where the fit parameter $n_{\rm ns}$ is determined mainly from the experimental data in the first clusters below $3.73$ GeV by adding the term $$\begin{aligned}
\label{chi2noncharm}
\chi^2_{\rm nc}(n_{\rm ns},\{d_k\}) & = &
\sum_{k=1}^{N_{\mathrm{exp}}}
\sum_{i=1}^{N^{k,1}}\left(\dfrac{
R_i^{k,1}-(1+\Delta f_i^{k,1}\, d_k)\, n_{ns}\,
R_{uds}(E_i^{k,1})}{\sigma_i^{k,1}}\right)^2\,,\end{aligned}$$ to the . The complete then has the form $$\begin{aligned}
\label{chi2total}
\chi^2(\{R_m\}, n_{\rm ns},\{d_k\}) & = &
\chi^2_{\rm corr}(\{d_k\}) \, + \,
\chi^2_{\rm nc}(n_{\rm ns},\{d_k\})\,+\,\chi^2_{\rm c}(\{R_m\}, n_{\rm ns},\{d_k\})
\,,\end{aligned}$$ where[^12] $$\begin{aligned}
\label{chi2charm}
&&\chi^2_{\rm c}(\{R_m\}, n_{\rm ns},\{d_k\}) = \\[2mm]
&&\sum_{k=1}^{N_{\mathrm{exp}}}
\sum_{m=2}^{N_{\mathrm{clusters}}}\sum_{i=1}^{N^{k,m}}
\left(\dfrac{R_{i}^{k,m}-(1+\Delta f_i^{k,m}\, d_k)\,
(R_m+n_{ns}\, R_{uds}(E_i^{k,m}))}{\sigma_i^{k,m}}\right)^{2}
\,.\nonumber\end{aligned}$$ Note that in our approach the non-charm normalization constant $n_{\rm ns}$ is obtained from a combined fit together with the cluster values $R_m$.
This form of the is an extended and adapted version of the ones used in Refs. [@D'Agostini:1993uj; @Takeuchi:1995xe]. A special characteristic of the in Eqs. (\[chi2noncharm\]) and (\[chi2charm\]) is that the relative correlated experimental uncertainties $\Delta f_i^{k,m}$ enter the by multiplying the fit value $R_m$ rather than the experimental values $R_i^{k,m}$. This leads to a non-bilinear dependence of the on the $d_m$ and the $R_m$ fit parameters and avoids spurious solutions where the best fit values for the $R_m$ are located systematically below the measurements. Such spurious solutions can arise for data points with substantial positive correlation when with strictly bilinear dependences are employed [@D'Agostini:1993uj; @Takeuchi:1995xe].[^13]
We also note that the implementation of the non-charm background subtraction given in Eq. (\[chi2charm\]) leads to a partial cancellation of systematical uncertainties for the $R_m$ best fit values for the charm cross section. Moreover, it is interesting to mention that in the limit where each cluster contains exactly one measurement (except below threshold, in which we always keep one cluster) the decouples, after performing the change of variables $R'_m=R_{m}+n_{ns}\, R_{uds}(E^{k,m})$, into the sum of two independent , one containing data below threshold and depending only on $n_s$ and $d_k$, and another one containing data above threshold $R^{k,m}$ (we drop the label $i$ because having only one data per cluster it can take only the value $1$) and depending only on $R'_m$. After minimizing the first one can obtain the best fit values for $n_s$ and $d_k$, denoted by $n_s^{(0)}$ and $d_k^{(0)}$, respectively. The second $\chi^2$ has a minimal value of $0$ and the best fit parameters read $R_m^{(0)}=R^{k,m}/(1+d_k^{(0)}\Delta^{k,m})-n_s^{(0)}R_{uds}(E^{k,m})$.
Close to the minimum the of Eq. (\[chi2total\]) can be written in the Gaussian approximation $$\begin{aligned}
\label{chi2Gauss}
\chi^2(\{p_i\}) & = & \chi^2_{\rm min}\,+\, \sum_{i,j} (p_i-p_i^{(0)})V_{i,j}^{-1}(p_j-p_j^{(0)})
\, + \, {\cal O}\Big((p-p^{(0)})^3\Big)
\,,\end{aligned}$$ where $p_i=(
\{R_m\}, n_{\rm ns}, \{d_k\})$ and the superscript $(0)$ indicates the respective best fit value. After determination of the correlation matrix $V_{ij}$ by numerically inverting $V^{-1}_{i,j}$ we can drop the dependence on the auxiliary variables $n_{\rm nc}$ and $d_k$ and obtain the correlation matrix of the $R_m$ from the of $V_{ij}$ which we call $V_{m m^\prime}^R$. In order to separate uncorrelated statistical and systematical uncertainties from correlated systematical ones we compute the complete $V_{m m^\prime}^R$ accounting for all uncertainties and a simpler version of the correlation matrix, $V_{m m^\prime}^{R,u}$ accounting only for uncorrelated uncertainties. The latter is obtained from dropping all correlated errors $\Delta_i^{k,m}$ from the (\[chi2total\]).[^14]
The outcome of the fit for the sum of the charm and the non-charm cross section in the threshold and the data continuum region using the standard data set explained above is shown in Figs. \[fig:fitcompilation\](a)-(f) together with the input data sets. The red line segments connect the best fit values and the brown band represents the total uncertainty. The clusters are indicated by dashed vertical lines. For completeness we have also given all numerical results for the $R_m$ values in Appendix A. There we also give results for the minimal and maximal data set selections.
Experimental Moments {#subsectionmoments}
--------------------
[**Narrow resonances**]{}\
For the $J/\psi$ and $\psi^\prime$ charmonium contributions to the experimental moments we use the narrow width approximation, $$\begin{aligned}
\label{Mnresonances}
M_n^{\rm res} & = & \dfrac{9\,\pi\,\Gamma_{ee}}{\alpha(M)^{2}M^{2n+1}}\,,\end{aligned}$$ with the input numbers given in Tab. \[tabpsidata\]. We neglect the tiny uncertainties in the charmonium masses as their effects are negligible.
[**Threshold and data continuum region**]{}\
For the determination of the moment contributions from the threshold and the continuum region between $3.73$ and $10.538$ GeV we use the results for the clustered $c\bar c$ cross section values $R_m$ determined in Sec. \[subsectioncombination\] and the trapezoidal rule. Using the relations ${\rm d}s={\rm d}E^2=2E{\rm d}E$ we thus obtain $$\begin{aligned}
\label{Mndata}
M_n^{\rm thr+cont} & = &\sum_{i=1}^{N_{\mathrm{clusters}}}\dfrac{R_{i}}{E_{i}^{2n+1}}(E_{i+1}-E_{i-1})+\dfrac{R_{0}}{E_{0}^{2n+1}}(E_{1}-E_{0})\\
&+&\dfrac{R_{N_{\mathrm{clusters}}+1}}{E_{N_{\mathrm{clusters}}+1}^{2n+1}}(E_{N_{\mathrm{clusters}}+1}-E_{N_{\mathrm{clusters}}})\,,\nonumber\end{aligned}$$ where $R_0$ and $E_0$ are the $R$ and energy values at the lower boundary of the smallest energy cluster, and $R_{N_{cl}+1}$ and $E_{M_{cl}+1}$ are the corresponding values of the upper boundary of the highest energy cluster. The values for $R_0$ and $R_{N_{cl}+1}$ are obtained from linear extrapolation using the $R_m$ values of the two closest lying clusters[^15] $m^\prime$ and $m^\prime+1$ with the formula $$\begin{aligned}
\label{REextrapol}
R(E) & = &
\frac{R_{m^\prime+1}-R_{m^\prime}}{E_{m^\prime+1}-E_{m^\prime}}\, (E-E_{m^\prime})+ R_{m^\prime}
\,.\end{aligned}$$ For the computation of the moment contributions from subintervals within the range between $3.73$ and $10.538$ GeV we also use Eq. (\[Mndata\]) using corresponding adaptations for the boundary values at $m=0$ and $m=N_{cl}+1$.
[**Perturbative QCD region**]{}\
For the region above $10.538$ GeV where we use the perturbative QCD input described in Eqs. (\[Rcchighdef1\]) and (\[Rnsdef1\]) for the charm the contribution to the experimental moment is obtained from the defining equation (\[momentdef2\]) with a lower integration limit of $10.538$ GeV: $$\begin{aligned}
\label{MnQCD}
M_n^{\rm QCD} & = & \gamma \times\int_{(10.538\,\rm GeV)^2}^\infty {\rm d}s \,\dfrac{R_{cc}^{\rm th}(s)}{s^{n+1}}
\,.\end{aligned}$$ The variable $\gamma$ is an auxiliary variable used to parametrize the $10\%$ uncertainty we assign to the perturbative QCD contribution, $$\begin{aligned}
\label{gammadef}
\gamma & = & 1.0 \,\pm \, 0.1
\,.\end{aligned}$$
[**Correlations**]{}\
The experimental moments are obtained from the sum of the resonance, threshold plus continuum and perturbative QCD contributions described just above, $$\begin{aligned}
\label{Mnsum}
M_n^{\rm exp} & = &
M_n^{\rm res} \, + \,
M_n^{\rm thr+cont} \, + \,
M_n^{\rm QCD}
\,.\end{aligned}$$ To determine the uncertainties we account for the errors in the $e^+e^-$ widths of $J/\psi$ and $\psi^\prime$ and in the cluster values $R_m$, and for the $10\%$ assigned uncertainty in $M_n^{\rm QCD}$. For the evaluation we use the usual error propagation based on a $\bar m\times\bar m$ correlation matrix with $\bar
m=N_{cl}+3$. The correlation matrix of the experimental moments thus has the form $$\begin{aligned}
\label{CMcorr}
C_{nn^\prime}^{\rm exp} & = & \sum\limits_{i,j=1}^{N_{cl}+3}
\Big(\frac{\partial M_n^{\rm exp}}{\partial \bar R_i}\Big)\,
\Big(\frac{\partial M_{n^\prime}^{\rm exp}}{\partial \bar R_j}\Big)
\, V_{ij}^{\bar R}
\,,\end{aligned}$$ where we have $\bar R_i =
(\{R_m\},\Gamma_{e^+e^-}(J/\psi),\Gamma_{e^+e^-}(\psi^\prime),\gamma)$. The entries of $V^{\bar R}$ in the $R_m$ subspace are just the entries of the correlation matrix $V^R$ obtained from the cluster fit. The diagonal entries in the $\Gamma_{e^+e^-}$ subspace are the combined statistical and systematical uncertainties of the $e^+e^-$ widths and the $\delta\gamma=0.1$ for $M_n^{\rm QCD}$, respectively. We treat the latter uncertainty as uncorrelated with all other uncertainties. So all non-diagonal entries of $V_{ij}^{\bar R}$ for $i$ or $j=N_{cl}+3$ are zero. For the uncertainty of the $e^+e^-$ widths we adopt a model where the (quadratic) half of the error is uncorrelated and the other (quadratic) half is positively correlated among the two narrow resonances and to the $R_n$ cluster values as well. Thus for the corresponding non-diagonal entries of $V_{i,j}^{\bar R}$ with $i\in\{1,N_{cl}\}$ and we have the entries $V_{ij}^{\bar R}=\delta\Gamma^{1,2}_{e^+e^-}\Delta R_m/\sqrt{2}$, and for $i=N_{cl}+1$ and $j=N_{cl}+2$ we have $V_{ij}^{\bar R}=\delta\Gamma^1_{e^+e^-}\delta\Gamma^2_{e^+e^-}/2$, where $\delta\Gamma^{1,2}_{e^+e^-}$ are the respective $e^+e^-$ width total uncertainties for $J/\psi$ and $\psi^\prime$, respectively, and $\Delta R_m$ the correlated uncertainty of the respective cluster $R_m$ value.
The results for the moments showing separately the contributions from the resonances, various energy subintervals and their total sum using the defaults data set collection (see Sec. \[subsectioncollections\]) are given in Tab. \[tab:momres1\]. Using Eq. (\[CMcorr\]) it is straightforward to compute the correlation matrix of the moments, and we obtain $$\begin{aligned}
\label{CMtotal}
C^{\rm exp} & = &\left(
\begin{array}{cccc}
0.250 & 0.167 & 0.147 & 0.142\\
0.167 & 0.120 & 0.107 & 0.103\\
0.147 & 0.107 & 0.095 & 0.092\\
0.142 & 0.103 & 0.092 & 0.090
\end{array}
\right)\,,\end{aligned}$$ for the total correlation matrix accounting for all correlated and uncorrelated uncertainties. We remind the reader that all numbers related to the moment $M_n^{\rm exp}$ are given in units of $10^{-(n+1)}\,\mbox{GeV}^{-2n}$. To quote correlated and uncorrelated uncertainties separately it is also useful to show the correlation matrix that is obtained when only uncorrelated uncertainties are accounted for. The result it $$\begin{aligned}
\label{CMuncorr}
C^{\rm exp}_{\rm uc} & = &
\left(
\begin{array}{cccc}
0.041 & 0.035 & 0.034 & 0.034\\
0.035 & 0.034 & 0.034 & 0.035\\
0.034 & 0.034 & 0.035 & 0.036\\
0.034 & 0.035 & 0.036 & 0.037
\end{array}\right)
\,.\end{aligned}$$ These results can be used to carry out combined simultaneous fits to several of the moments. This is, however, not attempted in this work.
Examination {#subsectionexaminations}
-----------
We conclude this section with an examination of some of the choices and assumptions we have implemented for the treatment of the experimental uncertainties. Our defaults choices include
- treating one (quadratic) half of $J/\psi$ and $\psi^\prime$ $e^+e^-$ partial width uncertainties as uncorrelated and the other half as positively correlated among themselves and with the correlated uncertainties of the $R_m$ cluster values;
- treating the entire systematical uncertainties of the measurements as correlated for the data sets where only total systematical uncertainties were quoted;
- defining the cluster energies $E_m$ through the weighed average of measurement energies falling into the clusters, see Eq. (\[Eclusterdef\]);
- using $N_{cl}=52+1$ clusters distributed in groups of (2,20,20,10) clusters in the energy ranges bounded by $(3.73, 3.75, 3.79, 4.55, 10.538)$ GeV (see Sec. \[subsectioncombination\]) and
- using the default data set collection consisting of all data sets discussed in Sec. \[subsectioncollections\] except for sets 16, 17 and 19 as defined in Tab. \[tab:datasets\].
In Tab. \[tab:compcorr1\] alternative correlation models are being studied. The second column shows, as a reference, the first four moments with our default settings. The third and fourth columns show the moments treating the uncertainties of the $J/\psi$ and $\psi^\prime$ partial widths being minimally correlated with the $R_m$ values[^16] and completely uncorrelated, respectively. In the fifth column we display the moments treating, for data sets 1, 6, 13, 15, 16 and 17, one (quadratic) half of systematical uncertainties for the uncorrelated and the other half correlated. In the sixth column all $R_m$ values of all data sets are treated as completely uncorrelated. We see that the central values depend only weakly on the correlation model for those data were the corresponding information is unknown. In particular, for the determination of the uncertainties the ignorance about the composition of the systematical uncertainties in the from data sets 1, 6, 13, 15, 16 and 17 is not essential. However, for quoting the final uncertainties it is important to account for all (known) correlations since treating all errors uncorrelated underestimates the uncertainties considerably.
In Tab. \[tab:compcorr2\] we examine the impact of modifying the definition of the cluster energy $E_m$ and of changes to the default cluster distribution (2,20,20,10). In the second column we display the resulting moments of the default setting. In the third and fourth columns we have shown the moments using for $E_m$ simply the mean of the energies and the center of the cluster, respectively. The resulting differences to the default definition is an order of magnitude smaller than the uncertainties and thus negligible. The fifth, sixth and seventh columns display the moments using some alternative cluster distributions. The deviations for the default choice illustrated in the table are much smaller than the uncertainties and typical for all modifications that satisfy the guidelines for viable cluster definitions we have formulated in Sec. \[subsectioncombination\]. This demonstrates that the choice of the cluster distribution does not result in a bias for the resulting experimental moments.
Finally, we also examine the dependence of the moments on the data set collections as described in Sec. \[subsectioncollections\]. In Tabs. \[tab:moments-results-minimal\] and \[tab:moments-results-maximal\] the results for the moments are displayed using the minimal and the maximal collections (with default choices for all other settings). We see that the differences in the central values to the default collection are about half the size of the systematical correlated uncertainties for the first moment $M_1^{\rm exp}$. For the higher moments the differences are much smaller than the uncertainties. Using, instead of the default, the minimal and maximal collections affects the uncertainties of $M_1^{\rm exp}$ by only about $10\%$. For the higher moments the differences decrease strongly and basically disappear for the fourth moment. Again, the results show that having an increased or decreased redundancy in the data set collection only has a minor impact on the final numbers for the experimental moments.
To summarize, we find that modifications to the choices and assumptions that go into the combined treatment of the experimental data from different publications and experiments lead to changes that are well within the experimental uncertainties we obtain from our combination method. We therefore consider these uncertainties as appropriate. An instructive comparison of the moments obtained in our analysis to those obtained in some previous publications is given in Tab. \[tab:mom-comparison\]. A graphical illustration of the results is shown in Fig. \[fig:Moment-comparison\].
Charm Quark Mass Analysis {#sectionanalysis}
=========================
$\alpha_s(m_Z)$ ${\overline m_c}({\overline m_c})$ $\Delta_{\rm pert}$ $\Delta_{\rm stat}$ $\Delta_{\rm syst}$ ${\overline m_c}(3\,{\rm GeV})$ $\Delta_{\rm pert}$ $\Delta_{\rm stat}$ $\Delta_{\rm syst}$
----------------- ------------------------------------ --------------------- --------------------- --------------------- --------------------------------- --------------------- --------------------- ---------------------
$0.113$ $1.255$ $0.013$ $0.006$ $0.013$ $1.010$ $0.013$ $0.006$ $0.014$
$0.114$ $1.259$ $0.014$ $0.006$ $0.013$ $1.006$ $0.015$ $0.006$ $0.014$
$0.115$ $1.263$ $0.015$ $0.006$ $0.013$ $1.002$ $0.016$ $0.006$ $0.014$
$0.116$ $1.267$ $0.016$ $0.006$ $0.013$ $0.999$ $0.017$ $0.006$ $0.014$
$0.117$ $1.271$ $0.017$ $0.006$ $0.013$ $0.995$ $0.019$ $0.006$ $0.014$
$0.118$ $1.275$ $0.019$ $0.006$ $0.013$ $0.990$ $0.020$ $0.006$ $0.014$
$0.119$ $1.280$ $0.020$ $0.006$ $0.013$ $0.986$ $0.022$ $0.006$ $0.014$
: Results for the central values of ${\overline m_c}({\overline m_c})$ and ${\overline m_c}(3\,{\rm GeV})$ (second and sixth columns) and their perturbative (third and seventh columns), experimental statistical (fourth and eight columns) and experimental systematical (fifth and ninth columns) errors.[]{data-label="tabmcalphas"}
Since it is theoretically most reliable, we use the first moment $M_1$ for our final numerical charm quark mass analysis. As ingredients for the analysis we use
- the iterative expansion method for the perturbative contribution of the theoretical moment at ${\cal O}(\alpha_s^3)$, see Eq. (\[eq:iterative-general\]),
- the gluon condensate correction with its Wilson coefficient determined at ${\cal O}(\alpha_s)$ as described in Sec. \[subsectioncondensate\],
- the first experimental moment $$\begin{aligned}
\label{M1exp}
M_1^{\rm exp} & = & 0.2138\pm 0.0020_{\rm stat}\pm 0.0046_{\rm syst}\end{aligned}$$ using our default settings as discussed in Sec. \[sectiondata\].
One important source of uncertainty we have not yet discussed is the value of the strong $\overline{\mbox{MS}}$ coupling $\alpha_s$. Since in the recent literature [@Abbate:2010xh; @Frederix:2010ne; @Gehrmann:2009eh; @Blumlein:2006be; @Blumlein:2010rn] $\alpha_s$ determinations with a spread larger than the current world average [@Bethke:2009jm] have been obtained, we carry out our numerical analysis for values of $\alpha_s(m_Z)$ between $0.113$ and $0.119$.[^17] The outcome of our analysis is shown in Tab. \[tabmcalphas\]. In Fig. \[figmcalphas\] the central values, perturbative, statistical and systematical uncertainties are displayed graphically. For the central value and the perturbative uncertainty, which show a significant dependence on $\alpha_s$, we can present a linear fit. For the statistical and systematical uncertainties the variation with $\alpha_s$ is smaller than $1$ MeV and we only quote constant values. We thus obtain $$\begin{aligned}
\label{mcfinalalphas}
\overline m_c(\overline m_c)& = &(0.788 + 4.13\times\alpha_s(m_Z))
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.013)_{\rm syst}\\ &&
\, \pm \, (-0.127+1.23\times\alpha_s(m_Z))_{\rm pert}
\, \pm \, (0.002)_{\langle GG\rangle}
\,,\nonumber
\\[2mm]
\overline m_c(3~\mbox{GeV})& = &(1.462 - 4.00\times\alpha_s(m_Z))
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.014)_{\rm syst}\\ &&
\, \pm \, (-0.148+1.43\times\alpha_s(m_Z))_{\rm pert}
\, \pm \, (0.002)_{\langle GG\rangle}\,.\nonumber\end{aligned}$$ Taking as an input $$\begin{aligned}
\label{alphaswa}
\alpha_s(m_Z) & = & 0.1184 \pm 0.0021\,,\end{aligned}$$ which is the current world Bethke average [@Bethke:2009jm] with a tripled uncertainty we obtain $$\begin{aligned}
\label{mcfinalalphaswa}
\overline m_c(\overline m_c) & = & 1.277
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.013)_{\rm syst}
\, \pm \, (0.019)_{\rm pert}
\, \pm \, (0.009)_{\alpha_s}
\, \pm \, (0.002)_{\langle GG\rangle}
\,,\nonumber
\\[2mm]
\overline m_c(3~\mbox{GeV}) & = & 0.998
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.014)_{\rm syst}
\, \pm \, (0.022)_{\rm pert}
\, \pm \, (0.010)_{\alpha_s}
\, \pm \, (0.002)_{\langle GG\rangle}
\,,\nonumber\\\end{aligned}$$ which represents, together with Eq. (\[mcfinalalphas\]), our final numerical result for the $\overline{\mbox{MS}}$ charm mass. Our result is in good agreement with other recent determinations of $\overline{m}_{c}(\overline{m}_{c})$. A summary of the numerical results is shown in Tab. \[tab:comparison-alpha\] and in Fig. \[fig:comparison-alpha\]. Compared to the analysis carried out in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] our experimental uncertainty is larger by $5$ MeV and our perturbative uncertainty is a factor of $10$ larger. Compared to Refs. [@Chetyrkin:2006xg; @Boughezal:2006px] the discrepancy in the perturbative error estimate is even larger. We note that the almost head on agreement in the central value of Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] is accidental, since the central value we obtain is based on a different perturbative expansion (iterative expansion in our work versus fixed-order expansion in theirs), on the way how the central value is defined (average of the lowest and highest values in our work versus value for $\mu=3$ GeV) and on a different central value for the experimental moment ($M_1^{\rm exp}=0.2138$ in our work versus $M_1^{\rm exp}=0.2166$ in theirs). Whereas our definition of the central value makes $\overline m_c$ smaller, our smaller value for $M_1^{\rm exp}$ makes it bigger by almost the same amount.
![ Comparison of recent determinations of $\overline{m}_{c}(\overline{m}_{c})$. Red corresponds to our result, black and gray correspond to ${\mathcal O}\alpha_s^3)$ and ${\mathcal O}(\alpha_s^2)$ charmonimum sum rules analyses, respectively, green labels other kind of sum rule analyses (weighted finite energy sum rules [@Bodenstein:2010qx] and ratios of moments [@Narison:2010cg]), and blue stands for lattice simulations. \[fig:comparison-alpha\] ](alpha-comparison.eps){width="90.00000%"}
Conclusions and Final Thoughts {#sectionconclusions}
==============================
In this work we have used state-of-the-art ${\cal O}(\alpha_s^3)$ input from perturbative QCD to determine the $\overline{\mbox{MS}}$ charm quark mass from relativistic ($n=1$) charmonium sum rules using experimental data on the total hadronic cross section in $e^+e^-$ annihilation. The main aims were (i) to carefully reexamine perturbative uncertainties in the charm mass extractions from the moments of the charm vector current correlator and (i) to fully exploit the available experimental data on the hadronic cross section.
We carried out this work having in mind recent ${\mathcal O}(\alpha_s^3)$ sum rule analyses [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp], where perturbative errors of $2$ MeV or smaller were quoted. Moreover in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] an experimental uncertainty of $9$ MeV was quoted. Given these numbers we found it appropriate to reexamine their analysis. In their work the perturbative uncertainty estimate was achieved using a specific choice to arrange the perturbative expansion and by setting the renormalization scales in $\alpha_s$ ($\mu_\alpha$) and of the $\overline{\mbox{MS}}$ charm quark mass ($\mu_m$) equal. We demonstrated that this results in an accidental cancellation of $\mu_\alpha$ and $\mu_m$ scale variations that is not observed in other alternative ways to treat the perturbative expansion. Moreover, concerning the experimental input their work relied on perturbative QCD predictions instead of available data for energies $\sqrt{s}>4.8$ GeV which results in a rather strong dependence of the final experimental uncertainty on the model-uncertainty assigned to the theory input.
Concerning a proper assessment of perturbative and experimental uncertainties we established in our work the following improvements:
- We demonstrated that for achieving an estimate of perturbative uncertainties based on scale variations that is independent of the perturbative expansion method one needs to vary $\mu_\alpha$ and $\mu_m$ independently, albeit with ranges that avoid large logs. As a result the perturbative uncertainty estimates using different ways to carry out the expansion in $\alpha_s$ become equivalent, which is not the case for $\mu_\alpha=\mu_m$.
- Using a data clustering method similar to Refs. [@D'Agostini:1993uj; @Takeuchi:1995xe; @Hagiwara:2003da] we combined available data on the total $e^+e^-$ hadronic cross section from many different experiments covering energies up to $\sqrt{s}=10.538$ GeV to fully exploit the existing experimental information for the experimental moments. This avoids a strong dependence of the experimental moments on ad-hoc assumptions on the “experimental” model uncertainty being associated to the QCD theory input used for the experimental moments. As a result we were also able to quantify the correlation between different experimental moments.
Using $\alpha_s(m_Z)$ as an unspecified variable and the theoretically more reliable first moment $M_1$ for the fits we have obtained $$\begin{aligned}
\overline m_c(\overline m_c)& = &(0.788 + 4.13\times\alpha_s(m_Z))
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.013)_{\rm syst}\\ &&
\, \pm \, (-0.127+1.23\times\alpha_s(m_Z))_{\rm pert}
\, \pm \, (0.009)_{\alpha_s}
\, \pm \, (0.002)_{\langle GG\rangle}
\,,\nonumber
\\[2mm]
\overline m_c(3~\mbox{GeV})& = &(1.462 - 4.00\times\alpha_s(m_Z))
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.014)_{\rm syst}\\ &&
\, \pm \, (-0.148+1.43\times\alpha_s(m_Z))_{\rm pert}
\, \pm \, (0.002)_{\langle GG\rangle}
\,,\nonumber\end{aligned}$$ for the $\overline{\mbox{MS}}$ charm mass. At the level of uncertainties obtained in our work excellent convergence of perturbation theory was observed. Adopting $\alpha_s(m_Z) = 0.1184 \pm 0.0021$ we then obtain $$\begin{aligned}
\overline m_c(\overline m_c) & = & 1.277
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.013)_{\rm syst}
\, \pm \, (0.019)_{\rm pert}
\, \pm \, (0.009)_{\alpha_s}
\, \pm \, (0.002)_{\langle GG\rangle}
\,,\nonumber
\\[2mm]
\overline m_c(3~\mbox{GeV}) & = & 0.998
\, \pm \, (0.006)_{\rm stat}
\, \pm \, (0.014)_{\rm syst}
\, \pm \, (0.022)_{\rm pert}
\, \pm \, (0.010)_{\alpha_s}
\, \pm \, (0.002)_{\langle GG\rangle}
\,.\nonumber\\\end{aligned}$$ Our perturbative error of $20$ MeV is a factor of ten larger, and the experimental uncertainty of $14$ MeV is by $5$ MeV larger than in the most recent analysis of Ref. [@Kuhn:2007vp]. For estimating the perturbative error a range of scale variations between $\overline m_c(\overline m_c)$ and $4$ GeV was employed. Adding all uncertainties quadratically we obtain $$\begin{aligned}
\label{conclusionmc}
\overline m_c(\overline m_c) & = & 1.277
\, \pm \, 0.026~\mbox{GeV}
\,,\nonumber
\\[2mm]
\overline m_c(3~\mbox{GeV}) & = & 0.998\,\pm \, 0.029~\mbox{GeV}
\,,\end{aligned}$$ giving an uncertainty that is twice the size of the one obtained in Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp]. This difference arises mainly from the more appropriate estimate of perturbative uncertainties obtained in this work. The agreement in the central value with Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp] is a coincidence, arising from an interplay of differences in the experimental input and theoretical treatment.
As a final thought one might ask which further improvements might be possible in the future. As can be seen from Tab. \[tab:momres1\], from the experimental side the biggest improvement could be made from more accurate measurements of the $J/\psi$ and $\psi^\prime$ electronic partial widths. The current relative uncertainties are $2.5$% and $2.4$%, respectively. Here some improvement might be conceivable with dedicated measurements. On theory side, viewing the uncertainties, it is not unreasonable to assume that the computation of ${\cal
O}(\alpha_s^4)$ moments of the vector current correlator might further reduce the perturbative error below the level of $20$ MeV. Using the $\overline{\mbox{MS}}$ scheme for the charm mass the OPE states that the remaining perturbative infrared renormalon ambiguity is of order $\Lambda_{\rm
QCD}^4/\overline m_c^3\sim {\cal O}(5 \mbox{\,-}15~\mbox{MeV})$. This expectation has also been confirmed by explicit bubble chain (large-$\beta_0$) computations [@Grozin:2004ez] and indicates that a further reduction of the perturbative uncertainty is not excluded.
However, to throw in some words of caution, at the level of the present perturbative uncertainties one should also remind oneself about possible loopholes still left in the charmonium sum rule method. An issue we would like to mention concerns the separation of charm and non-charm hadronic production rates needed to carry out the charmonium sum rule. On the theory side the issue is conceptually subtle due to the singlet and secondary radiation contributions which arise at ${\cal O}(\alpha_s^3)$ and ${\cal O}(\alpha_s^2)$, respectively. In this work (as well as in Ref. [@Chetyrkin:2009fv]) both contributions have been considered as non-charm although they contain terms belonging to the $c\bar c$ final state. This treatment might be justified since the size of the corresponding terms is quite small (see Tab. \[tabneglected\]) and since it is the common approach to determine the experimental charm production rate in the continuum region by subtracting theoretical results (or models) for the non-charm rate from the measured total hadronic . In our method to determine the experimental moments this subtraction involves a normalization constant multiplying the theoretical non-charm that is fitted within our clustering method as well accounting for data below and above the charm threshold. The result (see Eq. (\[eq:ns-results\])) reveals a disparity of $4$% between the theoretical non-charm and the data. Setting, in contrast, the normalization constant to unity results in a shift in the charm mass by $-15$ MeV.[^18] Since this shift is compatible with the overall systematical uncertainty in the experimental data we have not treated it as an additional source of uncertainty. On the other hand, the size of the shift could also be considered as an inherent conceptual uncertainty related to separating the charm from the non-charm , which is based on theory considerations rather than on experimental methods and which apparently cannot be improved simply by higher order perturbative computations. As an alternative, one might avoid the separation of the charm and the non-charm contributions altogether and use the total hadronic cross section for the charm mass fits. Apart from the shift mentioned above such an approach would, however, also lead to a substantially larger dependence on the uncertainties in $\alpha_s$. Given these considerations we believe that a substantial reduction of the uncertainties also relies on a resolution of the disparity mentioned above. This might certainly involve more precise measurements in the charm threshold and below-threshold regions, but also some deeper conceptual insight. Until then a substantial reduction of the uncertainties shown in Eq. (\[conclusionmc\]) appears hard to achieve without imposing ad-hoc assumptions.
This work was supported in part by the European Community’s Marie-Curie Research Networks under contract MRTN-CT-2006-035482 (FLAVIAnet), MRTN-CT-2006-035505 (HEPTOOLS) and PITN-GA-2010-264564 (LHCphenOnet), and by the U.S. Department of Energy under the grant FG02-94ER40818. V. Mateu has been partially supported by a DFG “Eigenen Stelle” under contract MA 4882/1-1 and by a Marie Curie Fellowship under contract PIOF-GA-2009-251174. S. Zebarjad thanks the MPI for hospitality while part of this work was accomplished. S. Zebarjad and V. Mateu are grateful to the MPI guest program for partial support. We thank S. Schutzmeier for confirmation of our numerical ${\cal O}(\alpha_s^3)$ fixed-order results. A. Hoang acknowledges discussion with H. Kühn and C. Sturm.
Appendix A: Results of the Fit Procedure {#appendix-a-results-of-the-fit-procedure .unnumbered}
========================================
\[ap:FitOutcome\]
In this appendix we present in some more detail the numerical results of our fit procedure. In Tabs. \[tab:FitFunction-Standard\], \[tab:FitFunction-Minimal\] and \[tab:FitFunction-Maximal\], the results for the cluster energies and the cluster charmed are shown for the standard, minimal and maximal selection of data sets, respectively, using our default setting for the correlations. We use the results for the standard data set selection for our final charm mass analysis. The numbers in the parentheses correspond to the statistical and systematical errors. The correlation matrices for the is available, but cannot be displayed due to lack of space. They can be obtained by the authors on request. For the three data selections, the fit gives the following minimal $\chi^2$ per degree of freedom, $$\dfrac{\chi_{\rm standard}^2}{\rm dof}=1.89\,, \qquad
\dfrac{\chi_{\rm minimal}^2}{\rm dof}=1.86\,,
\qquad \dfrac{\chi_{\rm maximal}^2}{\rm dof}=1.81\,,
\label{eq:chi2-results}$$ and the following normalization constants for the non-charm background $$\begin{aligned}
\!\!\!\!\!\!n^{\rm standard}_s&=&1.039 \pm 0.003_{\rm stat} \pm 0.012_{\rm syst},\; n^{\rm minimal}_s=1.029 \pm 0.003_{\rm stat} \pm
0.015_{\rm syst},
\label{eq:ns-results} \\
\!\!\!\!\!\!n^{\rm maximal}_s&=&1.023 \pm 0.003_{\rm stat} \pm 0.011_{\rm syst}.\nonumber\end{aligned}$$ Since the minimal $\chi^2/{\rm dof}$ values are not close to unity, one has to conclude that fit quality is not really very good. This is not at all visible from the agreement of the fit and the data for the total cross section (see Figs. \[fig:fitcompilation\]) and thus might be related to the disparity between the fits of charm versus non-charm production rates described in Sec. \[sectionconclusions\].
In Eq. (\[correlmatrices\]) we show for the correlation matrices of the first four experimental moments for the minimal and the maximal data set selection. The results for our standard selection are given in Eqs. (\[CMtotal\]) and (\[CMuncorr\]). All numbers are related to moments $M_n^{\rm exp}$ normalized to units of $10^{-(n+1)}\,\mbox{GeV}^{-2n}$. We show the results accounting for the full set of correlated and uncorrelated uncertainties and the correlation matrices accounting only for uncorrelated systematical and statistical uncertainties (subscript uc).
$$\begin{aligned}
\label{correlmatrices}
C^{\rm exp}_{\rm min}&=&\left(
\begin{array}{cccc}
0.304 & 0.194 & 0.165 & 0.157\\
0.194 & 0.133 & 0.115 & 0.109\\
0.165 & 0.115 & 0.100 & 0.096\\
0.157 & 0.109 & 0.096 & 0.092\\
\end{array}
\right)\,,\quad C^{\rm exp}_{\rm uc}\,=\,\left(
\begin{array}{cccc}
0.047 & 0.038 & 0.035 & 0.035\\
0.038 & 0.035 & 0.035 & 0.035\\
0.035 & 0.035 & 0.035 & 0.036\\
0.035 & 0.035 & 0.036 & 0.037
\end{array}\right)\,,\\\nonumber\\
C^{\rm exp}_{\rm max}&=&\left(
\begin{array}{cccc}
0.193 & 0.138 & 0.125 & 0.122\\
0.138 & 0.108 & 0.099 & 0.097\\
0.125 & 0.099 & 0.091 & 0.090\\
0.122 & 0.097 & 0.090 & 0.088\\
\end{array}
\right)\,,\quad
C^{\rm exp}_{\rm uc}\,=\,\left(
\begin{array}{cccc}
0.037 & 0.034 & 0.033 & 0.034\\
0.034 & 0.033 & 0.034 & 0.035\\
0.033 & 0.034 & 0.035 & 0.036\\
0.034 & 0.035 & 0.036 & 0.037
\end{array}\right)\,.\nonumber\end{aligned}$$
Appendix B: On the Equivalence of $\chi^2$-Functions {#appendix-b-on-the-equivalence-of-chi2-functions .unnumbered}
====================================================
In this appendix we demonstrate that a in which the auxiliary fit parameters $d_k$, which describe the correlated deviation off the experimental central value within experiment $k$, multiplies only the experimental systematical uncertainties,[^19] $$\chi^2 = \sum_k\left[d_k^2+\sum_{i,m}\left(\dfrac{R_i^{k,m}+d_k\,
\Delta_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2\right]\,, \label{additive}$$ is mathematically equivalent to the well known written solely in terms of the fit parameters $R_m$ and a correlation matrix, $$\bar \chi^2=\sum_{i,j,k,m,n}(R_i^{k,m}-R_m)(V_k^{-1})_{ij}^{mn}(R_j^{k,n}-R_n)\,,
\label{traditional}$$ where there is no correlation among the different experiments $k$ and $k^\prime$, and the correlation matrix within one experiment $k$ has the form $$(V_k)_{ij}^{mn}=\sigma_i^{k,m\,2}\delta_{ij}\delta^{mn}+\Delta_i^{k,m}\Delta_j^{k,n}
\,.
\label{tradcorr}$$ The matrix Eq. (\[tradcorr\]) can be inverted analytically giving $$(V_k^{-1})_{ij}^{mn} = \dfrac{\delta_{ij}\delta^{mn}}{(\sigma_i^{k,m})^2}-
\dfrac{1}{1+\dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle_k^2}}
\dfrac{\Delta_i^{k,m}}{(\sigma_i^{k,m})^2}\dfrac{\Delta_j^{k,n}}{(\sigma_j^{k,n})^2}\,,
\label{invtradcorr}$$ where we have defined the mean statistical error and the statistical average of the systematical error within one experiment as follows: $$\left\langle \sigma\right\rangle_k^2 \equiv
\left(\sum_{i,m}\dfrac{1}{(\sigma_i^{k,m})^2}\right)^{-1},\qquad
\left\langle \Delta^2
\right\rangle_k\equiv\left\langle \sigma\right\rangle_k^2\sum_{i,m}
\dfrac{(\Delta_i^{k,m})^2}{(\sigma_i^{k,m})^2}\,.
\label{averages}$$
In case of a sizable positive correlation between measurements in the same experiment (such that $\Delta_i^{k,m}/R_i^{k,m}$ is sizable and approximately constant) is it known that the form of the in Eq. (\[additive\]) leads to best fit values $R_m$ that are systematically below the measurements [@D'Agostini:1993uj]. Our proof demonstrates that the standard of Eq. (\[traditional\]) has the same property for correlation matrices with the form of Eq. (\[tradcorr\]). This motivates to use the so-called minimal-overlap correlation model where the second term on the RHS of Eq. (\[traditional\]) is replaced by $\mbox{min}^2(\Delta_i^{k,m},\Delta_j^{k,n})$. In general, within the minimal-overlap model, the correlations are sufficiently reduced such that the unphysical effect described above does not arise.
We proceed by showing that one can “integrate out" auxiliary parameters $d_k$ in Eq. (\[additive\]) obtaining a new function $\tilde \chi^2(R^i)$ which yields the same results for the best fit for the $R_m$, as long as one works in the Gaussian approximation. The minimum of $\chi^2(R_i,d_k)$ is located at the best fit values (indicated by superscripts $(0)$) defined by the conditions $$\left.\dfrac{\partial\chi^{2}}{\partial R_{i}}\right|_{R_i^{(0)},d_k^{(0)}}=\left.\dfrac{\partial\chi^{2}}{\partial d_{j}}\right|_{R_i^{(0)},d_k^{(0)}}=0\,.\label{Mincond}$$ To invert the matrix of second derivatives we proceed in blocks $$\left(\begin{array}{cc}
\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}} & \dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial d_{k}}\\
\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}} & \dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial b_{k}}\end{array}\right)_{\tilde{R},\tilde{b}}\left(\begin{array}{cc}
c_{jm} & b_{jr}\\
b_{mk} & a_{kr}\end{array}\right)=\left(\begin{array}{cc}
\delta_{im} & 0\\
0 & \delta_{lr}\end{array}\right).$$ where $c_{ij}$ and $a_{kr}$ are $N_{\rm cluster}\times N_{\rm cluster}$ and $N_{\rm exp}\times N_{\rm exp}$ square matrices, respectively, and $b_{jr}$ is a $N_{\rm cluster}\times N_{\rm exp}$ rectangular matrix. We find the following four matrix relations $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\, c_{jm}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial d_{k}}\, b_{mk}=\delta_{im}\,, \quad
\sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\,
b_{jr}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial d_{k}}\,
a_{kr}=0\,,
\label{eq:general_block_1}\\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}}\,
c_{jm}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}\,
b_{mk}=0\,,
\qquad\;\, \sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial
d_{l}\partial R_{j}}\, b_{jr}+\sum_{k}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}} a_{kr}=\delta_{lr}\,.\label{eq:general_block_2}\end{aligned}$$ Combining Eqs. (\[eq:general\_block\_1\]a) and (\[eq:general\_block\_2\]a) we find the inverse of the upper left , $$(c^{-1})_{ij}=2(V^{-1})_{ij}=\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}-\sum_{kl}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{i}}\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial
d_{k}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial R_{j}}\,,
\label{eq:noEOM_Rj_bk}$$ where $\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial
d_{k}}\right]^{-1}$ stands for the $(l,k)$ element of the inverse matrix of $\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}$ (and not the inverse of the element). Combining Eqs. (\[eq:general\_block\_1\]b) and (\[eq:general\_block\_2\]b) one can obtain relations for the $a$ and $b$ submatrices: $$\begin{aligned}
b_{jr} & = & -\sum_{i,k}\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial R_{i}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial R_{i}}\, a_{kr}\,,\\
\delta_{lr}&=&\sum_{k}\left(\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial d_{k}}-\sum_{i,j}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{j}}\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial
R_{i}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{i}\partial R_{k}}\right)a_{kr}\,,\\
\delta_{im} & = & \sum_{j}\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\, c_{jm}-\sum_{k,l,s}\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial R_{i}}\,\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{m}\partial
R_{s}}\right]^{-1}\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{s}}\, a_{lk}\,,\end{aligned}$$ where again $\left[\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial R_{i}}\right]^{-1}$ stands for the $(j,i)$ element of the inverse matrix of $\dfrac{\partial^{2}\chi^{2}}{\partial R_{j}\partial R_{i}}$. We “integrate out" the auxiliary parameters $d_k$ by substituting their minimum conditions $d^{(0)}_k=\tilde d_k(R^{(0)}_i)$ (which is analogous to using the equation of motion when integrating out heavy particles): $$\left.\dfrac{\partial\chi^{2}}{\partial d_{j}}\right|_{d_k=\tilde d_k(R_i)}=0\,,\qquad \tilde \chi^2(R_i)=\chi^2(R_i,\tilde d_k(R_i))\,.\label{EOM}$$ Their first derivatives with respect to $R_{j}$ read $$\begin{aligned}
\dfrac{\partial}{\partial R_{j}}\left.\dfrac{\partial\chi^{2}}{\partial d_{i}}\right|_{\tilde d_m(R_{k})} & = & \left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{i}\partial R_{j}}\right|_{\tilde
d_m(R_{k})}+\sum_{l}\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{i}\partial d_{l}}\right|_{{\tilde d}_{m}(R_{k})}\dfrac{\partial {\tilde d}_{l}(R_{n})}{\partial R_{j}}=0\,,\nonumber \\
\dfrac{\partial \tilde d_i(R_{n})}{\partial R_{j}} & = & -\sum_l\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_i\partial d_l}\right]_{\tilde d_m(R_{k})}^{-1}\,\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial
R_{j}}\right|_{\tilde d_m(R_{k})}
\,.\end{aligned}$$ The minimum of $\tilde \chi^2(R_i)$ is indeed located at $R^{(0)}_{i}$ because $$\dfrac{\partial\tilde\chi^2}{\partial R_{i}}=\sum_k\left.\dfrac{\partial\chi^{2}}{\partial d_k}\right|_{d_k=\tilde d_k(R_i)}\dfrac{{\rm d} \tilde d_k}{{\rm d} R_i}+
\left.\dfrac{\partial\chi^{2}}{\partial R_i}\right|_{d_k=\tilde
d_k(R_i)}=\left.\dfrac{\partial\chi^{2}}{\partial R_i}\right|_{d_k=\tilde
d_k(R_i)}
\,,\label{first_der}$$ and because the first term vanishes by the condition in Eq. (\[EOM\]). When evaluating Eq. (\[first\_der\]) for $R_i=R_i^{(0)}$ it vanishes because of Eq. (\[Mincond\]). Finally, let us calculate the inverse correlation matrix: $$\begin{aligned}
&&\left.\dfrac{\partial^{2}\tilde{\chi}^{2}}{\partial R_{i}\partial R_{j}}\right|_{\tilde{R}_{k}} =
\left.\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\right|_{d^{(0)}_{m},R^{(0)}_{k}}+
\sum_{k}\left.\dfrac{\partial^2\chi^{2}}{\partial d_k\partial R_j}\right|_{d^{(0)}_m,R^{(0)}_k}
\left.\dfrac{\partial \tilde d_{k}(R_{i})}{\partial R_{i}}\right|_{\tilde{R}_{k}}\nonumber \\
& = & \left.\dfrac{\partial^{2}\chi^{2}}{\partial R_{i}\partial R_{j}}\right|_{d^{(0)},R^{(0)}_{k}}-\sum_{k,l}\left[\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial
d_{l}}\right]_{d^{(0)}_{m},R^{(0)}_{k}}^{-1}\,\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{l}\partial R_{i}}\right|_{d^{(0)}_{m},R^{(0)}_{k}}\left.\dfrac{\partial^{2}\chi^{2}}{\partial d_{k}\partial
R_{j}}\right|_{d^{(0)}_{m},R^{(0)}_{k}},\label{eq:EOM_Rj_bk}\end{aligned}$$ which agrees with Eq. (\[eq:noEOM\_Rj\_bk\]).
We can now apply the previous results to Eq. (\[additive\]): $$\begin{aligned}
\chi^{2} & = & \sum_{k}\left[d_k^2+\sum_{i,m}\left(\dfrac{R_i^{k,m}+d_k\,
\Delta_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2\right]\label{eq:with_bk_Rj}\\
& = & \sum_k\left\{ d_k^2\left[1+
\dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle _k^2}\right]+2\, d_k\sum_{i,m}\dfrac{(R_i^{k,m}-R_m)\Delta_i^k}{\sigma_{i}^{k,m\,2}}+\sum_{i,m}
\left(\dfrac{R_i^{k,m}-R_{m}}{\sigma_i^{k,m}}\right)^2\right\}
\,.\nonumber\end{aligned}$$ The equation of motion for $d_{k}$ reads $$d_k(R)=-\,\dfrac{1}{1+\dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle_k^2}}
\sum_{i,m}\dfrac{(R_i^{k,m}-R_m)\Delta_i^{k,m}}{\sigma_i^{k,m\,2}}
\,.$$ This renders for $\tilde\chi^2$ the form $$\begin{aligned}
\tilde\chi^2 & = & \sum_k\left[d_k(R)\sum_{i,m}\dfrac{(R_i^{k,m}-R_m)\Delta_i^{k,m}}{\sigma_i^{k,m\,2}}+
\sum_{i,m}\left(\dfrac{R_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2\right]\\
& = & \sum_k\left[\sum_{i,m}\left(\dfrac{R_i^{k,m}-R_m}{\sigma_i^{k,m}}\right)^2-
\dfrac{1}{1+\dfrac{\left\langle \Delta^2\right\rangle_k}{\left\langle \sigma\right\rangle_k^2}}\sum_{i,j,m,n}\dfrac{(R_i^{k,m}-R_m)\Delta_i^{k,m}}
{\sigma_i^{k,m\,2}}\dfrac{(R_j^{k,n}-R_n)\Delta_j^{k,n}}{\sigma_j^{k,n\,2}}\right]\,,\nonumber \end{aligned}$$ which reproduces Eq. (\[traditional\]), as we wanted to demonstrate.
We conclude this appendix presenting an alternative way to write Eq. (\[chi2total\]) after using the equations of motion for $d_k$. We again concentrate on the simpler case without subtraction of the non-charm contribution: $$\begin{aligned}
\chi^2 & = & \sum_k\left\{ d_k^{2}+\sum_{i,m}\left[\dfrac{R_i^{k,m}-(1+d_k\Delta f_k^{i,m})\, R_m}{\sigma_k^{i,m}}\right]^{2}\right\} \nonumber \\
& = & \sum_k\left\{ \left[1+\sum_{i,m}
\left(\dfrac{\Delta f_k^{i,m}\, R_m}{\sigma_k^{i,m}}\right)^2\right]d_k^2-2d_k\sum_{i,m}\dfrac{\left(R_{k}^{i,m}-R_m\right)
\Delta f_k^{i,m}\, R_m}{(\sigma_k^{i,m})^2}\right.\nonumber \\
&+&\left.\sum_{i,m}\left[\dfrac{R_k^{i,m}-R_m}
{\sigma_k^{i,m}}\right]^2\right\} .\label{eq:chi2_auxiliary}\end{aligned}$$ The EOM for $d_k$ now reads $$\left[1+\sum_{i,m}\left(\dfrac{\Delta f_k^{i,m}\, R_m}{\sigma_k^{i,m}}\right)^2\right]
\tilde d_k(R_n)-\sum_{i,m}\dfrac{\left(R_k^{i,m}-R_m\right)\Delta f_k^{i,m}\,
R_m}{(\sigma_k^{i,m})^2}=0
\,,$$ which upon insertion into Eq. (\[eq:chi2\_auxiliary\]) renders $$\begin{aligned}
\widetilde{\chi}^2 & = & \sum_k\left\{
\sum_{i,m}\left[\dfrac{R_k^{i,m}-R_m}{\sigma_k^{i,m}}\right]^2-\tilde
d_k(R_n)\sum_{i,m}\dfrac{\left(R_k^{i,m}-R_m\right)\Delta f_k^{i,m}\, R_m}
{(\sigma_k^{i,m})^2}\right\} \nonumber\\
& = & \sum_k\Biggr\{ \sum_{i,m}\left[\dfrac{R_k^{i,m}-R_m}{\sigma_k^{i,m}}\right]^2\nonumber\\
&-&
\dfrac{1}{\left[1+\sum\left(\dfrac{\Delta f_k^{i,m}\,
R_m}{\sigma_k^{i,m}}\right)^2\right]}
\left[\sum_{i,m}\dfrac{\left(R_k^{i,m}-R_m\right)\Delta_k^{i,m}\, R_m}
{(\sigma_k^{i,m})^2}\right]^2\Biggr\}
\,.\label{eq:chi2_final} \end{aligned}$$ One can rewrite Eq. (\[eq:chi2\_final\]) in the matrix form $$\begin{aligned}
\widetilde{\chi}^2 & = & \sum_k\left\{ \sum_{i,j,m,n}\left[R_{k}^{im}-R_m\right]\left[V_{k}^{-1}\right]^{mn}_{ij}\left[R_{k}^{jn}-R_n\right]\right\} \,,\label{eq:Matrix-Form} \\
\left[V_{k}^{-1}\right]^{mn}_{ij} & = & \dfrac{\delta_{ij}\delta^{mn}}{(\sigma_{k}^{i,m})^{2}}-
\dfrac{\left(R_k^{i,m}-R_m\right)\Delta f_k^{i,m}\, R_m
\left(R_k^{jn}-R_n\right)\Delta f_k^{jn}\, R_n}
{\left[1+\sum\left(\dfrac{\Delta_k^{i,m}\, R_m}{\sigma_k^{i,m}}\right)^2\right]}\,.\nonumber\end{aligned}$$ In Eq. (\[eq:Matrix-Form\]) one can interpret the second term of the inverse correlation matrix, as a non-linear correlation among the measurements, where the correlation matrix itself depends on the value of the fit parameters. The total inverse correlation matrix is block diagonal, and the blocks correspond to $V_k^{-1}$.
[^1]: As a word of caution we mention that, except for the contributions of the $J/\psi$ and $\psi'$ resonances, no experimental separation of the charm and non-charm contributions in the hadronic cross section has been provided in available data, although charm-tagged measurements are possible, see e.g. Ref. [@CroninHennessy:2008yi] (CLEO collaboration). So the charm pair production rate from above the $J/\psi$ and $\psi^\prime$ that enters the charmonium sum rules in Eq. (\[momentdef2\]) is usually obtained partly from the measured total with theory motivated subtractions of the non-charm rate, and partly by using theory predictions for the charm production rate.
[^2]: Since the analyses of Refs. [@Chetyrkin:2006xg; @Boughezal:2006px] were based on outdated and less precise data for the $J/\psi$ and $\psi^\prime$ electronic partial widths [@Yao:2006px], we frequently only compare our numerical results with those of Refs. [@Chetyrkin:2009fv; @Kuhn:2007vp]. However the perturbative input of Refs. [@Chetyrkin:2006xg; @Boughezal:2006px; @Chetyrkin:2009fv; @Kuhn:2007vp] and ours is identical.
[^3]: This can to happen at any order for $n>1$ and for $\mu_m\sim 3$ GeV.
[^4]: This works in general as long as the path dependent $\mu_\alpha^c(q^2,\overline m_c^2)$ does not produce a spurious cut in $\alpha_s$ starting at $q^2=0$ and running towards $-\infty$. This condition is implemented into Eq. (\[mualphacontour\]) by $(\mu_\alpha^c)^2$ being negative along the physical cut of the vacuum polarization function above the charm pair threshold.
[^5]: In this examination and throughout our other analyses we use the $\overline{\rm
MS}$ renormalization group equations with $n_f=4$ active running flavors.
[^6]: The numerical value of $\overline m_c(\mu=3~\mbox{GeV})$ is actually irrelevant, since the running involves $\overline m_c$ only in a linear way and exactly cancels in the ratio.
[^7]: For the discussions in this section we use $\alpha_s^{(n_f=5)}(m_Z)=0.118$ ($\alpha_s^{(n_f=4)}(4.2~\mbox{GeV})=0.2245$) as an input using five-to-four flavor matching at $4.2$ GeV. For the first moment employed for the charm mass fits we use $M_1=0.2138\,{\rm GeV}^{-1}$, corresponding to the standard selection of data sets, as described in Sec. \[sectiondata\].
[^8]: Of course the extracted mass depends on the moment considered as well. Since in our analysis we focus on the first moment only, for simplicity we drop that label.
[^9]: We are grateful to Thomas Schutzmeier for confirming agreement of the results of our ${\cal O}(\alpha_s^3)$ fixed-order code with theirs of Ref. [@Boughezal:2006px].
[^10]: There are 18 references quoted since Ref. [@Edwards:1990pc] provides results from two independent runs that we treat as two different data sets.
[^11]: The situation is quite different if perturbative QCD is used already from $4.8$ GeV. Here the contribution to the moments $M_{1,2,3,4}^{\rm exp}$ from energies above $4.8$ GeV amounts to $(30, 10, 3,
1)\%$. For the first moment the $10$% penalty would then represent the largest source of uncertainty and correspond to an uncertainty in the charm mass of around $18$ MeV.
[^12]: We have checked that the effect of using $R_{uds}(E_m)$ instead of $R_{uds}(E_i^{k,m})$ is totally negligible.
[^13]: We prove in Appendix B the equivalence of a bilinear with fit auxiliary parameters to a bilinear without auxiliary fit parameters, but containing the standard correlation matrix.
[^14]: In a very good approximation, it can be also obtained by dropping in $V^{-1}_{i,j}$ the rows and columns corresponding to $d_k$ and inverting the resulting matrix. After that one also drops the raw and column corresponding to $n_{\rm nc}$. We adopt this simplified procedure for our numerical fits.
[^15]: For $R_0$ we have $m^\prime=1$ and for $R_{N_{cl}+1}$ we have $m^\prime=N_{cl}-1$.
[^16]: We use a modified version of the minimal correlation model. The non-diagonal entries of the correlation matrix are filled in with $\Gamma_i\, R_m\, {\rm Min}^2\left\lbrace
{\Delta{\Gamma_i}/\Gamma_i, \Delta R_m/R_m}\right\rbrace
$. Here $\Delta\Gamma_i$ and $\Delta R_m$ represent the systematical uncertainties of the width of the narrow resonance and the $R$ value of the cluster, respectively, and $i=1,2$ refer to $J/\psi$ and $\psi^\prime$.
[^17]: As our default we use $\alpha_s^{n_f=5}(m_Z)$ as the input, use the four-loop QCD beta-function for the renormalization group evolution and three-loop matching conditions to the $n_f=4$ theory at $\mu=4.2$ GeV.
[^18]: The same disparity was found in Ref. [@Kuhn:2007vp]. In their analysis the corresponding effect is $-5$ MeV since experimental data were used in the experimental moments only for energies $\sqrt{s}<4.8$ MeV. In this approach, however, the moments are strongly dependent on the uncertainty one assigns to the theory input used for $\sqrt{s}>4.8$ MeV.
[^19]: For the discussion in this appendix we do for simplicity of the presentation not account for fits of the non-charm contribution. It is, however, straightforward to generalize the presentation to this case.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $W,W''\subseteq G$ be nonempty subsets in an arbitrary group $G$. $W''$ is said to be a complement to $W$ if $WW''=G$ and it is minimal if no proper subset of $W''$ is a complement to $W$. We show that, if $W$ is finite then every complement of $W$ has a minimal complement, answering a problem of Nathanson. We also give necessary and sufficient conditions for the existence of minimal complements of a certain class of infinite subsets $W$ in finitely generated abelian groups, partially answering another problem of Nathanson.'
address:
- 'Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.'
- 'Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India'
author:
- Arindam Biswas
- Jyoti Prakash Saha
title: On minimal complements in groups
---
[^1]
[^2]
Introduction
============
Motivation
----------
Let $(G,.)$ be a group and $W\subseteq G$ be a nonempty subset. A nonempty set $W'\subseteq G$ is said to be a complement to $W$ if $$WW' = G.$$ Let $\mathcal{W}$ denote the set of all complements of $W$. Then it is clear that $\mathcal{W}\neq \emptyset$ (since $G\in \mathcal{W}$) and also the fact that the elements of $\mathcal{W}$ form a partially ordered set under inclusion.
A complement $W'$ to $W$ is minimal if no proper subset of $W'$ is a complement to $W$, i.e., $$WW' = G \,\text{ and }\, W.(W'\setminus \lbrace w'\rbrace)\subsetneq G \,\,\, \forall w'\in W'.$$
Given a minimal complement $W'$ of $W$, we see that the right translation $W'g$ is also a minimal complement of $W$ and $W'$ is a minimal complement of $gW$ for all $g \in G$. Thus, the existence of a minimal complement of a nonempty subset is equivalent to the existence of a minimal complement of any of its left translates.
It was shown by Nathanson (see [@NathansonAddNT4 Theorem 8]) that for a non-empty, finite subset $W$ in the additive group $\mathbb{Z}$, any complement to $W$ has a minimal complement. In the same paper he asked the following questions:
[@NathansonAddNT4 Problem 11] \[nathansonprob11\] “Let $W$ be an infinite set of integers. Does there exist a minimal complement to $W$? Does there exist a complement to $W$ that does not contain a minimal complement?”
[@NathansonAddNT4 Problem 12] \[nathansonprob12\] “Let $G$ be an infinite group, and let $W$ be a finite subset of $G$. Does there exist a minimal complement to $W$? Does there exist a complement to $W$ that does not contain a minimal complement?”
[@NathansonAddNT4 Problem 13] \[nathansonprob13\] “Let $G$ be an infinite group, and let $W$ be an infinite subset of $G$. Does there exist a minimal complement to $W$? Does there exist a complement to $W$ that does not contain a minimal complement?”
Since then the problems have generated considerable interest. Chen and Yang in 2012 gave examples of two infinite sets $W_1,W_2 \subset \mathbb{Z}$, such that $W_1$ has a complement that does not contain a minimal complement and every complement to $W_2$ contains a minimal complement (see [@ChenYang12]). They also gave certain necessary and certain sufficient conditions on the infinite set $W \subset \mathbb{Z}$ such that $W$ has a minimal complement (see [@ChenYang12 Theorem 1, 2]). Very recently, Kiss, Sándor and Yang [@KissSandorYangJCT19] succeeded in giving necessary and sufficient conditions for the existence of minimal complements of several other class of infinite sets in $\mathbb{Z}$ (which were not covered in the previous work of Chen and Yang). See [@KissSandorYangJCT19 Theorems 1, 2, 3].
Statement of results
--------------------
All the aforementioned progresses were in the setting of Question \[nathansonprob11\]. In this article, we deal with the Questions \[nathansonprob12\] and \[nathansonprob13\]. Specifically, we show that -
\[Theorem \[theorem1\]\] \[theorem1.1\] Let $G$ be an arbitrary group with $S$ a nonempty finite subset of $G$. Then every complement of $S$ in $G$ has a minimal complement.
See Section \[sec2\], Theorem \[theorem1\]. This answers Question \[nathansonprob12\] of Nathanson.\
We turn to Question \[nathansonprob13\]. Before commencing the discussion in detail, we state that it has a simple answer when the subset $W$ is a subgroup. Namely, in that case a minimal complement always exist. See Proposition \[prop4.1\]. For general infinite subsets, the situation is more delicate. To give an answer to Question \[nathansonprob13\], one needs to consider the infinite subsets which have less algebraic structure.
Our goal in this case is to establish the existence of minimal complements for a large class of infinite sets in finitely generated abelian groups. This will give the claimed partial solution to this question of Nathanson. In section \[Sec: Minimal complement\], we focus on the minimal complements of certain infinite subsets of free abelian groups of finite rank, which are of the form ${\ensuremath{\mathbb{Z}}}^d$ for some integer $d\geq 1$. It is interesting to consider the subsets $X$ of ${\ensuremath{\mathbb{Z}}}^d$ such that $x + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $X$ for any $x\in X$, i.e., $$\label{Eqn: preperiodic}
X \supseteq X + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d,$$ where $u_1, \cdots, u_d$ are element of ${\ensuremath{\mathbb{Z}}}^d$ satisfying no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation. However it turns out that such sets do not necessarily have a minimal complement. For instance, ${\ensuremath{\mathbb{Z}}}\times {\ensuremath{\mathbb{N}}}$ is a subset of ${\ensuremath{\mathbb{Z}}}^d$ which satisfies Equation , but do not have any minimal complement.
To obtain examples of infinite subsets of ${\ensuremath{\mathbb{Z}}}^d$ having minimal complements, we consider the periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ (these are subsets of ${\ensuremath{\mathbb{Z}}}^d$ satisfying Equation along with a finiteness condition, see Definition \[Defn: periodic\]). Unfortunately, there exist periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$, which do not admit any minimal complement (see Proposition \[Cor: Nd minimal complement\]). So we consider a more general class of subsets of ${\ensuremath{\mathbb{Z}}}^d$ satisfying a weaker version of Equation along with certain finiteness condition (which we call *eventually periodic subsets*, see Definition \[Defn: periodic\] - these are $d$-dimensional analogues of the eventually periodic sets in $\mathbb{Z}$ considered by Kiss-Sándor-Yang in [@KissSandorYangJCT19]). Given an eventually periodic subset $W$ of ${\ensuremath{\mathbb{Z}}}^d$ (with periods $u_1, \cdots, u_d$), by Theorem \[Thm: Structure of Eventually Periodic\], there exist subsets ${\ensuremath{\mathcal {W}}}, {\ensuremath{\mathscr{W}}}$ of $W$ such that $$W = {\ensuremath{\mathscr{W}}}\sqcup ({\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$$ holds. It turns out that certain eventually periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ have minimal complements. The following result provides a necessary condition for an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ to have a minimal complement. We prove the following results -
\[Theorem \[Thm: existence of min complement implies\]\] \[theorem1.2\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$. Then ${\ensuremath{\mathscr{W}}}_1$ is nonempty and there exists a nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ such that the following conditions hold.
1. The map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective.
2. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$.
This is a simplified version of Theorem \[Thm: existence of min complement implies\]. We refer to Theorem \[Thm: existence of min complement implies\] for a more general statement.
As a consequence, we obtain that no periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ has a minimal complement (Corollary \[Cor: No periodic has minimal\]). We also prove the result below, giving a sufficient condition for an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ to have a minimal complement.
\[Theorem \[Thm: Implies existence of min comple\]\] \[theorem1.3\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose there exists a nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ such that the following conditions hold.
1. The set ${\ensuremath{\mathscr{W}}}_1$ is nonempty.
2. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective.
3. The map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective.
4. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$.
Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$.
Using the above, we get a necessary and sufficient condition for certain type of eventually periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ to have a minimal complement.
\[Theorem \[Thm: Even peri 1 has min comp iff\]\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose ${\ensuremath{\mathscr{W}}}_1$ contains only one element. Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$ if and only if there exists nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ satisfying the following.
1. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective.
2. The map $\pi: ({\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective.
3. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$.
Finally, we conclude a general statement on the existence of minimal complements for certain infinite sets in finitely generated abelian groups.
\[Theorem \[sec5Thm\]\] Let $G$ be a finitely generated (infinite) abelian group with the decomposition $G\simeq \mathbb{Z}^{d}\times(\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$ with $d\geqslant 1$ and where $a_{1}|a_{2}|...|a_{s}$ are positive integers $>1$ (determined uniquely by the isomorphism type of $G$). Suppose $W\subset \mathbb{Z}^{d}$ such that either $W$ satisfies the conditions of Theorem \[theorem1.3\] (equivalently Theorem \[Thm: Implies existence of min comple\]) or $W$ is a product set of the form $W_{1}\times W_{2}\times \cdots \times W_{d}$ where each $W_{i}\subseteq \mathbb{Z}, \forall 1\leqslant i\leqslant d$ has minimal complements in $\mathbb{Z}$, then $W\times H$ will have a minimal complement in $G$ where $H \subseteq (\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$ is any arbitrary nonempty subset.
See also Remark \[Remark:MinCompEg\] for further discussion.
Plan of the paper
-----------------
The article is divided into $6$ main sections. These are\
1. Introduction - Here we give the background, motivation and results.\
2. Existence of minimal complements for finite sets in arbitrary groups - This section deals with the finite subset case in arbitrary groups, answering the Question \[nathansonprob12\] of Nathanson cf Theorem \[theorem1\].
The subsequent sections deal with Question \[nathansonprob13\].
3. Inexistence of minimal complements for certain infinite sets - This section deals with specific examples of sets which do not have minimal complements. It also provides insight on which sets to avoid in the search for sets having minimal complements in $\mathbb{Z}^{d}$. See Propositions \[prop3.1\], \[prop3.3\].\
4. Minimal complements of infinite sets in free abelian groups - This section is concerned with the necessary and sufficient conditions for a class of infinite sets to have minimal complements in $\mathbb{Z}^{d}$ cf Theorem \[Thm: existence of min complement implies\], Theorem \[Thm: Implies existence of min comple\] and Theorem \[Thm: Even peri 1 has min comp iff\].\
5. Minimal complements in finitely generated abelian groups - This section is concerned with the existence of minimal complements of a class of infinite sets in arbitrary finitely generated abelian groups cf Theorem \[sec5Thm\], giving a partial answer to Question \[nathansonprob13\] of Nathanson.\
6. Conclusion and further remarks - In this concluding section we give specific examples of sets which don’t fall in the purview of Theorem \[Thm: existence of min complement implies\] and Theorem \[Thm: Implies existence of min comple\] but have minimal complements. See Propositions \[prop6.1\], \[prop6.2\], \[prop6.3\] and \[prop6.4\].
Existence of minimal complements for finite sets in arbitrary groups {#sec2}
====================================================================
In this section we answer Question \[nathansonprob12\].
\[theorem1\] Let $G$ be an arbitrary group with $S$ a nonempty finite subset of $G$. Then every complement of $S$ in $G$ has a minimal complement.
We divide the proof into two propositions, see Proposition \[prop1\] (for the countable case) and Proposition \[prop2\] (for the uncountable case).
\[prop1\] Every complement of a nonempty finite subset of an at most countable group contains a minimal complement.
For finite groups, the proof of the proposition is clear. Now, let $S$ be a nonempty finite subset of a countably infinite group $G$ and $C$ be a complement of $S$ in $G$. Since $S$ is finite and $G$ is countably infinite, the complement $C$ is countably infinite. Hence the elements of $C$ can be enumerated using the positive integers. Write $C = \{c_i\}_{i=1}^\infty$. Define $C_1$ to be equal to $C$ and for any integer $i\geq 1$, define $$C_{i+1} :=
\begin{cases}
C_i\setminus\{c_i\} & \text{ if $C_i\setminus\{c_i\}$ is a complement to $S$},\\
C_i & \text{ otherwise}.
\end{cases}$$ Since each $C_i$ is a complement to $S$, for each $x\in G$ and any $i\geq 1$, there exist elements $c_{x, i}\in C_i, s_{x, i}\in S$ such that $x = s_{x, i}c_{x, i}$. Since $S$ is finite, by the Pigeonhole principle, for some element $s\in S$, the equality $s = s_{x,i}$ holds for infinitely many $i\geq 1$. Hence for infinitely many $i$, we obtain $s^{{-1}}x = c_{x, i}$, which is an element of $C_i$. Consequently, for each positive integer $k$, there exists an integer $i_k > k$ such that $s^{{-1}}x$ is an element of $C_{i_k}$, which is contained in $C_k$ (as $i_k > k$) and thus $C_k$ contains $s^{{-1}}x$. Define $M$ to be the intersection $\cap_{i\geq 1} C_i$. Then for each $x\in G$, there exists an element $s\in S$ such that $s^{{-1}}x$ belongs to $M$. Hence $M$ is a complement to $S$ in $G$.
We claim that $M$ is a minimal complement to $S$. On the contrary, assume that $M\setminus \{c_j\}$ is a complement to $S$ for some element $c_j$ in $M$. Since $C_j$ contains $M$, $C_j\setminus \{c_j\}$ is also a complement to $S$. Then $C_{j+1}$ is equal to $C_j\setminus \{c_j\}$. Hence $c_j$ cannot lie in $M$, which is absurd. So $M$ is a minimal complement of $S$ contained in $C$.
\[prop2\] Every complement of a nonempty finite subset of an uncountable group contains a minimal complement.
Let $S$ be a finite subset of a group $G$ and $C$ be a complement of $S$ in $G$. Let ${\ensuremath{\mathscr{C}}}$ denote the set of all complements of $S$ in $G$ which are contained in $C$. Note that ${\ensuremath{\mathscr{C}}}$ is partially ordered with respect to strict inclusion $\subsetneq$. Note also that the minimal elements of the partially ordered set $({\ensuremath{\mathscr{C}}}, \subsetneq)$ are the minimal complements of $S$ in $G$ which are contained in $C$. If every chain in ${\ensuremath{\mathscr{C}}}$ has a lower bound in ${\ensuremath{\mathscr{C}}}$, then by Zorn’s lemma, it would follow that ${\ensuremath{\mathscr{C}}}$ has a minimal element. We claim that every chain in ${\ensuremath{\mathscr{C}}}$ has a lower bound in ${\ensuremath{\mathscr{C}}}$. Let ${\ensuremath{\mathcal{C}}}= \{C_\lambda\}_{\lambda\in \Lambda}$ be a chain in ${\ensuremath{\mathscr{C}}}$. If a member of ${\ensuremath{\mathcal{C}}}$ is contained in all other members of ${\ensuremath{\mathcal{C}}}$, then it is a lower bound of ${\ensuremath{\mathcal{C}}}$ in ${\ensuremath{\mathscr{C}}}$. Henceforth we assume that no member of ${\ensuremath{\mathcal{C}}}$ is a lower bound of ${\ensuremath{\mathcal{C}}}$.
Note that if ${\ensuremath{\mathcal{C}}}_1, \cdots, {\ensuremath{\mathcal{C}}}_k$ are pairwise disjoint subsets of ${\ensuremath{\mathcal{C}}}$ such that their union is equal to ${\ensuremath{\mathcal{C}}}$, then for some integer $i$ with $1\leq i \leq k$, each member of ${\ensuremath{\mathcal{C}}}$ contains some member of ${\ensuremath{\mathcal{C}}}_i$. Otherwise, for each $i$, some member of $C_{\lambda_i}$ of ${\ensuremath{\mathcal{C}}}$ would be contained in every member of ${\ensuremath{\mathcal{C}}}_i$. Let $j$ be a positive integer $\leq k$ such that $C_{\lambda_j}$ is contained in $C_{\lambda_i}$ for any $i$ satisfying $1\leq i\leq k$. So $C_{\lambda_j}$ is contained in every member of ${\ensuremath{\mathcal{C}}}_i$ for each $i$, which is a contradiction to the assumption that no member of ${\ensuremath{\mathcal{C}}}$ is a lower bound of ${\ensuremath{\mathcal{C}}}$.
Since $SC_\lambda = G$, for each $x\in G$ and $\lambda\in \Lambda$, there exist elements $s_{x, \lambda}\in S, c_{x,\lambda}\in C_\lambda$ such that $x=s_{x, \lambda} c_{x, \lambda}$. Let $S_x$ denote the subset of elements of $S$ of the form $s_{x, \lambda}$ for some $\lambda\in \Lambda$. For $s\in S_x$, define ${\ensuremath{\mathcal{C}}}_s$ to be the subchain $\{C_\lambda\,|\, s_{x,\lambda} =s\}$ of $\{C_\lambda\}_{\lambda\in \Lambda}$. Then the sets ${\ensuremath{\mathcal{C}}}_s$, for $s\in S_x$, form a collection of finitely many pairwise disjoint subsets of ${\ensuremath{\mathcal{C}}}$ and their union is equal to ${\ensuremath{\mathcal{C}}}$. Hence by the observation made in the preceding paragraph, there exists an element $s'\in S$ such that each member of ${\ensuremath{\mathcal{C}}}$ contains some element of ${\ensuremath{\mathcal{C}}}_{s'}$. Since $s'^{{-1}}x$ belongs to each member of ${\ensuremath{\mathcal{C}}}_{s'}$, it also belongs to each member of ${\ensuremath{\mathcal{C}}}$ and hence $s'^{{-1}}x$ belongs to the intersection $\cap_{\lambda\in \Lambda} C_\lambda$. Consequently, $\cap_{\lambda\in \Lambda} C_\lambda$ is a complement of $S$ in $G$ contained in $C$. In other words, it is an element of ${\ensuremath{\mathscr{C}}}$. So each chain in ${\ensuremath{\mathscr{C}}}$ has a lower bound in ${\ensuremath{\mathscr{C}}}$. This proves the claim. So the proposition follows from Zorn’s lemma.
In particular, for finitely generated free abelian groups, e.g., $\mathbb{Z}^k$, every (additive) complement of a finite set contains a minimal complement.
Later on, we shall give specific examples of infinite subsets (both countably infinite subsets and uncountably infinite subsets) in certain groups which admit minimal complements. See section \[Sec:Conclusion\], Propositions \[prop6.1\], \[prop6.2\], \[prop6.3\] and \[prop6.4\]. Also see section \[Sec: Minimal complement\] for minimal complements of eventually periodic sets.
Inexistence of minimal complements for certain infinite sets
============================================================
In the following, $d$ denotes a positive integer and ${\ensuremath{\mathbb{N}}}$ denotes the set of all nonnegative integers.
\[prop3.1\] \[Prop: Complement iff\] Let $M$ be a subset of ${\ensuremath{\mathbb{Z}}}^d$. Then the following statements are equivalent.
1. The set $M$ is a complement of ${\ensuremath{\mathbb{N}}}^d$ in ${\ensuremath{\mathbb{Z}}}^d$.
2. $M$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d$ such that the maximum of their coordinates is arbitrarily small negative number.
3. $\liminf _{ (x_1, \cdots, x_d)\in M} \max\{x_1, \cdots, x_d\}=-\infty.$
Suppose statement (1) holds. We need to show that given any integer $n\in {\ensuremath{\mathbb{Z}}}$, $M$ contains a point all whose coordinates are less than or equal to $n$. On the contrary, suppose this is not true. Then for some $n\in {\ensuremath{\mathbb{Z}}}$, $M$ is contained in ${\ensuremath{\mathbb{Z}}}^d \setminus(n-{\ensuremath{\mathbb{N}}}^d)$. So $M+ {\ensuremath{\mathbb{N}}}^d$ would also be contained in ${\ensuremath{\mathbb{Z}}}^d \setminus(n-{\ensuremath{\mathbb{N}}}^d)$, which is absurd. So the second statement holds.
It is clear that statement (3) follows from statement (2).
Suppose statement (3) holds. Then for each integer $n$, the set $M$ contains a point $P_n=(x_{n1}, \cdots, x_{nd})$ with each coordinate $\leq n$, i.e., $x_{ni}\leq n$ for any $1\leq i\leq d$. So ${\ensuremath{\mathbb{N}}}^d$ contains the point $(n-x_{n1}, \cdots, n-x_{nd})$. Hence ${\ensuremath{\mathbb{N}}}^d$ contains $(n-x_{n1}, \cdots, n-x_{nd}) + {\ensuremath{\mathbb{N}}}^d$, and thus $$P_n + {\ensuremath{\mathbb{N}}}^d \supseteq P_n + ((n-x_{n1}, \cdots, n-x_{nd}) + {\ensuremath{\mathbb{N}}}^d)
=
(n, \cdots, n) + {\ensuremath{\mathbb{N}}}^d .$$ Consequently, $M+{\ensuremath{\mathbb{N}}}^d$ contains $\cup_{n\in {\ensuremath{\mathbb{Z}}}} ((n, \cdots, n) + {\ensuremath{\mathbb{N}}}^d) = {\ensuremath{\mathbb{Z}}}^d$. This proves statement (1).
\[cor3.2\] \[Cor: Nd minimal complement\]
1. The subset ${\ensuremath{\mathbb{N}}}^d$ has complements in ${\ensuremath{\mathbb{Z}}}^d$.
2. The subset ${\ensuremath{\mathbb{N}}}^d$ has no minimal complement in ${\ensuremath{\mathbb{Z}}}^d$.
3. No complement of ${\ensuremath{\mathbb{N}}}^d$ in ${\ensuremath{\mathbb{Z}}}^d$ contains a minimal complement of ${\ensuremath{\mathbb{N}}}^d$.
The first statement follows since $\{(-n,\cdots,-n)\,|\,n\in {\ensuremath{\mathbb{N}}}\}$ and any of its infinite subsets are complements of ${\ensuremath{\mathbb{N}}}^d$ in ${\ensuremath{\mathbb{Z}}}^d$ (which can be seen by applying Proposition \[Prop: Complement iff\]). For any complement $M$ of ${\ensuremath{\mathbb{N}}}^d$ and for any finite subset $F$ of $M$, it follows from Proposition \[Prop: Complement iff\] that $M\setminus F$ is also a complement of ${\ensuremath{\mathbb{N}}}^d$. Hence the second and the third statement hold.
We strengthen Proposition \[Prop: Complement iff\] and Corollary \[Cor: Nd minimal complement\] in Proposition \[Prop: AP Complement iff\] below.
\[prop3.3\] \[Prop: AP Complement iff\] Let $u_1, \cdots, u_d$ be elements of ${\ensuremath{\mathbb{Z}}}^d$ which satisfy no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation. Denote the subgroup ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$ of ${\ensuremath{\mathbb{Z}}}^d$ by ${\ensuremath{\mathcal {L}}}$.
1. A subset $M$ of ${\ensuremath{\mathbb{Z}}}^d$ is a complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ if and only if each fibre of the map $\pi: M \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number.
2. The subset ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ has complements in ${\ensuremath{\mathbb{Z}}}^d$.
3. The subset ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ has no minimal complement in ${\ensuremath{\mathbb{Z}}}^d$.
4. No complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ contains a minimal complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$.
Let ${\ensuremath{\mathfrak{L}}}$ be a subset of ${\ensuremath{\mathbb{Z}}}^d$ such that the map $\pi: {\ensuremath{\mathfrak{L}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is bijective. For any subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ and any ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$, let $X_{\ensuremath{\mathfrak{l}}}$ denote the set of all elements of $X$ which are congruent to ${\ensuremath{\mathfrak{l}}}{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$. Since each element of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is congruent to zero modulo ${\ensuremath{\mathcal {L}}}$, it follows that a subset $M$ of ${\ensuremath{\mathbb{Z}}}^d$ is a complement of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ if and only if $M_{\ensuremath{\mathfrak{l}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) = {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ for any ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$.
Choose an element ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$. Suppose $M_{\ensuremath{\mathfrak{l}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) = {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ holds. We claim that $M_{\ensuremath{\mathfrak{l}}}$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number. On the contrary, suppose this is false. Then for some $n\in {\ensuremath{\mathbb{Z}}}$, $$M_{\ensuremath{\mathfrak{l}}}\subseteq {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}\setminus({\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n))-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)).$$ So $M_{\ensuremath{\mathfrak{l}}}+ ( {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ would also be contained in ${\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}\setminus({\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n))-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$, which is absurd. So the claim follows. Conversely, assume that $M_{\ensuremath{\mathfrak{l}}}$ contains a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number. So for any $n\in {\ensuremath{\mathbb{Z}}}$, the set $M_{\ensuremath{\mathfrak{l}}}$ contains a point $P_n={\ensuremath{\mathfrak{l}}}+ x_{n1}u_1 + \cdots + x_{nd}u_d$ with $x_{ni} \leq n$ for any $1\leq i\leq d$. So ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ contains the point $(n-x_{n1})u_1 + \cdots + ( n-x_{nd})u_d$. Hence ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ contains $((n-x_{n1})u_1 + \cdots + ( n-x_{nd})u_d) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$, and thus $$\begin{aligned}
P_n + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d
&\supseteq P_n + (((n-x_{n1})u_1 + \cdots + ( n-x_{nd})u_d)\\
& \quad + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))\\
&={\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n)) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) .\end{aligned}$$ Consequently, $M_{\ensuremath{\mathfrak{l}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ contains $\cup_{n\in {\ensuremath{\mathbb{Z}}}} ({\ensuremath{\mathfrak{l}}}+ (n(u_1 + \cdots + u_n)) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) ) = {\ensuremath{\mathbb{Z}}}^d_{\ensuremath{\mathfrak{l}}}$. This proves statement (1).
From the first statement, it follows that the set ${\ensuremath{\mathfrak{L}}}+ (-{\ensuremath{\mathbb{N}}}(u_1+\cdots + u_d))$ and any of its subsets $C$ containing infinitely elements of ${\ensuremath{\mathfrak{l}}}+ (-{\ensuremath{\mathbb{N}}}(u_1+\cdots + u_d))$ for any ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$ is a complement to ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$. So the second statement holds.
For any complement $M$ of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ and any finite subset $F$ of $M$, it follows from the first statement that $M\setminus F$ is also a complement to ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$. This proves the third and the fourth statement.
Minimal complements of infinite sets in free abelian groups {#Sec: Minimal complement}
===========================================================
We start the section with the statement on existence of minimal complements of subgroups of arbitrary groups as stated in the introduction.
\[prop4.1\] Let $G$ be an arbitrary group with $S$ a subgroup of $G$. Then every complement of $S$ in $G$ has a minimal complement.
Let $C$ be a complement of $S$ in $G$. Let $M$ be a subset of $C$ such that $m_1m_2^{{-1}}\notin S$ for any two distinct elements $m_1, m_2$ of $M$, and for any $c\in C$, $S$ contains $m_c^{{-1}}c$ for some $m_c\in M$. Then $M$ is a minimal complement to $S$ contained in $C$.
With the subgroup case done, we shall study minimal complements of sets which are not subgroups. In this section we will be in free abelian groups of finite rank. For notational convenience, we will consider the free abelian group ${\ensuremath{\mathbb{Z}}}^d$ only. Note that a condition holds for sufficiently large elements of ${\ensuremath{\mathbb{Z}}}^d$ if and only if the condition holds for almost all elements of ${\ensuremath{\mathbb{Z}}}^d$. This enables us to use the terms “sufficiently large” and “almost all” interchangeably without any confusion. However, to refer to sufficiently large elements, the underlying space is certainly required to have a notion of a metric (which in general is not available in an arbitrary free abelian group).
We will prove Theorems \[Thm: existence of min complement implies\], \[Thm: Implies existence of min comple\] providing a necessary and a sufficient condition for the existence of minimal complements for eventually periodic sets. We also provide a necessary and sufficient condition for certain eventually periodic sets to have a minimal complement. See Theorem \[Thm: Even peri 1 has min comp iff\].
Periodic and eventually periodic sets
-------------------------------------
The union of two subsets $A, B$ of ${\ensuremath{\mathbb{Z}}}^d$ is denoted by $A\cup B$. When $A, B$ are disjoint, we write $A\sqcup B$ to denote the union $A\cup B$ and to indicate that $A, B$ are disjoint.
In the following, $u_1, \cdots, u_d$ denote elements of ${\ensuremath{\mathbb{Z}}}^d$ which satisfy no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation. Let ${\ensuremath{\mathcal {L}}}$ denote the subgroup ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$ of ${\ensuremath{\mathbb{Z}}}^d$. Two elements $x,y\in {\ensuremath{\mathbb{Z}}}^d$ are said to be *equivalent modulo ${\ensuremath{\mathcal {L}}}$* if $x-y\in {\ensuremath{\mathcal {L}}}$. Denote by $\pi$ the quotient map $\pi: {\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$. The image of an element $v\in {\ensuremath{\mathbb{Z}}}^d$ under the quotient map $\pi: {\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is denoted by $\bar v$. A typical element of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ would be denoted by $\bar v$ (which is legitimate since the quotient map $\pi: {\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective), and by $v$, we would denote a lift of $\bar v$ to ${\ensuremath{\mathbb{Z}}}^d$. For any subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ and any $\bar v\in {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, denote by $X_{\bar v}$ the intersection $X\cap \pi^{{-1}}(\bar v)$. For any subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$, the restriction of the map $\pi$ to $X$ is the composite map $X{\hookrightarrow}{\ensuremath{\mathbb{Z}}}^d {\xrightarrow}{\pi} {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, which we denote by $\pi_X$ (or, simply by $\pi$ to avoid cumbersome notation).
\[Defn: periodic\] A nonempty subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ is called [periodic with periods]{.nodecor} $u_1,\cdots, u_d$ if $X$ is contained in $F_X + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ for some nonempty finite subset $F_X\subset {\ensuremath{\mathbb{Z}}}^d$ and $x + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $X$ for any $x\in X$. A nonempty subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ is called [eventually periodic with periods]{.nodecor} $u_1,\cdots, u_d$ if $X$ is contained in $F_X + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ for some nonempty finite subset $F_X\subset {\ensuremath{\mathbb{Z}}}^d$ and $x + {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $X$ for any sufficiently large element $x\in X$.
For any integer $k\geq 0$, the set $\{(r, -r)\in {\ensuremath{\mathbb{Z}}}^2\,|\, -k\leq r\leq k\} + {\ensuremath{\mathbb{N}}}^2$ is a periodic subset of ${\ensuremath{\mathbb{Z}}}^2$ with periods $(1,0), (0,1)$. The set ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ appearing in Proposition \[Prop: AP Complement iff\] is periodic and also eventually periodic. Note that periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ are also eventually periodic with the same periods. The set ${\ensuremath{\mathbb{Z}}}^d$ is neither periodic nor eventually periodic.
First we prove Proposition \[Prop: Structure of Periodic\] and Theorem \[Thm: Structure of Eventually Periodic\], which describe the structure of periodic sets and eventually periodic sets.
\[Prop: Structure of Periodic\] Let $W$ be a periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathcal{Q}}}$ denote the image of $W$ under the map $\pi: W\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$.
1. The nonempty fibres of the map $\pi: W \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, i.e., the sets $W_{\bar v}$ for $\bar v\in {\ensuremath{\mathcal{Q}}}$ are periodic with periods $u_1, \cdots, u_d$.
2. For any $\bar v\in {\ensuremath{\mathcal{Q}}}$, denote by ${\ensuremath{\mathcal {W}}}'_{\bar v}$ the set of elements $w\in W_{\bar v}$ such that the intersection of the sets $w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W_{\bar v}$ contains no element other than $w$. Then for any $\bar v\in {\ensuremath{\mathcal{Q}}}$, the set ${\ensuremath{\mathcal {W}}}'_{\bar v}$ is a nonempty finite subset of $W_{\bar v}$ and the equality $$W_{\bar v} = {\ensuremath{\mathcal {W}}}'_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ holds.
3. Let ${\ensuremath{\mathcal {W}}}$ denote the set of elements $w\in W$ such that the intersection of the sets $w- ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W$ contains no element other than $w$. Then the equalities $$\begin{aligned}
{\ensuremath{\mathcal {W}}}&= \sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}, \\
{\ensuremath{\mathcal {W}}}_{\bar v}& = {\ensuremath{\mathcal {W}}}'_{\bar v}, \\
W
&= {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)\\
&= \sqcup _{\bar v\in {\ensuremath{\mathcal{Q}}}} ( {\ensuremath{\mathcal {W}}}_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d )) \end{aligned}$$ hold.
Since $W$ is periodic with periods $u_1, \cdots, u_d$, and ${\ensuremath{\mathcal {L}}}$ is equal to ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$, it follows that the nonempty fibres of the map $\pi: W\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ are periodic with periods $u_1, \cdots, u_d$. These fibres are precisely the sets $W_{\bar v}$ for $\bar v\in {\ensuremath{\mathcal{Q}}}$. This proves the first statement.
Let $\bar v$ be an element of ${\ensuremath{\mathcal{Q}}}$ and $v\in W_{\bar v}$ be a lift of $\bar v$ modulo ${\ensuremath{\mathcal {L}}}$. Note that $-v+ W_{\bar v}$ is periodic with periods $u_1, \cdots, u_d$ and there exists a nonempty finite subset $F$ of ${\ensuremath{\mathbb{Z}}}^d$ such that $-v+ W_{\bar v}$ is contained in $F+ {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$. Since $-v+ W_{\bar v}$ is contained in ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$, note that $F$ can be taken to be a subset of ${\ensuremath{\mathbb{Z}}}u_1 + \cdots + {\ensuremath{\mathbb{Z}}}u_d$. For $1\leq i\leq d$, let $\lambda_i$ denote the minimum of the $i$-th coordinates of the elements of $F$. Then it follows that $F+ {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is contained in $(\lambda_1 u_1 + \cdots + \lambda_d u_d)+({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$. So $-v+ W_{\bar v}$ is periodic with periods $u_1, \cdots, u_d$ and it is contained in $(\lambda_1 u_1 + \cdots + \lambda_d u_d)+({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$. Note that $$-v+{\ensuremath{\mathcal {W}}}'_{\bar v} =
\{u\in (-v+ W_{\bar v})\,|\, (u-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))\cap (-v+ W_{\bar v}) = \{u\} \}.$$ Then the second statement follows since if a periodic subset $U$ of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$ is contained in $(\mu_1 u_1 + \cdots + \mu_d u_d)+({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ for some integers $\mu_1, \cdots, \mu_d$, then ${\ensuremath{\mathcal{U}}}:= \{u\in U \,|\, (u-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))\cap U = \{u\} \}$ is a nonempty finite subset of $U$ and $$U = {\ensuremath{\mathcal{U}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ holds.
If $w$ is an element of ${\ensuremath{\mathcal {W}}}$ and $\pi(w)$ is equal to $\bar v$, then the intersection of the sets $w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W_{\bar v}$ contains no element other than $w$. Hence $w$ belongs to ${\ensuremath{\mathcal {W}}}'_{\bar v}$. So ${\ensuremath{\mathcal {W}}}$ is contained in $\sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}$. Note that for any $\bar v\in {\ensuremath{\mathcal{Q}}}$ and any $w\in W_{\bar v}$, $$(w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) )\cap W =
(w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) )\cap W_{\bar v} = \{w\}.$$ This implies that $\sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}$ is contained in ${\ensuremath{\mathcal {W}}}$, proving the first equality. Then the remaining equalities are immediate.
\[Thm: Structure of Eventually Periodic\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$.
1. Let ${\ensuremath{\mathscr{W}}}$ denote the set of all elements $w\in W$ such that $w+ {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ is not contained in $W$. Then ${\ensuremath{\mathscr{W}}}$ is a finite set.
2. The set $W\setminus {\ensuremath{\mathscr{W}}}$ is periodic with periods $u_1, \cdots,u_d$.
3. The following sets are equal.
1. The set of all elements of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ having infinite fibre under the composite map $W{\hookrightarrow}{\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$.
2. The set of all elements of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ having infinite fibre under the composite map $W\setminus {\ensuremath{\mathscr{W}}}{\hookrightarrow}{\ensuremath{\mathbb{Z}}}^d \twoheadrightarrow {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$.
3. The image $\pi(W\setminus {\ensuremath{\mathscr{W}}})$ of $W\setminus {\ensuremath{\mathscr{W}}}$ under $\pi$, to be denoted by ${\ensuremath{\mathcal{Q}}}$.
4. The nonempty fibres of the map $\pi: W\setminus {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, i.e., the sets $(W\setminus {\ensuremath{\mathscr{W}}})_{\bar v}$ for $\bar v\in {\ensuremath{\mathcal{Q}}}$ are periodic with periods $u_1, \cdots, u_d$.
5. For any $\bar v\in {\ensuremath{\mathcal{Q}}}$, denote by ${\ensuremath{\mathcal {W}}}'_{\bar v}$ the set of elements $w\in (W\setminus{\ensuremath{\mathscr{W}}})_{\bar v}$ such that the intersection of the sets $w - ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $(W\setminus{\ensuremath{\mathscr{W}}})_{\bar v}$ contains no element other than $w$. Then for any $\bar v\in {\ensuremath{\mathcal{Q}}}$, the equality $$(W\setminus{\ensuremath{\mathscr{W}}})_{\bar v} = {\ensuremath{\mathcal {W}}}'_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ holds. Moreover, ${\ensuremath{\mathcal {W}}}'_{\bar v}$ is a finite set.
6. Let ${\ensuremath{\mathcal {W}}}$ denote the set of elements $w\in W\setminus {\ensuremath{\mathscr{W}}}$ such that the intersection of the sets $w- ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ and $W\setminus {\ensuremath{\mathscr{W}}}$ contains no element other than $w$. Then the equalities $$\begin{aligned}
{\ensuremath{\mathcal {W}}}& = \sqcup_{\bar v\in {\ensuremath{\mathcal{Q}}}} {\ensuremath{\mathcal {W}}}'_{\bar v}, \\
{\ensuremath{\mathcal {W}}}_{\bar v} & = {\ensuremath{\mathcal {W}}}'_{\bar v}, \\
W \setminus {\ensuremath{\mathscr{W}}}&= {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)\\
&= \sqcup _{\bar v\in {\ensuremath{\mathcal{Q}}}} ( {\ensuremath{\mathcal {W}}}_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d )) \end{aligned}$$ hold.
7. The following $$\label{Eqn: Decomposition}
W
= {\ensuremath{\mathscr{W}}}\sqcup ({\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))
= \mathscr W \sqcup ( \sqcup _{\bar v\in {\ensuremath{\mathcal{Q}}}} ( {\ensuremath{\mathcal {W}}}_{\bar v} + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d )) )$$ expresses $W$ as a disjoint union of its subsets.
Since $W$ is eventually periodic with periods $u_1, \cdots, u_d$, it contains $w + {\ensuremath{\mathbb{N}}}u_1 + \cdots +{\ensuremath{\mathbb{N}}}u_d$ for almost all $w\in W$. So it follows that ${\ensuremath{\mathscr{W}}}$ is finite.
Since $W$ is nonempty, it is infinite. So $W\setminus{\ensuremath{\mathscr{W}}}$ is nonempty. Moreover, since $W$ is eventually periodic, it follows that $W\setminus{\ensuremath{\mathscr{W}}}$ is contained in $F + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ for some finite subset $F$ of ${\ensuremath{\mathbb{Z}}}^d$. By the definition of ${\ensuremath{\mathscr{W}}}$, it follows that $(W\setminus {\ensuremath{\mathscr{W}}}) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ is contained in $W$. To prove that $W\setminus {\ensuremath{\mathscr{W}}}$ is periodic with periods $u_1, \cdots, u_d$, it only remains to show that $(W\setminus {\ensuremath{\mathscr{W}}}) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ does not intersect with ${\ensuremath{\mathscr{W}}}$. Suppose ${\ensuremath{\mathscr{W}}}$ contains $w+ \lambda_1 u_1 + \cdots + \lambda_d u_d$ for some $w\in W\setminus {\ensuremath{\mathscr{W}}}$ and for some $\lambda_1, \cdots, \lambda_d\in {\ensuremath{\mathbb{N}}}$. So for some $\mu_1, \cdots, \mu_d\in {\ensuremath{\mathbb{N}}}$, the vector $(w + \lambda_1 u_1 + \cdots + \lambda_d u_d) + (\mu_1 u_1 + \cdots + \mu_d u_d)$ would not be contained in $W$, which is absurd since $w\in W\setminus {\ensuremath{\mathscr{W}}}$. Consequently, $(W\setminus {\ensuremath{\mathscr{W}}}) + ( {\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ is contained in $W\setminus {\ensuremath{\mathscr{W}}}$. Hence $W\setminus {\ensuremath{\mathscr{W}}}$ is periodic with periods $u_1, \cdots,u_d$.
Note $W= {\ensuremath{\mathscr{W}}}\sqcup (W\setminus {\ensuremath{\mathscr{W}}})$ is a decomposition of $W$ into two disjoint subsets where ${\ensuremath{\mathscr{W}}}$ is finite and $W\setminus {\ensuremath{\mathscr{W}}}$ is infinite. Since the set ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is finite, it follows that an element $\bar v$ of ${\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ has infinite fibre under $\pi: W \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ if and only if it has infinite fibre under $\pi: W\setminus {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$. Moreover, any element in the image of the map $\pi: W\setminus {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ has infinite fibre because $W\setminus {\ensuremath{\mathscr{W}}}$ contains the $({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$-translate of any of its points. Hence the three sets in part (a), (b), (c) are equal.
The statements (4), (5), (6), (7) follow from Proposition \[Prop: Structure of Periodic\].
Note that not all the nonempty subsets $X$ of ${\ensuremath{\mathbb{Z}}}^d$ satisfying $$\label{Eqn: Periodic}
X\supseteq X +({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$$ are of the form ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$ for some subset ${\ensuremath{\mathcal {W}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ (for instance, consider the subset ${\ensuremath{\mathbb{Z}}}\times {\ensuremath{\mathbb{N}}}$ of ${\ensuremath{\mathbb{Z}}}^2$). Moreover, there are subsets $X$ of ${\ensuremath{\mathbb{Z}}}^d$ which satisfy Equation and are of the form ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$, but are not periodic, i.e., not contained in the sum of ${\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d$ and a nonempty finite set (for instance, consider the subset $\{ (n,-n)\,|\, n\in {\ensuremath{\mathbb{N}}}\} + {\ensuremath{\mathbb{N}}}^2$ of ${\ensuremath{\mathbb{Z}}}^2$).
Necessary condition
-------------------
\[Thm: existence of min complement implies\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}, {\ensuremath{\mathcal {W}}}$ be as in Theorem \[Thm: Structure of Eventually Periodic\]. Let $M$ be a minimal complement of $W$ in ${\ensuremath{\mathbb{Z}}}^d$. Let $M_\infty$ denote the union of those fibres of the map $\pi: M \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ containing a sequence of elements of ${\ensuremath{\mathbb{Z}}}^d$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small negative number. Let ${\ensuremath{\mathcal{M}}}$ be a subset of $M_\infty$ such that the composite map ${\ensuremath{\mathcal{M}}}{\hookrightarrow}M_\infty \twoheadrightarrow \pi(M_\infty)$ is a bijection. Let ${\ensuremath{\mathscr{W}}}_0$ (resp. ${\ensuremath{\mathscr{W}}}_1$) denote the set of elements of ${\ensuremath{\mathscr{W}}}$ which are congruent to some element (resp. no element) of ${\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$[^3]. Then the following statements hold.
1. The set $M_\infty$ is an infinite set, ${\ensuremath{\mathcal{M}}}$ is a nonempty finite subset of $M_\infty$, ${\ensuremath{\mathscr{W}}}_1$ is nonempty, and the map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective.
2. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$.
Let $M_{\ensuremath{\mathrm{fin}}}$ denote $M\setminus M_\infty$. Let ${\ensuremath{\mathfrak{L}}}$ be a subset of ${\ensuremath{\mathbb{Z}}}^d$ such that the map $\pi: {\ensuremath{\mathfrak{L}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is bijective. By Theorem \[Thm: Structure of Eventually Periodic\] (7), it follows that for each ${\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}$, there exists a positive integer $\lambda_{\ensuremath{\mathfrak{l}}}$ such that the intersection of $M_{\ensuremath{\mathrm{fin}}}+ W$ and ${\ensuremath{\mathfrak{l}}}+(-\lambda_{\ensuremath{\mathfrak{l}}}(u_1 + \cdots + u_d)) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$ is empty. Hence $M_{\ensuremath{\mathrm{fin}}}+W$ contains no element of the set ${\ensuremath{\mathfrak{L}}}+ (-(\max_{{\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}}\lambda_{\ensuremath{\mathfrak{l}}}) (u_1 + \cdots + u_d)) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$. Since $M$ is a complement of $W$ in ${\ensuremath{\mathbb{Z}}}^d$ and $M$ is equal to the union of $M_{\ensuremath{\mathrm{fin}}}$ and $M_\infty$, it follows that $M_\infty$ is nonempty and $$\label{Eqn: Containment}
M_\infty + W \supseteq {\ensuremath{\mathfrak{L}}}+ (-(\max_{{\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}}\lambda_{\ensuremath{\mathfrak{l}}}(u_1 + \cdots + u_d))) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)).$$ So $M_\infty$ is infinite and ${\ensuremath{\mathcal{M}}}$ is nonempty. If ${\ensuremath{\mathscr{W}}}$ were empty, then Equation would imply $$\label{Eqn: Containment 2}
M_\infty + {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d) \supseteq {\ensuremath{\mathfrak{L}}}+ (-(\max_{{\ensuremath{\mathfrak{l}}}\in {\ensuremath{\mathfrak{L}}}}\lambda_{\ensuremath{\mathfrak{l}}}(u_1 + \cdots + u_d))) +(-({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)).$$ Then for any $m_0\in M_\infty$, $M\setminus\{m_0\}$ would be a complement to $W$. Indeed, given an element $x\in {\ensuremath{\mathbb{Z}}}^d$, it is equal to $m+w$ for some $m\in M$ and $w\in W$. If $m\neq m_0$, then $x$ belongs to $M\setminus\{m_0\}+W$. If $m=m_0$, then for some positive integers $\mu_1, \cdots, \mu_d$, $M_\infty$ would contain $m_0-(\mu_1 u_1 + \cdots + \mu_d u_d)$. Since $w$ belongs to $W = {\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$, it follows that $w + (\mu_1 u_1 + \cdots + \mu_d u_d)$ belongs to $W$. Hence $x$ belongs to $M\setminus\{m_0\}+W$. So $M\setminus\{m_0\}+W$ is equal to ${\ensuremath{\mathbb{Z}}}^d$, which is absurd. Hence ${\ensuremath{\mathscr{W}}}$ is nonempty. Now, using Theorem \[Thm: Structure of Eventually Periodic\] (7) again, we obtain from Equation that the map $$\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}})) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$$ is surjective. By the definition ${\ensuremath{\mathscr{W}}}_0$, it follows that the map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective. This proves the first statement.
Now we prove that the second statement is true. On the contrary, suppose the second statement is false. So there exists an element $m\in {\ensuremath{\mathcal{M}}}$ such that for each $w\in {\ensuremath{\mathscr{W}}}_1$, there exist elements $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$ such that $m+w \equiv m'+w'{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$. We prove that $M\setminus \{m\}$ is a complement to $W$, which would contradict the minimality of $M$, and thereby establish the second statement.
Let $x$ be an element of ${\ensuremath{\mathbb{Z}}}^d$. Since $M$ is a complement of $W$, it follows that $x = m_0 + w_0$ for some $m_0\in M, w_0\in W$. If $m_0\neq m$, then $x$ belongs to $M\setminus\{m\} + W$. Suppose $m_0$ is equal to $m$, i.e., $x = m+w_0$. If $w_0$ belongs to ${\ensuremath{\mathscr{W}}}_1$, then there exist elements $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$ such that $m+w_0\equiv m'+w'{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$. This gives $x = m'+w'+(n_1u_1 + \cdots + n_du_d)$ for some integers $n_1, \cdots, n_d$. Since $w'$ belongs to ${\ensuremath{\mathcal {W}}}$, for any integers $r_1, \cdots, r_d$ with $r_i\geq - n_i$, $W$ contains $w' + (n_1u_1 + \cdots + n_du_d) + (r_1u_1 + \cdots + r_du_d)$. Since $m'\in {\ensuremath{\mathcal{M}}}\subseteq M_\infty$, it follows that $m'-(\lambda_1 u_1 + \cdots + \lambda_d u_d)\in M_\infty \setminus\{m\}\subseteq M\setminus\{m\}$ for some integers $\lambda_1, \cdots, \lambda_d$ satisfying $\lambda_i\geq \max \{0, -n_i\}$ for $1\leq i\leq d$. Hence $x = (m'-(\lambda_1 u_1 + \cdots + \lambda_d u_d)) + (w'+ (n_1u_1 + \cdots + n_du_d) +( \lambda_1 u_1 + \cdots + \lambda_d u_d) )$ belongs to $M \setminus \{m\} + W$. On the other hand, if $w_0$ does not belong to ${\ensuremath{\mathscr{W}}}_1$, then $w_0$ belongs to ${\ensuremath{\mathscr{W}}}_0 \sqcup ({\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d))$. Since the image of ${\ensuremath{\mathscr{W}}}_0$ under $\pi: {\ensuremath{\mathscr{W}}}_0\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is contained in ${\ensuremath{\mathcal{Q}}}$ and the image of ${\ensuremath{\mathcal {W}}}$ under $\pi: {\ensuremath{\mathcal {W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is equal to ${\ensuremath{\mathcal{Q}}}$, there exist positive integers $\mu_1, \cdots, \mu_d$ such that $w_0 + \mu_1 u_1 + \cdots + \mu_d u_d$ belongs to ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)$. Moreover, since $m\in M\subseteq M_\infty$, there exist integers $\lambda_1, \cdots, \lambda_d$ with $\lambda_i\geq \mu_i$ for any $1\leq i\leq d$ such that $m - (\lambda_1 u_1 + \cdots + \lambda_d u_d)$ belongs to $M_\infty \setminus \{m\} \subseteq M\setminus\{m\}$. Also note that $w_0 + \lambda_1 u_1 + \cdots + \lambda_d u_d$ belongs to $W$ because $w_0 + \mu_1 u_1 + \cdots + \mu_d u_d$ belongs to ${\ensuremath{\mathcal {W}}}+ ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d)\subseteq W$ and $\lambda_i\geq \mu_i$ for any $1\leq i\leq r$. So $x = m+w_0$ belongs to $M\setminus\{m\} + W$. Consequently, $x$ belongs to $M\setminus\{m\} + W$ whether $w_0$ belongs to ${\ensuremath{\mathscr{W}}}_1$ or not. Hence $M\setminus\{m\}$ is a complement to $W$, which is a contradiction to the given condition that $M$ is a minimal complement to $W$. So the second statement is true.
Note that Theorem \[Thm: existence of min complement implies\] is in accordance with Corollary \[Cor: Nd minimal complement\](2), (3) and Proposition \[Prop: AP Complement iff\](3), (4).
\[Cor: No periodic has minimal\] By Theorem \[Thm: existence of min complement implies\](2), periodic subsets of ${\ensuremath{\mathbb{Z}}}^d$ do not have minimal complements in ${\ensuremath{\mathbb{Z}}}^d$.
Sufficient condition
--------------------
\[Thm: Implies existence of min comple\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}, {\ensuremath{\mathcal {W}}}$ be as in Theorem \[Thm: Structure of Eventually Periodic\], and ${\ensuremath{\mathscr{W}}}_0, {\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose there exists a nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ such that the following conditions hold.
1. The set ${\ensuremath{\mathscr{W}}}_1$ is nonempty.
2. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective.
3. The map $\pi: ({\ensuremath{\mathcal{M}}}+ ({\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective.
4. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$.
Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$.
Let $C$ denote the collection of all elements in ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to some element of ${\ensuremath{\mathcal{M}}}$ modulo ${\ensuremath{\mathcal {L}}}$, and $C'$ denote the collection of all elements in ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to no element of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. Note that $C+{\ensuremath{\mathscr{W}}}_1$ contains all elements of ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to some element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathscr{W}}}_1$. By the second condition, it follows that each element of $C'$ is congruent to some element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathscr{W}}}_1$. So $C+{\ensuremath{\mathscr{W}}}_1$ contains $C'$.
We claim that $C$ has a subset $M$ such that $M$ is minimal among the subsets of $C$ with respect to the property that $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Note that $C$ is countably infinite. Hence its elements can be enumerated by the positive integers. Write $C=\{c_i\,|\, i\geq 1\}$. Define $C_1$ to be equal to $C$ and for any integer $i\geq 1$, define $$C_{i+1}=
\begin{cases}
C_i\setminus\{c_i\} & \text{ if $C_i\setminus\{c_i\}+{\ensuremath{\mathscr{W}}}_1$ contains }C',\\
C_i & \text{ otherwise}.
\end{cases}$$ Let $M$ denote the intersection $\cap_{i\geq 1} C_i$. Note that for each $x\in C'$ and any integer $i\geq 1$, there exist elements $c_{x,i}\in C_i$ and $w_{x,i}\in {\ensuremath{\mathscr{W}}}_1$ such that $x = c_{x,i} + w_{x,i}$ holds. Since ${\ensuremath{\mathscr{W}}}_1$ is a nonempty finite set, by the Pigeonhole principle, for some element $t\in {\ensuremath{\mathscr{W}}}_1$, the equality $t=w_{x,i}$ holds for infinitely many positive integers $i$. Hence for infinitely many $i\geq 1$, we obtain $x-t=c_{x,i}$, which is an element of $C_i$. Consequently, for each positive integer $k$, there exists an integer $i_k > k$ such that $x-t$ is an element of $C_{i_k}$, which is contained in $C_k$ (as $i_k > k$) and thus $C_k$ contains $x-t$. We conclude that for each $x\in C'$, there exists an element $t\in {\ensuremath{\mathscr{W}}}_1$ such that $x-t$ belongs to $C_k$ for any $k\geq 1$, i.e., $x-t\in M$. In other words, $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Moreover, $M$ is minimal with respect to the property that $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. On the contrary, assume that for some integer $j\geq 1$, $M$ contains $c_j$ and $M\setminus\{c_j\}+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Since $M$ is contained in $C_j$, it follows that $C_j\setminus\{c_j\}+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. So $C_{j+1}$ does not contain $c_j$ and hence $c_j$ does not belong to $M= \cap _{i\geq 1} C_i$, which is absurd. This proves the claim.
We claim that $M$ is a minimal additive complement to $S$ in ${\ensuremath{\mathbb{Z}}}^d$. For $m\in {\ensuremath{\mathcal{M}}}$, define $M_m$ to be the set of all elements of $M$ congruent to $m$ modulo ${\ensuremath{\mathcal {L}}}$, i.e., $M_m$ is the fibre of the map $M\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ at the point $\bar m$. Note that for any $m\in {\ensuremath{\mathcal{M}}}$, the set $M_m$ contains a sequence of points such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrary small negative number. Indeed, if we fix $m\in {\ensuremath{\mathcal{M}}}$, then note that by condition (4), there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m+w \not \equiv m' + w'{\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}$ and $w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$. Then it follows that $m+w$ is congruent to no element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$, and hence $m+w+{\ensuremath{\mathcal {L}}}$ is contained in $$C'\subseteq M+{\ensuremath{\mathscr{W}}}_1=(M_m+{\ensuremath{\mathscr{W}}}_1)\sqcup (\sqcup_{m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}} (M_{m'}+{\ensuremath{\mathscr{W}}}_1)).$$ Furthermore, condition (4) implies that $\sqcup_{m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}} (M_{m'}+{\ensuremath{\mathscr{W}}}_1)$ contains no element of $m+w+ {\ensuremath{\mathcal {L}}}$. So $m+w+{\ensuremath{\mathcal {L}}}$ is contained in $M_m+{\ensuremath{\mathscr{W}}}_1$. Thus $$m+w+(-{\ensuremath{\mathbb{N}}}(u_1 + \cdots + u_d))\subseteq M_m+{\ensuremath{\mathscr{W}}}_1=\cup_{w'\in {\ensuremath{\mathscr{W}}}_1}(w'+M_m).$$ Since ${\ensuremath{\mathscr{W}}}_1$ is finite, it follows that for some $w'\in {\ensuremath{\mathscr{W}}}_1$, $w'+M_m$ contains a sequence of points of $m+w+(-{\ensuremath{\mathbb{N}}}(u_1 + \cdots + u_d))$ such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrarily small number. Consequently, $M_m$ contains a sequence of points such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrary small.
Let $x$ be an element of ${\ensuremath{\mathbb{Z}}}^d$ which is congruent to some element of ${\ensuremath{\mathcal{M}}}+{\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$, i.e., $x$ is equal to $m+w+\ell$ for some $m\in {\ensuremath{\mathcal{M}}}, w\in {\ensuremath{\mathcal {W}}}, \ell\in {\ensuremath{\mathcal {L}}}$. Since the set $M_m$ contains a sequence of points such that the maximum of their coordinates with respect to $u_1, \cdots, u_d$ is arbitrary small, it follows that for some large enough positive integers $\lambda_1, \cdots, \lambda_d$, the set $M_m$ contains $m-(\lambda_1 u_1 + \cdots + \lambda_d u_d)$ and $w+\ell+(\lambda_1 u_1 + \cdots + \lambda_d u_d) = w + (\ell + \lambda_1 u_1 + \cdots + \lambda_d u_d)$ belongs to $({\ensuremath{\mathbb{N}}}u_1 + \cdots+{\ensuremath{\mathbb{N}}}u_d)+{\ensuremath{\mathcal {W}}}$. Hence $$x = m + w+\ell = (m-(\lambda_1 u_1 + \cdots + \lambda_d u_d)) + (w+\ell+(\lambda_1 u_1 + \cdots + \lambda_d u_d))$$ belongs to $$M_m+ (({\ensuremath{\mathbb{N}}}u_1 + \cdots+{\ensuremath{\mathbb{N}}}u_d)+{\ensuremath{\mathcal {W}}})\subseteq M+W.$$ So $M+W$ contains all elements of ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to some elements of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. Moreover, $M+W$ contains $C'$, i.e., it contains all elements of ${\ensuremath{\mathbb{Z}}}^d$ which are congruent to no element of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. Hence $M+W$ is equal to ${\ensuremath{\mathbb{Z}}}^d$.
Let $M'$ be a subset of $M$ such that $M' + W = {\ensuremath{\mathbb{Z}}}^d$. Then each element of $M'+(W\setminus {\ensuremath{\mathscr{W}}}_1)$ is congruent to some element of ${\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}$ modulo ${\ensuremath{\mathcal {L}}}$. So $M'+(W\setminus {\ensuremath{\mathscr{W}}}_1)$ and $C'$ has no common element. Thus $M'+{\ensuremath{\mathscr{W}}}_1$ contains $C'$. Since $M$ is minimal among the subsets of $C$ with respect to the property that $M+{\ensuremath{\mathscr{W}}}_1$ contains $C'$, we conclude that $M'$ is equal to $M$. Hence $M$ is a minimal additive complement to $W$ in ${\ensuremath{\mathbb{Z}}}^d$.
Note that the set $M$ constructed in the proof of Theorem \[Thm: Implies existence of min comple\] satisfies $M= M_\infty$, i.e., $M_{\ensuremath{\mathrm{fin}}}= \emptyset$.
Equivalent condition
--------------------
\[Thm: Even peri 1 has min comp iff\] Let $W$ be an eventually periodic subset of ${\ensuremath{\mathbb{Z}}}^d$ with periods $u_1, \cdots, u_d$. Let ${\ensuremath{\mathscr{W}}}_1$ be as in Theorem \[Thm: existence of min complement implies\]. Suppose ${\ensuremath{\mathscr{W}}}_1$ contains only one element. Then $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$ if and only if there exists nonempty finite subset ${\ensuremath{\mathcal{M}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ satisfying the following.
1. The map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective.
2. The map $\pi: ({\ensuremath{\mathcal{M}}}+ {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1)) \to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is surjective.
3. For any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}\cup {\ensuremath{\mathscr{W}}}_1$.
If ${\ensuremath{\mathbb{Z}}}^d$ has a subset ${\ensuremath{\mathcal{M}}}$ such that the conditions (1), (2), (3) hold, then by Theorem \[Thm: Implies existence of min comple\], $W$ has a minimal complement in ${\ensuremath{\mathbb{Z}}}^d$.
Suppose $W$ has a minimal complement $M$ in ${\ensuremath{\mathbb{Z}}}^d$. Let ${\ensuremath{\mathcal{M}}}$ be as in Theorem \[Thm: existence of min complement implies\]. Then ${\ensuremath{\mathcal{M}}}$ is a nonempty finite set. Since the composite map ${\ensuremath{\mathcal{M}}}{\hookrightarrow}M_\infty \twoheadrightarrow \pi(M_\infty)$ is bijective, it follows that the map $\pi: {\ensuremath{\mathcal{M}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$ is injective. By Theorem \[Thm: existence of min complement implies\], condition (2) follows, and for any $m\in {\ensuremath{\mathcal{M}}}$, there exists $w\in {\ensuremath{\mathscr{W}}}_1$ such that $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}, w'\in {\ensuremath{\mathcal {W}}}$, which gives $m + w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ for any $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathcal {W}}}$. Note that for $m'\in {\ensuremath{\mathcal{M}}}\setminus\{m\}, w'\in {\ensuremath{\mathscr{W}}}_1$, we obtain $m+w \not\equiv m' + w' {\ensuremath{\mathrm{\;mod\;}}}{\ensuremath{\mathcal {L}}}$ (since ${\ensuremath{\mathscr{W}}}_1$ contains only one element). This proves condition (3).
Minimal complements in finitely generated abelian groups {#Sec: Minimal complement fngen}
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In this section $G$ is a finitely generated abelian group. We have already seen in Theorem \[theorem1\] that for finite sets $W\subset G$, a minimal complement of $W$ always exists. So we shall consider infinite subsets $W\subset G$ throughout this section. By the structure theorem for finitely generated abelian groups, $$G\simeq \mathbb{Z}^{d}\times(\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$$ with $d\geqslant 0$ and where $a_{1}|a_{2}|...|a_{s}$ are positive integers $>1$, determined uniquely by the isomorphism type of the group $G$. $\mathbb{Z}^{d}$ is said to be the torsion free part and $F :=(\mathbb{Z}/a_{1}\mathbb{Z})\times \cdots \times(\mathbb{Z}/a_{s}\mathbb{Z})$ is the torsion part. $G$ is finite if and only if $d=0$. Since we are in the realm of infinite subsets $W\subset G$, so $G$ is infinite. Hence, $d>0$ in this section. We first show the following proposition which describes the behaviour of minimal complements of product sets.
\[MinCompProd\] Let $n$ be a positive integer. Let $G_1, \cdots, G_n$ be groups. For each $1\leq i\leq n$, let $A_i$ be a subset of $G_i$ with minimal complement $M_i$. Then $\prod_{1\leqslant i\leqslant n}M_{i}$ is a minimal complement of $\prod_{1\leqslant i\leqslant n}A_{i}$ in $\prod_{1\leqslant i\leqslant n}G_{i}$.
Let $A_{1}$ and $A_{2}$ be two given sets in the groups $G_{1}$ and $G_{2}$ with minimal complements $M_{1}$ and $M_{2}$ respectively. Then $$\begin{aligned}
A_{1}.M_{1} & = G_{1}, \,\, A_{1}.\big\lbrace M_{1}\setminus \lbrace m\rbrace \big\rbrace \subsetneq G_{1} \,\,\forall m\in M_{1} ,\\
A_{2}.M_{2} & = G_{2}, \,\, A_{2}.\big\lbrace M_{2}\setminus \lbrace m\rbrace \big\rbrace \subsetneq G_{2} \,\,\forall m\in M_{2} .
\end{aligned}$$ Now $A_{1}\times A_{2}\subset G_{1}\times G_{2}$. First we show that $M_{1}\times M_{2}$ is a complement of $A_{1}\times A_{2}$ in $G_{1}\times G_{2}$. Pick any element $(g_{1},g_{2})\in G_{1}\times G_{2}$. Then $\exists a_{1}\in A_{1},a_{2}\in A_{2}, m_{1}\in M_{1},m_{2}\in M_{2}$ with $a_{1}m_{1} = g_{1}$ and $a_{2}m_{2} = g_{2}$. Thus any $(g_{1},g_{2})\in G_{1}\times G_{2}$ can be represented as $(a_{1}m_{1},a_{2}m_{2})$ for some $a_{1}\in A_{1},a_{2}\in A_{2}, m_{1}\in M_{1},m_{2}\in M_{2}$, i.e., $$(A_{1}\times A_{2}). (M_{1}\times M_{2}) = G_{1}\times G_{2},$$ thus it is a complement of $A_{1}\times A_{2}$.
Now, we show that $M_{1}\times M_{2}$ is minimal. Remove an element $(m,n)$ from $M_{1}\times M_{2}$ and look at the set $ M :=M_{1}\times M_{2} \setminus \lbrace (m,n)\rbrace $. We show that $M$ is not a complement of $A_{1}\times A_{2}$ in $G_{1}\times G_{2}$, i.e., $(A_{1}\times A_{2}).M\subsetneq G_{1}\times G_{2}$. Since $M_{1}$ is a minimal complement of $A_{1}$, $\exists a_{1}\in A_{1},g_{1}\in G_{1}$ such that the only way of representing $g_{1}$ in $A_{1}.M_{1}$ is $a_{1}m$. Similarly, $\exists a_{2}\in A_{2},g_{2} \in G_{2}$ with $g_{2} = a_{2}n$. We show that this $(g_{1},g_{2}) \notin (A_{1}\times A_{2}).M$. Indeed, this is clear from the fact that $(g_{1},g_{2}) $ can only be represented in $(A_{1}\times A_{2}). (M_{1}\times M_{2})$ as $(a_{1}m,a_{2}n)$.
To prove the general case we use induction.
Without loss of generality, let us assume that the statement is true for $k$ groups $G_{1},G_{2},\cdots, G_{k}$ with $k<n$. We show that the statement holds for $(k+1)$-groups. By hypothesis, $$\begin{aligned}
A_{1} . M_{1} = G_{1}, \,\, A_{1} . M_{1}\setminus & \lbrace m_{1}\rbrace \subsetneq G_{1} \,\,\forall m_{1}\in M_{1} \\
A_{2} . M_{2} = G_{2}, \,\, A_{2} . M_{2}\setminus & \lbrace m_{2}\rbrace \subsetneq G_{2} \,\,\forall m_{2}\in M_{2} \\
\vdots \\
A_{k} . M_{k} = G_{k}, \,\, A_{k} . M_{k}\setminus & \lbrace m_{k}\rbrace \subsetneq G_{k} \,\,\forall m_{k}\in M_{k}\\
A_{k+1} . M_{k+1} = G_{k+1}, \,\, A_{k+1} . M_{k+1}\setminus & \lbrace m_{k+1}\rbrace \subsetneq G_{k+1} \,\,\forall m_{k+1}\in M_{k+1}. \\
\end{aligned}$$ By the inductive assumption $N :=M_{1}\times \cdots \times M_{k}$ is a minimal complement of $A_{1}\times \cdots \times A_{k}$ in $G_{1}\times \cdots \times G_{k}$. To show that $N\times M_{k+1}$ is a minimal complement of $A_{1}\times \cdots \times A_{k+1}$ in $G_{1}\times \cdots \times G_{k+1}$. It is clear that $N\times M_{k+1}$ is a complement. To show that it is minimal, we remove an arbitrary point $(x,y)$ from $N\times M_{k+1}$ and argue as above to get the required statement.
An immediate consequence of the above proposition (combined with previous work of Chen–Yang [@ChenYang12] and Kiss–Sándor–Yang [@KissSandorYangJCT19]) is the construction of infinite sets in $\mathbb{Z}^{d}$ having minimal complements. We first state their results -
\[Chen-Yang\] Let $W\subset \mathbb{Z}$. Suppose either of the following holds
1. $\inf W = -\infty, \,\,\sup W = +\infty$.
2. $\limsup_{i\rightarrow \infty} (w_{i+1}-w_{i}) = +\infty$.
Then $W$ has a minimal complement in $\mathbb{Z}$.
\[Kiss-Sandor-Yang\] Let $W\subset \mathbb{Z}$ be of the following form, $$W = (m\mathbb{N} + X_{m}) \cup Y^{(0)} \cup Y^{(1)},$$ where $X_{m}\subseteq \lbrace 0, 1,\cdots, m-1\rbrace, Y^{(0)}\subseteq \mathbb{Z}^{-}, Y^{(1)} $ are finite sets with $Y^{(0)} \mod m \subseteq X_{m}$ and $(Y^{(1)}\mod m) \cap X_{m} = \emptyset .$ There exists a minimal complement to $W$ if and only if there exists $T\in \mathbb{Z}^{+}, m |T$, and $C\subseteq \lbrace 0, 1, \cdots, T-1\rbrace $ such that
1. $C+(X_{T}\cup Y^{(1)} ) \mod T = \lbrace 0, 1,\cdots, T-1\rbrace$ where $X_{T}=\cup_{i=0}^{\frac{T}{m}-1}\lbrace im +X_m\rbrace ;$
2. for any $c\in C$, there exists $y\in Y^{(1)} $ for which $c +y \not\equiv c'+x \,(mod\, T),$ where $c'\in C\setminus{c} $ and $x \in X_{T}$.
Using Theorem \[Chen-Yang\], Theorem \[Kiss-Sandor-Yang\] and Proposition \[MinCompProd\] we have the following immediate result.
\[MinProdThm\] Fix $d$ a positive integer. Let $W_{i}\subset \mathbb{Z}$ be of the form described in Theorem \[Chen-Yang\] or Theorem \[Kiss-Sandor-Yang\] or finite, for each $1\leqslant i\leqslant d$. Then $W_{1}\times W_{2}\times \cdots \times W_{d}$ has a minimal complement in $\mathbb{Z}^{d}$.
Each $W_{i}\subset \mathbb{Z}$ has a minimal complement $M_{i}$ in $\mathbb{Z}$ (using Theorem \[Chen-Yang\] or Theorem \[Kiss-Sandor-Yang\] or Theorem \[theorem1\]). Using Proposition \[MinCompProd\], we see that $M_{1}\times M_{2}\times \cdots \times M_{d}$ is a required minimal complement.
However, not all infinite sets in $\mathbb{Z}^{d}$ are product sets. We have seen in the previous section a sufficient condition for the existence of minimal complements for eventually periodic sets in $\mathbb{Z}^{d}$ (which are not product sets, when they are not periodic). We shall now exploit the structure of the finitely generated abelian group $G$.
\[lemmasec5\] Let $G\simeq \mathbb{Z}^{d}\times F$ be any finitely generated abelian group ($F$ is the finite torsion part). Let $\emptyset\neq A\subset \mathbb{Z}^{d}$. Suppose minimal complement of $A$ in $\mathbb{Z}^{d}$ exist. Then a minimal complement of $A\times H $ exists $\forall \,\emptyset\neq H\subseteq F$.
We have $G\simeq \mathbb{Z}^{d}\times F$, with $F$ a finite group. Suppose $\emptyset\neq A\subset \mathbb{Z}^{d}, \emptyset\neq H\subset F$.
Let $B$ be a minimal complement of $A$ in $\mathbb{Z}^{d}$. Since $F$ is finite, $H$ is also finite. By Theorem \[theorem1\], we know that a minimal complement of $H$ exists. Let it be $H'$. Now using the previous proposition, we have that $B\times H'$ is a minimal complement of $A\times H$ in $G$.
We come to the main theorem which describes a large class of infinite sets having minimal complements in finitely generated abelian groups.
\[sec5Thm\] Let $G\simeq \mathbb{Z}^{d}\times F$ be any finitely generated abelian group. Suppose $W\subseteq \mathbb{Z}^{d}$ be either of the form given in Theorem \[Thm: Implies existence of min comple\] or a product set $W_{1}\times W_{2}\times \cdots \times W_{d}$ as described in Theorem \[MinProdThm\]. Then $W\times H$ will have a minimal complement in $G$ where $H \subseteq F$ is any arbitrary nonempty subset.
The form of $W\subseteq \mathbb{Z}^{d}$ ensures that $W$ has a minimal complement in $\mathbb{Z}^{d}$. After this we apply Lemma \[lemmasec5\] to get the desired conclusion.
\[Remark:MinCompEg\] The upshot of the discussion in Section \[Sec: Minimal complement fngen\] is to provide examples of subsets of finitely generated abelian groups admitting minimal complements. Some such examples were already given in Section \[Sec: Minimal complement\] (see Theorem \[Thm: Implies existence of min comple\]), where we considered the group ${\ensuremath{\mathbb{Z}}}^d$, i.e., free abelian groups only. The immediate question is look for examples of such subsets in finitely generated abelian groups having nontrivial torsion. By Proposition \[MinCompProd\], a subset $W$ of a group $G$ admits a minimal complement if $G$ is isomorphic to the direct product of groups $G_1\times \cdots \times G_n$ and under such an isomorphism $W$ corresponds to the product of subsets $W_i$ of $G_i$ having minimal complements. Hence, as a consequence of Theorems \[Chen-Yang\], \[Kiss-Sandor-Yang\], \[Thm: Implies existence of min comple\], \[theorem1\] and Propositions \[prop4.1\], \[Prop:Hypersurf\], it follows that a subset $W$ of a finitely abelian group $G$ admits a minimal complement in $G$ if there exists finitely generated free abelian groups $G_1, \cdots, G_n$ and an isomorphism $\psi: G{\xrightarrow}{\sim} G_{\ensuremath{\mathrm{tors}}}\times G_1\times \cdots \times G_n$ (where $G_{\ensuremath{\mathrm{tors}}}$ denotes the torsion part of $G$) such that $\psi(W)$ is equal to the product $W_0 \times W_1 \times \cdots \times W_n$ where $W_0$ is a nonempty subset of $G_{\ensuremath{\mathrm{tors}}}$, $W_1, \cdots, W_n$ are subsets of $G_1, \cdots, G_n$ and for each $1\leq i\leq n$, one of the following conditions hold.
1. $G_i={\ensuremath{\mathbb{Z}}}$ and $W_i$ satisfies the conditions of Theorem \[Chen-Yang\].
2. $G_i = {\ensuremath{\mathbb{Z}}}$ and $W_i$ satisfies the conditions of Theorem \[Kiss-Sandor-Yang\].
3. $G_i = {\ensuremath{\mathbb{Z}}}^{d_i}$ for some integer $d_i\geq 1$ and $W_i$ satisfies the conditions of Theorem \[Thm: Implies existence of min comple\].
4. $G_i = {\ensuremath{\mathbb{Z}}}^{d_i}$ for some integer $d_i\geq 1$ and $W_i$ is a nonempty finite subset of $G_i$.
5. $G_i$ is a free abelian group of finite rank and $W_i$ is a subgroup of $G_i$.
6. $G_i = {\ensuremath{\mathbb{Z}}}^{d_i}$ for some integer $d_i\geq 1$ and $W_i$ is equal to $\{(\pm n,\cdots, \pm n)\in {\ensuremath{\mathbb{Z}}}^{d_i}\,|\, n\in {\ensuremath{\mathbb{Z}}}\}$ (see Proposition \[Prop:Hypersurf\]).
To conclude this section, we see that a combination of Theorem \[sec5Thm\], Theorem \[MinProdThm\] and Proposition \[MinCompProd\] gives us infinite sets in arbitrary finitely generated abelian groups having minimal complements, providing a partial answer to Nathanson’s Question \[nathansonprob13\].\
Conclusion and further remarks {#Sec:Conclusion}
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Finally, we conclude the article with a class of examples of infinite sets which are not eventually periodic (i.e., they don’t fall in the class of Theorem \[Thm: existence of min complement implies\] and Theorem \[Thm: Implies existence of min comple\]) but have minimal complements.
\[prop6.1\] \[Prop:Hypersurf\] Let $d\geq 2$ be a positive integer. Then the subset $${\ensuremath{\mathcal{D}}}:= \{(\pm n,\cdots, \pm n)\in {\ensuremath{\mathbb{Z}}}^d\,|\, n\in {\ensuremath{\mathbb{Z}}}\}$$ of ${\ensuremath{\mathbb{Z}}}^d$ is not eventually periodic. For each $1\leq i\leq d$, the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{Z}}}^d\,|\, x_i=0\}$$ is a minimal complement of ${\ensuremath{\mathcal{D}}}$.
Suppose ${\ensuremath{\mathcal{D}}}$ is eventually periodic. So there exist elements $u_1, \cdots, u_d$ in ${\ensuremath{\mathbb{Z}}}^d$ satisfying no nontrivial ${\ensuremath{\mathbb{Z}}}$-linear relation and an integer $n_0\geq 1$ such that for any integer $n$ with $|n|> n_0$, $${\ensuremath{\mathcal{D}}}\supseteq (\pm n, \cdots, \pm n) + ({\ensuremath{\mathbb{N}}}u_1 + \cdots + {\ensuremath{\mathbb{N}}}u_d).$$ Let $n'$ denote the integer $\max \{n_0 , ||u_1||, \cdots, ||u_d||\}$. For any positive integer $n$ with $n>\sqrt d n'$ and any $1\leq i\leq d$, the vector $(n, \cdots, n)+u_i$ is contained in the open ball $B_{(n, \cdots, n), n/\sqrt d}$ in ${\ensuremath{\mathbb{R}}}^d$ with center at the point $(n,\cdots, n)$ with radius $n/\sqrt d$, and it is also contained in ${\ensuremath{\mathcal{D}}}$. Hence the vectors $u_1, \cdots, u_d$ are scalar multiples of $(1, \cdots, 1)$, which is absurd. So ${\ensuremath{\mathcal{D}}}$ is not eventually periodic.
For any element $(x_1, \cdots, x_d)$ of ${\ensuremath{\mathbb{Z}}}^d$, $(x_i, \cdots, x_i)$ belongs to ${\ensuremath{\mathcal{D}}}$ and ${\ensuremath{\mathcal{H}}}_i$ contains $(x_1, \cdots, x_d) - (x_i, \cdots, x_i)$ and hence ${\ensuremath{\mathcal{H}}}_i$ is a complement of ${\ensuremath{\mathcal{D}}}$.
Suppose $M_i$ is a subset of ${\ensuremath{\mathcal{H}}}_i$ such that $M_i + {\ensuremath{\mathcal{D}}}= {\ensuremath{\mathbb{Z}}}^d$. Note that the $i$-th coordinate of any element of $M_i + {\ensuremath{\mathcal{D}}}\setminus\{(0,\cdots,0)\}$ is nonzero. So it contains no element of ${\ensuremath{\mathcal{H}}}_i$. Thus ${\ensuremath{\mathcal{H}}}_i$ is contained in $(0,\cdots, 0) + M_i$. So $M_i$ is equal to ${\ensuremath{\mathcal{H}}}_i$. Hence ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement to ${\ensuremath{\mathcal{D}}}$.
\[prop6.2\] Let $d\geq 2$ be a positive integer. Let $f_1, \cdots, f_d$ be elements of ${\ensuremath{\mathbb{Z}}}[X_1, \cdots, X_m]$. Define the subset ${\ensuremath{\mathcal{S}}}$ of ${\ensuremath{\mathbb{Z}}}^d$ by $${\ensuremath{\mathcal{S}}}:= \{( f_1(n_1, \cdots, n_m),\cdots, f_d(n_1, \cdots, n_m))\,|\, (n_1, \cdots, n_m)\in {\ensuremath{\mathbb{Z}}}^m\}.$$ Let $1\leq i\leq d$ be an integer such that the map $f_i:{\ensuremath{\mathbb{Z}}}^m\to {\ensuremath{\mathbb{Z}}}$ is surjective. Then the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{Z}}}^d\,|\, x_i=0\}$$ is a complement of ${\ensuremath{\mathcal{S}}}$. Moreover, if ${\ensuremath{\mathcal{S}}}\cap {\ensuremath{\mathcal{H}}}_i$ contains only one element, then ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement of ${\ensuremath{\mathcal{S}}}$.
For each element $(x_1, \cdots, x_d)$ of ${\ensuremath{\mathbb{Z}}}^d$, there exists an element $(n_1, \cdots, n_m)\in {\ensuremath{\mathbb{Z}}}^m$ such that $f_i(n_1, \cdots, n_m) = x_i$. So ${\ensuremath{\mathcal{S}}}$ contains the point $(f_1(n_1, \cdots, n_m), \cdots,$ $ f_d(n_1, \cdots, n_m))$ and ${\ensuremath{\mathcal{H}}}_i$ contains the difference $(x_1, \cdots, x_d)-(f_1(n_1, \cdots, n_m), \cdots,$ $ f_d(n_1, \cdots, n_m))$. Hence ${\ensuremath{\mathcal{H}}}_i$ is a complement to ${\ensuremath{\mathcal{S}}}$.
Suppose ${\ensuremath{\mathcal{S}}}\cap {\ensuremath{\mathcal{H}}}_i$ contains only one element $P$ of ${\ensuremath{\mathbb{Z}}}^d$. Let $M_i$ be a subset of ${\ensuremath{\mathcal{H}}}_i$ such that $M_i + {\ensuremath{\mathcal{S}}}= {\ensuremath{\mathbb{Z}}}^d$. Note that the $i$-th coordinate of any point of $M_i + {\ensuremath{\mathcal{S}}}\setminus\{P\}$ is nonzero. So ${\ensuremath{\mathcal{H}}}_i$ is contained in $P + M_i$. Hence ${\ensuremath{\mathcal{H}}}_i - P = {\ensuremath{\mathcal{H}}}_i$ is contained in $M_i$. Hence ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement of ${\ensuremath{\mathcal{S}}}$.
In fact, the ambient group (and even the set, the existence/inexistence of the minimal complement of which we seek to investigate) can be taken to be uncountable. The following propositions shed light on this fact -
\[prop6.3\] \[Prop:Hypersurf R\] Let $d\geq 2$ be a positive integer. For each $1\leq i\leq d$, the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{R}}}^d\,|\, x_i=0\}$$ is a minimal complement of $${\ensuremath{\mathcal{D}}}:= \{(\pm x,\cdots, \pm x)\,|\, x\in {\ensuremath{\mathbb{R}}}\}$$ in ${\ensuremath{\mathbb{R}}}^d$.
The proof is similar to the proof of Proposition \[Prop:Hypersurf\].
\[prop6.4\] \[Prop:Hypersurfaces R\] Let $d\geq 2$ be a positive integer. Let $f_1, \cdots, f_d$ be elements of ${\ensuremath{\mathbb{R}}}[X_1, \cdots, X_m]$. Define the subset ${\ensuremath{\mathcal{S}}}$ of ${\ensuremath{\mathbb{R}}}^d$ by $${\ensuremath{\mathcal{S}}}:= \{( f_1(n_1, \cdots, n_m),\cdots, f_d(n_1, \cdots, n_m))\,|\, (n_1, \cdots, n_m)\in {\ensuremath{\mathbb{R}}}^m\}.$$ Let $1\leq i\leq d$ be an integer such that the map $f_i:{\ensuremath{\mathbb{R}}}^m\to {\ensuremath{\mathbb{R}}}$ is surjective. Then the hypersurface defined by $x_i=0$, i.e., the set $${\ensuremath{\mathcal{H}}}_i:=\{(x_1, \cdots, x_d)\in {\ensuremath{\mathbb{R}}}^d\,|\, x_i=0\}$$ is a complement of ${\ensuremath{\mathcal{S}}}$. Moreover, if ${\ensuremath{\mathcal{S}}}\cap {\ensuremath{\mathcal{H}}}_i$ contains only one element, then ${\ensuremath{\mathcal{H}}}_i$ is a minimal complement of ${\ensuremath{\mathcal{S}}}$.
The proof is similar to the proof of Proposition \[prop6.2\].
Acknowledgements
================
The first author would like to thank the Fakultät für Mathematik, Universität Wien where a part of the work was carried out. The second author would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPIRE Faculty Award from the Department of Science and Technology, Government of India.
[KSY19]{}
Yong-Gao Chen and Quan-Hui Yang, *On a problem of [N]{}athanson related to minimal additive complements*, SIAM J. Discrete Math. **26** (2012), no. 4, 1532–1536. [MR ]{}[3022150]{}
Sándor Z. Kiss, Csaba Sándor, and Quan-Hui Yang, *On minimal additive complements of integers*, J. Combin. Theory Ser. A **162** (2019), 344–353. [MR ]{}[3875615]{}
Melvyn B. Nathanson, *Problems in additive number theory, [IV]{}: [N]{}ets in groups and shortest length [$g$]{}-adic representations*, Int. J. Number Theory **7** (2011), no. 8, 1999–2017. [MR ]{}[2873139]{}
[^1]:
[^2]:
[^3]: Equivalently, ${\ensuremath{\mathscr{W}}}_0$ is the inverse image of ${\ensuremath{\mathcal{Q}}}$ under the map $\pi: {\ensuremath{\mathscr{W}}}\to {\ensuremath{\mathbb{Z}}}^d/{\ensuremath{\mathcal {L}}}$, and ${\ensuremath{\mathscr{W}}}_1$ is its complement in ${\ensuremath{\mathscr{W}}}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper introduces and analyses the new grid-based tensor approach for approximate solution of the eigenvalue problem for linearized Hartree-Fock equation applied to the 3D lattice-structured and periodic systems. The set of localized basis functions over spatial $(L_1,L_2,L_3)$ lattice in a bounding box (or supercell) is assembled by multiple replicas of those from the unit cell. All basis functions and operators are discretized on a global 3D tensor grid in the bounding box which enables rather general basis sets. In the periodic case, the Galerkin Fock matrix is shown to have the three-level block circulant structure, that allows the FFT-based diagonalization. The proposed tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. We describe fast numerical algorithms for the block circulant representation of the core Hamiltonian in the periodic setting based on low-rank tensor representation of arising multidimensional functions. Lattice type systems in a box with open boundary conditions are treated by our previous tensor solver for single molecules, which makes possible calculations on large $(L_1,L_2,L_3)$ lattices due to reduced numerical cost for 3D problems. The numerical simulations for box/periodic $(L,1,1)$ lattice systems in a 3D rectangular “tube” with $L$ up to several hundred confirm the theoretical complexity bounds for the tensor-structured eigenvalue solvers in the limit of large $L$.'
author:
- 'V. KHOROMSKAIA, [^1]'
- 'B. N. KHOROMSKIJ [^2]'
title: 'Tensor Numerical Approach to Linearized Hartree-Fock Equation for Lattice-type and Periodic Systems'
---
*AMS Subject Classification:* ** 65F30, 65F50, 65N35, 65F10
*Key words:* Hartree-Fock equation, tensor-structured numerical methods, 3D grid-based tensor approximation, Fock operator, core Hamiltonian, periodic systems, lattice summation, block circulant matrix, Fourier transform.
Introduction {#sec:introduct}
============
The efficient numerical simulation of periodic and perturbed periodic systems is one of the most challenging computational tasks in quantum chemistry calculations of crystalline, metallic and polymer-type compounds. The reformulation of the nonlinear Hartree-Fock equation for periodic molecular systems based on the Bloch theory [@Bloch:1925] has been addressed in the literature for more than forty years ago, and nowadays there are several implementations mostly relying on the analytic treatment of arising integral operators [@CRYSTAL:2000; @CRYSCOR:12; @GAUSS:09]. The mathematical analysis of spectral problems for PDEs with the periodic-type coefficients was an attractive topic in the recent decade, see [@CancesDeLe:08; @CanEhrMad:2012; @Ortn:ArX] and the references therein. However, the systematic developments and optimization of the basic numerical algorithms in the Hartree-Fock calculations for large lattice structured compounds still are largely unexplored.
Grid-based approaches for single molecules and moderate size systems based on the locally adaptive grids and multiresolution techniques have been discussed (see [@HaFaYaBeyl:04; @SaadRev:10; @Frediani:13; @CanEhrMad:2012; @Ortn:ArX; @BiVale:11; @RahOsel:13] and references therein).
In this paper, we consider the Hartree-Fock equation for extended systems composed of atoms or molecules, determined by means of an $(L_1, L_2, L_3)$ lattice in a box, both for open boundary conditions and in the periodic setting (supercell). The grid-based tensor-structured method is applied (see [@KhKhFl_Hart:09; @VKH_solver:13; @KhorSurv:10; @VeBoKh:Ewald:14] and references therein) to calculate the core Hamiltonian in the localized Gaussian-type basis sets living on a box/periodic spatial lattice. To perform numerical integration by using low-rank tensor formats we represent all basis functions on the fine global grid covering the whole computational box (supercell). The Hartree-Fock equation for periodic systems is reformulated as the eigenvalue problem for large block circulant matrices which are diagonalizable in the Fourier space, that allows efficient computations on large lattices of size $L=\max\{L_1,L_2,L_3\}$. In the following we consider the model problem for the Fock operator confined to the core Hamiltonian part.
One of the severe difficulties in the Hartree-Fock calculations for lattice-structured periodic or box-restricted systems is the computation of 3D lattice sums of a large number of long-distance Coulomb interaction potentials. This problem is traditionally treated by the so-called Ewald-type summation techniques [@Ewald:27] combined with the fast multipole expansion or/and FFT methods. Notice that the traditional approaches for lattice summation by the Ewald-type methods scale as $O(L^3 \log L)$ at least, for both periodic and box-type lattice sums. We apply the recent lattice summation method [@VeBoKh:Ewald:14] by assembled rank-structured tensor decomposition, which reduces the asymptotic cost at this computational step to linear scaling in $L$, i.e. $O(L)$.
In the presented approach the Fock matrix is calculated directly by 3D grid-based tensor numerical methods in the basis set of localized Gaussian-type-orbitals (GTO) specified by $m_0$ elements in the 3D unit cell [@VeKh_Diss:10; @VKH_solver:13]. Hence, we do not impose explicitly the periodicity-like features of the solution by means of the approximation ansatz that is normally the case in the Bloch formalism. Instead, the periodic properties of the considered system appear implicitly through the block structure in the Fock matrix. In periodic case this matrix is proved to inherit the three-level symmetric block circulant form, that allows its efficient diagonalization in the Fourier basis [@KaiSay_book:99; @Davis]. In the case of $d$-dimensional lattice ($d=1,2,3$), the weak overlap between lattice translated basis functions improves the block sparsity thus reducing the storage cost to $O(m_0^2 L)$, while the FFT-based diagonalization procedure amounts to $O(m_0^2 L^d \log L)$ operations. Introducing the low-rank tensor structure into the diagonal blocks of the Fock matrix represented in the Fourier space, and using the initial block-circulant structure it becomes possible to further reduce the numerical costs to linear scaling in $L$, $O(m_0^2 L \log L)$. We present numerical tests in the case of a rectangular 3D “tube” composed of $(L, 1, 1)$ cells with $L$ up to several hundred.
In the new approach one can potentially benefit from the additional flexibility that allows to treat slightly perturbed periodic systems in a straightforward way. Such situations may arise, for example, in the case of finite extended systems in a box (open boundary conditions) also considered in this paper, or for slightly perturbed periodic compounds, say for quasi-periodic systems with vacancies [@BGKh:12]. The proposed numerical scheme can be applied in the framework of self-consistent Hartree-Fock calculations, in particular, in the reduced Hartree-Fock model [@CancesDeLe:08], where the similar block-structure in the Fock matrix can be observed. The Wannier-type basis constructed by the lattice translation of the initial localized molecular orbitals precomputed on the reference unit cell, can be also adapted to our framework.
Furthermore, the arising block-structured matrix representing the stiffness matrix $H$ of the core Hamiltonian, as well as some auxiliary function-related tensors, can be shown to be well suited for further optimization by imposing the low-rank tensor formats, and in particular, the quantics-TT (QTT) tensor approximation [@KhQuant:09] of long vectors, which especially benefits in the limiting case of large $L$-periodic systems. In the QTT approach the algebraic operations on the 3D $n\times n\times n$ Cartesian grid can be implemented with logarithmic cost $O(\log n)$. Literature surveys on tensor algebra and rank-structured tensor methods for multi-dimensional PDEs can be found in [@Kolda; @KhorSurv:10; @GraKresTo:13], see also [@HaKhSaTy:08; @DoKhSavOs_mEIG:13].
The rest of the paper is organized as follows. Section \[sec\_MLBlock-circ\] recalls the main properties of the multilevel block circulant matrices with special focus on their diagonalization by FFT. Section \[sec:core\_H\] includes the main results on the analysis of core Hamiltonian on lattice structured compounds. In particular, section \[Core\_Hamil\] describes the tensor-structured calculation of the core Hamiltonian for large lattice-type molecular/atomic systems. We recall tensor-structured calculation of the Laplace operator and fast summation of lattice potentials by assembled canonical tensors. The complexity reduction due to low-rank tensor structures in the matrix blocks is discussed (see Proposition \[prop:low\_rank\_coef\]). Section \[sec:Core\_Ham\_period\_FFT\] discusses in detail the block circulant structure of the core Hamiltonian and presents numerical illustrations for $(L,1,1)$ lattice systems. Appendix recalls the classical results on the properties of block circulant/Toeplitz matrices.
Diagonalizing multilevel block circulant matrices {#sec_MLBlock-circ}
=================================================
The direct Hartree-Fock calculations for lattice structured systems in the localized GTO-type basis lead to the symmetric block circulant/Toeplitz matrices (see Appendix \[sec\_Append:block-circ\]), where the first-level blocks, $A_0,...,A_{L-1}$, may have further block structures. In particular, the Galerkin approximation of the 3D Hartree-Fock core Hamiltonian in periodic setting leads to the symmetric, three-level block circulant matrix.
Multilevel block circulant/Toeplitz matrices {#ssec:MLblock-circ}
--------------------------------------------
In this section we consider the extension of (one-level) block circulant matrices described in Appendix. First, we recall the main notions of multilevel block circulant (MBC) matrices with the particular focus on the three-level case. Given a multi-index ${\bf L}=(L_1, L_2, L_3)$, we denote $|{\bf L}|=(L_1,\, L_2,\, L_3)$. A matrix class ${\cal BC} (d,{\bf L},m_0)$ ($d=1,2,3$) of $d$-level block circulant matrices can be introduced by the following recursion.
\[def:Bcirc\] For $d=1$, define a class of one-level block circulant matrices by ${\cal BC} (1,{\bf L},m)\equiv {\cal BC} (L_1,m)$ (see Appendix), where ${\bf L}=(L_1,1,1)$. For $d=2$, we say that a matrix $A\in \mathbb{R}^{|{\bf L}|m_0 \times |{\bf L}| m_0}$ belongs to a class ${\cal BC} (d,{\bf L},m_0)$ if $$A = \operatorname{bcirc}(A_1,...,A_{L_1})\quad \mbox{with}\quad
A_j\in {\cal BC}(d-1,{\bf L}_{[1]},m_0),\; j=1,...,L_1,$$ where ${\bf L}_{[1]}=(L_2,L_3)\in \mathbb{N}^{d-1} $. Similar recursion applies to the case $d=3$.
Likewise to the case of one-level BC matrices (see Appendix), it can be seen that a matrix $A \in {\cal BC} (d,{\bf L},m_0)$, $d=1,2,3$, of size $|{\bf L}| m_0 \times |{\bf L}| m_0$ is completely defined (parametrized) by a $d$th order matrix-valued tensor ${\bf A}=[A_{k_1 ... k_d}]$ of size $L_1\times ... \times L_d $, ($k_\ell=1,...,L_\ell$, $\ell=1,...,d$), with $m_0\times m_0$ matrix entries $A_{k_1 ... k_d}$, obtained by folding of the generating first column vector in $A$. A diagonalization of a MBC matrix is based on representation via a sequence of cycling permutation matrices $\pi_{L_1}, ...,\pi_{L_d}$, $d=1,2,3$. Recall that the $q$-dimensional Fourier transform (FT) can be defined via the Kronecker product of the univariate FT matrices (rank-$1$ operator), $$F_{\bf L}=F_{L_1}\otimes \cdots \otimes F_{L_d}.$$ The block-diagonal form of a MBC matrix is well known in the literature. Here we prove the diagonal representation in a form that is useful for the description of numerical algorithms. To that end we generalize the notations ${\cal T}_L$ and $\widehat{A}$ (see Section \[sec\_Append:block-circ\]) to the class of multilevel matrices. We denote by $\widehat{A}\in \mathbb{R}^{|{\bf L}|m_0\times m_0}$ the first block column of a matrix $A\in {\cal BC} (d,{\bf L},m_0)$, with a shorthand notation $$\widehat{A}=[A_0,A_1,...,A_{L-1}]^T,$$ so that a $|{\bf L}|\times m_0 \times m_0$ tensor ${\cal T}_{\bf L} \widehat{A}$ represents slice-wise all generating $m_0\times m_0$ matrix blocks. Notice that in the case $m_0=1$, $\widehat{A}\in \mathbb{R}^{|{\bf L}|}$ represents the first column of the matrix $A$. Now the Fourier transform $F_{\bf L}$ applies to ${\cal T}_{\bf L} \widehat{A}$ columnwise, and the backward reshaping of the resultant tensor, ${\cal T}_{\bf L}'$, returns an $|{\bf L}|m_0 \times m_0$ block matrix column.
\[lem:DiagMLCirc\] A matrix $A\in {\cal BC} (d,{\bf L},m_0)$, is block-diagonalysed by the Fourier transform, $$\label{eqn:DiagMLcirc}
A= (F_{\bf L}^\ast \otimes I_{m_0}) \operatorname{bdiag} \{ \bar{A}_{\bf 0}, \bar{A}_{\bf 1},\ldots ,
\bar{A}_{\bf L-1}\}(F_{\bf L} \otimes I_{m_0}),$$ where $$\left[ \bar{A}_{\bf 0}, \bar{A}_{\bf 1},\ldots , \bar{A}_{\bf L-1}\right]^T =
{\cal T}_{\bf L}'(F_{\bf L} ({\cal T}_{\bf L} \widehat{A})).$$
First, we confine ourself to the case of three-level matrices, i.e. $d=3$, and proceed $$\begin{aligned}
A & = \sum\limits^{L_1 -1}_{k_1=0} \pi_{L_1}^{k_1} \otimes {\bf A}_{k_1} \\ \nonumber
& = \sum\limits^{L_1 -1}_{k_1=0} \pi_{L_1}^{k_1}\otimes
(\sum\limits^{L_2 -1}_{k_2=0} \pi_{L_2}^{k_2}\otimes {\bf A}_{k_1 k_2} )=
\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes {\bf A}_{k_1 k_2} \\ \nonumber
& = \sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}\sum\limits^{L_3 -1}_{k_3=0}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3}, \nonumber\end{aligned}$$ where ${\bf A}_{k_1}\in \mathbb{R}^{L_2\times L_3 \times m_0\times m_0}$, ${\bf A}_{k_1 k_2} \in \mathbb{R}^{L_3 \times m_0\times m_0}$ and $A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}$.
Diagonalizing the periodic shift matrices $\pi_{L_1}^{k_1}, \pi_{L_2}^{k_2}$, and $\pi_{L_3}^{k_3}$ via the Fourier transform (see Appendix), we arrive at $$\begin{aligned}
\label{eqn:MLbcircDiag2}
A & = (F_{\bf L}^\ast \otimes I_{m_0}) \left[\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}
\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3} \right]
(F_{\bf L}\otimes I_{m_0})
\\
& = (F_{\bf L}^\ast \otimes I_{m_0})
\mbox{bdiag}_{m_0\times m_0} \{{\cal T}_{\bf L}'(F_{\bf L} ({\cal T}_{\bf L} \widehat{A}))\}
(F_{\bf L}\otimes I_{m_0}),\nonumber
\end{aligned}$$ with the monomials of diagonal matrices $D_{L_\ell}^{k_\ell}\in \mathbb{R}^{L_\ell \times L_\ell}$, $\ell=1,2,3$ are defined by (\[eqn:diagshift\]).
The generalization to the case $d >3$ can be proven by the similar argument.
Taking into account representation (\[eqn:symBc\]), the multilevel symmetric block circulant matrix can be described in form (\[eqn:DiagMLcirc\]), such that all real-valued diagonal blocks remain symmetric.
Similar to Definition \[def:Bcirc\], a matrix class ${\cal BT}_s (d,{\bf L},m_0)$ of symmetric $d$-level block Toeplitz matrices can be introduced by the following recursion.
\[def:BToepl\] For $d=1$, ${\cal BT}_s (1,{\bf L},m_0)\equiv {\cal BT}_s (L_1,m_0)$ is the class of one-level symmetric block circulant matrices with ${\bf L}=(L_1,1,1)$. For $d=2$ we say that a matrix $A\in \mathbb{R}^{|{\bf L}|m \times |{\bf L}| m_0}$ belongs to a class ${\cal BT}_s (d,{\bf L},m_0)$ if $$A= \operatorname{btoepl}_s(A_1,...,A_{L_1})\quad
\mbox{with}\quad A_j\in {\cal BT}_s(d-1,{\bf L_{[1]}},m_0),\; j=1,...,L_1.$$ Similar recursion applies to the case $d=3$.
The following remark compares the properties of circulant and Toeplitz matrices.
\[rem:BToepl\] A block Toeplitz matrix does not allow diaginalization by FT as it was the case for block circulant matrices. However, it is well known that a block Toeplitz matrix can be extended to the double-size (at each level) block circulant that makes it possible the efficient matrix-vector multiplication, and, in particular, the efficient application of power method for finding the senior eigenvalues.
Low-rank tensor structure in matrix blocks {#ssec:Tensor_bcirc}
------------------------------------------
In the case $d=3$, the general block-diagonal representation (\[eqn:DiagMLcirc\]) - (\[eqn:MLbcircDiag2\]) takes form $$A= (F_{\bf L}^\ast \otimes I_{m_0}) (\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}
\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3} )
(F_{\bf L}\otimes I_{m_0}),$$ that allows the reduced storage costs of order $O(|{\bf L}| m_0^2)$, where $|{\bf L}|=L^3$. For large $L$ the numerical cost may become prohibitive. However, the above representation indicates that the further storage and complexity reduction becomes possible if the third-order coefficients tensor ${\bf A}= [A_{k_1 k_2 k_3}]$, $k_\ell=0,...,L_\ell-1$, with the matrix entries $A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}$, allows some low-rank tensor representation (approximation) in the multiindex ${\bf k}$ described by a small number of parameters.
To fix the idea, let us assume the existence of rank-$1$ separable matrix factorization, $$A_{k_1 k_2 k_3} = A_{k_1}^{(1)}\odot A_{k_2}^{(2)} \odot A_{k_3}^{(3)},
\quad A_{k_1}^{(1)}, A_{k_2}^{(2)},A_{k_3}^{(3)} \in \mathbb{R}^{m_0\times m_0},
\quad \mbox{for} \quad k_\ell=0,...,L_\ell-1,$$ where $\odot$ denotes the Hadamard (pointwise) product of matrices. The latted representation can be written in the factorized tensor-product form $$\begin{aligned}
\label{eqn:bcirc_R1}
& D_{L_1}^{k_1} \otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3} \\
= &
((D_{L_1}^{k_1}\otimes A_{k_1}^{(1)}) \otimes I_{L_2}\otimes I_{L_3} )\odot
(I_{L_1} \otimes (D_{L_2}^{k_2} \otimes A_{k_2}^{(2)})\otimes I_{L_3} ) \odot
(I_{L_1} \otimes I_{L_2}\otimes (D_{L_3}^{k_3} \otimes A_{k_3}^{(3)})). \end{aligned}$$ Given $\ell \in \{1,...,d\}$ and a matrix $A\in \mathbb{R}^{L_\ell \times L_\ell}$, define the [*tensor prolongation*]{} mapping, ${\cal P}_\ell: \mathbb{R}^{L_\ell\times L_\ell}\to \mathbb{R}^{|{\bf L}|\times |{\bf L}|}$, by $$\label{eqn:Tensor_prolong}
{\cal P}_\ell(A):= \bigotimes_{i=1}^{\ell-1}I_{L_i}\otimes A \bigotimes_{i=\ell+1}^{d}I_{L_i}.$$ This leads to the powerful matrix factorization $$\begin{aligned}
A=& (F_{\bf n}^\ast \otimes I_m)
\left[\sum\limits^{L_1 -1}_{k_1=0} {\cal P}_1(D_{L_1}^{k_1}\otimes A_{k_1}^{(1)}) \odot
\sum\limits^{L_2 -1}_{k_2=0} {\cal P}_2(D_{L_2}^{k_2} \otimes A_{k_2}^{(2)}) \odot
\sum\limits^{L_3 -1}_{k_3=0} {\cal P}_3(D_{L_3}^{k_3} \otimes A_{k_3}^{(3)})\right]
(F_{\bf L}\otimes I_{m_0}),\\
=& (F_{\bf L}^\ast \otimes I_m)
\left[ {\cal P}_1(\sum\limits^{L_1-1 -1}_{k_1=0} D_{L_1}^{k_1}\otimes A_{k_1}^{(1)}) \odot
{\cal P}_2(\sum\limits^{L_2 -1}_{k_2=0} D_{L_2}^{k_2} \otimes A_{k_2}^{(2)}) \odot
{\cal P}_3(\sum\limits^{L_3 -1}_{k_3=0} D_{L_3}^{k_3} \otimes A_{k_3}^{(3)})\right]
(F_{\bf n}\otimes I_{m_0}),\\
= & (F_{\bf L}^\ast \otimes I_{m_0})
\left[ {\cal P}_1(\mbox{bdiag} F_{L_1} A^{(1)}) \odot
{\cal P}_2(\mbox{bdiag} F_{L_2} A^{(2)})\odot
{\cal P}_3(\mbox{bdiag} F_{L_3} \otimes A^{(3)})\right]
(F_{\bf L}\otimes I_{m_0}),\end{aligned}$$ where the tensor $A^{(\ell)}\in \mathbb{R}^{L_\ell\times m_0 \times m_0}$ is defined by concatenation $A^{(\ell)}=[A_{0}^{(\ell)},...,A_{L_\ell-1}^{(\ell)}]^T$, and the tensor prolongation ${\cal P}_\ell$ is defined by (\[eqn:Tensor\_prolong\]). This representation requires only 1D Fourier transforms thus reducing the numerical cost to $$O(m_0^2 {\sum}_{\ell=1}^d L_\ell \log L_\ell).$$ Moreover, and it is even more important, that the eigenvalue problem for the large matrix $A$ now reduces to only $L_1+L_2+L_3 \ll L_1 L_2 L_3$ small $m_0 \times m_0$ matrix eigenvalue problems.
The above block-diagonal representation for $d=3$ generalizes easily to the case of arbitrary dimension $d$. Finally, we prove the following general result.
\[thm:tens\_FFT\] Introduce the notation $D_{\bf L}^{\bf k}= D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes \cdots \otimes D_{L_d}^{k_d}$, then we have $$A= (F_{\bf L}^\ast \otimes I_{m_0})(\sum\limits^{\bf L -1}_{\bf k=0}
D_{\bf L}^{\bf k}\otimes A_{\bf k}) (F_{\bf L} \otimes I_{m_0}).$$ Assume the separability of a tensor $[A_{\bf k}]$ in the ${\bf k}$ space, we arrive at the factorized block-diagonal form of $A$ $$A= (F_{\bf L}^\ast \otimes I_{m_0})
\left[ {\cal P}_1(F_{L_1} A^{(1)}) \odot
{\cal P}_2(F_{L_2} A^{(2)})\odot \dots \odot
{\cal P}_d (F_{L_d} A^{(d)})\right]
(F_{\bf L}\otimes I_{m_0}).$$
The rank-$1$ decomposition was considered for the ease of exposition only. The above low-rank representations can be easily generalized to the case of canonical or Tucker formats in ${\bf k}$ space (see Proposition \[prop:low\_rank\_coef\] below).
Notice that in the practically interesting 3D case the use of MPS/TT type factorizations does not take the advantage over the Tucker format since the Tucker and MPS ranks in 3D appear to be close to each other. Indeed, the HOSVD for a tensor of order $3$ leads to the same rank estimates for both the Tucker and MPS/TT tensor formats.
Core Hamiltonian for lattice structured compounds {#sec:core_H}
=================================================
In this section we analyze the structure of the Galerkin matrix for the core Hamiltonian part in the Fock operator with respect to the localized GTO basis replicated over a lattice, $\{g_m (x) \}_{1\leq m \leq N_b}, x \in {\mathbb{R}^3}$ in a box, or in a supercell with the priodic boundary conditions.
The core Hamiltonian in a GTO basis set {#Core_Hamil}
---------------------------------------
The nonlinear Fock operator ${\cal F}$ in the governing Hartree-Fock eigenvalue problem, describing the ground state energy for $2N_b$-electron system, is defined by $$\left[-\frac{1}{2} \Delta - v_c(x) +
\int_{\mathbb{R}^3} \frac{\rho({ y})}{\|{ x}-{ y}\|}\, d{ y}\right] \varphi_i({ x})
- \int_{\mathbb{R}^3} \; \frac{\tau({ x}, { y})}{\|{x} - { y}\|}\, \varphi_i({ y}) d{ y}
= \lambda_i \, \varphi_i({ x}), \quad x\in \mathbb{R}^3,$$ where $i =1,...,N_{orb}$. The linear part in the Fock operator is presented by the core Hamiltonian $$\label{eqn:HFcore}
{\cal H}=-\frac{1}{2} \Delta - v_c,$$ while the nonlinear Hartree potential and exchange operators, depend on the unknown eigenfunctions (molecular orbitals) comprising the electron density, $\rho({ y})= 2 \tau(y,y)$, and the density matrix, $
\tau(x,y) =\sum\limits^{N_{orb}}_{i=1} \varphi_i (x)\varphi_i (y),\quad x,y\in \mathbb{R}^3,
$ respectively. The electrostatic potential in the core Hamiltonian is defined by a sum $$\label{eqn:ElectrostPot}
v_c(x)= \sum_{\nu=1}^{M}\frac{Z_\nu}{\|{x} -a_\nu \|},\quad
Z_\nu >0, \;\; x,a_\nu\in \mathbb{R}^3,$$ where $M$ is the total number of nuclei in the system, $a_\nu$, $Z_\nu$, represent their Cartesian coordinates and the respective charge numbers. Here $\|\cdot \|$ means the distance function in $\mathbb{R}^3$. Given a general set of localized GTO basis functions $\{g_\mu\}$ ($\mu=1,...,N_b$), the occupied molecular orbitals $\psi_i$ are approximated by $$\label{expand}
\psi_i=\sum\limits_{\mu=1}^{N_b} C_{\mu i} g_\mu, \quad i=1,...,N_{orb},$$ with the unknown coefficients matrix $C=\{C_{ \mu i} \}\in \mathbb{R}^{N_b \times N_{orb}}$ obtained as the solution of the discretized Hartree-Fock equation with respect to $\{g_\mu\}$, and described by $N_b\times N_b$ Fock matrix. Since the number of basis functions scales cubically in $L$, $N_b = m_0 L^3$, the calculation of the Fock matrix may become prohibitive as $L$ increases ($m_0$ is the number of basis functions in the unit cell).
In what follows we describe the grid-based tensor method for the block-structured representation of the core Hamiltonian in the Fock matrix in a box and in a supercell subject to the periodic boundary conditions. The stiffness matrix $H=\{h_{\mu \nu}\}$ of the core Hamiltonian (\[eqn:HFcore\]) is represented by the single-electron integrals, $$\label{eqn:Core_Ham}
h_{\mu \nu}= \frac{1}{2} \int_{\mathbb{R}^3}\nabla g_\mu \cdot \nabla g_\nu dx -
\int_{\mathbb{R}^3} v_c(x) g_\mu g_\nu dx, \quad 1\leq \mu, \nu \leq N_b,$$ such that the resulting $N_b\times N_b$ Galerkin system of equations governed by the reduced Fock matrix $ H$ reads as follows $$\begin{aligned}
\label{eqn:HF discr}
H C &= SC \Lambda, \quad \Lambda= diag(\lambda_1,...,\lambda_{N_{orb}}), \\
C^T SC &= I_N, \nonumber\end{aligned}$$ where the mass (overlap) matrix $S=\{s_{\mu \nu} \}_{1\leq \mu, \nu \leq N_b}$, is given by $
s_{\mu \nu}=\int_{\mathbb{R}^3} g_\mu g_\nu dx.
$
The numerically extensive part in (\[eqn:Core\_Ham\]) is related to the integration with the large sum of lattice translated Newton kernels. Indeed, let $M_0$ be the number of nuclei in the unit cell, then the expensive calculations are due to the summation over $M_0 L^3$ Newton kernels, and further spacial integration of this sum with the large set of localized atomic orbitals $\{g_\mu\}$, ($\mu=1,...,N_b$), where $N_b$ is of order $m_0 L^3$.
The present approach solves this problem by using the fast and accurate grid-based tensor method for evaluation of the electrostatic potential $v_c$ defined by the lattice sum in (\[eqn:ElectrostPot\]), see [@VeBoKh:Ewald:14], and subsequent efficient computation and structural representation of the stiffness matrix $V_c$, $$V_c=[V_{\mu \nu}]:\quad V_{\mu \nu}= \int_{\mathbb{R}^3} v_c(x) g_\mu g_\nu dx,
\quad 1\leq \mu, \nu \leq N_b,$$ by numerical integration by using the low-rank tensor representation on the grid of all functions involved.
This approach is applicable to the large $L\times L \times L$ lattice. In the next sections, we show that in the periodic case the resultant stiffness matrix $H=\{h_{\mu \nu}\}$ of the core Hamiltonian can be parametrized in the form of a symmetric, three-level block circulant matrix. In the case of lattice system in a box the block structure of $H$ is a small perturbation of the block Toeplitz matrix.
Low-rank tensor form of the nuclear potential in a box {#ssec:nuclear}
------------------------------------------------------
We consider the nuclear (core) potential operator describing the Coulomb interaction of the electrons with the nuclei, see (\[eqn:ElectrostPot\]). In the scaled unit cell $\Omega=[-b/2,b/2]^3$, we introduce the uniform $n \times n \times n$ rectangular Cartesian grid $\Omega_{n}$ with the mesh size $h=b/n$. Let $\{ \psi_\textbf{i}\}$ be the set of tensor-product piecewise constant basis functions, $ \psi_\textbf{i}(\textbf{x})=\prod_{\ell=1}^d \psi_{i_\ell}^{(\ell)}(x_\ell)$ for ${\bf i}=(i_1,i_2,i_3)\in I \times I \times I $, $i_\ell \in I=\{1,...,n\}$. The Newton kernel is discretized by the projection/collocation method in the form of a third order tensor of size $n\times n \times n$, defined point-wise as $$\begin{aligned}
\mathbf{P}:=[p_{\bf i}] \in \mathbb{R}^{n\times n \times n}, \quad
p_{\bf i} =
\int_{\mathbb{R}^3} \frac{\psi_{{\bf i}}({x})}{\|{x}\|} \,\, \mathrm{d}{x},
\label{galten}\end{aligned}$$ see [@KhKhFl_Hart:09; @BeHaKh:08; @VeKh_Diss:10; @VKH_solver:13]. Our low-rank canonical decomposition of the $3$rd order tensor $\mathbf{P}$ is based on using exponentially convergent $\operatorname*{sinc}$-quadratures for approximation of the Laplace-Gauss transform, see [@Stenger; @GHK:05; @HaKhtens:04I], $$\frac{1}{z}= \frac{2}{\sqrt{\pi}}\int_{\mathbb{R}_+} e^{- z^2 t^2 } dt,$$ which can be adapted to the Newton kernel by substitution $z=\sqrt{x_1^2 + x_2^2 + x_3^2}$. Rational type approximation by exponential sums have been addressed in [@Braess:95; @Braess:BookApTh]. We denote the resultant $R$-term canonical representation by $$\label{eqn:sinc_general}
\mathbf{P} \approx \mathbf{P}_R
= \sum\limits_{q=1}^{R} {\bf p}^{(1)}_q \otimes {\bf p}^{(2)}_q \otimes {\bf p}^{(3)}_q
\in \mathbb{R}^{n\times n \times n}.
$$ In a similar way, we also introduce the “master tensor”, $\widetilde{\bf P}_R \in \mathbb{R}^{\widetilde{n}\times \widetilde{n} \times \widetilde{n}}$, approximating the Newton kernel in the extended (accompanying) domain $\widetilde{\Omega} \supset \Omega$, and associated with the grid parameter $\widetilde{n}=n_0+n$ (say, $n_0=n$), $$\label{eqn:master_pot}
\widetilde{\bf P}_R=
\sum\limits_{q=1}^{R} \widetilde{\bf p}^{(1)}_q \otimes
\widetilde{\bf p}^{(2)}_q \otimes \widetilde{\bf p}^{(3)}_q
\in \mathbb{R}^{\widetilde{n}\times \widetilde{n} \times \widetilde{n}}.$$
The core potential for the molecule is approximated by the canonical tensor $${\bf P}_{c} = \sum_{\nu=1}^{M_0} Z_\nu {\bf P}_{{c},\nu}\approx \widehat{\bf P}_{c}
\in \mathbb{R}^{n\times n \times n},$$ with the rank bound $rank({\bf P}_{c})\leq M_0 R$, where the rank-$R$ tensor ${\bf P}_{{c},\nu}$ represents the single Coulomb potential shifted according to coordinates of the corresponding nuclei, [@VeBoKh:Ewald:14], $$\label{eqn:core_tens}
{\bf P}_{c,\nu} = {\cal W}_{\nu} \widetilde{\bf P}_R =
\sum\limits_{q=1}^{R} {\cal W}_{\nu}^{(1)} \widetilde{\bf p}^{(1)}_q \otimes
{\cal W}_{\nu}^{(2)} \widetilde{\bf p}^{(2)}_q
\otimes {\cal W}_{\nu}^{(3)} \widetilde{\bf p}^{(3)}_q\in \mathbb{R}^{n\times n \times n},
$$ such that every rank-$R$ canonical tensor ${\cal W}_{\nu} \widetilde{\bf P}_R \in \mathbb{R}^{n\times n \times n}$ is thought as a sub-tensor of the master tensor obtained by a shift and restriction (windowing) of $\widetilde{\bf P}_R$ onto the $n \times n \times n$ grid $\Omega_{n}$ in the unit cell $\Omega$, $\Omega_{n} \subset \Omega_{\widetilde{n}}$. A shift from the origin is specified according to the coordinates of the corresponding nuclei, $a_\nu$, counted in the $h$-units. Here $\widehat{\bf P}_{c}$ is the rank-$R_c$ ($R_c\leq M_0 R$, actually $R_c \approx R$) canonical tensor obtained from ${\bf P}_{c}$ by the rank optimization procedure (see [@VeBoKh:Ewald:14], Remark 2.2).
For the tensor representation of the Newton potentials, ${\bf P}_{{c},\nu}$, we make use of the piecewise constant discretization on the equidistant tensor grid, where, in general, the univariate grid size $n$ can be noticeably smaller than that used for the piecewise linear discretization applied to the Laplace operator. Indeed, since we use the global basis functions for the Galerkin approximation to the eigenvalue problem, the grid-based representation of these basis functions can be different in the calculation of the kinetic and potential parts in the Fock operator. The corresponding choice is the only controlled by the respective approximation error and by the numerical efficiency depending on the separation rank parameters. The error $\varepsilon >0$ arising due to the separable approximation of the nuclear potential is controlled by the rank parameter $R_{P}= rank({\bf P}_{c})$. Now letting $rank({\bf G}_m) = R_m$ implies that each matrix element is to be computed with linear complexity in $n$, $O(R_kR_m R_{P} \, n)$. The almost exponential convergence of the rank approximation in $R_{P}$ allows us the choice $R_{P}=O(|\log \varepsilon |)$.
Let us discuss the lattice structured systems. Low-rank tensor decomposition of the Coulomb interaction defined by the large lattice sum is proposed in [@VeBoKh:Ewald:14]. Given the potential sum $v_c$ in the scaled unit cell $\Omega=[-b/2,b/2]^3$, of size $b\times b \times b$, we consider an interaction potential in a symmetric box (supercell) $$\Omega_L =B_1\times B_2 \times B_3,
$$ consisting of a union of $L_1 \times L_2 \times L_3$ unit cells $\Omega_{\bf k}$, obtained from $\Omega$ by a shift proportional to $ b$ in each variable, and specified by the lattice vector $b {\bf k}$, where ${\bf k}=(k_1,k_2,k_3)\in \mathbb{Z}^3$, $-(L_\ell-1)/2 \leq k_\ell\leq (L_\ell-1)/2 $, ($\ell=1,2,3$), such that, without loss of generality, we assume $L_\ell= 2 p_\ell +1, p_\ell\in \mathbb{N}$. Hence, we have $$B_\ell = \frac{b}{2}[- L_\ell ,L_\ell ], \quad \mbox{for} \quad
L_\ell \in \mathbb{N},$$ where $L_\ell=1$ corresponds to one-layer systems in the respective variable. Recall that $b=n h$, where $h$ is the spacial grid size that is the same for all spacial variables. To simplify the discussion, we often consider the case $L_\ell = L$. We also introduce the accompanying domain $\widetilde{\Omega}_L$. In the case of extended system in a box, further called case (B), the summation problem for the total potential $v_{c_L}$ is formulated in the box $\Omega_L= \bigcup_{k_1,k_2,k_3=-(L-1)/2}^{(L-1)/2} \Omega_{\bf k}$ as well as in the accompanying domain $\widetilde{\Omega}_L$. On each $\Omega_{\bf k}\subset \Omega_L$, the potential sum of interest, $v_{\bf k}(x)=(v_{c_L})_{|\Omega_{\bf k}}$, is obtained by summation over all unit cells in $\Omega_L$, $$\label{eqn:EwaldSumE}
v_{\bf k}(x)= \sum_{\nu=1}^{M_0} \sum\limits_{k_1,k_2,k_3=-(L-1)/2}^{(L-1)/2}
\frac{Z_\nu}{\|{x} -a_\nu (k_1,k_2,k_3)\|}, \quad x\in \Omega_{\bf k},
$$ where $a_\nu (k_1,k_2,k_3)=a_\nu + b {\bf k}$. This calculation is performed at each of $L^3$ elementary cells $\Omega_{\bf k}\subset \Omega_L$, which is implemented by the tensor summation method described in [@VeBoKh:Ewald:14]. The resultant lattice sum is presented by the canonical tensor ${\bf P}_{c_L}$ with the rank $R_0 \leq M_0 R$, $$\label{eqn:EwaldTensorGl}
{\bf P}_{c_L}= \sum\limits_{\nu=1}^{M_0} Z_\nu \sum\limits_{q=1}^{R}
(\sum\limits_{k_1=0}^{L-1}{\cal W}_{\nu({k_1})} \widetilde{\bf p}^{(1)}_{q}) \otimes
(\sum\limits_{k_2=0}^{L-1} {\cal W}_{\nu({k_2})} \widetilde{\bf p}^{(2)}_{q}) \otimes
(\sum\limits_{k_3=0}^{L-1}{\cal W}_{\nu({k_3})} \widetilde{\bf p}^{(3)}_{q}).$$ The numerical cost and storage size are bounded by $O(M_0 R L N_L )$, and $O(M_0 R N_L)$, respectively (see [@VeBoKh:Ewald:14], Theorem 3.1), where $N_L= nL$. The lattice sum is also computed in the accompanying domain $\widetilde{\Omega}_L$, $\widetilde{\bf P}_{c_L}$, where the grid size is equal to $N_L +2 n_0$.
The lattice sum in (\[eqn:EwaldTensorGl\]) converges only conditionally as $L\to \infty$. This aspect will be addressed in Section (\[ssec:Complexity\_EigPr\]) following the approach introduced in [@VeBoKh:Ewald:14].
Nuclear potential operator in a box {#ssec:Core_Ham_gener}
-----------------------------------
First, consider the case of a single molecule in the unit cell. Given the GTO-type basis set $\{{g}_k\}$, $k=1,...,m_0$, i.e. $N_b=m_0$, associated with the scaled unit cell and extended to the local bounding box $\widetilde{\Omega}$. The corresponding rank-$1$ coefficients tensors ${\bf G}_k={\bf g}_k^{(1)}\otimes{\bf g}_k^{(2)} \otimes{\bf g}_k^{(3)}$ representing their piecewise constant approximations $\{\overline{g}_k\}$ on the fine $\widetilde{n}\times \widetilde{n}\times \widetilde{n}$ grid. Then the entries of the respective Galerkin matrix for the core potential operator $v_c$ in (\[eqn:ElectrostPot\]), ${V}_c=\{{V}_{km}\}$, are represented (approximately) by the following tensor operations, $$\label{eqn:nuc_pot}
{V}_{km} \approx \int_{\widetilde{\Omega}_L} V_c(x) \overline{g}_k(x) \overline{g}_m(x) dx \approx
\langle {\bf G}_k \odot {\bf G}_m , {\bf P}_{c}\rangle =: {v}_{km} ,
\quad 1\leq k, m \leq m_0.$$ In the case of lattice syastem in a box we define the basis set on a supercell $\Omega_{L}$ (and on $\widetilde{\Omega}_L$) by translation of the generating basis by the lattice vector $\delta {\bf k}$, i.e., $\{g_{\mu}({x})\} \mapsto \{g_{\mu}({x+\delta {\bf k} })\}$, where ${\bf k}=(k_1,k_2,k_3)$, $0 \leq k_\ell\leq L_\ell -1$, ($\ell=1,2,3$), assuming zero extension of $\{g_{\mu}({x+\delta {\bf k} })\}$ beyond each local bounding box $\widetilde{\Omega}_{\bf k}$. In this construction the total number of basis functions is equal to $N_b=m_0 L_1 L_2 L_3$. In practically interesting case of localized atomic orbital basis functions, the matrix $V_{c_L}$ exhibits the special block sparsity pattern since the effective support of localized atomic orbitals associated with every unit cell $\Omega_{\bf k} \subset \widetilde{\Omega}_{\bf k}$ overlaps only fixed (small) number of neighboring cells.
In the following, the matrix block entries will be numbered by a pair of multi-indicies, $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$, ${\bf k}=(k_1,k_2,k_3)$, where the $m_0\times m_0$ matrix block $V_{{\bf k}{\bf m}}$ is defined by $$\label{eqn:nuc_MatrSparsP}
V_{{\bf k}{\bf m}} = \langle {\bf G}_{\bf k} \odot {\bf G}_{\bf m} , {\bf P}_{c_L}\rangle,
\quad - L/2 \leq k_\ell, m_\ell \leq L/2,\quad \ell=1,2,3,$$ where the canonical tensors ${\bf G}_{\bf k}$ inherit the same block numbering.
We denote by $L_0$ the number of cells measuring the overlap in basis functions in each spacial direction (overlap constant).
\[lem:SparseCaseE\] Assume that the number of overlapping cells in each spacial direction does not exceed $L_0$, then in case (B): (a) the number of non-zero blocks in each block row (column) of the symmetric Galerkin matrix $V_{c_L}$ does not exceed $(2 L_0 + 1)^3$, (b) the required storage is bounded by $m_0^2 [(L_0 + 1)L]^3$.
In case (B), the matrix representation $V_{c_L}=\{v_{km}\}\in \mathbb{R}^{N_b\times N_b}$ of the tensor as in (\[eqn:nuc\_pot\]) is obtained elementwise by the following tensor operations $$\label{nuc_potMatrTot}
\overline{v}_{km}= \int_{\mathbb{R}^3} v_c(x) \overline{g}_k(x) \overline{g}_m(x) dx
\approx \langle {\bf G}_k \odot {\bf G}_m , {\bf P}_{c_L}\rangle =: v_{km},
\quad 1\leq k, m \leq N_b,$$ where $\{\overline{g}_k\}$ denotes the piecewise constant representations to the respective Galerkin basis functions. This leads to the final expression $$\begin{split}
{v}_{km} & = \sum\limits_{\nu=1}^{M_0} Z_\nu \sum\limits_{q=1}^{R_{\cal N}}
\langle {\bf G}_k \odot {\bf G}_m ,
(\sum\limits_{k_1=0}^{L_1-1} {\cal W}_{\nu({k_1})} \widetilde{\bf p}^{(1)}_{q}) \otimes
(\sum\limits_{k_2=0}^{L_2-1} {\cal W}_{\nu({k_2})} \widetilde{\bf p}^{(2)}_{q}) \otimes
(\sum\limits_{k_3=0}^{L_3-1} {\cal W}_{\nu({k_3})} \widetilde{\bf p}^{(3)}_{q}) \rangle \\
&=
\sum\limits_{\nu=1}^{M_0} Z_\nu \sum\limits_{q=1}^{R_{\cal N}}
\prod\limits_{\ell=1}^3
\langle {\bf g}_k^{(\ell)} \odot { \bf g}_m^{(\ell)},
\sum\limits_{k_\ell=1}^{L_\ell} {\cal W}_{\nu({k_\ell})} \widetilde{\bf p}^{(\ell)}_{q} \rangle.
\end{split}$$ Taking into account the block representation (\[eqn:nuc\_MatrSparsP\]) and the overlapping property $$\label{eqn:Overlap_Basis}
{\bf G}_{\bf k} \odot {\bf G}_{\bf m}=0 \quad \mbox{if} \quad | k_\ell - m_\ell| \geq L_0,$$ we analyze the block sparsity pattern in the Galerkin matrix $V_{c_L}$. Given $3 M_0 R_{\cal N} $ vectors $\sum\limits_{k_\ell=1}^{L_\ell} {\cal W}_{\nu({k_\ell})} \widetilde{\bf p}^{(\ell)}_{q}\in \mathbb{R}^{N_L}$, where $N_L$ denotes the total number of grid points in $\Omega_L$ in each space variable. Now the numerical cost to compute ${v}_{km}$ for every fixed index $(k,m)$ is estimated by $O(M_0 R_{\cal N} N_L)$ indicating linear scaling in the large grid parameter $N_L$ (but not cubic).
Fixed the row index in $(k,m)$, then (b) follows from the bound on the total number of cells $\Omega_{\bf k}$ in the effective integration domain in (\[nuc\_potMatrTot\]), that is $(2 L_0 + 1)^3$, and the symmetry of $V_{c_L}$.
Figure \[fig:3DCorePerScellErr\] illustrates the sparsity pattern of the nuclear potentail operator $V_{c_L}$ in the matrix $H$, for $(L, 1, 1)$ lattice in a supercell with $L=32$ and $m_0=4$, corresponding to the overlapping parameter $L_0=3$. One can observe the nearly-boundary effects due to the non-equalized contributions from the left and from the right (supercell in a box).
Figure \[fig:3DCorePerScellErr\] shows the difference between matrices $V_{c_L}$ in periodic (see §\[ssec:Core\_Ham\_period\] for more details) and non-periodic cases. The relative norm of the difference is vanishing if $L\to \infty$.
![ Matrix $V_{c_L}$ in a supercell for $L_0=3, L=32$ (left). Difference between matrices $V_{c_L}$ in periodic and single-box cases (middle). Block-sparsity in the matrix $V_{c_L}$ in periodic case (right).[]{data-label="fig:3DCorePerScellErr"}](NM_Lx64_1_1.eps "fig:"){width="5.0cm"}![ Matrix $V_{c_L}$ in a supercell for $L_0=3, L=32$ (left). Difference between matrices $V_{c_L}$ in periodic and single-box cases (middle). Block-sparsity in the matrix $V_{c_L}$ in periodic case (right).[]{data-label="fig:3DCorePerScellErr"}](NM_err_Lx64_1_1.eps "fig:"){width="5.0cm"} ![ Matrix $V_{c_L}$ in a supercell for $L_0=3, L=32$ (left). Difference between matrices $V_{c_L}$ in periodic and single-box cases (middle). Block-sparsity in the matrix $V_{c_L}$ in periodic case (right).[]{data-label="fig:3DCorePerScellErr"}](VH_P_64_1_1_matr.eps "fig:"){width="5.0cm"}
Notice that the quantized approximation of canonical vectors involved in ${\bf G}_k$ and ${\bf P}_{c_L}$ reduces this cost to the logarithmic scale, $O(M_0 R_{\cal N} \log N_L)$, that is important in the case of large $L$ in view of $N_L=O(L)$.
The block $L_0$-diagonal structure of the matrix $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$, ${\bf k}\in \mathbb{Z}^3$ ($- L/2 \leq {k}_\ell L/2$) described by Lemma \[lem:SparseCaseE\] allows the essential saving in the storage costs.
However, the polynomial complexity scaling in $L$ leads to severe limitations on the number of unit cells. These limitations can be relax if we look more precisely on the defect between matrix ${V}_{c_L}$ and its block-circulant version corresponding to the periodic boundary conditions (see §\[ssec:Core\_Ham\_period\]). This defect can be split into two components with respect to their local and non-local features:
1. Non-local effect due to the asymmetry in the interaction potential sum on the lattice in a box.
2. The near boundary (local) defect that effects only those blocks in $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$ lying in the $L_0$-width of $\partial\Omega_L$, $$L_0+1 -(L-1)/2\leq k_\ell, m_\ell \leq (L-1)/2-1 -L_0.$$
Item (A) is related to a slight modification of the core potential to the shift invariant Toeplitz-type form $V_{{\bf k}{\bf m}} = V_{|{\bf k}-{\bf m}|}$ by replication of the central block corresponding to $k=0$, as considered in the next section. In this way the overlap condition (\[eqn:Overlap\_Basis\]) for the tensor ${\bf G}_{\bf k}$ will impose the block sparsity.
The boundary effect in item (B) becomes relatively small for large number of cells so that the block-circulant part of the matrix $V_{c_L}$ is getting dominating as $L\to \infty$.
The full diagonalization for above mentioned matrices is impossible. However the efficient storage and fast matrix-times-vector algorithms can be applied in the framework of iterative methods for calculation of a small subset of eigenvalues.
Discrete Laplacian and the mass matrix {#ssec:Lap_Op}
--------------------------------------
The Laplace operator part included in eigenvalue problem for a single molecule is posed in the unit cell $ \Omega=[-b/2,b/2]^3 \in \mathbb{R}^3 $, subject to the homogeneous Dirichlet boundary conditions on $\partial \Omega$. In periodic case they should be substituted by the periodic boundary conditions. For given discretization parameter $\overline{n} \in \mathbb{N}$, we use the equidistant $\overline{n}\times \overline{n} \times \overline{n}$ tensor grid $\omega_{{\bf 3},\overline{n}}=\{x_{\bf i}\} $, ${\bf i} \in {\cal I} :=\{1,...,\overline{n}\}^3 $, with the mesh-size $h=2b/(\overline{n} + 1)$, which might be different from the grid $\omega_{{\bf 3},n} $ introduced for representation of the interaction potential (usually, $n\leq \overline{n}$).
Define a set of piecewise linear basis functions $\overline{g}_k := {\bf I}_1 g_k $, $k=1,...,N_b$, by linear tensor-product interpolation via the set of product hat functions, $\{\xi_{\bf i}\}= \xi_{i_1} (x_1) \xi_{i_2} (x_2)\xi_{i_3} (x_3)$, ${\bf i} \in {\cal I}$, associated with the respective grid-cells in $\omega_{{\bf 3},N}$. Here the linear interpolant ${\bf I}_1= {I}_1\times {I}_1 \times {I}_1$ is a product of 1D interpolation operators, $\overline{g}_k^{(\ell)}= {I}_1 {g}_k^{(\ell)}$, $\ell=1,...,3$, where ${I}_1:C^0([-b,b])\to W_h:=span\{\xi_i\}_{i=1}^{\overline{n}}$ is defined over the set of piecewise linear basis functions by $$({I}_1 \, w)(x_\ell):=\sum_{i=1}^{\tilde{n}} w(x_{i_\ell})\xi_{i}(x_\ell),
\quad x_{\bf i} \in \omega_{{\bf 3},\tilde{n}}.$$
With these definitions, the rank-$3$ tensor representation of the standard FEM Galerkin stiffness matrix for the Laplacian, $A_3$, in the tensor basis $\{\xi_i(x_1) \xi_j (x_2)\xi_k (x_3) \} $, $i,j,k = 1,\ldots \overline{n}$, is given by $$A_3 := A^{(1)} \otimes S^{(2)} \otimes S^{(3)} + S^{(1)} \otimes A^{(2)} \otimes S^{(3)}
+ S^{(1)} \otimes S^{(2)} \otimes A^{(3)}\in
\mathbb{R}^{\overline{n}^{\otimes 3}\times \overline{n}^{\otimes 3}},$$ where the 1D stiffness and mass matrices $A^{(\ell)}, S^{(\ell)} \in \mathbb{R}^{\overline{n}\times \overline{n}}$, $\ell=1,\,2,\,3$, are represented by $$A^{(\ell)} := \{ \langle \frac{d}{d x_\ell} \xi_i(x_\ell) , \frac{d}{d x_\ell} \xi_j(x_\ell)
\rangle \}^{\overline{n}}_{i,j=1} = \frac{1}{h} \mbox{tridiag} \{-1,2,-1\},$$ $$S^{(\ell)}=\{ \langle \xi_i ,\xi_j\rangle \}^{\overline{n}}_{i,j=1} = \frac{h}{6}\;
\mbox{tridiag} \{1,4,1\},$$ respectively.
This leads to the separable grid-based approximation of the initial basis functions $g_k(x)$, $$\label{eq. Gaus pwl}
g_k (x) \approx \overline{g}_k (x)=\prod^3_{\ell=1}
\overline{g}_k^{(\ell)} (x_{\ell})=\prod^3_{\ell=1}
\sum\limits^{\overline{n}}_{i=1} g_{k}^{(\ell)}(x_{i_\ell}) \xi_i (x_{\ell}),$$ where the rank-$1$ coefficients tensor ${\bf G}_k$ is given by ${\bf G}_k= {\bf g}_k^{(1)} \otimes {\bf g}_k^{(2)} \otimes {\bf g}_k^{(3)}$, with the canonical vectors ${\bf g}_k^{(\ell)}=\{g_{k_i}^{(\ell)}\}\equiv \{g_{k}^{(\ell)}(x_{i_\ell})\}$. Let us agglomerate the rank-$1$ tensors ${\bf G}_k\in \mathbb{R}^{{\overline{n}}^{\otimes 3}}$, ($k=1,...,N_b$) in a tensor-valued matrix $G\in \mathbb{R}^{N^{\otimes 3}\times N_b}$, the Galerkin matrix in the basis set ${\bf G}_k $ can be written in a matrix form $$A_G= G^T A_3 G\in \mathbb{R}^{N_b \times N_b},$$ corresponding to the standard matrix-matrix transform under the change of basis. The matrix entries in $A_G=\{a_{k m}\}$ can be represented by $$a_{k m}= \langle A_3 {\bf G}_k, {\bf G}_m\rangle, \quad k,m = 1,...,N_b.$$ Likewise, for the entries of the stiffness matrix we have $s_{k m}= \langle {\bf G}_k, {\bf G}_m\rangle$.
It is easily seen that in the periodic case both matrices, $A_G$ and $S$, take the multilevel block circulant structure.
Linearized spectral problem by FFT-diagonalization {#sec:Core_Ham_period_FFT}
==================================================
There are two possibilities for mathematical modeling of the $L$-periodic molecular systems, composed, of $(L, L, L)$ elementary unit cells. In the first approach, the system is supposed to contain an infinite set of equivalent atoms that map identically into itself under any translation by $L$ units in each spacial direction. The other model is based on the ring-type periodic structures consisting of $L$ identical units in each spacial direction, where every unit cell of the periodic compound will be mapped to itself by applying a rotational transform from the corresponding rotational group symmetry [@SzOst:1996].
The main difference between these two concepts is in the treatment of the lattice sum of Coulomb interactions, thought, in the limit of $L\to \infty$ both models approach each other. In this paper we mainly follow the first approach with the particular focus on the asymptotic complexity optimization for large lattice parameter $L$. The second concept is useful for understanding the block structure of the Galerkin matrices for Laplacian and the identity operators.
Block circulant structure of core Hamiltonian in periodic case {#ssec:Core_Ham_period}
--------------------------------------------------------------
Inthis section we consider the periodic case, further called case (P), and derive the more refined sparsity pattern of the matrix $V_{c_L}$ using the $d$-level ($d=1,2,3$) tensor structure in this matrix. The matrix block entries are numbered by a pair of multi-indices, $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$, ${\bf k}=(k_1,k_2,k_3)$, where the $m_0\times m_0$ matrix block $V_{{\bf k}{\bf m}}$ is defined by (\[eqn:nuc\_MatrSparsP\]). Figure \[fig:3DPeriodStruct\] illustrates an example of 3D lattice-type structure of size $(4, 4, 2)$.
![Example of the 3D lattice-type structure of size $(4, 4, 2)$. []{data-label="fig:3DPeriodStruct"}](periodic42.eps){width="7.0cm"}
Following [@VeBoKh:Ewald:14] we introduce the periodic cell ${\cal R}= \mathbb{Z}^d$, $d=1,2,3$ for the $\bf k$ index, and consider a 3D $B$-periodic supercell $\Omega_L= B\times B\times B$, with $B= \frac{b}{2}[-L,L]$. The total electrostatic potential in the supercell $\Omega_L$ is obtained by, first, the lattice summation of the Coulomb potentials over $\Omega_L$ for (rather large) $L$, but restricted to the central unit cell $\Omega_0$, and then by replication of the resultant function to the whole supercell. Hence, that the total potential sum $v_{c_L}(x)$ is designated at each elementary unit-cell in $\Omega_L$ by the same value (${\bf k}$-translation invariant). The effect of the conditional convergence of the lattice summation can be treated by using the extrapolation to the limit (regularization) on a sequence of different lattice parameters $L$ as described in [@VeBoKh:Ewald:14].
The electrostatic potential in any of $B$-periods can be obtained by copying the respective data from $\Omega_L$. The basis set in $\Omega_L$ is constructed by replication from the the master unit cell $\Omega_0$ over the whole periodic lattice.
Consider the case $d=3$ in more detail. Recall that the reference value $v_{c_L}(x)$ will be computed at the central cell $\Omega_0$, indexed by $(0,0,0)$, by summation over all contributions from $L^3$ elementary sub-cells in $\Omega_L$. For technical reasons here and in the following we vary the summation index in $k_\ell=0,..., L-1$ and obtain $$\label{eqn:EwaldSumP}
v_0(x)= \sum_{\nu=1}^{M_0} \sum\limits_{k_1,k_2,k_3=0}^{L-1}
\frac{Z_\nu}{\|{x} -a_\nu (k_1,k_2,k_3)\|},
\quad x\in \Omega_0.$$ The local lattice sum on the index set $n\times n \times n$ corresponding to $\Omega_0$, is represented by $${\bf P}_{\Omega_0} = \sum_{\nu=1}^{M_0} Z_\nu \sum\limits_{k_1,k_2,k_3=0}^{L-1} \sum\limits_{q=1}^{R_{\cal N}}
{\cal W}_{\nu({\bf k})} \widetilde{\bf p}^{(1)}_{q}
\otimes \widetilde{\bf p}^{(2)}_{q} \otimes \widetilde{\bf p}^{(3)}_{q}
\in \mathbb{R}^{n\times n \times n},$$ for the corresponding local projected tensor of small size $n\times n \times n$. Here the $\Omega$-windowing operator, $
{\cal W}_{\nu({\bf k})}={\cal W}_{\nu(k_1)}^{(1)}\otimes {\cal W}_{\nu(k_2)}^{(2)}
\otimes {\cal W}_{\nu(k_3)}^{(3)},
$ restricts onto the small $n\times n \times n$ unit cell by shifting by the lattice vector ${\bf k}=(k_1,k_2,k_3)$. This reduces both the computational and storage costs by factor $L$.
In the 3D case, we set $q=3$ in the notation for multilevel BC matrix. Similar to the case of one-level BC matrices, we notice that a matrix $A\in {\cal BC} (3,{\bf L},m)$ of size $|{\bf L}| m \times |{\bf L}| m$ is completely defined by a $3$-rd order coefficients tensor ${\bf A}=[A_{k_1 k_2 k_3}]$ of size $L_1 \times L_2 \times L_3 $, ($k_\ell=0,...,L_\ell-1$, $\ell=1,2,3$), with $m\times m$ block-matrix entries, obtained by folding of the generating first column vector in $A$.
\[lem:SparseCaseP\] Assume that in case (P) the number of overlapping unit cells (in the sense of supports of basis functions) in each spatial direction does not exceed $L_0$. Then the Galerkin matrix $V_{c_L}$ exhibits the symmetric, three-level block circulant Kronecker tensor-product form. i.e. $V_{c_L} \in {\cal BC} (3,{\bf L},m_0)$, (${\bf L}=(L_1,L_2,L_3)$) $$\label{eqn:BC-Core}
V_{c_L}= \sum\limits_{k_1=0}^{L_1-1} \sum\limits_{k_2=0}^{L_2-1} \sum\limits_{k_3=0}^{L_3-1}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3},
\quad A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0},$$ where the number non-zero matrix blocks $A_{k_1 k_2 k_3}$ does not exceed $(L_0+1)^3$.
The required storage is bounded by $m_0^2 [(L_0 + 1)]^3$ independent of $L$. The set of non-zero generating matrix blocks $\{ A_{k_1 k_2 k_3}\}$ can be calculated in $O(m_0^2 [(L_0 + 1)]^3 n)$ operations.
Furthermore, assume that the QTT ranks of the assembled canonical vectors do not exceed $r_0$. Then the numerical cost can be reduced to the logarithmic scale, $O( m_0^2 [(L_0 + 1)]^3\log n)$.
First, we notice that the shift invariance property in the matrix $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$ is a consequence of the translation invariance in the canonical tensor ${\bf P}_{c_L}$ (periodic case), and in the basis-tensor ${\bf G}_{\bf k}$ (by construction), $$\label{eqn:Basis_shift}
{\bf G}_{\bf k m}:= {\bf G}_{\bf k} \odot {\bf G}_{\bf m}= {\bf G}_{|{\bf k} -{\bf m}|} \quad
\mbox{for} \quad | k_\ell|, |m_\ell| \leq L-1,$$ so that we have $$\label{nuc_BCirculantP}
V_{{\bf k}{\bf m}} = V_{|{\bf k}-{\bf m}|}, \quad 0\leq k_\ell, m_\ell \leq L-1.
$$ This ensures the perfect three-level block-Toeplitz structure of $V_{c_L}$ (compare with the case of a box). Now the block circulant pattern in ${\cal BC} (3,{\bf L},m_0)$ is imposed by the periodicity of a lattice-structured basis set.
To prove the complexity bounds we observe that a matrix $V_{c_L} \in {\cal BC} (3,{\bf L},m_0)$ can be represented in the Kronecker tensor product form (\[eqn:BC-Core\]), obtained by an easy generalization of (\[eqn:bcircPol\]). In fact, we apply (\[eqn:bcircPol\]) by successive slice-wise and fiber-wise splitting to obtain $$\begin{split}
V_{c_L}
& = \sum\limits_{k_1=0}^{L_1-1}\pi_{L_1}^{k_1}\otimes {\bf A}_{k_1} \\
&= \sum\limits_{k_1=0}^{L_1-1}\pi_{L_1}^{k_1}\otimes
\left( \sum\limits_{n_2=0}^{L_2-1} \pi_{L_2}^{k_2}\otimes {\bf A}_{k_1 k_2} \right)\\
&=\sum\limits_{k_1=0}^{L_1-1}\pi_{L_1}^{k_1}\otimes
\left( \sum\limits_{k_2=0}^{L_2-1} \pi_{L_2}^{k_2}\otimes
\left(\sum\limits_{k_3=0}^{L_3-1} \pi_{L_3}^{k_3}\otimes {A}_{k_1 k_2 k_3} \right) \right),
\end{split}$$ where ${\bf A}_{k_1}\in \mathbb{R}^{L_2\times L_3 \times m_0\times m_0}$, ${\bf A}_{k_1 k_2} \in \mathbb{R}^{L_3 \times m_0\times m_0}$, and $A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}$. Now the overlapping assumption ensures that the number of non-zero matrix blocks $A_{k_1 k_2 k_3}$ does exceed $(L_0+1)^3$.
Furthermore, the symmetric mass matrix, $S_{c_L}=\{{s}_{\mu \nu} \}\in \mathbb{R}^{N_b\times N_b}$, for the Galerkin representation of the identity operator reads as follows, $$\label{Ident_pot}
{s}_{\mu \nu}= \langle {\bf G}_\mu , {\bf G}_\nu \rangle
=\langle S^{(1)}{\bf g}_\mu^{(1)},{\bf g}_\nu^{(1)} \rangle
\langle S^{(2)} {\bf g}_\mu^{(2)},{\bf g}_\nu^{(2)} \rangle
\langle S^{(3)} {\bf g}_\mu^{(3)},{\bf g}_\nu^{(3)} \rangle,
\quad 1\leq \mu, \nu \leq N_b,$$ where $N_b=m_0 L^3$. It can be seen that in the periodic case the block structure in the basis-tensor ${\bf G}_{\bf k} $ imposes the three-level block circulant structure in the mass matrix $S_{c_L}$ $$\label{eqn:BC-mass}
S_{c_L}= \sum\limits_{k_1=0}^{L_1-1} \sum\limits_{k_2=0}^{L_2-1} \sum\limits_{k_3=0}^{L_3-1}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes S_{k_1 k_2 k_3},
\quad S_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}.$$ By the previous arguments we conclude that $S_{k_1 k_2 k_3}=S^{(1)}_{k_1} S^{(2)}_{k_2} S^{(3)}_{k_3}$ implying the rank-$1$ separable representation in (\[eqn:BC-mass\]).
Likewise, it is easy to see that the stiffness matrix representing the (local) Laplace operator in the periodic setting has the similar block circulant structure, $$\label{eqn:BC-Laplace}
\Delta_{c_L}= \sum\limits_{k_1=0}^{L_1-1} \sum\limits_{k_2=0}^{L_2-1} \sum\limits_{k_3=0}^{L_3-1}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes B_{k_1 k_2 k_3},
\quad B_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0},$$ where the number non-zero matrix blocks $B_{k_1 k_2 k_3}$ does not exceed $(L_0+1)^3$. In this case the matrix block $B_{k_1 k_2 k_3}$ admits a rank-$3$ product factorization.
This proves the sparsity pattern of our tensor approximation to $H$.
In the Hartree-Fock calculations for lattice structured systems we deal with the multilevel, symmetric block circulant/Toeplitz matrices, where the first-level blocks, $A_0,...,A_{L_1-1}$, may have further block structures. In particular, Lemma \[lem:SparseCaseP\] shows that the Galerkin approximation of the 3D Hartree-Fock core Hamiltonian in periodic setting leads to the symmetric, three-level block circulant matrix.
Figure \[fig:3DCoreHamPer\] represents the block-sparsity in the core Hamiltonian matrix in a box for $L=8$ (left), and the rotated matrix profile (right).
![Block-sparsity in the core Hamiltonian matrix in a box for $L=8$ (left); Rotated matrix profile (right).[]{data-label="fig:3DCoreHamPer"}](Core_4_128_CHd.eps "fig:"){width="6.0cm"}![Block-sparsity in the core Hamiltonian matrix in a box for $L=8$ (left); Rotated matrix profile (right).[]{data-label="fig:3DCoreHamPer"}](Core_4_128_CH_rotd.eps "fig:"){width="6.0cm"}
In the next section we discuss computational details of the FFT-based eigenvalue solver on the example of 3D linear chain of molecules.
Regularized spectral problem and complexity analysis {#ssec:Complexity_EigPr}
----------------------------------------------------
Combining the block circulant representations (\[eqn:BC-Core\]), (\[eqn:BC-Laplace\]) and (\[eqn:BC-mass\]), we are able to represent the eigenvalue problem for the Fock matrix in the Fourier space as follows $$\label{eqn:HF-FSpace}
\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes
(B_{k_1 k_2 k_3} + A_{k_1 k_2 k_3}) U
= \lambda
\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3} S_{k_1 k_2 k_3} U,$$ with the diagonal matrices $D_{L_\ell}^{k_\ell}\in \mathbb{R}^{L_\ell \times L_\ell}$, $\ell=1,2,3$, where $U=F_{\bf L}\otimes I_m C$. The equivalent block-diagonal form reads $$\label{eqn:HF-FSpace-Block}
\mbox{bdiag}_{m_0\times m_0}
\{{\cal T}_{\bf L}'[F_{\bf L} ({\cal T}_{\bf L}\widehat{B})
+ F_{\bf L} ({\cal T}_{\bf L}\widehat{A})] -
\lambda {\cal T}_{\bf L}'(F_{\bf L} [{\cal T}_{\bf L} \widehat{S})] \} U =0.
$$ The block structure specified by Lemma \[lem:SparseCaseP\] allows to apply the efficient eigenvalue solvers via FFT based diagonalization in the framework of Hartree-Fock calculations, in general, with the numerical cost $O(m_0^2 L^d \log L)$.
![Molecular orbitals, i.e. the eigenvectors represented in GTO basis: the $4$th orbital (left), the $8$th orbital (right). []{data-label="fig:3DCoreEigVect"}](Vec_128_3_4.eps "fig:"){width="6.0cm"} ![Molecular orbitals, i.e. the eigenvectors represented in GTO basis: the $4$th orbital (left), the $8$th orbital (right). []{data-label="fig:3DCoreEigVect"}](Vec_128_3_8.eps "fig:"){width="6.0cm"}
\[prop:low\_rank\_coef\] The low-rank structure in the coefficients tensor mentioned above (see Section \[ssec:Tensor\_bcirc\]) allows to reduce the factor $L^d \log L$ to $L \log L$ for $d=2,3$. It was already observed in the proof of Lemma \[lem:SparseCaseP\] that the respective coefficients in the overlap and Laplacian Galerkin matrices can be treated as the rank-$1$ and rank-$3$ tensors, respectively. Clearly, the factorization rank for the nuclear part of the Hamiltonian does not exceed $R_{\cal N}$. Hence, Theorem \[thm:tens\_FFT\] can be applied in generalized form.
Figure \[fig:3DCoreEigVect\] visualizes molecular orbitals on fine spatial grid with $n=2^{14}$: the $4$th orbital (left), the $8$th orbital (right). The eigenvectors are computed in GTO basis for $(L,1,1)$ system with $L=128$ and $m_0=4$.
Table \[Table\_Times\_SupSvsPer\] compares CPU times in sec. (Matlab) for the full eigenvalue solver on a 3D $(L, 1, 1)$ lattice in a box, and for the FTT-based diagonalization in the periodic supercell, all computed for $m_0=4$, $L=2^p$ ($p=7,8,...,15$). The number of basis function (problem size) is given by $N_b=m_0 L$.
![Spectrum of the core Hamiltonian.[]{data-label="fig:3DCoreEig"}](Spectr_H4_128_1_1.eps "fig:"){width="6.0cm"} ![Spectrum of the core Hamiltonian.[]{data-label="fig:3DCoreEig"}](Spectr_H4_256_1_1.eps "fig:"){width="6.0cm"}
\[c\][|r|r|r|r|r|r|r|r|r|r|]{}Problem size $N_b=n_0 L $ & $512$ & $1024$ & $2048$ & $4096$ & $8192$ & $16384$ & $32768$ & $65536$ & $131072$\
Full EIG-solver & $0.67$ & $5.49$ & $48.6 $ & $497.4$ & $--$ & $--$ & $--$ & $--$ & $--$\
FFT diagonalization & $0.10$ & $0.09$ & $0.08 $ & $0.14$ & $0.44$ & $1.5$ & $5.6$ & $22.9$ & $89.4$\
Figure \[fig:3DCoreEig\] represents the spectrum of the core Hamiltonian in a box vs. those in a periodic supercell for different number of cells $L=128,256$, where $m_0=4$. The systematic difference between the eigenvalues in both cases can be observed even for very large $L$. This spectral pollution effects have been discussed and theoretically analyzed in [@CancesDeLe:08].
![Spectrum of the core Hamiltonian for a $(L,1,1)$ lattice with $L=256$, and $m_0=4$, in a box (left) and for periodic case (right).[]{data-label="fig:3DSpectBand"}](Spectrum_EVP_512_b.eps "fig:"){width="6.0cm"} ![Spectrum of the core Hamiltonian for a $(L,1,1)$ lattice with $L=256$, and $m_0=4$, in a box (left) and for periodic case (right).[]{data-label="fig:3DSpectBand"}](Spectrum_EVP_512_p.eps "fig:"){width="6.0cm"}
Figure \[fig:3DSpectBand\] presents the spectral bands for a $(L,1,1)$ lattice system in a box and in the periodic setting, for $L=256$, and $m_0=4$.
![Average energy per unit cell vs. $L$ for a $(L,1,1)$ lattice in a 3D rectangular “tube“.[]{data-label="fig:3DAveEig"}](Average_Lambda_512.eps){width="7.0cm"}
Figure \[fig:3DAveEig\] demonstrates the relaxation of the average energy per unit cell with $m_0=4$, for a $(L,1 ,1)$ lattice structure in a 3D rectangular “tube‘ up to $L=512$, for both periodic and open boundary conditions.
Conclusions {#sec:Conclusions}
===========
We have introduced and analyzed the grid-based tensor product approach to discretization and solution of the Hartree-Fock equation in ab initio modeling of the lattice-structured molecular systems. In this presentation we consider the case of core Hamiltonian. All methods and algorithms developed in this paper are implemented and tested in Matlab.
The proposed tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D Fourier transforms. The main contributions include:
- Fast computation of Fock matrix by 1D matrix-vector operations using low-rank tensors represented on a 3D spacial grid.
- Analysis and numerical implementation of the multilevel block circulant representation of the Fock matrix in the periodic setting.
- Investigation of the low-rank tensor structure in the diagonal blocks of the Fock matrix represented in the Fourier space, that allows to reduce the conventional 3D FFT to the product of 1D FFTs.
- Numerical tests illustrating the computational efficiency of the tensor-structured methods applied to the reduced Hartree-Fock equation for lattice-type and periodic systems. Numerical experiments on verification of the theoretical results on the asymptotic complexity estimated of the presented algorithms.
Here we confine ourself to the case of core Hamiltonian part in the full Fock matrix (linear part in the Fock operator). The rigorous study of the fully nonlinear self-consistent Hartree-Fock eigenvalue problem for periodic and lattice-structured systems in a box is a matter of future research.
Appendix: Overview on block circulant matrices {#sec_Append:block-circ}
==============================================
We recall that a one-level block circulant matrix $A\in {\cal BC} (L,m_0)$ is defined by [@Davis], $$\label{eqn:block_c}
A=\operatorname{bcirc}\{A_0,A_1,...,A_{L-1}\}=
\begin{bmatrix}
A_0 & A_{L-1} & \cdots & A_{2} & A_{1} \\
A_{1} & A_0 & \cdots & \vdots & A_{2} \\
\vdots & \vdots & \ddots & A_0 & \vdots \\
A_{L-1} & A_{L-2} & \cdots & A_{1} & A_0 \\
\end{bmatrix}
\in \mathbb{R}^{L m_0\times L m_0},$$ where $A_k \in \mathbb{R}^{m_0\times m_0}$ for $k=0,1, \ldots ,L-1$, are matrices of general structure. The equivalent Kronecker product representation is defined by the associated matrix polynomial, $$\label{eqn:bcircPol}
A= \sum\limits^{L -1}_{k=0} \pi^k \otimes A_k =:p_A(\pi),$$ where $\pi=\pi_L\in \mathbb{R}^{L \times L}$ is the periodic downward shift (cycling permutation) matrix, $$\pi_L:=
\begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & \cdots & 0 & 0 \\
\vdots &\vdots & \ddots & \vdots & \vdots \\
0 & \cdots & 1 & 0 & 0 \\
0 & 0 & \cdots & 1 & 0 \\
\end{bmatrix}
,$$ and $\otimes$ denotes the Kronecker product of matrices.
In the case $m_0=1$ a matrix $A\in {\cal BC} (L,1)$ defines a circulant matrix generated by its first column vector $\widehat{a}=(a_0,...,a_{L-1})^T$. The associated scalar polynomial then reads $$p_A(z):= a_0 + a_1 z + ... +a_{L-1} z^{L-1},$$ so that (\[eqn:bcircPol\]) simplifies to $$A=p_A(\pi_L).$$ Let $\omega= \omega_L= \exp(-\frac{2\pi i}{L})$, we denote by $$F_L=\{f_{k\ell}\}\in \mathbb{R}^{L\times L}, \quad \mbox{with} \quad
f_{k\ell}=\frac{1}{\sqrt{L}}\omega_L^{(k-1)(\ell-1)},\quad
k,l=1,...,L,$$ the unitary matrix of Fourier transform. Since the shift matrix $\pi_L$ is diagonalizable in the Fourier basis, $$\label{eqn:diagshift}
\pi_L=F_L^\ast D_L F_L,\quad D_L= \mbox{diag}\{1,\omega,...,\omega^{L-1} \},$$ the same holds for any circulant matrix, $$\label{eqn:circDiag}
A = p_A(\pi_L) = F_L^\ast p_A(D_L) F_L,$$ where $$p_A(D_L)=\mbox{diag}\{p_A(1),p_A(\omega),...,p_A(\omega^{L-1})\}= \mbox{diag}\{F_L a\}.$$
Conventionally, we denote by $\mbox{diag}\{x\}$ a diagonal matrix generated by a vector $x$. Let $X$ be an $L m_0\times m_0$ matrix obtained by concatenation of $m_0\times m_0$ matrices $X_k$, $k=0,...,L-1$, $X=\operatorname{conc}(X_0,...,X_{L-1})=[X_0,...,X_{L-1}]^T$. For example, the first block column in (\[eqn:block\_c\]) has the form $\operatorname{conc}(A_0,...,A_{L-1})$. We denote by $\mbox{bdiag}\{X\}$ the $L m_0\times L m_0$ block-diagonal matrix of block size $L$ generated by $m_0\times m_0$ blocks $X_k$.
It is known that similarly to the case of circulant matrices (\[eqn:circDiag\]), block circulant matrix in ${\cal BC} (L,m_0)$ is unitary equivalent to the block diagonal one by means of Fourier transform via representation (\[eqn:bcircPol\]), see [@Davis]. In the following, we describe the block-diagonal representation of a matrix $A\in {\cal BC} (L,m_0)$ in the form that is convenient for generalization to the multi-level block circulants as well as for the description of FFT based implementational schemes. To that end, let us introduce the reshaping (folding) transform ${\cal T}_L$ that maps a $L m_0\times m_0$ matrix $X$ (i.e., the first block column in $A$) to $L\times m_0\times m_0$ tensor $B={\cal T}_L X$ by plugging the $i$th $m_0\times m_0$ block in $X$ into a slice $B(i,:,:)$. The respective unfolding returns the initial matrix $X={\cal T}_L' B$. We denote by $\widehat{A}\in \mathbb{R}^{L m_0\times m_0}$ the first block column of a matrix $A\in {\cal BC} (L,m_0)$, with a shorthand notation $$\widehat{A}=[A_0,A_1,...,A_{L-1}]^T,$$ so that the $L\times m_0\times m_0$ tensor ${\cal T}_L \widehat{A}$ represents slice-wise all generating $m_0\times m_0$ matrix blocks.
\[prop:eig\_bcmatr\] For $A\in {\cal BC} (L,m_0)$ we have $$\label{eqn:bcircDiag}
A= (F_L^\ast \otimes I_{m_0}) \operatorname{bdiag}
\{ \bar{A}_0, \bar{A}_1,\ldots , \bar{A}_{L-1}\}
(F_L \otimes I_{m_0}),$$ where $$\bar{A}_j = \sum\limits^{L -1}_{k=0} \omega_L^{jk} A_k \in \mathbb{C}^{m_0 \times m_0},$$ can be recognized as the $j$-th $m_0\times m_0$ matrix block in block column ${\cal T}_L'(F_L ({\cal T}_L \widehat{A}))$, such that $$\left[ \bar{A}_0, \bar{A}_1,\ldots , \bar{A}_{L-1}\right]^T =
{\cal T}_L'(F_L ({\cal T}_L \widehat{A})).$$ A set of eigenvalues $\lambda$ of $A$ is then given by $$\label{eqn:lambdAbc}
\{\lambda | Ax = \lambda x, \; x\in \mathbb{C}^{L m_0}\}=
\bigcup\limits_{j=0}^{L-1} \{ \lambda |\bar{A}_j u = \lambda u, \; u \in \mathbb{C}^{m_0} \}.$$ The eigenvectors corresponding to the spectral sets $$\Sigma_j= \{\lambda_{j, m} |\bar{A}_j u_{j,m} = \lambda_{j,m} u_{j,m},
\; u_{j,m} \in \mathbb{C}^{m_0}\},
\quad j= 0,1,\ldots , L-1,\quad m=1,...,m_0,$$ can be represented in the form $$\label{eqn:eigvecA}
U_{j,m}=(F_L^\ast \otimes I_{m}) \bar{U}_{j,m},\quad \mbox{where} \quad
\bar{U}_{j,m}= E_{[j]} \operatorname{vec}\, [u_{0,m},u_{1,m},...,u_{L-1,m}],$$ with $E_{[j]}=\operatorname{diag}\{e_j\}\otimes I_{m_0} \in \mathbb{R}^{L m_0\times L m_0} $, and $e_j\in \mathbb{R}^{L}$ being the $j$th Euclidean basis vector.
We combine representations (\[eqn:bcircPol\]) and (\[eqn:diagshift\]) to obtain $$\begin{aligned}
\label{eqn:Bcircdiag}
A & = \sum\limits^{L -1}_{k=0} \pi^k \otimes A_k =
\sum\limits^{L -1}_{k=0} (F_L^\ast D^k F_L) \otimes A_k \\ \nonumber
& = (F_L^\ast \otimes I_{m_0}) (\sum\limits^{L -1}_{k=0} D^k
\otimes A_k)(F_L \otimes I_{m_0}) \\ \nonumber
& = (F_n^\ast \otimes I_m)(\sum\limits^{L -1}_{k=0}
\mbox{bdiag}\{A_k,\omega_L^k A_k,...,\omega_L^{k(L-1)}A_{k} \} )
(F_L \otimes I_{m_0})\\ \nonumber
& = (F_L^\ast \otimes I_{m_0})
\mbox{bdiag}\{\sum\limits^{L -1}_{k=0} A_k,\sum\limits^{L -1}_{k=0} \omega_L^k A_k,...,
\sum\limits^{L -1}_{k=0} \omega_L^{k(L-1)}A_{k} \} (F_L \otimes I_{m_0})\\ \nonumber
& = (F_L^\ast \otimes I_{m_0}) \mbox{bdiag}_{m_0 \times m_0}
\{{\cal T}_L'(F_L ({\cal T}_L \widehat{A}))\} (F_L \otimes I_{m_0}), \nonumber\end{aligned}$$ where the final step follows by the definition of FT matrix and by the construction of ${\cal T}_L$. The structure of eigenvalues and eigenfunctions then follows by simple calculations with block-diagonal matrices.
The next statement describes the block-diagonal form for a class of symmetric BC matrices, ${\cal BC}_s (L,m_0)$, that is a simple corollary of [@Davis], Proposition \[prop:eig\_bcmatr\]. In this case we have $A_0=A_0^T$, and $A_k^T=A_{L-k}$, $k=1,...,L-1$.
\[cor:eig\_symbcmatr\] Let $A\in {\cal BC}_s (L,m_0)$ be symmetric, then $A$ is unitary similar to a Hermitian block-diagonal matrix, i.e., $A$ is of the form $$\label{eqn:F_bc}
A= (F_L \otimes I_{m_0}) \operatorname{bdiag} (\tilde{A}_0, \tilde{A}_1,\ldots , \tilde{A}_{L-1})
(F_L^\ast \otimes I_{m_0}),$$ where $I_{m_0}$ is the $m_0\times m_0$ identity matrix. The matrices $\tilde{A}_j \in \mathbb{C}^{m_0\times m_0}$, $j= 0,1,\ldots , L-1$, are defined for even $n\geq 2$ as $$\label{eqn:symBc}
\tilde{A}_j =A_0 + \sum\limits^{L/2-1}_{k=1} (\omega^{kj}_L A_k +
\overline{\omega}^{kj}_L A^T_k) + (-1)^j A_{L/2}.$$
Corollary \[cor:eig\_symbcmatr\] combined with Proposition \[prop:eig\_bcmatr\] describes a simplified structure of eigendata in the symmetric case. Notice that the above representation imposes the symmetry of each real-valued diagonal blocks $\tilde{A}_j \in \mathbb{R}^{m_0\times m_0}, \; j= 0,1,\ldots , L-1$, in (\[eqn:F\_bc\]).
Finally, we recall that a one-level symmetric block Toeplitz matrix $A\in {\cal BT}_s (L,m_0)$ is defined by [@Davis], $$\label{eqn:block_SToepl}
A=\operatorname{BToepl}_s\{A_0,A_1,...,A_{L-1}\}=
\begin{bmatrix}
A_0 & A_{1}^T & \cdots & A_{L-2}^T & A_{L-1}^T \\
A_{1} & A_0 & \cdots & \vdots & A_{L-2}^T \\
\vdots & \vdots & \ddots & A_0 & \vdots \\
A_{L-1} & A_{L-2} & \cdots & A_{1} & A_0 \\
\end{bmatrix}
\in \mathbb{R}^{L m_0\times L m_0},$$ where $A_k \in \mathbb{R}^{m_0\times m_0}$ for $k=0,1, \ldots ,L-1$, are matrices of a general structure.
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[^1]: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany ([vekh@mis.mpg.de]{}).
[^2]: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany ([bokh@mis.mpg.de]{}).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities. All such metric spaces have an “onion-shell” geometry. We demonstrate the power of this approach by considering many-body systems immersed in a magnetic field, with a finite ground state current. In the associated metric spaces we find regions of allowed and forbidden distances, a “band structure” in metric space directly arising from the conservation of the $z$ component of the angular momentum.'
author:
- 'P. M. Sharp and I. D’Amico'
bibliography:
- 'References.bib'
title: Metric Space Formulation of Quantum Mechanical Conservation Laws
---
Introduction
============
Conservation laws are a central tenet of our understanding of the physical world. Their tight relationship to natural symmetries was demonstrated by Noether in 1918 [@Noether1918] and has since been a fundamental tool for developing theoretical physics. In this paper we demonstrate how these laws induce appropriate “natural” metrics on the related physical quantities. Conservation laws are central to the behavior of physical systems and we show how this relevant physics is translated into the metric analysis. We argue that this alternative picture provides a new powerful tool to study certain properties of many-body systems, which are often complex and hardly tractable when considered within the usual coordinate space-based analysis, while may become much simpler when analyzed within metric spaces. We exemplify this concept by considering functional relationships fundamental to current density functional theory (CDFT) [@Vignale1987; @Vignale1988].
We will first introduce a way to derive appropriate “natural” metrics from a system’s conservation laws. Second, as an example application of the approach, we will explicitly consider an important class of systems – systems with applied external magnetic fields. In contrast with those to which standard density functional theory (DFT) [@Dreizler1990] can be applied, systems subject to external magnetic fields are not simply characterized by their particle densities as even their ground states may display a finite current [@Vignale1987; @Vignale1988]. These systems are of great importance, e.g., due to the emerging quantum technologies of spintronics and quantum information where, for example, few electrons in nano- or microstructures immersed in magnetic fields are proposed as hardware units [@Takahashi2010; @daSilva2009; @Brandner2013; @Amaha2013; @Castellanos-Beltran2013].
To analyze systems immersed in a magnetic field, we will introduce a metric associated with the paramagnetic current density, which can be associated with the angular momentum components. We will show that, at least for systems which preserve the $z$ component of the angular momentum, the paramagnetic current density metric space displays an “onion-shell” geometry, directly descending from the related conservation law. In recent work [@D'Amico2011; @Artacho2011; @D'Amico2011b] appropriate metrics for characterizing wavefunctions and particle densities within quantum mechanics were introduced. It was shown that wavefunctions and their particle densities both form metric spaces with an “onion-shell” structure [@D'Amico2011]. We will show that, within the same general procedure used for the paramagnetic current, these metrics descend from the respective conservation laws. We will then focus on ground states and characterize them not only through the mapping between wavefunctions and particle densities, but importantly through mappings involving the paramagnetic current density. In fact, for systems with an applied magnetic field, ground state wavefunctions are characterized uniquely only by knowledge of both particle *and* paramagnetic current densities (and vice versa), as demonstrated within CDFT [@Vignale1987; @Vignale1988].
The rest of this paper is organized as follows: In Sec. \[metric\] we introduce our general approach to derive metric spaces from conservation laws. We demonstrate the application of this approach to wavefunctions, particle densities, and paramagnetic current densities in Sec. \[apply\]. We consider systems subject to magnetic fields in Sec. \[cdft\]. Here we use the metrics derived from our approach to study the fundamental theorem of CDFT. We present our conclusions in Sec. \[conclusion\].
Derivation of Metric Spaces from Conservation Laws {#metric}
==================================================
A metric or distance function $D$ over a set $X$ satisfies the following axioms for all $x,y,z \in X$ [@Megginson1998; @Sutherland2009]: $$\begin{aligned}
D(x,y) &\geqslant 0\ \text{and}\ D(x,y)=0 \iff x=y, \label{axiom1}\\
D(x,y) &= D(y,x), \label{axiom2}\\
D(x,y) &\leqslant D(x,z)+D(z,y), \label{axiom3}\end{aligned}$$ with (\[axiom3\]) known as the triangle inequality. The set $X$ with the metric $D$ forms the metric space $(X,D)$. It can be seen from the axioms (\[axiom1\]) - (\[axiom3\]) that many metrics could be devised for the same set, some trivial. Here we introduce “natural” metrics associated to conservation laws: this will avoid arbitrariness and in turn will ensure that the proposed metrics stem from core characteristics of the systems analyzed and contain the related physics.
In quantum mechanics, many conservation laws take the form $$\label{conservation}
\int {\left|f(x)\right|}^{p} dx = c$$ for $0<c<\infty$. For each value of $1\leqslant p<\infty$, the entire set of functions that satisfy (\[conservation\]) belong to the $L^p$ vector space, where the standard norm is the $p$ norm [@Megginson1998] $$\label{lp_norm}
{\left|\left|f(x)\right|\right|}_p =\left[\int {\left|f(x)\right|}^{p} dx \right]^{\frac{1}{p}}.$$ From any norm a metric can be introduced in a standard way as $D(x,y)={\left|\left|x-y\right|\right|}$ so that with $p$ norms we get $$\label{lp_metric}
D_{f}(f_1,f_2):={\left|\left|f_1-f_2\right|\right|}_p.$$ However before assuming this metric for the physical functions related to the conservation laws, an important consideration must be made: Eq. (\[lp\_metric\]) has been derived assuming the ensemble $\{f\}$ to be a vector space; this is in fact necessary to introduce a norm. If we want to retain the metric (\[lp\_metric\]), but restrict it to the ensemble of *physical* functions satisfying (\[conservation\]), which does not necessarily form a vector space, we must show that (\[lp\_metric\]) is a metric for this restricted function set. This can be done using the general theory of metric spaces: given a metric space $(X,D)$ and $S$ a non empty subset of $X$, $(S,D)$ is itself a metric space with the metric $D$ inherited from $(X,D)$. The metric axioms (\[axiom1\]) - (\[axiom3\]) automatically hold for $(S,D)$ because they hold for $(X,D)$ [@Megginson1998; @Sutherland2009]. Hence, we have a metric for the functions of interest, as their sets are non empty subsets of the respective $L^p$ sets.
The metric (\[lp\_metric\]) is then the one that *directly descends* from the conservation law (\[conservation\]). Conversely any conservation law which can be recast as (\[conservation\]) (for example conservation of quantum numbers) can be interpreted as inducing a metric on the appropriate, physically relevant, subset of $L^{p}$ functions. This provides a general procedure to derive “natural” metrics from physical conservation laws.
Applications of the Metric Space Approach {#apply}
=========================================
We now consider specific quantum mechanical functions and conservation laws. Following Ref. [@D'Amico2011] we use a convention where wavefunctions are normalized to the particle number $N$ [^1]. Then the particle density of an $N$-particle system and its paramagnetic current density are defined as $$\begin{aligned}
\rho({\mathbf{r}})&=\int {\left|\psi\left({\mathbf{r}},{\mathbf{r}}_{2},\ldots,{\mathbf{r}}_{N}\right)\right|}^{2} d{\mathbf{r}}_{2}\ldots d{\mathbf{r}}_{N},\label{density}\\
{\mathbf{j}}_{p}({\mathbf{r}})&=-\frac{i}{2}\int \left(\psi^{\ast}\nabla\psi - \psi\nabla\psi^{\ast}\right) d{\mathbf{r}}_{2}\ldots d{\mathbf{r}}_{N}.\label{current}\end{aligned}$$ First of all we note that $\psi\left({\mathbf{r}}_1,{\mathbf{r}}_{2},\ldots ,{\mathbf{r}}_{N}\right)$ and $\rho({\mathbf{r}})$ are subject to the following conservation laws (wavefunction norm and particle conservation): $$\begin{aligned}
&\int{\left|\frac{\psi\left({\mathbf{r}}_1,{\mathbf{r}}_{2},\ldots ,{\mathbf{r}}_{N}\right)}{\sqrt{N}}\right|}^{2}d{\mathbf{r}}_{1}\ldots d{\mathbf{r}}_{N} = 1,\label{psi_cons}\\
&\int\rho({\mathbf{r}}) d{\mathbf{r}} = N.\label{rho_cons}\end{aligned}$$ Similarly the paramagnetic current density ${\mathbf{j}}_{p}({\mathbf{r}})$ obeys $$\label{Lz}
\int \left[{\mathbf{r}}\times{\mathbf{j}}_{p}({\mathbf{r}})\right]_z d{\mathbf{r}} = \langle\psi|\hat{L}_z|\psi\rangle.$$ For eigenstates of systems for which the $z$ component of the angular momentum is preserved we then have $\langle\hat{L}_z\rangle=m$, with $m$ an integer, and (\[Lz\]) can be recast as $$\label{j_p_cons}
\int {\left|\left[{\mathbf{r}}\times{\mathbf{j}}_{p}({\mathbf{r}})\right]_z\right|} d{\mathbf{r}} = {\left|m\right|}.$$ For wavefunctions and particle densities our procedure leads to the metrics introduced in Ref. [@D'Amico2011] ($N$ fixed) [@Artacho2011; @D'Amico2011b] $$\begin{aligned}
D_{\psi}(\psi_{1},\psi_{2})=&\left[\int \left({\left|\psi_{1}\right|}^{2}+{\left|\psi_{2}\right|}^{2}\right)d{\mathbf{r}}_1\ldots d{\mathbf{r}}_{N}\right.\nonumber\\
&- \left. 2{\left|\int\psi_{1}^{*}\psi_{2}d{\mathbf{r}}_1\ldots d{\mathbf{r}}_{N}\right|}\right]^{\frac{1}{2}},\label{dpsi}\\
D_{\rho}(\rho_{1},\rho_{2})=&\int{\left|\rho_{1}({\mathbf{r}})-\rho_{2}({\mathbf{r}})\right|} d{\mathbf{r}}; \label{drho}\end{aligned}$$ for the paramagnetic current density, our procedure introduces the following metric: $$\label{dj_p}
D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}_{p,1},{\mathbf{j}}_{p,2})=\int{\left|\left\{{\mathbf{r}}\times\left[{\mathbf{j}}_{p,1}({\mathbf{r}})-{\mathbf{j}}_{p,2}({\mathbf{r}})\right]\right\}_{z}\right|} d{\mathbf{r}}.$$ We note that $D_{{\mathbf{j}}_{p_{\perp}}}$ will be a distance between equivalence classes of paramagnetic currents, each class characterized by current densities having the same transverse component ${\mathbf{j}}_{p_{\perp}}\equiv(j_{p,x},j_{p,y})$. $D_{{\mathbf{j}}_{p_{\perp}}}$ is gauge invariant provided that ${\mathbf{j}}_{p,1}$ and ${\mathbf{j}}_{p,2}$ are within the same gauge and $[\hat{L_{z}},\hat{H}]=0$.
Next we show that conservation laws naturally build within the related metric spaces a hierarchy of concentric spheres, or “onion-shell” geometry. If we set as the center of each sphere the zero function $f^{(0)}(x)\equiv 0$, and consider the distance between it and any other element in the metric space, we recover the $p$-norm expressions (\[lp\_norm\]) directly descending from the related conservation laws. This procedure induces in the related metric spaces a structure of concentric spheres with radii, in the cases considered here, of natural numbers to the power of $1/p$: all functions corresponding to the same value of a certain conserved quantity will lay on the surface of the same sphere. Specifically, for systems of $N$ particles, wavefunctions lie on spheres of radius $\sqrt{N}$, and particle densities on spheres of radius $N$; for the metric space of paramagnetic current densities, all paramagnetic current densities with a $z$ component of the angular momentum equal to $\pm m$ lie on spheres of radius ${\left|m\right|}$.
The first axiom of a metric (\[axiom1\]) guarantees that the minimum value for all distances is $0$, and that this value is attained for two identical states. The onion-shell geometry guarantees that, for functions on the surface of the same sphere, i.e., which satisfy a certain conservation law with the same value, there is also an upper limit for their distance associated with the diameter of the sphere. From (\[dj\_p\]) we see that for paramagnetic current densities this upper limit is achieved in the limit of currents which do not spatially overlap. This is also the case for particle densities, as seen in (\[drho\]).
![(Color online) For the ISI system energy is plotted against the confinement frequency for several values of the angular momentum quantum number $m$ (as labeled), and with constant cyclotron frequency and interaction strength. Arrows indicate where the value of $m$ for the ground state changes.[]{data-label="energy"}](energy.pdf){width="\columnwidth"}
Interestingly, and in contrast to wavefunctions and particle densities [@D'Amico2011], even when considering systems with the same number of particles it may be necessary to consider paramagnetic current densities with different values of $m$; in terms of their metric space geometry, current densities that have different values of ${\left|m\right|}$ lie on different spheres. Therefore, the maximum value for the distance between paramagnetic current densities of a system of $N$ particles is related to the upper limit of the number of spheres in the onion-shell geometry. Using the triangle inequality we have in fact $$\begin{aligned}
D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}_{p,m_{1}},{\mathbf{j}}_{p,m_{2}})&\leqslant D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}_{p,m_{1}},{\mathbf{j}}^{(0)}_{p})+D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}^{(0)}_{p},{\mathbf{j}}_{p,m_{2}})\nonumber\\
&={\left|m_1\right|}+{\left|m_2\right|}\leqslant l_{1}+l_{2},\end{aligned}$$ where $l_i$ is the quantum number related to the total angular momentum of system $i$.
Study of Model Systems {#cdft}
======================
We now concentrate on the sets of ground state wavefunctions, related particle densities, and related paramagnetic current densities. Since ground states are non empty subsets of all states, ground-state-related functions form metric spaces with the metrics (\[dpsi\]), (\[drho\]), and (\[dj\_p\]). The importance of characterizing ground states and their properties has been highlighted by the huge success of DFT (in all its flavors) as a method to predict devices’ and material properties [@Dreizler1990; @Ullrich2013]. Standard DFT is built on the Hohenberg-Kohn (DFT-HK) theorem [@HK1964], which demonstrates a one-to-one mapping between ground state wavefunctions and their particle densities. This theorem is highly complex and nonlinear in coordinate space. However, Ref. [@D'Amico2011] showed that the DFT-HK theorem is a mapping between metric spaces, and may be very simple when described in these terms, becoming monotonic and almost linear for a wide range of parameters and for the systems there analyzed. CDFT is a formulation of DFT for systems in the presence of an external magnetic field. In CDFT [@Vignale1987; @Vignale1988] the original HK mapping is extended (CDFT-HK theorem) to demonstrate that $\psi$ is uniquely determined only by knowledge of both $\rho({\mathbf{r}})$ and ${\mathbf{j}}_p({\mathbf{r}})$ (and vice versa). This is the theorem we will consider in this section.
To further our analysis, we now explicitly examine two model systems with applied magnetic fields. They both consist of two electrons parabolically confined that interact via different potentials, Coulomb (magnetic Hooke’s atom) [@Taut2009] and inverse square interaction (ISI) [@Quiroga1993], respectively. Both systems may be used to model electrons confined in quantum dots. The Hamiltonians for the magnetic Hooke’s atom and the ISI system are $$\begin{aligned}
\hat{H}_{HA}&=\sum_{i=1}^{2}\left\{\frac{1}{2}\left[\hat{{\mathbf{p}}}_{i}+{\mathbf{A}}\left({\mathbf{r}}_{i}\right)\right]^{2}+\frac{1}{2}\omega_{0}^{2}r_{i}^{2}\right\}+\frac{1}{{\left|{\mathbf{r}}_{2}-{\mathbf{r}}_{1}\right|}}, \label{Hooke_H}\\
\hat{H}_{ISI}&=\sum_{i=1}^{2}\left\{\frac{1}{2}\left[\hat{{\mathbf{p}}}_{i}+{\mathbf{A}}\left({\mathbf{r}}_{i}\right)\right]^{2}+\frac{1}{2}\omega_{0}^{2}r_{i}^{2}\right\}+\frac{\alpha}{\left({\mathbf{r}}_{1}-{\mathbf{r}}_{2}\right)^{2}}, \label{ISI_H}\end{aligned}$$ (atomic units, $\hbar=m_e=e=1$). Here $\alpha$ is a positive constant, ${\mathbf{A}}=\frac{1}{2}{\mathbf{B}}\times{\mathbf{r}}$ (symmetric gauge), and ${\mathbf{B}}=\omega_{c}c{\mathbf{\hat{z}}}$ is a homogeneous, time-independent external magnetic field. For these systems $\langle\hat{L}_z\rangle$ is a conserved quantity. Following Refs. [@Vignale1987; @Taut2009] we disregard spin to concentrate on the features of the orbital currents. For Hooke’s atom, we obtain highly precise numerical solutions following the method in Ref. [@Coe2008]. The ISI system is solved exactly [@Quiroga1993].
{width="\textwidth"}
To produce families of ground states, for each system we systematically vary the value of $\omega_0$ (while keeping all other parameters constant), and for each value we calculate the ground state wavefunction, particle density, and paramagnetic current density. A reference state is determined by choosing a specific $\omega_0$ value, and the appropriate metric is then used to calculate the distances between it and each member of the family. To ensure that we select ground states, varying $\omega_0$ may require varying the quantum number $m$ [@Taut2009; @Quiroga1993]. This is shown for the ISI system in Fig. \[energy\]. Here, as $\omega_{0}$ increases, we must decrease the value of ${\left|m\right|}$ in order to remain in the ground state. As a result of this property, within each family of ground states, paramagnetic current densities will “jump” from one sphere of the onion-shell geometry to another \[see Fig. \[spheres\](a), where the reference state is the ‘north pole’ of its sphere\]. To obtain ground states with nonzero paramagnetic currents, we must use $\omega_0$ values corresponding to $m<0$ [@Taut2009; @Quiroga1993].
![(Color online) (a) Sketch of the onion-shell geometry of the metric space for paramagnetic current densities, where ${\left|m_q\right|}>{\left|m_r\right|}>{\left|m_{ref}\right|}$ (left) and ${\left|m_{ref}\right|}>{\left|m_s\right|}>{\left|m_t\right|}$ (right). The reference state is at the north pole on the reference sphere. The dark gray areas denote the regions where ground state currents are located (‘bands’), with dashed lines indicating their widths. (b) Results of the angular displacement of ground state currents for the ISI system. Lines are a guide to the eye. Inset: Definition of relevant angles.[]{data-label="spheres"}](sphere_fig.pdf){width="\columnwidth"}
In Fig. \[results\], we plot each pair of distances for the two systems. The reference states have been chosen so that most of the available distance range can be explored both for the case of increasing and for the case of decreasing values of $\omega_0$. When considering the relationship between ground state wavefunctions and related particle densities, Figs. \[results\](a) and \[results\](b), our results confirm the findings in Ref. [@D'Amico2011]: a monotonic mapping, linear for low to intermediate distances, and where vicinities are mapped onto vicinities; also curves for increasing and decreasing $\omega_0$ collapse onto each other. However closer inspection reveals a fundamental difference with Ref. [@D'Amico2011], the presence of a “band structure.” By this we mean regions of allowed (“bands”) and forbidden (“gaps”) distances, whose widths depend, for the systems considered here, on the value of ${\left|m\right|}$. This structure is due to the changes in the value of the quantum number $m$, which result in a substantial modification of the ground state wavefunction (and therefore density) and a subsequent large increase in the related distances.
When we focus on the plots of paramagnetic current densities’ against wavefunctions’ distances, Figs. \[results\](c) and \[results\](d), we find that the “band structure” dominates the behavior. Here the change in ${\left|m\right|}$ has an even stronger effect, in that $dD_{{\mathbf{j}}_{p_{\perp}}}/dD_{\psi}$ is noticeably discontinuous when moving from one sphere to the next in ${\mathbf{j}}_p$ metric space. This discontinuity is more pronounced for the path ${\left|m\right|}<{\left|m_{ref}\right|}$ than for the path ${\left|m\right|}>{\left|m_{ref}\right|}$. Similarly to Figs. \[results\](a) and \[results\](b), the mapping of $D_{\psi}$ onto $D_{{\mathbf{j}}_{p_{\perp}}}$ maps vicinities onto vicinities and remains monotonic, but for small and intermediate distances it is only piecewise linear. In contrast with $D_{\rho}$ vs $D_{\psi}$, curves corresponding to increasing and decreasing $\omega_0$ do not collapse onto each other.
Figures \[results\](e) and \[results\](f) show the mapping between particle and paramagnetic current density distances: this has characteristics similar to the one between $D_{\psi}$ and $D_{{\mathbf{j}}_{p_{\perp}}}$, but remains piecewise linear even at large distances.
We will now concentrate on the ${\mathbf{j}}_{p}$ metric space to characterize the “band structure” observed in Fig. \[results\]. Within the metric space geometry, we consider the polar angle $\theta$ between the reference ${\mathbf{j}}_{p,ref}$ and the paramagnetic current density ${\mathbf{j}}_{p}$ of angular momentum ${\left|m\right|}$. Using the law of cosines, $\theta$ is given by $$\label{angle}
\cos{\theta}=\frac{m_{ref}^2+m^2-D_{{\mathbf{j}}_{p_{\perp}}}^2({\mathbf{j}}_{p,ref},{\mathbf{j}}_{p}) }{2{\left|m_{ref}\right|}{\left|m\right|}}.$$ We define the polar angles corresponding to the two extremes of a given band as $\theta_{min}$ and $\theta_{max}$ (inset of Fig. \[spheres\]). The width of each band is then ${\Delta}{\theta}={\theta}_{max}-{\theta}_{min}$, and its position defined by $\theta_{min}$. Now we can calculate the bands’ widths and positions by sweeping, for each ${\left|m\right|}$, the values of $\omega_0$ corresponding to ground states (Fig. \[spheres\]).
For both systems under study, we find that as ${\left|m\right|}$ increases from ${\left|m_{ref}\right|}$, both ${\theta}_{max}$ and ${\theta}_{min}$ increase. This has the effect of the bands moving from the north pole to the south pole as we move away from the reference. Additionally, we find that the bandwidth ${\Delta}{\theta}$ decreases as ${\left|m\right|}$ increases \[sketched in Fig. \[spheres\](a), left\]. As ${\left|m\right|}$ decreases from ${\left|m_{ref}\right|}$, we again find that both ${\theta}_{max}$ and ${\theta}_{min}$ increase, with the bands moving from the north pole to the south pole. However, this time, as ${\left|m\right|}$ decreases, ${\Delta}{\theta}$ increases, meaning that the bands get wider as we move away from the reference \[sketched in Fig. \[spheres\](a), right\].
Quantitative results for the ISI system are shown in Fig. \[spheres\](b). We obtain similar results for Hooke’s atom (not shown). The band on the surface of each sphere indicates where all ground state paramagnetic current densities lie within that sphere. In contrast with particle densities or wavefunctions, we find that, at least for the systems at hand, ground state currents populate a well-defined, limited region of each sphere, whose size and position display monotonic behavior with respect to the quantum number $m$. This regular behavior is not at all expected, as the CDFT-HK theorem does not guarantee monotonicity in metric space, and not even that the mapping of $D_{\psi}$ to $D_{{\mathbf{j}}_{p_{\perp}}}$ is single valued. In the CDFT-HK theorem ground state wavefunctions are uniquely determined only by particle and paramagnetic current densities *together*. In this sense we can look at the panels in Fig. \[results\] as projections on the axis planes of a 3-dimensional $D_{\psi}D_{\rho}D_{{\mathbf{j}}_{p_{\perp}}}$ relation. The complexity of the mapping due to the application of a magnetic field – the changes in quantum number $m$ – is fully captured by $D_{{\mathbf{j}}_{p_{\perp}}}$ only, as this is related to the relevant conservation law. However the mapping from $D_{\rho}$ to $D_{\psi}$ inherits the “band structure,” showing that the two mappings $D_{{\mathbf{j}}_{p_{\perp}}}$ to $D_{\psi}$ and $D_{\rho}$ to $D_{\psi}$ are not independent.
Conclusion
==========
In conclusion we showed that conservation laws induce related metric spaces with an “onion-shell” geometry and that they may induce a “band structure” in ground state metric spaces, a signature of the enhanced constraints due to the system conservation laws on the relation between wavefunctions and the relevant physical quantities.
The method proposed may help with understanding extended HK theorems, such as, in the case at hand, the CDFT-HK theorem. In this respect we find that in metric spaces and for the systems considered, the relevant mappings display distinctive signatures, including (piecewise) linearity at short and medium distances, the mapping between ground state $\psi$ and ${\mathbf{j}}_{p}$ resembling the one between $\rho$ and ${\mathbf{j}}_{p}$, and the mapping between ground state $\psi$ and ${\mathbf{j}}_{p}$ showing different trajectories for increasing or decreasing Hamiltonian parameters, in contrast with the mapping between $\psi$ and $\rho$. Features like this could be used to build or test (single-particle) approximate solutions to many-body problems, e.g., within DFT schemes.
Our results show that using conservation laws to derive metrics makes these metrics a powerful tool to study many-body systems governed by integral conservation laws.
We thank M. Taut, K. Capelle, and C. Verdozzi for helpful discussions. P. M. S. acknowledges EPSRC for financial support. I. D. and P. M. S. gratefully acknowledge support from a University of York - FAPESP combined grant.
[^1]: This allows the description of Fock space as a set of concentric spheres
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We examine the convexity and tractability of the two-sided linear chance constraint model under Gaussian uncertainty. We show that these constraints can be applied directly to model a larger class of nonlinear chance constraints as well as provide a reasonable approximation for a challenging class of quadratic chance constraints of direct interest for applications in power systems. With a view towards practical computations, we develop a second-order cone outer approximation of the two-sided chance constraint with provably small approximation error.'
author:
- Miles Lubin
- Daniel Bienstock
- Juan Pablo Vielma
bibliography:
- 'refs.bib'
date: February 2016
title: 'Two-sided linear chance constraints and extensions'
---
Introduction
============
Chance constraints (or probabilistic constraints) were among the first extensions proposed to linear programming as a natural formulation for treating constraints where some of the coefficients are uncertain at the time of optimization [@CharnesCooper]. In the chance constraint model, we suppose that the uncertain values follow a known distribution and enforce that the constraint holds with high probability as a function of the decision variables.
Nemirovski and Shapiro [@NS2007] observe that, in general, convexity and tractability results in chance constraints are a rare combination. When the corresponding deterministic constraint is convex, the chance constraint may be nonconvex. And even for those chance constraints which are in fact convex, the authors [@NS2007] cite examples where such constraints remain computationally intractable because it is NP-Hard to test if the constraint is satisfied. For linear chance constraints of the form $$\label{eq:onesidechance}
\mathbb{P}(x^T\xi \le b) \ge 1-\epsilon,$$ where $x \in \mathbb{R}^n$ and $b \in \mathbb{R}$ are decision variables, the constraint is known to be convex (that is, the set $\{ (x,b) : \mathbb{P}(x^T\xi \le b) \ge 1-\epsilon\}$ is convex) and computationally tractable when $\xi$ has an *elliptical log-concave* distribution [@Lagoa05], examples of which include the multivariate Gaussian distribution and few others. The computational challenges presented by chance constraints have motivated approximation schemes [@NS2007] and alternative formulations such as robust optimization [@BenTalNemirovskiRobust2000].
Even more challenging than linear chance constraints, *joint chance constraints* require that a set of linear constraints hold jointly with high probability. Prékopa [@PrekopaBook] reviews many of the standard results. In particular, he proves convexity of the constraint $\mathbb{P}(x \ge \xi) \ge 1-\epsilon$ with respect to $x\in \mathbb{R}^n$ when $\xi$ follows a multivariate continuous log-concave distribution and of the constraint $\mathbb{P}(Tx \ge 0) \ge 1-\epsilon$ when some elements of the matrix $T$ are random with a joint Gaussian distribution and have a specialized covariance structure between the rows of $T$ (further generalized by [@Copulas]). Van Ackooij et al. [@VanAckooij10] consider *rectangular* chance constraints of the form $\mathbb{P}(a \le \xi \le b) \ge 1-\epsilon$ with respect to vectors $a$ and $b$ where $\xi$ follows a multivariate Gaussian distribution. Their model does not allow for products between random variables and decision variables.
The basic model we consider in this work, which is a special case of a joint chance constraint, is the two-sided chance constraint $$\label{eq:twosideintro}
\mathbb{P}(a \le x^T\xi \le b) \ge 1-\epsilon,$$ where $a \in \mathbb{R}, b \in \mathbb{R}$, and $x \in \mathbb{R}^n$ are decision variables, and $\xi$ is jointly Gaussian with known mean and covariance. In Section \[sec:cvx2side\], we prove that this constraint is in fact convex in $a$, $b$ and $x$ given $\epsilon \le \frac{1}{2}$. The proof, which we believe is the first, follows from a geometrical insight combined with standard tools for chance constraints such as log-concavity. The major methodological contributions of this work lie in the subsequent generalizations of the model and in our analysis of the computational tractability of the chance constraint. In Section \[sec:exact\_extensions\] we show that a number of seemingly more complex and nonlinear constraints can be formulated by using the two-sided constraint . In Section \[sec:tractability\], we demonstrate computational tractability of these constraints under a modern mathematical optimization lens. In addition to an exact derivative-based nonlinear formulation, we develop an approximate second-order cone (SOC) formulation for with provable approximation quality. This SOC formulation permits one to incorporate such constraints into large-scale models solvable by state-of-the-art commercial and open-source software.
Using as a primitive, we develop an approximation for the more challenging chance constraint $$\label{eq:quadintro}
\mathbb{P}((a^T\xi + b)^2 + (c^T\xi + d)^2 \le k) \ge 1-\epsilon,$$ where $a,c \in \mathbb{R}^n$, $b,d,k \in \mathbb{R}$ are decision variables, and $\xi$ is jointly Gaussian with known mean and covariance. This constraint is motivated by applications in power systems which we discuss in Section \[sec:motivation\]. In Section \[sec:quadapprox\], we study the constraint in detail and compare a number of approximation schemes, ultimately demonstrating that our approximation based on two-sided constraints is reasonable and of practical interest for its tractability.
Motivation {#sec:motivation}
==========
The basic question which motivates this work is the short-term planning problem, known as *optimal power flow* (OPF), which is solved as part of the real-time operation of the power grid to determine the minimum-cost production levels of controllable generators subject to reliably delivering electricity to customers across a large geographical area [@OPFreview; @BergenBook]. Conceptually, OPF is similar to a network flow problem with the additional complication that power flows according to the nonlinear Kirchhoff laws. On top of the nonlinear power flow laws, we aim to consider the uncertainty in production levels of renewable energy sources such as wind and solar photovoltaic.
In its traditional, deterministic form, OPF seeks to minimize total production costs $$\operatorname*{minimize}_{p,\theta,f} \sum_{i \in \mathcal{G}} c_{i}p_i\label{eq:Det_OPF}$$ $$\begin{gathered}
\sum_{n : \{b,n\} \in \mathcal{L}} f_{bn} - \sum_{m : \{m,b\} \in \mathcal{L}} f_{mb} = \sum_{i \in G_b} p_i + w_b - d_b, \quad \forall b \in \mathcal{B},\label{eq:balance} \\
\label{eq:gencapacity} p_{i}^{min} \leq p_i \leq p_i^{max}, \quad \forall i \in \mathcal{G},\\
\label{eq:flowdef} f_{mn} = \beta_{mn}(\theta_m - \theta_n), \quad \forall \{m,n\} \in \mathcal{L}, \\
\label{eq:Det_OPF_end} -f_{mn}^{max} \leq f _{mn} \leq f_{mn}^{max}, \quad \forall \{m,n\} \in \mathcal{L}, \end{gathered}$$ where $\mathcal{B}$ is the set of nodes (buses) in the grid, $\mathcal{G}$ is the set of generators, $G_b$ is the set of generators located at node $b$, and $\mathcal{L}$ is the set of edges (transmission lines). Decision variables $p_i$ denote the production levels of generator $i$, and the variables $f_{mn}$ denote the flow from node $m$ to node $n$. The value $d_b$ is the demand at each node (assumed to be known), and the value $w_b$ is the forecast production level from renewable energy sources (again assumed to be known). Constraint is the familiar flow balance constraint which balances supply with demand at each node. Constraints and enforce the capacities of the generators and transmission lines, respectively. The constraint links the flows to the bus angles $\theta$ and arises from the standard “DC” linearization of the nonlinear power flow laws; hence, this formulation is often called DCOPF. The formulation as stated above is efficiently solvable by linear programming on large-scale systems with tens of thousands of nodes within real-time operational constraints.
Our motivation is to address two major deficiencies in the standard DCOPF model. The first major deficiency is the deterministic nature of the model. In particular, the amount of power generated by renewable energy sources such as wind is highly variable and must be accounted for in short-term planning.
The line of work by [@ccopf-sirev; @JuMPChanceCaseStudy] addresses this deficiency by introducing chance constraints. More specifically, Bienstock et al. [@ccopf-sirev] propose to model the deviations from the forecast wind production levels as zero-mean Gaussian random variables $\boldsymbol\omega_b$, combined with a proportional response policy for the generators. Letting $\Omega$ be the total, real-time deviation from the forecast (a positive value if there is more renewable generation than expected), each generator has a proportional response coefficient $\alpha_i$ and adjusts its real-time production to match $p_i - \alpha_i\Omega$. If $\sum_i \alpha_i = 1$, then this response policy guarantees balance of supply and demand, although it does not guarantee that output capacities or transmission capacities are always satisfied. Both $p_i$ and $\alpha_i$ are decision varibles. Transmission capacities, in practice, are soft constraints, and hence [@ccopf-sirev] propose to enforce them as chance constraints $$\label{eq:chanceabsflow}
\mathbb{P}(|\boldsymbol f_{mn}| \le f_{mn}^{max}) \ge 1-\epsilon,$$ where $\boldsymbol f_{mn}$ is the random flow driven by the deviations $\boldsymbol\omega_b$. Bienstock et al. [@ccopf-sirev] then approximate by splitting it into two constraints $$\mathbb{P}(\boldsymbol f_{mn} \le f_{mn}^{max}) \ge 1-\epsilon \text{ and } \mathbb{P}(\boldsymbol f_{mn} \ge -f_{mn}^{max}) \ge 1-\epsilon,$$ both of which can be expressed as simple linear Gaussian chance constraints . The assumption that deviations from the forecast follow a Gaussian distribution is made for tractability. This assumption can be further refined, with practical gain, without loss of tractability by introducing uncertainty sets on the parameters of the Gaussian distribution [@JuMPChanceCaseStudy].
The second major deficiency in the standard DCOPF model is the crude approximation it provides of the true, nonlinear, nonconvex power flow laws. In particular, the linearized model assumes constant voltage and therefore neglects so-called *reactive* power flow, which is the imaginary component of complex-valued power flow. The real component is referred to as *active* power. Although we cannot directly treat the nonconvex case, we propose to consider more accurate linearizations which account for reactive power, such as those which arise from linearizing around a current operating solution [@bolognani2015fast]. When extending the model of [@ccopf-sirev] to account for reactive power, we obtain chance constraints of the form $$\label{eq:quadpower}
\mathbb{P}((\boldsymbol f_{mn}^{active})^2 +(\boldsymbol f_{mn}^{reactive})^2 \le (f_{mn}^{max})^2) \ge 1-\epsilon,$$ because transmission capacities are limited by the magnitude of the complex-valued power flow across a line.
Our first attempt at studying the constraint led us to study the simpler two-sided form . These results, in turn, provided us with a means to approximate , as we discuss in Section \[sec:quadapprox\]. The approximation we derive here has already yielded a practical implementation in the JuMPChance modeling package [@JuMPChance] which is being used to study the value of the model we propose in ongoing work [@working].
Convexity of two-sided Gaussian linear chance constraints {#sec:cvx2side}
=========================================================
The main result in this section is the convexity of the two-sided chance constraint .
Let $\varphi(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$ be the standard Gaussian density and $\Phi(x) = \int_{-\infty}^x \varphi(t)\,dt$ the Gaussian integral.
Let $\xi \sim N(0,1)$ be a standard Gaussian random variable. Let $\epsilon \in (0,1)$. We define the set $S_\epsilon := \{ (x,y) \in \mathbb{R}^2 : \mathbb{P}(x \le \xi \le y) \ge 1-\epsilon \}.$
Note that $S_\epsilon$ has two equivalent representations as $\{ (x,y) : \int_x^y \varphi(t)\, dt \ge 1-\epsilon \}$ and $\{ (x,y) : \Phi(y) - \Phi(x) \ge 1-\epsilon \}$.
We will proceed to prove that $S_\epsilon$ is convex, but first we define *log-concavity* and recall some basic properties. See Boyd [@BoydBook] for further discussion and proofs of these properties.
A non-negative function $f : \mathbb{R}^n \to \mathbb{R}$ is log-concave if $\forall\, x,y \in \operatorname{dom} f \text{ and } \lambda \in (0,1)$ $$f(\lambda x + (1-\lambda)y) \geq f(x)^\lambda f(y)^{1-\lambda}.$$
For strictly positive functions $f$, this definition is equivalent to the condition that $\log f$ is concave. It is easy to verify, therefore, that the Gaussian density $\varphi$ is log-concave. Lemma \[lem:logconcave\] recalls basic properties of log-concave functions.
\[lem:logconcave\] The following properties hold for log-concave functions:
- If $f : \mathbb{R}^n \to \mathbb{R}$ and $g : \mathbb{R}^n \to \mathbb{R}$ are log-concave, then the product $h(x) = f(x)g(x)$ is log-concave.
- If $f : \mathbb{R}^n \to \mathbb{R}$ is the indicator function of a convex set, then $f$ is log-concave.
- If $f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ is log concave, then $g(x) = \int f(x,y)\,dy$ is log-concave on $\mathbb{R}^n$.
See Boyd [@BoydBook].
With these basic properties, we can proceed to prove the following lemma.
\[lem:seps\] The set $S_\epsilon$ is convex.
Let $I(s,r) = 1$ if $s \le r$ and zero otherwise. That is, $I$ is the indicator function for the convex set $\{(a,b) : a \le b\}$. Therefore the function $g(t,x,y) = \varphi(t)I(t,y)I(x,t)$ is log concave, because it is a product of log concave functions. Then for $y \geq x, f(x,y) = \int_x^y \varphi(t)\,dt = \int \varphi(t)I(t,y)I(x,t)\,dt$ is log concave, because it is the marginal of a log concave function. Hence $S_\epsilon$ is convex because it is an upper level set of a log-concave function.
Convexity of $S_\epsilon$ proves convexity of the very simple chance constraint $\mathbb{P}(x \le \xi \le y) \ge 1-\epsilon$ for all $\epsilon \in (0,1)$ with respect to $(x,y) \in \mathbb{R}^2$. Note that this convexity result is a special case of the rectangular constraints considered by [@VanAckooij10]. In order to account for products between the decision variables and the random variables, we require the following additional developments.
Let $\bar S_\epsilon = \operatorname{cl} \{ (x,y,z) : (x/z,y/z) \in S_\epsilon, z > 0 \}$ be the conic hull of $S_\epsilon$ (where $\operatorname{cl}$ is the closure operator).
By standard results [@HiriartLemarechal93book2], $\bar S_\epsilon$ is convex. The following lemma, in which we prove monotonicity properties of the set $\bar S_\epsilon$, is key to our main result.
\[lem:monotonicity\] Let $\epsilon \in (0,\frac{1}{2}]$. Then $(x,y,z) \in \bar S_\epsilon$ iff $z \ge 0$ and $\exists\, x' \ge x, y' \le y,$ and $z' \ge z$ such that $(x',y',z') \in \bar S_\epsilon$.
Suppose we are given $(x',y',z') \in \bar S_\epsilon$ and $(x,y,z)$ with $x \le x', y \ge y',$ and $0 < z \le z'$. We will show that $(x,y,z) \in \bar S_\epsilon$. By symmetry of the Gaussian density and $\epsilon \le \frac{1}{2}$, $(x',y',z') \in \bar S_\epsilon$ implies $x' < 0$ and $y' > 0$, so $x/z \le x'/z \le x'/z'$ and $y/z \ge y'/z \ge y'/z'$. By increasing the upper limit of integration or decreasing the lower limit of integration, we can only increase the value of the integral, so $$\int_{x/z}^{y/z} \phi(t)\, dt \ge \int_{x'/z'}^{y'/z'} \phi(t)\, dt \ge 1-\epsilon.$$ For the case of $z=0$, take a sequence of decreasing iterates $z_1 = z', z_2, z_3, \ldots$ with $z_i \to 0$. For each $i$, the above argument shows $(x,y,z_i) \in \bar S_\epsilon$, which implies $(x,y,0) \in \bar S_\epsilon$ since $\bar S_\epsilon$ is a closed set.
With these properties, we now prove the main result of this section.
\[thm:chance\] Let $\xi$ be a vector of $n$ i.i.d. standard Gaussian random variables, $0 < \epsilon \le \frac{1}{2}$ and $${C}:= {\left\{(a,b,x) \in \mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\,:\, \mathbb{P}(a \le x^T\xi \le b) \geq 1 - \epsilon \right\}}.$$ Then ${C}$ is a projection of the convex set $${\left\{(a,b,x,t) \in \mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\times \mathbb{R}\,:\, ||x||_2 \leq t,\quad (a,b,t) \in \bar S_\epsilon \right\}},$$ and hence ${C}$ is convex.
$$\mathbb{P}(a \leq x^T\xi \leq b) \geq 1-\epsilon$$
iff $$\label{eq:probrange}
\mathbb{P}\left(\frac{a}{||x||_2} \leq \frac{x^T\xi}{||x||_2} \leq \frac{b}{||x||_2}\right) \geq 1-\epsilon$$ iff $$\label{eq:transform_to_s_eps}
(a,b,||x||_2) \in \bar S_\epsilon$$ iff (by Lemma \[lem:monotonicity\]) $$\exists\, t \ge ||x||_2 \text{ such that } (a,b,t) \in \bar S_\epsilon.$$
Where the equivalence between and holds because $\frac{x^T\xi}{||x||_2}$ is a standard Gaussian random variable. The above proof assumes $x \neq 0$. For the case of $x = 0$, $$\mathbb{P}(a \le x^T\xi \le b) \geq 1 - \epsilon$$ iff $$a \le 0 \le b$$ iff $$(a,b,0) \in \bar S_\epsilon.$$ The justification for the final equivalence is as follows. If the strict inequality $a < 0 < b$ holds, then $\lim_{z \to 0+} \int_{a/z}^{b/z} \varphi(t)\, dt = 1$, so membership holds in $\bar S_\epsilon$. If $a=0$, $b=0$, or both, then we can construct a sequence of points $(a_i,b_i,0) \to (a,b,0)$ with each $(a_i,b_i,0) \in \bar S_\epsilon$, so the statement holds because $\bar S_\epsilon$ is closed.
More generally,
\[lem:generalcc\] Let $\xi \sim N(\mu,\Sigma)$ be a jointly distributed Gaussian random vector with mean $\mu$ and positive definite covariance matrix $\Sigma$ and $0 < \epsilon \le \frac{1}{2}$, and let $${C}_{\mu,\Sigma} := {\left\{(a,b,x) \in \mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\,:\, \mathbb{P}(a \le x^T\xi \le b) \geq 1 - \epsilon \right\}}.$$ Then ${C}_{\mu,\Sigma}$ convex.
Let $LL^T = \Sigma$ be the Cholesky decomposition of the covariance matrix $\Sigma$. Then $\xi = L\zeta + \mu$ where $\zeta$ is a vector of i.i.d. standard Gaussian random variables. The point $(a,b,x)$ satisfies $$\mathbb{P}(a \leq x^T\xi \leq b) \geq 1-\epsilon$$ iff $$\mathbb{P}(a \leq x^T(L\zeta + \mu) \leq b) \geq 1-\epsilon$$ iff $$(a-\mu^Tx,b-\mu^Tx,L^Tx) \in C.$$ That is, the set ${C}_{\mu,\Sigma}$ is an affine transformation of the convex set $C$ representing the i.i.d. case, and hence ${C}_{\mu,\Sigma}$ is convex.
Exact extensions of two-sided constraints {#sec:exact_extensions}
=========================================
In this section, we generalize the basic result in Section \[sec:cvx2side\] to a number of cases in which a seemingly more complex chance constraint can be represented exactly by using two-sided chance constraints.
Nonlinear chance constraints
----------------------------
The simplest nonlinear constraint we consider, which will be used in formulating the approximation of the quadratic chance constraint in Section \[sec:quadapprox\], is the absolute value constraint.
\[eq:absconvex\] Let $\xi \sim N(\mu,\Sigma)$ be a jointly distributed Gaussian random vector with mean $\mu$ and positive definite covariance matrix $\Sigma$ and $0 < \epsilon \le \frac{1}{2}$. Then the set $$\{ (a,b,x) \in \mathbb{R}\times\mathbb{R}\times\mathbb{R}^n: \mathbb{P}(|x^T\xi + a| \leq b) \geq 1 - \epsilon \}$$
is convex.
$\mathbb{P}(|x^T\xi + a| \leq b) \geq 1 - \epsilon$ $\mathbb{P}(-b - a \le x^T\xi \le b - a) \geq 1 - \epsilon$.
The above lemma is a special case of the following significantly more general theorem:
\[thm:nonlinearchance\] Let $f: \mathbb{R} \to \mathbb{R}$ be a convex function which attains its minimum at $x = c$, let $g : \mathbb{R}^m \to \mathbb{R}$ be an arbitrary convex function, and let $\xi$ be a standard Gaussian random vector (without loss of generality, we can assume independence and zero mean). Let $\epsilon \le \frac{1}{2}$. Then the set $$\label{eq:nonlinconstr}
D := \left\{ (x,z,b) \in \mathbb{R}^n\times\mathbb{R}^m\times \mathbb R : \mathbb{P}(f(x^T\xi + b) + g(z) \leq 0) \geq 1-\epsilon \right\}$$ is a projection of the convex set $$\begin{aligned}
\{ (x,z,b,k,x',y',t) \in \mathbb{R}^{n+m+5}: &t \ge ||x||_2, k \le -g(z),
x' \ge l(k) - b - c, \\&y' \le u(k) - b - c,
(x',y',t) \in \bar S_\epsilon \}\end{aligned}$$ where $l$ and $u$, are explicitly computable convex and concave functions, respectively, which we define below depending on $f$. And hence, $D$ is convex.
Let $l(k)$ and $u(k)$ be functions such that $f(x-c) \leq k$ iff $x \in [l(k),u(k)]$. We can obtain $l$ and $u$ by shifting the graph of $f$ so that the minimum is at zero and then reflecting the graph along $y = x$, and since $f(\cdot-c)$ is decreasing up to zero and increasing after zero, we have in particular that $u(k)$ is concave and increasing and $l(k)$ is convex and decreasing. Then $$\mathbb{P}(f(x^T\xi + b) + g(z) \leq 0) \geq 1-\epsilon$$ iff $$\mathbb{P}(l(-g(z)) \leq x^T\xi + b + c \leq u(-g(z))) \geq 1-\epsilon$$ iff $$\mathbb{P}(l(-g(z)) - b - c \leq x^T\xi \leq u(-g(z)) - b - c) \geq 1-\epsilon$$ iff (by Theorem \[thm:chance\]) $$\label{eq:nonlinear_extended}
\exists\, t \ge ||x||_2 \text{ and } x' \ge l(-g(z)) - b - c \text{ and } y' \le u(-g(z)) - b - c \text{ such that } (x',y',t) \in \bar S_\epsilon.$$ Finally, $$x' \ge l(-g(z)) - b - c \text{ and } y' \le u(-g(z)) - b - c$$ iff (by $l$ decreasing and $u$ increasing) $$\label{eq:nonlinear_extended2}
\exists\, k \le -g(z) \text{ such that } x' \ge l(k) - b - c \text{ and } y' \le u(k) - b - c.$$ Since $l(k)$ is convex and $u(k)$ is concave, conditions and give a convex formulation, in an extended set of variables, for the chance constraint .
Theorem \[thm:nonlinearchance\] is sufficiently general to shed light on the quadratic chance constraint which motivated our original work. If one of the terms in the chance constraint is deterministic, then the constraint is indeed convex, as the following lemma shows. This simpler form of the quadratic constraint itself can be useful for the motivating application in power systems, if, for example, the reactive power flow across a transmission line is not subject to randomness.
Let $\xi \sim N(\mu,\Sigma)$ be a jointly distributed Gaussian random vector with mean $\mu$ and positive definite covariance matrix $\Sigma$ and $0 < \epsilon \le \frac{1}{2}$. Then the set $$\{ (x,b,k,z) \in \mathbb{R}^n\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}: \mathbb{P}((x^T\xi + b)^2 + z^2 \leq k) \geq 1 - \epsilon \}$$
is convex.
Set $f(y) = y^2$, $g(z,k) = z^2 - k$ and apply Theorem \[thm:nonlinearchance\].
A similar proof technique as used in Theorem \[thm:nonlinearchance\] can also be applied in other cases. In the following lemma, we demonstrate convexity of the quadratic chance constraint in another special case when the random variable is univariate.
Let $\xi$ be a scalar standard Gaussian random variable and let $\epsilon \in (0, 1)$. Then the set $$\left\{ (b,d,k) \in \mathbb{R}^3 : \mathbb{P}((\xi+b)^2 +(\xi+d)^2 \leq k) \geq 1-\epsilon \right\}$$ is convex.
By applying the quadratic formula, we see that $(\xi+b)^2 +(\xi+d)^2 \leq k$ iff $\xi \in [l(b,d,k),u(b,d,k)]$ where $$l(b,d,k) = \frac{1}{2}\left(-(d+b) - \sqrt{2k - (d-b)^2}\right)$$ and $$u(b,d,k) = \frac{1}{2}\left(-(d+b) + \sqrt{2k - (d-b)^2}\right).$$ By analogy with the proof of Theorem \[thm:nonlinearchance\], it suffices to show that $l$ is convex and $u$ is concave. To prove this, it suffices to show that $\sqrt{2k - (d-b)^2}$ is concave, which holds since $\sqrt{\cdot}$ is concave increasing and $2k-(d-b)^2$ is concave. Note we allow $\epsilon > \frac{1}{2}$ because this proof requires only monotonicity properties of $S_\epsilon$, not $\bar S_\epsilon$.
We also observe that when $f$ and $g$ in Theorem \[thm:nonlinearchance\] are piecewise linear, e.g., as in Lemma \[eq:absconvex\], then we have demonstrated the convexity of a special family of joint linear chance constraints.
Distributionally robust two-sided chance constraints {#sec:distrobust}
----------------------------------------------------
So far we have left unquestioned the assumption that the parameters $\mu$ and $\Sigma$ of the Gaussian distribution are known with certainty, when often they are subject to measurement error. For the case of linear chance constraints, Bienstock et al. [@ccopf-sirev] propose a tractable model that enforces robustness with respect to deviations of the parameters $\mu$ and $\Sigma$ within a known uncertainty set $U$. Lubin et al. [@JuMPChanceCaseStudy] implement this model and demonstrate significant cost savings in the context of short-term operational planning of power systems when tested against out-of-sample realizations of uncertainty. Here, we define and demonstrate tractability of a similar distributionally robust model in the context of two-sided chance constraints.
Let $\xi \sim N(\mu,\Sigma)$ be a jointly distributed Gaussian random vector with mean $\mu$ and positive definite covariance matrix $\Sigma$ and $0 < \epsilon \le \frac{1}{2}$, and let $LL^T = \Sigma$ be the Cholesky decomposition of $\Sigma$.
From Lemma \[lem:generalcc\], recall $$\mathbb{P}(a \le x^T\xi \le b) \geq 1 - \epsilon$$ iff $$\exists\, t \ge ||L^T x||_2 \text{ such that } (a-\mu^T x,b - \mu^T x,t) \in \bar S_\epsilon.$$
We define the *distributionally robust* (or *ambiguous*) two-sided chance constraint as: $$\label{eq:robustcc}
\mathbb{P}_{\xi \sim N(\mu,\Sigma)}(a \le x^T\xi \leq b) \geq 1 - \epsilon \quad \forall (\mu,\Sigma) \in U$$
For $\epsilon \leq \frac{1}{2}$ and under the assumption that the uncertainty set decomposes by $\mu$ and $\Sigma$, i.e., $U = U_\mu \times U_\Sigma$, then the constraint is tractable if we can tractably optimize a linear objective over the sets $U_\mu$ and $U_\Sigma$.
Note that is a convex constraint, because it is the intersection of (infinitely) many convex constraints. We will prove tractability by demonstrating that we can easily separate, i.e., find the worst-case $\mu$ and $\Sigma$ given $(a,b,c)$.
We have that holds iff $\exists\, t$ s.t. $$\begin{aligned}
t &\geq ||L^T_{\Sigma}x|| &\forall \Sigma \in U_\Sigma \label{eq:conicrobust}\\
(a-\mu^Tx,b-\mu^Tx,t) &\in \bar S_\epsilon &\forall \mu \in U_\mu \label{eq:cdfrobust}\end{aligned}$$ Constraint can be reformulated as $t \geq \sqrt{\max_{\Sigma \in U_\Sigma}x^T\Sigma x}$, so we can separate by optimizing a linear objective over $U_\Sigma$. For $t>0$, the separation problem corresponding to constraint is $$\label{eq:cdfseparate}
\min_{\mu\in U_\mu}\Phi((b-\mu^Tx)/t) - \Phi((a-\mu^Tx)/t),$$ which is a minimization of a log-concave function. However, observe that it is essentially a one-dimensional problem depending on $\mu^Tx$, so it can be solved by testing with the values $\min_{\mu \in U_\mu} \mu^Tx$ and $\max_{\mu \in U_\mu} \mu^Tx$.
Computational tractability of $S_\epsilon$ and $\bar S_\epsilon$ {#sec:tractability}
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We have demonstrated applications of the set $S_\epsilon$ and its conic hull $\bar S_\epsilon$ to represent a number of classes of convex chance constraints. So far, we have used these sets as a theoretical tool in order to prove convexity. In practice, we are also interested in computationally tractable representations of these sets, which ideally can be used within off-the-shelf solvers.
Representation using convex functions {#sec:nlp_representation}
-------------------------------------
Recall from Lemma \[lem:seps\] that the function $f(x,y) = \int_x^y \varphi(t)\,dt = \Phi(y)-\Phi(x)$ is log-concave. Therefore the equivalence $(x,y) \in S_\epsilon$ iff $\log f(x,y) \ge \log(1-\epsilon)$ provides a representation which can essentially be used directly within derivative-based nonlinear solvers, which expect constraints in the form $f(x) \le 0$ where $f$ is smooth and convex. Furthermore, the *perspective function* $$g(x,y,z) = z(\log(\Phi(y/z)-\Phi(x/z))-\log(1-\epsilon))$$ is concave [@HiriartLemarechal93book2], and one can see that $(x,y,z) \in \bar S_\epsilon$ iff $g(x,y,z) \ge 0$ and $z \ge 0$, which provides a potentially useful representation of $\bar S_\epsilon$.
The above representations are valid for the interval $\epsilon \in (0,1)$. For the special case of $\epsilon \in (0,\frac{1}{2})$, we note that for $x > 0$, $\Phi(x)$ is concave, and for $x < 0$, $\Phi(x)$ is convex. Since $\epsilon < \frac{1}{2}$ and $f(x,y) = \Phi(y)-\Phi(x) \ge 1-\epsilon$ imply $x < 0$ and $y > 0$, we note that $f$ itself is concave over the domain of $S_\epsilon$ because it is a sum of two concave functions. This observation provides an alternative convex representation of $(x,y,z) \in \bar S_\epsilon$ with the constraints $$\label{eq:convexfunc}
z(\Phi(y/z)-\Phi(x/z) - (1-\epsilon)) \ge 0 \text{ and } z \ge 0.$$ With either convex representation, the derivatives are easy to compute when $z > 0$. However, derivative-based solvers may fail as $z \to 0$.
Separation oracles
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A functional, derivative-based representation of $\bar S_\epsilon$ may be directly applicable in many situations, but alternative solution methods exist. For example, algorithms for convex mixed-integer nonlinear optimization typically make use of a combination of continuous nonlinear relaxations and iteratively generated polyhedral outer approximations [@StubbsMehrotra99; @Bonmin]. In this section we discuss how separation oracles could be implemented to generate such polyhedral outer approximations. Our focus is on developing separation oracles which lead to polyhedral approximations which are “better” than the more common approach which follows from the functional representation of Section \[sec:nlp\_representation\], in the sense of producing hyperplanes which are tangent to the set $\bar S_\epsilon$.
In brief, when a separation oracle for the set $S_\epsilon$ is given a point $(x,y)$, it first determines if $(x,y) \in S_\epsilon$. If $(x,y) \not \in S_\epsilon$, it returns a hyperplane $(a,b) \in \mathbb{R}^2 \times \mathbb{R}$ such that $a_1x + a_2y > b$ and $S_\epsilon$ is contained in the halfspace defined by $\{ (x,y) : a_1 x + a_2 y \le b\}$. Hence, the hyperplane separates the point $(x,y)$ from the set $S_\epsilon$.
First we note that a separation oracle for $S_\epsilon$ immediately provides a separation oracle for the conic hull $\bar S_\epsilon$. Suppose $(x,y,z) \not \in \bar S_\epsilon$ and $z > 0$. Then $(x/z,y/z) \not \in S_\epsilon$ so we take a hyperplane $a_1 x + a_2 y = b$ which separates $(x/z,y/z)$ from $S_\epsilon$, then the hyperplane $a_1 x + a_2 y - b z = 0$ separates $(x,y,z)$ from $\bar S_\epsilon$. If $z = 0$, note that, assuming $\epsilon \le \frac{1}{2}$, $(x,y,0) \in \bar S_\epsilon$ iff $y \ge 0$ and $x \le 0$, so these two constraints serve as the separating hyperplanes in this case. Thus we restrict our discussion to separation oracles for $S_\epsilon$.
The most straightforward separation oracle for a convex set described by a smooth convex function is as follows. For any smooth, convex function $f$, if we are given $x'$ with $f(x') > 0$, then the hyperplane $f(x') + \nabla f(x')(x-x') \le 0$ separates $x'$ from the feasible set of $\{ x : f(x) \le 0 \}$ [@Bonmin][^1]. For the case of $S_\epsilon$, however, this hyperplane is weak. More specifically, we have $$\label{eq:sepnlp}
(x,y) \in S_\epsilon \text{ iff } f(x,y) := 1-\epsilon-\Phi(y)+\Phi(x) \le 0.$$
If we use this representation to separate the point $(0,0)$, then $f(0,0) = 1-\epsilon$ and $\nabla f(0,0) = (\frac{1}{\sqrt{2\pi}},-\frac{1}{\sqrt{2\pi}})$, and our separating hyperplane is $$\label{eq:badhyperplane}
x-y \le -\sqrt{2\pi}(1-\epsilon).$$ Figure \[fig:sepnlp\] shows the set $S_\epsilon$ together with this separating hyperplane. Observe that the hyperplane is not tangent to $S_\epsilon$, which means that it may serve poorly as an outer approximation.
![In blue, the set $S_\epsilon$, with $\epsilon=0.05$. In orange, the half space corresponding to the separating hyperplane . The hyperplane separates the point $(0,0)$ but is not tangent to $S_\epsilon$.[]{data-label="fig:sepnlp"}](derivcut)
Instead of using this hyperplane, we might consider computing the best possible separating hyperplane with the same slope by evaluating $\max_{(x,y) \in S_\epsilon} x - y$. By a symmetry argument which we omit here, this value is $2 \Phi^{-1}(\epsilon/2)$, so we can strengthen the previous hyperplane to $$\label{eq:betterhyperplane}
x-y \le 2 \Phi^{-1}(\epsilon/2).$$
More generally, the *support function* $\sigma_{S_\epsilon}(a,b) = \max_{(x,y)\in S_\epsilon} ax+by$ enables one to compute the best possible separating hyperplane of a given slope [@HiriartLemarechal93book2]. Another approach to generating tangent separating hyperplanes is to compute an orthogonal projection of the point $(x,y)$ onto the set $S_\epsilon$ and then add a hyperplane which is tangent to the projected point. In the following section, we provide simple representations of the support function and orthogonal projection operators for the set $S_\epsilon$. These developments may enable practical implementations.
Support function of $S_\epsilon$ and orthogonal projection onto $S_\epsilon$
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We begin with a lemma which characterizes the boundary of $S_\epsilon$.
Let $\epsilon \in (0,\frac{1}{2})$. Then the point $(x,y)$ lies on the boundary of the set $S_\epsilon$, i.e., $\Phi(y)-\Phi(x) = 1-\epsilon$ iff $\exists\, \lambda \in (0,1)$ such that $x = \Phi^{-1}(\lambda\epsilon)$ and $y = \Phi^{-1}(1-(1-\lambda)\epsilon)$.
First, let $\lambda \in (0,1)$. Then $\Phi(\Phi^{-1}(1-(1-\lambda)\epsilon)) - \Phi(\Phi^{-1}(\lambda\epsilon)) = (1-(1-\lambda)\epsilon) - \lambda\epsilon = 1-\epsilon$, and so the point $(\Phi^{-1}(\lambda\epsilon),\Phi^{-1}(1-(1-\lambda)\epsilon))$ lies on the boundary. In the other direction, suppose the point $(x,y)$ is on the boundary of $S_\epsilon$. Set $\lambda = \Phi(x)/\epsilon$. Note that since $x$ and $y$ are finite, we must have $0 < \Phi(x) < \epsilon$ and $1-\epsilon < \Phi(y) < 1$, and hence $0 < \lambda < 1$. Trivially $x = \Phi^{-1}(\lambda\epsilon)$. Then $\Phi(y) = 1-\epsilon+\Phi(x)$, and we see that $y = \Phi^{-1}(1-\epsilon+\lambda\epsilon) = \Phi^{-1}(1-(1-\lambda)\epsilon)$.
This result provides an explicit univariate parameterization of the boundary of $S_\epsilon$ in terms of $\lambda$, which is quite useful for computational purposes. For example, suppose we wanted to minimize a function $g(x,y)$ along the boundary of $S_\epsilon$. Then this problem can be formulated as a one-dimensional search problem, $$\label{eq:optboundary}
\min_{\lambda \in (0,1)} g(\Phi^{-1}(\lambda\epsilon), \Phi^{-1}(1-(1-\lambda)\epsilon)).$$
The following lemma uses the formulation to demonstrate that optimization of some linear functions over $S_\epsilon$ can be expressed as a univariate convex optimization problem.
\[lem:linear\] Suppose $g(x,y) = ax+by$ with $a < 0$ and $b > 0$. Then is a smooth, strictly convex optimization problem.
Define $h(\lambda) := g(\Phi^{-1}(\lambda\epsilon), \Phi^{-1}(1-(1-\lambda)\epsilon))$. We explicitly calculate the derivatives using the following basic formulas:
$$\frac{d}{dx} \Phi^{-1}(x) = \sqrt{2\pi} e^{\frac{\Phi^{-1}(x)^2}{2}}$$ $$\frac{d^2}{dx^2} \Phi^{-1}(x) = 2\pi\Phi^{-1}(x) e^{\Phi^{-1}(x)^2}$$
so $$\frac{d^2}{d\lambda^2} h(\lambda) = 2\pi a\epsilon^2 \Phi^{-1}(\lambda\epsilon) e^{\Phi^{-1}(\lambda\epsilon)^2} + 2\pi b\epsilon^2\Phi^{-1}(1-(1-\lambda)\epsilon) e^{\Phi^{-1}(1-(1-\lambda)\epsilon)^2}.$$
Note $\Phi^{-1}(\lambda\epsilon) < 0$ and $\Phi^{-1}(1-(1-\lambda)\epsilon) > 0$, so given $a < 0$ and $b > 0$, we have that $\frac{d^2}{d\lambda^2} h(\lambda) > 0$.
Following Lemma \[lem:linear\] we have an efficient way to evaluate the support function $$\sigma_{S_\epsilon}(a,b) = \max_{(x,y)\in S_\epsilon} ax+by.$$ Specifically, when $a > 0$ and $b < 0$, we solve a one-dimensional convex minimization problem. If $a < 0$ or $b > 0$, then $\sigma_{S_\epsilon}(a,b) = \infty$. If $a = 0$ and $b < 0$, $\sigma_{S_\epsilon}(a,b) = b\Phi(1-\epsilon)$. If $b = 0$ and $a > 0$, $\sigma_{S_\epsilon}(a,b) = a\Phi(\epsilon)$. These last two cases follow from taking the limit when $\lambda = 0$ and $\lambda = 1$, respectively.
We can compute an orthogonal projection onto $S_\epsilon$ by solving a one-dimensional strictly convex minimization problem.
Similar to Lemma \[lem:linear\], we will use the parameterization of the boundary, but solving over a restricted domain. Given $(a,b) \not \in S_\epsilon$, the orthogonal projection is the solution to with $g(x,y) = \frac{1}{2}(x-a)^2+\frac{1}{2}(y-b)^2$. Actually we do not need to optimize over all $\lambda \in (0,1)$; note that the orthogonal projection always lies on the boundary of $S_\epsilon$ between the projections along the x and y axes. More specifically, we need only consider $$\label{eq:proj_domain}
\lambda \in \left(1-\frac{1}{\epsilon}(1-\Phi(b)), \frac{1}{\epsilon}\Phi(a)\right),$$ and within this interval, by construction, the inequalities $$\label{eq:proj_ineq}
\Phi^{-1}(\lambda\epsilon) \leq a \text{ and }\Phi^{-1}(1-(1-\lambda)\epsilon) \geq b$$ hold.
Define $h(\lambda) := \frac{1}{2}(\Phi^{-1}(\lambda\epsilon)-a)^2+\frac{1}{2}(\Phi^{-1}(1-(1-\lambda)\epsilon)-b)^2$. We will prove strict convexity of $h$ within the domain by showing that $\frac{d^2h}{d\lambda^2} > 0$. From the chain rule (for arbitrary $f$), $$\frac{d^2}{dx^2} \frac{1}{2}(f(x)-a)^2 = \left(\frac{df}{dx}(x)\right)^2 + (f(x)-a)\frac{d^2f}{dx^2}(x).$$ Discarding the squared first derivative terms, we have $$\frac{d^2h}{d\lambda^2}(\lambda) \geq (\Phi^{-1}(\lambda\epsilon)-a) \frac{d^2}{d\lambda^2} \Phi^{-1}(\lambda\epsilon) +
(\Phi^{-1}(1-(1-\lambda)\epsilon)-b) \frac{d^2}{d\lambda^2}(\Phi^{-1}(1-(1-\lambda)\epsilon)$$ The result follows from noting that $\frac{d^2}{d\lambda^2} \Phi^{-1}(\lambda\epsilon) < 0$ and $\frac{d^2}{d\lambda^2}(\Phi^{-1}(1-(1-\lambda)\epsilon) > 0$ combined with the inequalities .
An approximate polyhedral representation of $S_\epsilon$ {#sec:polyapprox}
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In this section, we develop an approximate polyhedral representation of $S_\epsilon$.
A polyhedron $P_\epsilon$ is an *outer approximation* of $S_\epsilon$ if $S_\epsilon \subset P_\epsilon$.
While polyhedral outer approximations are straightforward to generate, either through an iterative cutting-plane procedure or by preselecting a number of tangent hyperplanes, we are interested in outer approximations with a provable approximation guarantee, in the sense which we now define.
A family of polyhedral outer approximations $P_\epsilon$ forms an $\alpha$-approximation of $S_\epsilon$ if $\forall \epsilon \in (0,\frac{1}{2}]$, $$\Phi(y)-\Phi(x) \ge 1-\alpha\epsilon\quad \forall (x,y) \in P_\epsilon.$$ Or equivalently, when $\alpha\epsilon < 1$, $S_\epsilon \subset P_\epsilon \subset S_{\alpha\epsilon}$.
We restrict $\epsilon \le \frac{1}{2}$ for notational convenience and because this is the case of direct interest, although many of the results here generalize for $\epsilon \in (0,1)$.
Note that although our development is from the perspective of outer approximation, a family of polyhedral outer approximations may be used to generate conservative approximations as well, since if $P_\epsilon$ is an $\alpha$-approximation, then $$P_{\epsilon/\alpha} \subset S_\epsilon\quad \forall \epsilon \in (0,\frac{1}{2}].$$
We begin with a very simple 2-approximation of $S_\epsilon$ with the axis-aligned polyhedra: $$A_\epsilon = \{ (x,y) \in \mathbb{R}^2 : x \le \Phi^{-1}(\epsilon), y \ge \Phi^{-1}(1-\epsilon) \}.$$ For $(x,y) \in A_\epsilon$, by monotonicity of the cumulative density function $\Phi$ we conclude $$\Phi(y) - \Phi(x) \ge (1-\epsilon) - \epsilon \ge 1-2\epsilon.$$
This 2-approximation of $S_\epsilon$ is equivalent to representing $\mathbb{P}(a \le x^T\xi \le b) \ge 1-\epsilon$ by using the two standard linear chance constraints $\mathbb{P}(a \le x^T\xi) \ge 1-\epsilon$ and $\mathbb{P}(x^T\xi \le b) \ge 1-\epsilon$. Bienstock et al. [@ccopf-sirev] employ this approximation citing improved computational tractability.
The 2-approximation model is the best one can achieve with two linear constraints in the following sense. The set $S_\epsilon$ has two extreme rays: $(-1,0)$ and $(0,1)$, which follow from the fact that $\Phi$ is monotonic increasing. Therefore, any outer approximation of $S_\epsilon$ must contain these rays. If, in addition, these are not the extreme rays of the outer approximation, then the approximation *cannot* be an $\alpha$-approximation for any $\alpha$, because $S_{\alpha\epsilon}$ cannot contain the set.
![In blue, the set $S_\epsilon$. In orange, the polyhedral outer approximation $A_\epsilon$ (left) and $B_\epsilon$ (right). By adding a single additional inequality, we strengthen the relaxation significantly.[]{data-label="fig:twothreecut"}](2epsilon.pdf "fig:") ![In blue, the set $S_\epsilon$. In orange, the polyhedral outer approximation $A_\epsilon$ (left) and $B_\epsilon$ (right). By adding a single additional inequality, we strengthen the relaxation significantly.[]{data-label="fig:twothreecut"}](125epsilon.pdf "fig:")
The main result of this section is that with a single additional linear constraint, one may improve the above 2-approximation to a 1.25-approximation. The axis-aligned approximation performs poorly at the “corner” where $x = \Phi^{-1}(\epsilon)$ and $y = \Phi^{-1}(1-\epsilon)$. If we add a hyperplane to separate this point, from the previous discussion we obtain the hyperplane .
Therefore we define the family of polyhedra as $$\label{eq:3cutdefn}
B_\epsilon := \{ (x,y) \in \mathbb{R}^2 : x \le \Phi^{-1}(\epsilon), y \ge \Phi^{-1}(1-\epsilon), x-y \le 2 \Phi^{-1}(\epsilon/2) \}.$$ The family $B_\epsilon$ forms a valid outer approximation because $A_\epsilon$ is a valid family, and we’ve added a valid separating hyperplane. Figure \[fig:twothreecut\] displays the two families of approximations for a fixed $\epsilon$.
The following lemma simplifies the task of proving the $\alpha$-approximation.
For $\alpha < 2$, a family of polyhedral outer approximations $P_\epsilon$ forms an $\alpha$-approximation of $S_\epsilon$ iff $\forall \epsilon \in (0,\frac{1}{2}]$
1. $\forall$ vertices $(x,y)$ of $P_\epsilon,$ we have $\Phi(y) - \Phi(x) \ge 1-\alpha\epsilon$, and
2. the extreme rays of $P_\epsilon$ are $(-1,0)$ and $(0,1)$.
That is, it is sufficient to verify the approximation quality at the vertices.
Fix $\epsilon$ and suppose that the two above conditions hold. Then all vertices, by definition, are contained in the set $S_{\alpha\epsilon}$ (our assumptions imply $\alpha\epsilon < 1$). By convexity of $S_{\alpha\epsilon}$, this implies that all convex combinations of the vertices of $P_\epsilon$ are contained in $S_{\alpha\epsilon}$. All elements of the polyhedron $P_\epsilon$ can be represented as a convex combination of its vertices plus a conic combination of its extreme rays. Since the extreme rays $(-1,0)$ and $(0,1)$ are also extreme rays of $S_{\alpha\epsilon}$, it follows that $P_{\epsilon} \subset S_{\alpha\epsilon}$.
The vertices of $B_\epsilon$ are $(\Phi^{-1}(\epsilon),\Phi^{-1}(\epsilon)-2\Phi^{-1}(\epsilon/2))$ and $(2\Phi^{-1}(\epsilon/2)+\Phi^{-1}(1-\epsilon),\Phi^{-1}(1-\epsilon))$. By the identities $\Phi^{-1}(1-\epsilon) = -\Phi^{-1}(\epsilon)$ and $\Phi(-x) = 1-\Phi(x)$ we see that these vertices are in fact symmetric, so it is sufficient to consider only one of them.
We first establish a simple bound that does not use any deep properties of the Gaussian distribution.
The “three-cut” family of outer approximations $B_\epsilon$ forms a 1.5-approximation of $S_\epsilon$.
Consider the vertex $(\Phi^{-1}(\epsilon),\Phi^{-1}(\epsilon)-2\Phi^{-1}(\epsilon/2))$. It is sufficient to show that it is contained in the set $S_{1.5\epsilon}$. $$\begin{aligned}
\Phi(\Phi^{-1}(\epsilon)-2\Phi^{-1}(\epsilon/2)) - \Phi(\Phi^{-1}(\epsilon)) &=
\Phi((\Phi^{-1}(\epsilon)-\Phi^{-1}(\epsilon/2))-\Phi^{-1}(\epsilon/2)) - \epsilon\\
&> \Phi(-\Phi^{-1}(\epsilon/2)) - \epsilon \\
&= \Phi(\Phi^{-1}(1-\epsilon/2)) - \epsilon\\
&= 1-1.5\epsilon,\end{aligned}$$ where the inequality follows from $\Phi^{-1}(\epsilon) > \Phi^{-1}(\epsilon/2)$ and monotonicity of $\Phi$.
We can improve this bound by using properties of the Gaussian distribution. Lemmas \[lem:tail2\] and \[lem:4bound\] below develop the necessary properties, and Theorem \[thm:125\] states the final result.
\[lem:tail2\] $\Phi^{-1}(1-\frac{\epsilon}{2}) - \Phi^{-1}(1-\epsilon) \geq \Phi^{-1}(1-\frac{\epsilon}{4}) - \Phi^{-1}(1-\frac{\epsilon}{2})$ for $\epsilon \in (0,\frac{1}{2}]$.
Let $f(\epsilon) = \Phi^{-1}(1-\frac{\epsilon}{2}) - \Phi^{-1}(1-\epsilon)$. Then we intend to show $f(\epsilon) \geq f(\epsilon/2)\, \forall \epsilon \in (0,\frac{1}{2}]$. It suffices to show that $f$ is monotonic increasing over the interval.
Recalling $$\frac{d}{dx} \Phi^{-1}(x) = \sqrt{2\pi} \exp\left(\frac{\Phi^{-1}(x)^2}{2}\right),$$ we have $$f'(\epsilon)= -\frac{1}{2}\sqrt{2\pi}\exp\left(\frac{\Phi^{-1}(1-\frac{\epsilon}{2})^2}{2}\right)+\sqrt{2\pi}\exp\left(\frac{\Phi^{-1}(1-\epsilon)^2}{2}\right).$$ We will show that $f'$ is always positive for $\epsilon \in (0,\frac{1}{2}]$. At $\epsilon = \frac{1}{2}$, $\Phi^{-1}(\frac{1}{2}) = 0$, so $$f'\left(\frac{1}{2}\right) = \sqrt{2\pi} - \frac{1}{2}\sqrt{2\pi} \exp\left(\Phi^{-1}\left(\frac{3}{4}\right)^2/2\right) \approx 0.93 > 0.$$ Suppose, for contradiction, $f'(\epsilon') = 0$ for some $\epsilon'$. Then $$\exp\left(\frac{\Phi^{-1}(1-\epsilon')^2}{2}\right) = \frac{1}{2}\exp\left(\frac{\Phi^{-1}(1-\frac{\epsilon'}{2})^2}{2}\right)$$ which implies $$\label{eq:fprimezero}
\Phi^{-1}(1-\epsilon')^2 = -2\log(2) + \Phi^{-1}\left(1-\frac{\epsilon'}{2}\right)^2.$$
Note that $g(\epsilon) := \Phi^{-1}(1-\epsilon)^2$ is strictly convex for $\epsilon \in (0,1)$ by examination of the second derivative. This means that $g'(\epsilon)$ is strictly monotonic increasing. We’re looking for a solution to $g(\epsilon/2) - g(\epsilon) = 2\log(2)$. Note that $g(\epsilon/2) - g(\epsilon)$ is strictly decreasing over the interval because $(1/2)g'(\epsilon/2) - g'(\epsilon) < 0$. One can verify the limit $$\lim_{\epsilon\to 0+} \Phi^{-1}\left(1-\frac{\epsilon}{2}\right)^2 -\Phi^{-1}(1-\epsilon)^2 = 2\log(2),$$ which implies in fact that there can be no solution to . This proves our original claim.
\[lem:4bound\] $\Phi^{-1}(\epsilon) - 2\Phi^{-1}(\frac{\epsilon}{2}) \geq \Phi^{-1}(1-\frac{\epsilon}{4})$ for $\epsilon \in (0,\frac{1}{2}]$
Applying Lemma \[lem:tail2\], we have: $$\begin{aligned}
\Phi^{-1}(\epsilon) - 2\Phi^{-1}(\frac{\epsilon}{2}) &= \Phi^{-1}(1-\frac{\epsilon}{2}) + (\Phi^{-1}(1-\frac{\epsilon}{2}) - \Phi^{-1}(1-\epsilon))\\
&\geq \Phi^{-1}(1-\frac{\epsilon}{2}) + (\Phi^{-1}(1-\frac{\epsilon}{4}) - \Phi^{-1}(1-\frac{\epsilon}{2}))\\
&=\Phi^{-1}(1-\frac{\epsilon}{4})\end{aligned}$$
\[thm:125\] The “three-cut” family of outer approximations $B_\epsilon$ forms a 1.25-approximation of $S_\epsilon$.
Consider the vertex $(\Phi^{-1}(\epsilon),\Phi^{-1}(\epsilon)-2\Phi^{-1}(\epsilon/2))$. It is sufficient to show that it is contained in the set $S_{1.25\epsilon}$. $$\begin{aligned}
\Phi(\Phi^{-1}(\epsilon)-2\Phi^{-1}(\epsilon/2)) - \Phi(\Phi^{-1}(\epsilon)) &=\\
&\ge \Phi(\Phi^{-1}(1-\epsilon/4)) - \epsilon \\
&= 1-1.25\epsilon,\end{aligned}$$ where the inequality follows from Lemma \[lem:4bound\].
With an additionally highly technical argument which we omit for brevity, it is possible to show that the 1.25 value is tight; that is, the “three-cut“ family of outer approximations $B_\epsilon$ is *not* an $\alpha$-approximation for any $\alpha < 1.25$.
We summarize the results of this section with a succinct statement of an SOC outer approximation of the two-sided chance constraint based on $B_\epsilon$.
\[lem:generalcc\_1.25\] Let $\xi \sim N(\mu,\Sigma)$ be a jointly distributed Gaussian random vector with mean $\mu$ and positive definite covariance matrix $\Sigma$ and $0 < \epsilon \le \frac{1}{2}$. Let $LL^T = \Sigma$ be the Cholesky decomposition of $\Sigma$. The following extended formulation, with the additional variable $t$, $$\begin{aligned}
t \ge & ||L^Tx||_2,\label{eq:outer1}\\
a-\mu^Tx & \le \Phi^{-1}(\epsilon)t,\label{eq:outer2}\\
b - \mu^T x & \ge \Phi^{-1}(1-\epsilon)t,\label{eq:outer3}\\
a-b & \le 2 \Phi^{-1}(\epsilon/2)t.\label{eq:outer4}\end{aligned}$$ is an SOC outer approximation of the constraint $$\mathbb{P}(a \le x^T\xi \leq b) \geq 1 - \epsilon$$ which in fact guarantees $$\mathbb{P}(a \le x^T\xi \leq b) \geq 1 - 1.25\epsilon.$$
From Lemma \[lem:generalcc\], $$\mathbb{P}(a \leq x^T\xi \leq b) \geq 1-\epsilon$$ iff $$\exists\, t \ge ||L^T x||_2 \text{ such that } (a-\mu^T x,b - \mu^T x,t) \in \bar S_\epsilon.$$ We take the conic hull of the polyhedral representation $B_\epsilon$ of $S_\epsilon$ in order to represent $\bar S_\epsilon$.
Approximation of quadratic chance constraints {#sec:quadapprox}
=============================================
Having extensively discussed the tractability of the two-sided chance constraint model and its extensions to represent more complex nonlinear chance constraints exactly, we return to our original motivation as discussed in Section \[sec:motivation\]. In this section, we will investigate the use of two-sided chance constraints to *approximately* represent a family of challenging *quadratic chance constraints*. These sets are of the form, $$\label{eq:quadprob}
H_\epsilon = \left\{ (a,b,c,d,k) \in \mathbb{R}^n\times \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \times \mathbb{R} : \mathbb{P}((a^T\xi + b)^2 + (c^T\xi + d)^2 \le k) \ge 1-\epsilon \right\},$$ where $a$ and $c$, and $b$, $d$, and $k$ are (vector and scalar, resp.) decision variables and $\xi$ follows a multivariate Gaussian distribution with known mean and covariance matrix.
Convexity of the quadratic chance constraint
--------------------------------------------
We are unaware of any existing results on the convexity of the set $H_\epsilon$ . We present here a proof of nonconvexity for the case of $\epsilon = 0.455$. The counterexample, while not as strong as a proof of nonconvexity for *all* $\epsilon \in (0,\frac{1}{2}]$, suggests that convexity is, at the least, not a simple extension of existing results such as those presented in Section \[sec:exact\_extensions\] which hold for all $\epsilon \in (0,\frac{1}{2}]$. We leave the question of convexity of $H_\epsilon$ over the full range of $\epsilon$ for future work. Nevertheless, we take this counterexample as a justification for seeking tractable, convex approximations of $H_\epsilon$ in subsequent sections.
Consider the constraint $$\label{eq:nonconvex_example}
\mathbb{P}((x\xi_1)^2 + (y\xi_2)^2 \le 1) \ge 1-\epsilon,$$ where $\xi_1$ and $\xi_2$ are independent, standard Gaussian random variables. The constraint is a special case of with $\xi = (\xi_1,\xi_2)$, $a = (x,0), b = 0, c = (0,y), d = 0$, and $k = 1$.
Figure \[fig:nonconvex\_example\] traces the value of the left-hand side of along the line $y = -x + 1.6$. We see that the upper level sets of the function $f(x,y) = \mathbb{P}((x\xi_1)^2 + (y\xi_2)^2 \le 1)$ are not convex. In particular, the points $(0.6,1.0)$ and $(1.0,0.6)$ belong to $H_{0.455}$ (by numerical integration with reported error bounds of $10^{-7}$) but the point $(0.8,0.8)$, their average, does not. We can evaluate $\mathbb{P}((0.8\xi_1)^2 + (0.8\xi_2)^2 \le 1)$ more explicitly as $F_{\chi_2}(1/0.8) \approx 0.542$ where $F_{\chi_2}$ is the cumulative distribution function of the chi distribution with two degrees of freedom.
![On the vertical axis, the value of the left-hand side of evaluated at the point $(x,-x+1.6)$ by numerical integration (with approximate error bounds of $10^{-7}$). We see that the set of points along this line that satisfy the quadratic chance constraint with probability 0.545 or greater, for example, is not convex. This proves nonconvexity of $H_\epsilon$ with $\epsilon = 0.455$. []{data-label="fig:nonconvex_example"}](nonconvex_example)
Approximation using two-sided constraints
-----------------------------------------
We propose an approximation of the quadratic chance constraint by two absolute value constraints, essentially splitting up the squared terms into separate constraints. We use the union bound to enforce a conservative approximation, hence we introduce a parameter $\beta \in (0,1)$ to balance the trade-off between violations in the two separate constraints. In Lemma \[lem:quadapprox\] below, we state this formulation formally and prove that it is a valid convex, conservative approximation of in an extended set of variables.
\[lem:quadapprox\] Fix $\epsilon < \frac{1}{2}$, fix $\beta \in (0,1)$, and let
\[eq:quadapproxall\] $$\begin{aligned}
G_{\epsilon,\beta} = \biggl\{ (a,b,c,d,k, f_1,f_2) \in \mathbb{R}^{2n+5} :&
\mathbb{P}(|a^T\xi + b| \le f_1) \ge 1-\beta\epsilon \label{eq:abs1}\\
&\mathbb{P}(|c^T\xi + d| \le f_2) \ge 1-(1-\beta)\epsilon \label{eq:abs2} \\
&f_1^2 + f_2^2 \le k & \biggr\}.\label{eq:quadquad}\end{aligned}$$
Let $G_{\epsilon,\beta}^{proj}$ be the projection of the set $G_{\epsilon,\beta}$ onto the variables $(a,b,c,d,k)$. Then $G_{\epsilon,\beta}^{proj}$ is convex and $G_{\epsilon,\beta}^{proj} \subseteq H_\epsilon$. That is, the set $G_{\epsilon,\beta}^{proj}$ is a conservative, convex approximation of the quadratic chance constraint .
Let $(a,b,c,d,k,f_1,f_2) \in G_{\epsilon,\beta}$. To simplify the proof, let $\chi = a^T\xi + b$ and $\psi = c^T\xi + d$ be random variables. Then $\chi^2 \le f_1^2$ and $\psi^2 \le f_2^2$ implies $\chi^2 + \psi^2 \le k$, which gives us the inequality $$\mathbb{P}(\chi^2 + \psi^2 \le k) \ge \mathbb{P}(\chi^2 \le f_1^2 \text{ and } \psi^2 \le f_2^2).$$ From the union bound, $$\begin{aligned}
\mathbb{P}(\chi^2 \le f_1^2 \text{ and } \psi^2 \le f_2^2) &\ge \mathbb{P}(\chi^2 \le f_1^2) + \mathbb{P}(\psi^2 \le f_2^2) - 1 \\
&\ge 1-\beta\epsilon + 1 - (1-\beta)\epsilon - 1\\
&=1 -\epsilon,\end{aligned}$$ which proves $G_{\epsilon,\beta}^{proj} \subseteq H_\epsilon$. Convexity of $G_{\epsilon,\beta}$ (and therefore $G_{\epsilon,\beta}^{proj}$) follows from Lemma \[eq:absconvex\] and the fact that $f_1^2 + f_2^2 \le k$ is a convex quadratic constraint.
Replacing constraints and with the outer approximation -, one obtains an SOC-representable approximation of $H_\epsilon$. Note that this approximation is no longer conservative, but can be made so by instead using the polyhedral conservative approximation of $S_\epsilon$ as discussed in Section \[sec:polyapprox\].
Approximation via robust optimization
-------------------------------------
An alternative conservative approximation which we consider is based on robust optimization [@RobustBook].
Let $\epsilon \in (0,1)$ and suppose that $\xi$ follows an $n$-dimensional multivariate Guassian distribution. Without loss of generality, we assume each component is independent standard Gaussian with zero mean and unit variance. Let $\Gamma = F_{\chi_n}^{-1}(1-\epsilon)$, where $F_{\chi_n}^{-1}$ is the inverse cumulative distribution function of the chi distribution with $n$ degrees of freedom. Let $$\begin{aligned}
\label{eq:robustsdp}
R_\epsilon = \left\{(a,b,c,d,k,\lambda) :
\left[\begin{array}{cccc}
\lambda I & & a & c \\
& k - \lambda\Gamma^2 & b & d\\
a^T & b& 1\\
c^T&d&&1
\end{array}\right] \succeq 0
\right \},\end{aligned}$$ where the notation $A \succeq 0$ means that the symmetric matrix $A$ is positive semidefinite, and blank entries represent zero blocks. Let $R_\epsilon^{proj}$ be the projection of the set $R_\epsilon$ onto the variables $(a,b,c,d,k)$. Then $R_\epsilon^{proj}$ is convex and $R_\epsilon^{proj} \subseteq H_\epsilon$. That is, the set $R_\epsilon^{proj}$ is a conservative, convex approximation of the chance constraint .
It is sufficient to show that if $(a,b,c,d,k,\lambda) \in R_\epsilon$, then there exists a set $U$ such that $P(\xi \in U) \ge 1-\epsilon$ such that $$\label{eq:robustquad}
(a^T\eta + b)^2 + (c^T\eta+d)^2 \leq k, \, \forall \, \eta \in U.$$ Instead of allowing $U$ to vary for any point in the set, which is equivalent to the original chance constraint, we fix $U = \{\eta : ||\eta||_2 \le \Gamma\}$ and therefore obtain a conservative approximation. By the definition of the chi distribution, $P(\xi \in U) = 1-\epsilon$. In the terminology of robust optimization, $U$ is an uncertainty set. It is a standard result, which follows from the S-lemma and Schur complement lemmas, that $R_\epsilon^{proj}$ is precisely the set of points satisfying for this choice of the uncertainty set $U$ [@RobustBook]. Convexity follows since $R_\epsilon$ is the set of points satisfying a linear matrix inequality (LMI), which is tractable by semidefinite programming (SDP).
Note that the choice of $\Gamma = F_{\chi_n}^{-1}(1-\epsilon)$ may be overly conservative, especially when $n$ is large, although we are not aware of any theoretical guidance on choosing a smaller value of $\Gamma$ such that the chance constraint remains satisfied.
Nemirovski-Shapiro CVaR approximation
-------------------------------------
The third approximation we consider is based on the so-called CVaR approximation proposed by Nemirovski and Shapiro [@NS2007]. Let $I(z)$ be the indicator function of the interval $[0,\infty)$, i.e., $I(z) = 1$ if $z \ge 0$ and $I(z) = 0$ otherwise. We can rewrite the quadratic chance constraint in the following equivalent expected-value form: $$\label{eq:expvalform}
\mathbb{E}_\xi\left[I((a^T\xi + b)^2 + (c^T\xi+d)^2-k)\right] \le \epsilon.$$
Nemirovski and Shapiro propose to upper bound the indicator function $I$ with the convex increasing function $\psi(z) = \max(1+z,0)$, which, up to rescaling ($z \to z/\alpha$ for some $\alpha$), is the best possible convex upper bound on the indicator function in the sense that if $\omega(z)$ is another convex increasing upper bound, then there exists $\alpha > 0$ such that $\psi(z/\alpha) \le \omega(z)$ for all $z \in \mathbb{R}$. Then the constraint $$\inf_{\alpha > 0}\left[\mathbb{E}_\xi\left[\psi( ((a^T\xi + b)^2 + (c^T\xi+d)^2-k)/\alpha)\right] -\epsilon\right] \le 0,$$ is a conservative approximation of the quadratic chance constraint which is furthermore convex in $(a,b,c,d,k)$, which motivates the following lemma.
Let $\epsilon \in (0,1)$ and suppose that $\xi$ follows an $n$-dimensional multivariate Guassian distribution. Let $$NS_\epsilon = \left\{(a,b,c,d,k,\alpha) :
\mathbb{E}_\xi\left[\max((a^T\xi + b)^2 + (c^T\xi+d)^2-k+\alpha,0)\right] \le \alpha\epsilon, \alpha \ge 0 \right\}$$ Let $NS_\epsilon^{proj}$ be the projection of the set $NS_\epsilon$ onto the variables $(a,b,c,d,k)$. Then $NS_\epsilon^{proj}$ is convex and $NS_\epsilon^{proj} \subseteq H_\epsilon$. That is, the set $NS_\epsilon^{proj}$ is a conservative, convex approximation of the chance constraint .
See [@NS2007].
A comparison of approximations
------------------------------
We have presented three convex, conservative formulations of the quadratic chance constraint: one based on two-sided chance constraints, one based on robust optimization, and one based on convex approximation of the indicator function. All three have different tractability properties. In order of increasing computational difficulty, the two-sided approximation can be implemented, with small additional approximation error, by second-order cone programming (SOCP) following the developments presented in this work. The approximation based on robust optimization has an SDP formulation which may not be practical on large-scale problems, although we note the work of [@ChordalSDP] where specialized methods were developed to exploit the block structure. The CVaR approximation is the most computationally challenging; it has no known reformulation in terms of standard problem classes and requires multidimensional integration to evaluate.
One might expect that the more computationally challenging approaches could yield tighter approximations. In this section, we examine a two-dimensional example in order to gain some understanding of the relative strengths of the approximations. We find, perhaps surprisingly, that no one approximation strictly dominates another. Hence, the two-sided approximation we propose has value in both its strength and ease of implementation.
As an example we will recall the simple case of $$\label{eq:nonconvex_example2}
\mathbb{P}((x\xi_1)^2 + (y\xi_2)^2 \le 1) \ge 1-\epsilon,$$ where $\xi_1$ and $\xi_2$ are independent, standard Gaussian random variables.
Note that $||(\xi_1,\xi_2)||_2$ follows the chi distribution with 2 degrees of freedom, so in the robust approximation we can pick the uncertainty set $U = \{ (\eta_1,\eta_2) : ||(\eta_1,\eta_2)||_2 \leq F_{\chi_2}^{-1}(1-\epsilon) \}$ where $F_{\chi_2}^{-1}$ is the inverse cumulative distribution function of the chi distribution with two degrees of freedom.
In this example, reduces to $$\label{eq:psdexample}
\left[\begin{array}{ccccc}
\lambda & & & x & \\
&\lambda & & & y\\
& & 1 - \lambda\Gamma^2 & & \\
x & & & 1 & \\
&y&&&1
\end{array}\right] \succeq 0.$$ By a Schur complement argument, the matrix is positive semidefinite iff $1-\lambda \Gamma^2 \ge 0, \lambda - x^2 \ge 0,$ and $\lambda - y^2 \ge 0$, which holds iff $x \in [-1/\Gamma,1/\Gamma]$ and $y \in [-1/\Gamma,1/\Gamma]$, a simple box constraint.
An interesting observation is that for $n = 2$, the choice of $\Gamma = F_{\chi_2}^{-1}(1-\epsilon)$ is *minimal* in the sense that any smaller value no longer corresponds to a conservative approximation of the chance constraint : $$\mathbb{P}(((1/\Gamma)\xi_1)^2 + ((1/\Gamma)\xi_2)^2 \le 1) = \mathbb{P}(\xi_1^2 + \xi_2^2 \le \Gamma^2) = F_{\chi_2}(\Gamma).$$ In other words, the robust approximation to touches the boundary of the exact feasible set at the corners of the box. This observation eliminates the possibility of relaxing the overconservatism of the robust approximation by decreasing the size of the uncertainty set for the case of $n = 2$.
A point is feasible to the two-sided approximation for $\beta = \frac{1}{2}$ iff $\exists f_1, f_2$ such that $f_1^2 + f_2^2 \leq 1$, $\mathbb{P}(|x\xi_1|\leq f_1) \geq 1-\frac{\epsilon}{2}$, and $\mathbb{P}(|y\xi_2|\leq f_2) \geq 1-\frac{\epsilon}{2}$. By symmetry, these two chance constraints hold iff $f_1/|x| \geq \Phi^{-1}(1-\frac{\epsilon}{4})$ and $f_2/|y| \geq \Phi^{-1}(1-\frac{\epsilon}{4})$. Therefore, the point $(x,y)$ feasible to the two-sided approximation iff $$x^2 + y^2 \leq \frac{1}{\Phi^{-1}(1-\frac{\epsilon}{4})^2},$$ a simple ball constraint.
The CVaR approximation, to our knowledge, does not yield a closed-form algebraic representation, although in this simple case we are able to evaluate it by numerical integration.
Figures \[fig:eps05\] and \[fig:eps005\] compare the three approximations with the exact feasible set for $\epsilon = 0.5$ and $\epsilon = 0.05$, respectively. For $\epsilon = 0.5$, both the robust and the two-sided approximations dominate the CVaR approximation. For $\epsilon = 0.05$, no approximation is a strict subset of another. Curiously, for this particular case the exact set $H_{0.05}$ appears to be convex.
![ Outlined in black, the exact nonconvex feasible set $(x,y)$ satisfying $\mathbb{P}((x\xi_1)^2 + (y\xi_2)^2 \le 1) \ge 1-\epsilon$ for $\epsilon = 0.5$. We compare the three different convex approximations.[]{data-label="fig:eps05"}](e05.pdf)
![ Outlined in black, the exact (seemingly convex) feasible set $(x,y)$ satisfying $\mathbb{P}((x\xi_1)^2 + (y\xi_2)^2 \le 1) \ge 1-\epsilon$ for $\epsilon = 0.05$. We compare the three different convex approximations. On the right, a zoomed-in view of the top-right corner shows that no approximation strictly dominates another. []{data-label="fig:eps005"}](e005.pdf "fig:") ![ Outlined in black, the exact (seemingly convex) feasible set $(x,y)$ satisfying $\mathbb{P}((x\xi_1)^2 + (y\xi_2)^2 \le 1) \ge 1-\epsilon$ for $\epsilon = 0.05$. We compare the three different convex approximations. On the right, a zoomed-in view of the top-right corner shows that no approximation strictly dominates another. []{data-label="fig:eps005"}](e005topright.pdf "fig:")
Conclusion
==========
Building on top of the basic convexity result for two-sided chance constraints developed in Section \[sec:cvx2side\], we have shown, perhaps surprisingly, that a large class of more general nonlinear chance constraints is in fact convex (Theorem \[thm:nonlinearchance\]). In addition, our analysis of the computational tractability of the two-sided chance constraint, and in particular the polyhedral approximation of the set $S_\epsilon$ with provable approximation quality, develops practical methodologies which we believe are novel in the chance constraint literature. Finally, we have demonstrated that the two-sided chance constraint yields a useful approximation of the quadratic chance constraint which originally motivated this work.
We believe that our convexity results in Section \[sec:cvx2side\] can be easily extended to elliptical log-concave distributions following [@Lagoa05]. Extensions to more general distributions are not at all obvious, although the distributionally robust model in Section \[sec:distrobust\] may serve as a useful approximation. The conditions under which the quadratic chance constraint set $H_\epsilon$ is convex is left as an open question, although based on our computational experiments we conjecture that the set is convex for $\epsilon$ sufficiently small.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Michael (Misha) Chertkov of Los Alamos National Laboratory for discussions which inspired this work. M. Lubin was supported by the DOE Computational Science Graduate Fellowship, which is provided under grant number DE-FG02-97ER25308.
[^1]: Taking for granted that we can compute $\Phi(\cdot)$ efficiently, this gradient-based separation oracle may also be of theoretical use in proving tractability via the ellipsoid algorithm [@schrijver2003].
| {
"pile_set_name": "ArXiv"
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Why QCD Explorer stage of the LHeC should have high(est) priority
S. A. Çetin$^{a}$, S. Sultansoy$^{b}$, G. Ünel$^{c}$
$^{a}$[Doğuş University, Istanbul, Turkey]{}
$^{b}$[TOBB University of Economics and Technology, Ankara, Turkey\
and ANAS Institute of Physics, Baku, Azerbaijan ]{}
$^{c}$[University of California, Irvine, USA]{}
*Abstract: The QCD Explorer will give opportunity to enlighten the origin of the 98.5% portion of the visible universe’s mass, clarify the nature of the strong intecartions from parton to nuclear level and provide precision pdf’s for the LHC. Especially the $\gamma{}$-nucleus option seems to be very promising for QCD studies.*
Linac-ring type colliders have two main goals: to explore TeV scale with lepton-hadron and photon-hadron collisions and to achieve highest luminosity at flavor factories (the history of corresponding proposals can be found in \[1\]). This note concentrates on the first goal which is represented by the linac-ring option of the Large Hadron electron Collider (LHeC), proposed to explore the highest energy proton and ion beams available at the LHC probed by energetic electron or gamma beams from a linac tangent to the LHC. The Conceptual Design Report (CDR) of the LHeC project which is published in \[2\], investigates two options for the collider: Linac-Ring (LR) type collider where electrons are provided by linac and Ring-Ring (RR) option which assumes an additional electron ring in the LHC tunnel.
The idea of the LHC based linac-ring type ep/$\gamma{}$p collider includes two stages: QCD Explorer (E$_{e }$= 60, 140 GeV) and Energy Frontier (E$_{e }$$\geq{}$ 500 GeV). The first stage is mandatory for a deeper understanding of the strong interactions and an adequate interpretation of the LHC data which requires precision pdf’s. The second stage, which actually depends on the outcomes of the LHC, hence called provisional, will mainly have great potential for BSM physics complementary to LHC and exceeding the possible ILC. It should be noted that the Energy Frontier as well as $\gamma{}$p options of both QCD Explorer and Energy Frontier can only be realised with the linac-ring option.
Today, LR option is considered as the basic one for the LHeC. Actually this decision was almost obvious from the beginning due to the complications in constructing by-pass tunnels around the existing experimental caverns and installing the e-ring in the already commisioned tunnel. Let us remind that the CDR sthgeaof the LHC assumed also ep collisions using the already existed LEP ring; but it turned out that LHC installation required dismantling of LEP from the tunnel.
Now that LR is the choice for the LHeC, Energy Recovery Linac (ERL) is being pushed as a basic choice instead of the single-pass option using the argument that it could provide an order of magnitude higher luminosity. Nevertheless, keeping in mind that such higher luminosity is not necesseary for the QCD Explorer, it is likely that the single-pass option will become dominant soon, however we believe the sooner the better. It should ne mentioned that a very important advantage of the LR option, namely the opportunity to construct $\gamma{}$p/$\gamma{}$A collider loses its strength at rhe ERL based LHeC, moreover the single-pass option will give the opportunity to increase the energy of the electrons by lengthening the linac further.
Concerning the physics program of the QCD Explorer, putting forward the search for SUSY or other BSM physics or even detailed study of the Higgs boson as the main goal would have serious drawbacks. The uniqueness of such a machine lies in its potential to probe the nature of the strong interactions from parton to nuclear level and provide precision pdf’s for the adequate interpretation of the LHC results. It is well known that big challenges still exist in the QCD part of the Standard Model like understanding confinement and quark-gluon plasma. QCD Explorer will give the opportunity to reach very small x$_{g}$ region \[3\] shedding light on confinement. Then according to vector meson dominance the $\gamma{}$A collider will act as a $\rho{}$A collider which will give an opportunity to investigate formation of the quark gluon plasma at very high temperatures and low densities.
In light of the discussions presented above we propose the following phases for QCD Explorer based on single-pass linac option. First phase: ep collider with luminosity of 10$^{32}$cm$^{-2}$s$^{-1}$ and eA collider with luminosity of AxL$_{eA}$=10$^{31}$cm$^{-2}$s$^{-1}$ which seems sufficient for QCD studies. Second Phase: $\gamma{}$p and $\gamma{}$A collider with similar luminosities. Third Phase: construction of a second single-pass linac for energy recovery \[4\] to achieve much higher luminosities. Fourth Phase: lengthening the single-pass linac to switch to Energy Frontier stage.
With the discovery of the long sought Higgs boson, the electroweak sector of the Standard Model has filled its gaps. At this point it is worth mentioning that the Higgs Mechanism accounts for only $\sim$1.5% of the mass of the visible universe and the rest, $\sim$98.5% is provided by the QCD. Hence another strong motivation of the QCD Explorer is to better understand the formation of the visible universe.
In conclusion, we hope that the presented qualitative arguments justify the necessity of the QCD Explorer for the future of the high energy physics.
**References**
1\. A.N. Akay, H. Karadeniz and S. Sultansoy, *Review of Linac-Ring Type Collider Proposals*, Int. J. Mod. Phys. A25 (2010) 4589-4602; e-Print: arXiv:0911.3314 \[physics.acc-ph\]
2\. J L Abelleira Fernandez et al. (LHeC Study Group), *A Large Hadron Electron Collider at CERN: Report on the Physics and Desigon Concepts for Machine and Detector*, J. Phys. G. 39 (2012) 075001; e-Print: arXiv:1206.2913 \[physics.acc-ph\]
3\. U. Kaya, S. Sultansoy, G. Unel, *Probing small x(g) region with the LHeC based gamma-p colliders*, Nov 2012, e-Print: arXiv:1211.5061 \[hep-ph\]
4\. V. Litvinenko, *LHeC with \~100% energy recovery linac*, 2nd CERN-ECFA-NuPECC workshop on LHeC, Divonne-les-Bains, 1-3 Sep (2009).
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"pile_set_name": "ArXiv"
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---
abstract: 'New mechanism of magnetoresistivity in itinerant metamagnets with a structural disorder is introduced basing on analysis of experimental results on magnetoresistivity, susceptibility, and magnetization of structurally disordered alloys (Y$_{1-x}$Gd$_{x}$)Co$_{2}$. In this series, YCo$_{2}$ is an enhanced Pauli paramagnet, whereas GdCo$_{2}$ is a ferrimagnet (T$_{\rm c}$=400 K) with Gd sublattice coupled antiferromagnetically to the itinerant Co-3d electrons. The alloys are paramagnetic for $x < 0.12$. Large positive magnetoresistivity has been observed in the alloys with magnetic ground state at temperatures T$<$T$_{\rm c}$. We show that this unusual feature is linked to a combination of structural disorder and metamagnetic instability of itinerant Co-3d electrons. This new mechanism of the magnetoresistivity is common for a broad class of materials featuring a static magnetic disorder and itinerant metamagnetism.'
author:
- 'A. T. Burkov, A. Yu. Zyuzin'
- 'T. Nakama, K. Yagasaki'
title: 'Anomalous magnetotransport in (Y$_{1-x}$Gd$_{x}$)Co$_{2}$ alloys: interplay of disorder and itinerant metamagnetism.'
---
Introduction
============
Interplay of structural disorder and magnetic interactions opens a rich field of new physical phenomena. Among them are the actively discussed possibility of disorder-induced Non-Fermi Liquid (NFL) behavior near a magnetic Quantum Critical Point (QCP) as well as a broader scope of effects of disorder on magnetotransport. [@Hertz76; @Millis93; @Varma2001] Structurally disordered alloys Y$_{1-x}$Gd$_{x}$Co$_{2}$ are quasi-binary solid solutions of Laves phase compounds YCo$_2$ and GdCo$_2$. The compounds belong to a large family of isostructural composites RCo$_2$. YCo$_2$ is an enhanced Pauli paramagnet whose itinerant Co-3d electron system is close to magnetic instability. In external magnetic field of about 70 T this system undergoes a metamagnetic transition into ferromagnetic (FM) ground state.[@Goto89] GdCo$_2$ is, on the other hand, a ferrimagnet with a Curie temperature of 400 K in which the spontaneous magnetization of 4f moments is anti-parallel to the induced magnetization of the Co-3d band. Compounds of this family and their alloys provide a convenient ground for experimental studies of magnetotransport phenomena. The electronic structure in the important for the transport vicinity of the Fermi energy is composed mainly of Co-3d states and is, to the first approximation, the same for all compounds of RCo$_2 $ family. It has been found that the main contribution to the resistivity of RCo$_2$ comes from the scattering of conduction electrons on magnetic fluctuations due to strong s–d exchange coupling, [@Gratz95] therefore the transport properties are expected to be especially sensitive to the magnetic state of the sample. GdCo$_2$ occupies a special place in RCo$_2$ family since Gd 4f magnetic moment has no orbital contribution and, therefore crystal-field effects are not important for this compound.
The experimental results on the transport properties of Y$_{1-x}$Gd$_{x}$Co$_{2}$ alloys has been published partly in our previous article. [@Nakama2001] Here we analyze these and new experimental results in order to reveal the physical mechanism of anomalous megnetotransport properties observed in the alloys. In this paper we will discuss the magnetotransport properties of the FM alloys. The properties of paramagnetic alloys will be published elsewhere.
Experimental
============
Samples of Y$_{1-x}$Gd$_{x}$Co$_{2}$ were prepared from pure components by melting in an arc furnace under a protective Ar atmosphere and were subsequently annealed in vacuum at 1100 K for about one week. An X-ray analysis showed no traces of impurity phases. A four–probe dc method was used for electrical resistivity measurements. Magnetoresistivity (MR) was measured with longitudinal orientation of electrical current with respect to the magnetic field. The size of the samples was typically about 1$\times $ 1$ \times $10 mm$^{3}$. Magnetization was measured by a SQUID magnetometer for samples from the same ingot as that used for the resistivity and AC susceptibility measurements.
Experimental results
====================
The magnetic phase diagram of the Y$_{1-x}$Gd$_{x}$Co$_{2}$ system inferred from the transport and magnetic measurements [@Nakama2001] is shown in Fig. \[PasDiag\].
![The upper panel shows the ordering temperature T$_{\rm
c}$ $\blacksquare $ (right y-axis), and the MR $\bigoplus $ (left axis) of the Y$_{1-x}$Gd$_{x}$Co$_{2}$ system [@Nakama2001]. The MR was measured at T = 2 K in magnetic field of 15 T. The dotted vertical lines indicate phase boundaries at zero temperature. The lower panel displays normalized resistivity $\frac{\rho(2~K)}{\rho(300~K)}$.[]{data-label="PasDiag"}](fig1.eps){width="1.0\linewidth"}
Curie temperature $T_{\mathrm{c}}$ decreases with increasing content of Y and eventually drops to zero. A precise determination of the critical concentration $x_c$ which separates the magnetically ordered ground state and the paramagnetic region is difficult, since on the onset of the long range order its signatures in the magnetic and transport properties are very weak. The first firm evidence of the long range order are found for alloy with x=0.14 in ac susceptibility at T=27 K, Fig. \[Sus\].
![The ac susceptibility of the Y$_{1-x}$Gd$_{x}$Co$_{2}$ alloys. Note, the experimental data for YCo$_2$ and for the alloy with $x=0.14$ are multiplied by factor 20.[]{data-label="Sus"}](fig2.eps){width="1.0\linewidth"}
Quantum critical scaling theory predicts that when Curie temperature of a FM system continuously depends on an external parameter $x$, this dependence is expressed as [@Millis93]: $$T_{\rm c}\varpropto \left|
x-x_{\rm c}\right| ^{\frac{z}{d+z-2}}$$ with critical index $z=3$ for a FM system of spatial dimension $d=3.$ The experimental T$_{\rm c}$ vs. $x$ dependency does follow this relation, but with additional kink at $x=x_{\rm t}$. A possible origin of this kink will be discussed later. Linear extrapolation of the phase separation line on the phase diagram Fig. \[PasDiag\] to $T_{\rm
c} =0$ gives as the critical concentration $x_c=0.12$. We do not claim however that QCP exists in this alloy system. Direct experimental verification that T$_{\rm c}$ $ \rightarrow$ 0 as $x$ approaches $x_{\rm c}$ from magnetically ordered state is difficult for a disordered alloy system.
The very surprising result is the positive MR in the FM phase at low temperatures, see Figs. \[PasDiag\] and \[DRvsT\]. The well known theoretical result for MR of a localized moment ferromagnet was derived long ago by Kasuya and De Gennes. [@Kasuya56] As it follows from their theory, MR of a metallic ferromagnet should be negative, having a maximum absolute value at Curie temperature, and approaching zero as T $\rightarrow$ 0, and in the limit of high temperatures. Qualitatively this behavior has been supported by experiment, as well as by later more detailed theoretical calculations. The present experimental results are in a qualitative agreement with this theoretical behavior only for alloys with large Gd content ($x \geqslant 0.4$) (Fig. \[DRvsT\]). MR of the FM alloys with composition $0.3>x>0.14$ fundamentally differs from this theoretical behavior. Let us note that this composition range falls into the region of the phase diagram between the paramagnetic phase and the additional phase boundary indicated by the kink in the T$_c$ vs $x$ dependency, see Fig. \[PasDiag\]. MR of these alloys is positive below Curie temperature and is very large.
![The upper panel shows the temperature dependence of the MR of the Y$_{1-x}$Gd$_{x}$Co$_{2} $ alloys, measured in field of 15 T. Large positive MR of FM alloys ($x\leq0.3$) is observed at low temperatures. The field dependencies of MR, measured at T=2 K, are presented on the lower panel.[]{data-label="DRvsT"}](fig3.eps){width="1.1\linewidth"}
The known mechanisms of a positive MR can not explain the experimental data. A rough estimate of Lorenz force-driven MR one can get from a comparison with the MR of pure YCo$_2$. [@Burkov98] In the most pure samples of YCo$_2$ (with residual resistivity of about 2 $\mu \Omega $cm) the Lorenz force-driven positive contribution to the total MR does not exceed 5%. On the other hand the resistivity of the FM alloys at low temperatures falls into the region from 30 to 100 $\mu \Omega $ cm, i.e. at least one order of magnitude larger than the resistivity of pure YCo$_2$. Therefore, according to Koehler’s rule, this mechanism can give MR of only about 0.5%, this has to be compared with the experimental MR of almost 40%.
Weak localization effect is known to give positive MR. However our estimates show that this mechanism can give a contribution which is at least two order of magnitude smaller than the observed MR.
Discussion
==========
The key for understanding the mechanism of the positive MR is a combination of strong dependence of the magnetic susceptibility $\chi$ of metamagnetic Co-3d subsystem on the effective magnetic field and of the structural disorder in the R - sublattice of the alloys. In case of GdCo$_2$ as well as in the case of other magnetic RCo$_2$ compounds with heavy R-elements, the 4f–3d exchange interaction is described by introducing an effective field which acts on 3d electrons as: $$B_{\rm eff}=n_{\rm
fd}M_{\rm f}-B$$ where B is external field, M$_f$ is the uniform magnetization of R - sublattice, and $n_{\rm fd}$ - is the f–d coupling constant (in case of GdCo$_2$ $n_{\rm fd}\approx 50$ T/f.u.$\mu_{\rm B}$[@Goto2001]). In Y$_{1-x}$Gd$_{x}$Co$_{2}$ alloys the Gd moments are randomly distributed over the R-sites of the crystal lattice. Therefore, the effective field acting on 3d electrons depends on the local distribution of Gd moments and is therefore a random function of coordinate. This random field can be characterized by a distribution function $P\{B_{\rm eff}(r)\}$. The spatially fluctuating effective field induces an inhomogeneous magnetization of Co-3d electron system: $$m(r)=\chi \left( B_{\rm
eff}\right) B_{\rm eff}\left( r\right).$$ Therefore even at zero temperature in the ferromagnetic ground state, there are two kind of static magnetic fluctuations in the system: i. $M_{\rm f}(r)$; ii. $m(r)$. These fluctuations give an additional contribution to the resistivity. Since at T=0 K in ferromagnetic phase the 4f magnetic moments are saturated the corresponding contribution to the resistivity does not depend on external magnetic field. On the other hand, the 3d magnetic moment, as we will see later, is not saturated even in the ferromagnetic ground state. Therefore the 3d magnetization does depend on the external field. However, as long as 3d susceptibility is field independent and uniform, the external magnetic field will change only the mean (non-fluctuating part) value of the magnetization, whereas the magnitude of the fluctuations of $m$ and corresponding contribution to the resistivity remain unchanged. However actually, the 3d system is close to the metamagnetic instability. Therefore the 3d susceptibility $\chi $ is field dependent and gives rise to static magnetic fluctuations which are dependent on magnetic field resulting in non-zero magnetoresistivity.
For a qualitative analysis of this new mechanism of MR we make the following assumptions.\
1.We consider the system only in its ground state.\
2.Correlations between potential and spin-dependent scattering are neglected.\
3.Only the contributions which may be strongly dependent on external magnetic field are retained, e.g. we do not include the effects related to a change of d-density of states in magnetic field, and contributions due to potential scattering and scattering on 4f moments. In fact only the contribution related to the scattering on fluctuations of Co-3d magnetization will be considered.\
Figure \[Co-magnet\] shows schematically the dependency of the Co-3d magnetization of RCo$_{2}$ compounds on effective magnetic field [@Goto2001] and the distribution function $P\{B_{\rm
eff}(r)\}$. The metamagnetic transition is indicated by the rapid increase of the magnetization around 70 T.
![Dependence of the magnetization of Co–subsystem in RCo$_2$ compounds vs effective magnetic field (left axis) [@Goto2001] (shown schematically). The dotted line is a schematic representation of the distribution function $P(B_{\bf
eff})$ (right axis), the shaded area indicates the position of $P(B)$ in external field of 15 T.[]{data-label="Co-magnet"}](fig4.eps){width="1.0\linewidth"}
For the further discussion it is important to have an estimate of the magnitude of the fluctuations of the effective field, i.e. the width of the distribution function $P\{B_{\rm eff}(r)\}$. We can get a hint considering experimental data on field (0 to 7 T) dependency of the magnetization, Fig.\[Magnz\].
![The magnetization of the Y$_{1-x}$Gd$_{x}$Co$_{2}$ alloys vs magnetic field.[]{data-label="Magnz"}](fig5.eps){width="1.0\linewidth"}
Two important points follow from these data: i. there is a non-saturating para-process in these field dependencies above about 2 T; ii. the estimated susceptibility of this para-process ($\approx$ 0.016 $\mu_B$/T) is larger than the susceptibility of 3d system below and above metamagnetic transition ($\approx $ 0.002 $\mu_B$/T), however it is smaller than the susceptibility in the transition region ($\approx$ 0.04 $\mu_B$/T).[@Goto2001] This indicates, that the scale of the fluctuations is larger than the width of the metamagnetic transition. Therefore we can not treat the fluctuating part of the effective field as a perturbation and have to resort to a phenomenological analysis.
The distribution function $P(B_{\bf eff})$ describing the fluctuating effective field depends on the alloy composition. For diluted alloys ($x \approx 0$) the most probable value $B _{\rm
av}$ of $B_{\rm eff}$ is close to zero. As $x$ increases $B _{\rm
av}$ shifts to higher values and in a certain range of the concentrations, the function will have essentially non-zero weight for both $B_{\bf eff} < B_0$ and $B_{\bf eff} > B_0$, see Fig. \[Co-magnet\]. In this case there shall be regions with low and with high 3d magnetization in the sample. The resistivity resulting from this static disorder in 3d magnetization can be expressed as: $$\rho_m = \rho_{sd}\cdot y\left(1-y\right).$$ Where $ y=\int_{B_0}^{\infty} P\left(B_{\rm eff}\right)dB$ is the volume fraction of the high magnetization component. Parameter $y$ depends on the alloy composition $x$, and on external magnetic field $B$. In zero field, point $y=0$ corresponds to $x=0.$ As the content of Gd increases $B _{\rm av}$ shifts to larger effective fields, and finally, at some alloy composition $x_t$, it becomes larger than $B_0$, i.e. at this composition almost whole volume is occupied by the high magnetization component, i.e. $y\approx 1$. With a further increase of Gd content the mean magnetization of Co should increase with a smaller rate, determined by the slope of the $m(B)$ dependency above $B_0$, however $y=1$ in this region. According to this scenario $\rho_m$ will increase with $x$ at first, reach a maximum value at $x$ which corresponds to $y=0.5$ and will decrease with further increase of $x$ approaching to zero at $x\approx x_t$. We believe that $x_t$ corresponds to the kink on the phase diagram, Fig. \[PasDiag\]. The expected variation of $\rho _m$ with the alloy composition is schematically shown in Fig. \[decom\]. The total experimental resistivity includes additionally contributions coming from potential scattering and from scattering on 4f moments ($\rho_0$), which both are proportional to $x(1-x)$ and also are depicted in Fig. \[decom\]. For comparison the experimental resistivity is shown in this picture too. We present here the resistivity measured at T=2 K in field of 15 T to exclude the contributions related to spin-flip scattering.
![Schematic of contributions to the low-temperature resistivity of Y$_{1-x}$Gd$_{x}$Co$_{2}$ alloys. $\bullet $ -experimental normalized resistivity measured at T=2 K in field of 15 T. Solid line represents schematically $\rho _m$, broken line - $\rho _0$.[]{data-label="decom"}](fig6.eps){width="1.0\linewidth"}
The experimental low-temperature resistivity shows the expected variation with $x$ (one needs to keep in mind that the relation between $x$ and $y$ is in general non-linear, especially around magnetic phase boundary $x_c$). The resistivity attains the maximum value of about 100 $\mu \Omega\, cm$ in the region, which corresponds to maximum static magnetic disorder at $y \approx 0.5$ (room temperature resistivity of the alloys weakly depends on $x$ and is about 150 $\mu \Omega\, cm$). About the same value was obtained for the high temperature limit (maximum magnetic disorder) of magnetic part of the resistivity, arising from scattering on 3d temperature-induced magnetic fluctuations in YCo$_2$.[@Gratz95] This suggests that the main part of the experimental low temperature resistivity at $x<x_t$ originates from the scattering on the magnetic fluctuations of 3dmagnetization, i.e. is identical to $\rho _{\rm m}$.
Let us show that $\rho _{m}$ is of order of the spin-fluctuation contribution to resistivity at large temperatures, i.e. is of order of 100 $\mu \Omega$cm. In case of YCo$_{2}$ the spin-fluctuation contribution is the most important and is nearly independent of temperature above about 200 K. Hamiltonian of s-d exchange interaction is given by
$$H_{sd}=G\int
d\mathbf{rs}(\mathbf{r})\mathbf{S}_{d}(\mathbf{r})$$
here $\mathbf{s}(\mathbf{r})$ and $\mathbf{S}_{d}(\mathbf{r})$ are spin density of s- and d- electrons, correspondingly.[^1]
Spin fluctuation contribution to resistivity has form [@Ueda75]:
$$\rho =\frac{3m}{4ne^{2}}G^{2}N_{s}\frac{1}{T}\int\limits_{0}^{1}dqq^{3}\int%
\limits_{-\infty }^{\infty }\frac{d\omega \omega }{\sinh
^{2}\left( \omega /2T\right) }Im\chi \left(
q\frac{2k_{F}}{k_{F}^{\ast }},\omega \right) \label{Ueda}$$
where $N_{s}$ is density of states of s-electrons, $\frac{k_{F}}{%
k_{F}^{\ast }}$ is ratio of Fermi momentum of s- and d- electrons. Dynamic susceptibility $\chi \left( q,\omega \right)$ is given by the equation:
$$\chi ^{-1}\left( q,\omega \right) =\chi ^{-1}\left( q\right)
\left( 1-i\omega /\Gamma _{q}\right)$$
here $\Gamma _q$ is damping of the spin-fluctuations, whereas static nonlocal susceptibility $\chi ^{-1}\left( q\right)$ is given by
$$\chi ^{-1}\left( q\right) =\chi ^{-1}+Aq^{2}/N_{d}$$
with $\chi =N_{d}/\zeta(T)$ , where $\zeta(T)$ is inverse Stoner factor, renormalized by spin fluctuations, $N_{d}$ is density of states of d-electrons, and $A < 1$ is a dimensionless constant.
At large temperatures $T>\Gamma _q$ the expression (\[Ueda\]) reduces to:
$$\rho =\frac{3\pi
m}{ne^{2}}G^{2}N_{s}T\int\limits_{0}^{1}dqq^{3}\chi \left( q
\frac{2k_{F}}{k_{F}^{\ast }}\right) \simeq \frac{3\pi m}{4ne^{2}}
G^{2}N_{s}T\chi \label{rred}$$
The last equality is valid when $2Ak_{F}/k_{F}^{\ast }<\zeta(T)<1$ which must be the case for YCo$_{2}$. Note that neglecting momentum dependence of susceptibility means that s-electrons see the d-spin fluctuations as point scatterers.
Scattering of conducting s – electrons by the static random distribution of spin density of d-electrons we can estimate in the following way.
Due to random distribution of magnetic moments of Gd, s-electrons experience scattering by $$\frac{G}{2}\left\langle
S_d\left(\mathbf{r}\right)\right\rangle= \frac{G}{2}\int
\chi_{m}(\mathbf{r-r}^{\prime
})\delta B_{eff}(\mathbf{r}^{\prime })$$ random potential. Here $\chi _{m}(%
\mathbf{r-r}^{\prime })$ is nonlocal susceptibility near metamagnetic transition. We estimate correlation function of fluctuating effective field as $$\left\langle \delta
B_{eff}(\mathbf{r})\delta B_{eff}(\mathbf{r}^{\prime
})\right\rangle =\left(2 S_{\rm Gd}n_{\rm fd}\right)^2\delta \left(\mathbf{r-r}^{\prime }\right)x\left(
1-x\right) a^{3}.$$ Here $a^{3}$ is volume of the formula unit. In Born approximation the corresponding contribution to the resistivity is given by the expression:
$$\rho _{m}=\frac{m}{ne^{2}}G^{2}N_{s}\left(2 S_{\rm Gd}n_{\rm fd}\right)^2x\left( 1-x\right)
a^{3}\int\limits_{0}^{1}dqq^{3}\chi _{m}^{2}\left( q\frac{2k_{F}}{%
k_{F}^{\ast }}\right)
\label{stat}$$
Both expressions (\[Ueda\]) and (\[stat\]) give the contributions to the resistivity due to scattering on spin fluctuations, however in the first case they are of thermal origin, whereas in the second case the fluctuations are due to randomness of the effective field.
Assuming that nonlocality of the susceptibility is not important we obtain from (\[rred\]) and (\[stat\]):
$$\frac{\rho _{m}}{\rho }=\frac{\left(2 S_{\rm Gd}n_{\rm fd}\right)^2x\left( 1-x\right) a^{3}\chi _{m}^{2}%
}{3\pi T\chi }$$
Using experimental results for $\chi _{\rm m}$ and $\chi$ [@Goto89; @Goto2001; @Burzo72] we find $\frac{\rho _{m}}{\rho }$ in the range from 0.5 to 3, i.e. the resistivity caused by the static magnetic fluctuations is of the same order as the temperature–induced spin fluctuation resistivity. The uncertainty is mainly due to determination of $\chi _{\rm m}$ near the metamagnetic transition. Taking $\chi _{\rm m}$ as the susceptibility of YCo$_2$ at B=70 T (at the field of metamagnetic transition)[@Goto89] gives the upper bound for $\frac{\rho
_{m}}{\rho }$. Whereas $\chi _{\rm m}$ for the disordered alloy of $x$=0.18, estimated from our results on M(B), Fig. \[Magnz\] gives the lower bound.
In external magnetic field, the effective field $B_{\bf eff}$ decreases [^2], therefore $y$ also decreases. Depending on the value of $y_0$ – the volume fraction in zero field, $\rho _{\rm m}$ will either increase or decrease, resulting in positive or negative magnetoresistivity: for $0.5<y_0<1$ it will be positive, whereas for $0<y_0<0.5$ we will have a negative MR. In agreement with the model, the experimental MR is positive at $0.15<x<x_t$ and quickly decreases at $x > x_t =
0.3$ where $y \approx 1$. Nearly linear field dependencies of MR, observed for $ x<0.3$, see Fig. \[DRvsT\], implies that the width of $P(B_{\rm eff})$ in this composition range is larger than our experimental field limit of 15 T. The region $y < 0.5$ almost coincides with the paramagnetic region of the phase diagram. The model predicts negative MR for this region, and this prediction agrees with the experimental result. The model also gives a satisfactory description of the residual resistivity behaviour in this region, see Fig. \[decom\]. This agreement suggests that at $x < x_c$ the system is actually in spin-glass state. In this region additional essential contributions to MR are present. First, in the paramagnetic region there is a negative MR due to suppression of magnetic disorder in 4f magnetic moment system by external magnetic field. This negative contribution may be the reason why the cross-over point from the positive to negative magnetoresistivity does not coincide with the maximum of resistivity, see Fig. \[PasDiag\]. Secondly, there can be additional contributions, both positive and negative, near to zero-temperature magnetic phase boundary due to closeness to QCP.
An independent test of the model is based on the observation that the critical magnetic field $B_0$ of the metamagnetic transition in 3d subsystem increases under external hydrostatic pressure (P). [@Saito99; @Yamada95] Therefore, basing on the model, we expect that the resistivity of the alloys with the composition left of the resistivity maximum (see Fig. \[PasDiag\]) will decrease with pressure, whereas for the alloys right of the maximum it will increase with the increasing pressure. The experimental results for three alloy compositions are shown in Fig. \[Press\].
![The resistivity of the Y$_{1-x}$Gd$_{x}$Co$_{2}$ alloys vs external hydrostatic pressure at T = 2 K.[]{data-label="Press"}](fig7.eps){width="1.0\linewidth"}
The sign of the pressure effect is in agreement with the model prediction: the resistivity decreases with pressure for $x$=0.1, whereas it increases with P for $x$=0.18 and $x$=0.3. Moreover, there is a good scaling of pressure and magnetic field dependencies of the resistivity, Figs. \[DRvsT\] and \[Press\] (P$\rightarrow\alpha $B) with scaling parameter $\alpha$ which is close to literature data on the pressure dependence of B$_0$: $\frac{dB_0}{dP}\approx
1.5$ T/kBar.[@Saito99; @Yamada95]
Conclusion
==========
It has been found that at low temperatures there is a large contribution to the resistivity related to scattering on magnetic fluctuations in metamagnetic itinerant 3d system, induced by fluctuating effective field of 4f moments. Large positive MR, found in the FM Y$_{1-x}$Gd$_x$Co$_2$ alloys and strong pressure dependence of the resistivity are explained as arising from a combination of static magnetic disorder and strong magnetic field dependence of magnetic susceptibility.
We want to emphasize that this mechanism of resistivity (and of MR) is not material specific, rather it should be common for a broad class of disordered itinerant metamagnets with strong coupling of conduction electrons to the magnetic fluctuations. In Y$_{1-x}$Gd$_x$Co$_2$ the relevant disorder originates from random distribution of d–f exchange fields, however a similar effect should arise when there is a random distribution of local susceptibilities (a corresponding treatment for Kondo systems was recently developed in [@Wilhelm2001]). Positive MR observed in FM alloys Y(Co$_{1-x}$Al$_x$)$_2$ [Nakama2000]{} and in Er$_{x}$Y$_{1-x}$Co$_2$ [@Hauser2001] may be explained by this mechanism.
ACKNOWLEDGMENT {#acknowledgment .unnumbered}
==============
This work is supported by grants 02-02-17671 and 01-03-17794 of Russian Science Foundation, Russia. We thank Dr. P. Konstantinov for stimulating discussions.
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[^1]: Hereafter we take $\mu _B \equiv 1$.
[^2]: The effective field decreases for antiferromagnetic 4f-3d exchange, for ferromagnetic exchange (like that in RCo$_2$ compounds with light R-elements) the effective field increases with the external field.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Direct evaluation of the 1-loop fluctuation determinant of non-static degrees of freedom in a complete static background is advocated to be more efficient for the determination of the effective three-dimensional model of the electroweak phase transition than the one-by-one evaluation of Feynman diagrams. The relation of the couplings and fields of the effective model to those of the four-dimensional finite temperature system is determined in the general ’t Hooft gauge with full implementation of renormalisation effects. Only field renormalisation constants display dependence on the gauge fixing parameter. Characteristics of the electroweak transition are computed from the effective theory in Lorentz-gauge. The dependence of various physical observables on the three-dimensional gauge fixing parameter is investigated.'
author:
- |
[A. Jakovác$^{1}$ and A. Patkós$^{2}$]{}\
[Department of Atomic Physics]{}\
[Eötvös University, Budapest, Hungary]{}\
title: |
Finite Temperature Reduction\
of the SU(2) Higgs-Model\
with Complete Static Background\
---
A new wave of investigations of finite temperature gauge theories is driven by the challenge of the matter-antimatter asymmetry of the Universe. Anomalous baryon number violating processes thermally excited near the electroweak phase transition certainly have had impact on any [*a priori*]{} asymmetry. Additional non-equilibrium and CP-violating effects, occuring during the transition, might have contributed to the generation of the present day value of the symmetry.
Temperature introduces a natural mass-scale into the relevant field theory. It builds up a hierarchy among the fluctuations, which should be exploited in the evaluation of the partition function. Heavy modes with non-zero Matsubara index are important for the accurate determination of the couplings between the (almost) T-scale independent static modes, which drive the phase transition. This physical picture is the content of the dimensional reduction [@pisarski; @landsman] of finite temperature field theories. The validity of the assumed mass-hierarchy should be checked carefully after each reduction step.
A correctly reduced 3-d effective model offers important advantages from the point of view of the application of standard methods of statistical physics to the electroweak phase transition [@arnold1; @march; @gleiser]. Also lattice simulations are greatly facilitated if the full 4-d system is replaced by the coresponding 3-d effective model [@kajantie; @farakos1; @karsch], since the extreme weak coupling situation makes the simulation of the 4-d system a particularly involved task [@bunk; @montvay].
We emphasize, that for the success of the above strategies the most faithful possible mapping of the 4-d couplings on the temperature dependent 3-d ones is essential. For instance, in the renormalisation group flow of the\
3-d model [*dim 6*]{} operators might play important role. The determination of their weights in the Lagrangian of the effective model with help of the usual Feynman diagram technique requires calculations of increased complexity.
The first complete determination of the reduced model up to [*dim 4*]{} operators in the 1-loop approximation, including field renormalisation effects has been published very recently [@farakos2]. The authors evaluate all relevant Feynman diagrams with two and four, zero-momentum external field insertions. The computation has been performed in the Landau gauge using dimensional regularisation, followed by the application of the $\overline {{\rm MS}}$ renormalisation scheme.
In this note we present evidence that the evaluation of the functional fluctuation determinant in a complete static background ($A_i^a({\bf
x}), A_{4}^{a}({\bf x}), \Phi ({\bf x})$) offers a simpler and more compact calculational scheme. It allows the unified determination of all renormalisation constants of the 4-d theory, and in principle it is easily extendable also to the computation of the higher dimensional operators. (After the completion of our investigation we received a paper by Chapman [@chapman], where an analogous calculation has been performed for SU(N) pure gauge theories up to [*dim 6*]{} operators. The calculational technique, however, was fully different.)
Since the method of symbolic evaluation of the functional determinant with constant complete background is of equal difficulty for any member of a certain gauge class, without any extra complication one is able to study the dependence of the action of the effective theory on the gauge fixing parameter. Specifically, we shall perform the reduction with general ’t Hooft gauge fixing, applying 3-d momentum cut-off regularisation. The normalisation of the scalar potential piece of the effective action will be fixed by imposing Linde’s conditions [@linde].
We shall show, that the effective theory and the expressions of the 3-d couplings do not depend on the gauge fixing parameter. The 1-loop effective potential of the 3-d theory will be determined next in the general (three-dimensional) Lorentz gauge, and the dependence of the critical data ($T_c$, order parameter discontinuity, etc...) on the parameter of the 3-d gauge fixing be discussed. This point essentially follows [@arnold2], going beyond it in the implementation of the detailed relation between the couplings of the 3-d theory to the 4-d ones, and the numerical evaluation of the physical characteristics of the transition, not restricting the discussion to analytic perturbative considerations. 1truecm 1. The model under consideration is the SU(2) gauge+scalar theory at finite temperature $$S=\int_{0}^{\beta}d\tau \int d^{3}x\bigl [{1\over 4}F_{mn}^{a}F_{mn}^{a}+
{1\over 2}(D_{m}\Phi)^{\dagger}(D_{m}\Phi)+V(\Phi )\bigr ],$$ $$V(\Phi )= {1\over 2}m^{2}\Phi^{\dagger}\Phi\
+{\lambda\over 24}(\Phi^{\dagger}\Phi)^{2},$$ $$D_{m}\Phi =(\partial_{m}+igA_{m}^{a}\tau^{a}/2)\Phi,$$ m=1,...,4; a=1,2,3. (In eqs. (1-3) the renormalised parameters appear, the counterterms are not displayed explicitly, also Euclidean metrics is understood). The 1-loop integration over non-static modes will be peformed with full background, that is all fields are split into a non-zero static and a non-static part: $$\begin{aligned}
&
A_m=A_m({\bf x})+a_m({\bf x},\tau),\nonumber\\
&
\Phi=\left(\matrix{ 0 \cr \Phi_{0}({\bf x})\cr}\right)
+\left(\matrix{\xi_{1}({\bf x},\tau )+i\xi_{2}({\bf x},\tau )\cr
\xi_{3}({\bf x},\tau )+i\xi_{4}({\bf x},\tau )\cr}
\right).
\label{eq4}\end{aligned}$$
We shall demonstrate that the full renormalised reduced action can be recovered by choosing the static background [*constant*]{} (with the most general orientation in the isospace). Upon substituting the decomposition (4) into (1) one separates terms containing the non-static fields up to second power, for the 1-loop integration. The piece depending only on the constant background takes the form: $$U^{(0)}=\beta V\bigl [{1\over 4}g^2(A_i\times A_j)^2+{1\over 2}
g^2(A_i\times A_4)^2+{1\over 8}g^2(A_i^2+A_4^2)\Phi_0^{\dagger}
\Phi_0+V(\Phi_0)\bigr ]$$ (i=1,2,3). The part quadratic in the non-static components will not be displayed explicitly, since its expression is lengthy and not enlightening. The only important point for us is, that the fluctuations are characterised by a $16\times 16$ matrix, because the 12 gauge field components and 4 real Higgs scalar components are fully coupled in the most general constant background.
The gauge fixing conditions imposed on the fluctuations $a_{m},
\xi_{\alpha} $ are $$\begin{aligned}
&
F^1 =(D_{\mu}(A)a_{\mu})^{1}-\alpha{g\Phi_0\over 2}\xi_2,\nonumber \\ &
F^2 =(D_{\mu}(A)a_{\mu})^2-\alpha{g\Phi_0\over 2}\xi_1,\nonumber \\ &
F^3 =(D_{\mu}(A)a_{\mu})^3+\alpha{g\Phi_0\over 2}\xi_4,\end{aligned}$$ ($D_{\mu}(A)$ is the covariant derivative in the background field $A$, $\alpha$ is the gauge fixing parameter). The corresponding Faddeev-Popov determinant is $$\det \{ [K^2 +g^2(A_i^2+A_4^2)+{\alpha \over 4}g^2\Phi_0^2 ]\delta^{a,b}-
2ig\epsilon^{abc}K_mA_m^c-g^2A_m^aA_m^b)\},$$ where $K^2 ={\bf k}^2+\omega_n^2$.
Since the distinguishing difference of the proposed method relative to the conventional Feynman diagrams consists of the explicit evaluation of the fluctuation determinant in constant background, we are going to elaborate on certain technical details of this computation.
The general structure of the 1-loop non-static contribution to the reduced action looks very simple $$\begin{aligned}
&
U^{(1)}[A_i ,A_4 ,\Phi_0 ]=
T\sum_{n\neq 0}\int{d^3k\over (2\pi)^3}\bigl\{{1\over 2}\ln [K^{32}+
\alpha^2 a_2 K^{30}+\alpha^3a_4K^{28}+...]\nonumber \\ &
{}~~~~~~~~~~~~~~~~~~~~~-\ln [K^6+...]\bigr\}\end{aligned}$$ (The second logarithm stands for the contribution of the Faddeev-Popov determinant). The coefficients $a_2 ,a_4$ depend on the background fields qudratically ($\Phi_0^2, (\hat K_m A_m)^2, A_m^2)$ and quartically $(\Phi_0^4 , (\hat K_m A_m)^2\Phi_0^2,
(\hat K_m A_m)^4,...)$, respectively. Here $\hat K_m$ is the Euclidean unit vector pointing in the direction of $K_m$. If one restricts the calculation to finding the coefficients in the effective action of all operators up to [*dim 4*]{}, one expands the first logarithm in (8) up to terms $a_2^2, a_4$, and similar expansion is applied to the contribution of the Faddeev-Popov determinant. After throwing away (divergent) field independent terms, the coefficients we are interested in, turn out to be proportional to various infrared safe sum-integrals of the type $$T\sum_{n\neq 0}\int{d^3k\over (2\pi)^3}\bigl( {1\over K^2}, {{\bf
k}^2\over K^4},{\omega_n^2\over K^4}, {1\over K^4},...\bigr ).$$ In these integrals, where it is necessary, a three-dimensional ultraviolet cut-off has been introduced.
The complete evaluation of the fluctuation determinant with help of symbolic programming met considerable computer memory problems. Fortunately, in the basis, where the gauge field components are explicitly separated into longitudinal and transversal ones, the diagonal elements are ${\cal O}(K^2)$, while the off-diagonals are at most ${\cal O}(K)$. Therefore, for the first few leading K-powers in the argument of the first logarithm of (8) it is convenient to use a decompositon of the determinant of a certain $n\times n$ matrix $M$ into a sum of subsequent contributions containig products of decreasing number of diagonal elements:
$$\det M=\prod\limits_{i=1}^n M_{ii}\,\,\, + \sum\limits_{k_1>k_2}^n\!
{\rm det}_2 N(k)\!\!\!\prod\limits_{i\neq k_1,k_2}^n\!\! M_{ii} \,\,\,
+\! \sum\limits_{k_1>k_2>k_3}^n\!\!\!\!
{\rm det}_3 N(k)\!\!\!\prod\limits_{i\neq k_1\dots k_3}^n\!\!\!\! M_{ii}
\,\,\,+\dots,$$ where $$N(k)_{ij}=M_{k_i k_j}(1-\delta_{k_i k_j})$$ Therefore, for $a_2$ and $a_4$ it is sufficient to consider the set of $2\times 2$ up to $4 \times 4$ minors of the full $16\times 16$ matrix, corresponding to the first four terms of the above decomposition.
The resulting cut-off regularised expression is the central computational result of the present note: $$\begin{aligned}
&
U^{(1)}[A_i, A_4, \Phi_0]=\beta V\bigl\{{1\over 2}\Phi_0^2\bigl [
({3g^2\over 16}
+{\lambda\over 12})T^2-({9g^2\over 4}+\lambda)
{\Lambda T\over 2\pi^2}\nonumber \\ &
+{m^2\over 8\pi^2}({3\alpha\over 4}g^2+\lambda)(1-D_0)\bigr ]
+{1\over 96\pi^2}\Phi_0^4(1-D_0)({27g^4\over 16}+{3\alpha g^2\lambda\over 8}
+\lambda^2)+{17g^4\over 192\pi^2}A_4^4
\nonumber \\ &
+{1\over 2}A_4^2({5g^2\over 6}T^2+{m^2g^2\over 8\pi^2}-
{5g^2\Lambda T\over 2\pi^2})
+{1\over 64\pi^2}g^2A_4^2\Phi_0^2((9-
{5\alpha \over 4})g^2+\lambda-{9+3\alpha\over 4}g^2D_0)\nonumber \\ &
{1\over 64\pi^2}g^2A_i^2\Phi_0^2({17\alpha g^2\over 12}-
{\lambda\over 3}-{9+3\alpha\over 4}g^2D_0)+{1\over 12\pi^2}g^4
(A_i\times A_4)^2({163\over 24}-\alpha-{43\over 8}D_0)\nonumber \\ &
+{1\over 24\pi^2}g^4(A_i\times A_j)^2({509\over 120}+\alpha-{43\over 8}
D_0)\bigr\}
\label{eq9}\end{aligned}$$ with the logarithmically divergent quantity: $D_0=\ln{\Lambda\over T}-\ln 2\pi +\gamma_{Euler}.$
It is important to call the attention of the reader to the inconvenient fact that blindly following the calculational scheme outlined above, two additional terms would appear in (10): $$U^{(1)}_{fake}=-\beta V({17g^4\over 960\pi^2}(A_i^2)^2+
{m^2g^2\over 48\pi^2}A_i^2).$$ These terms clearly violate the invariance of the final reduced theory under spatial gauge transformations. In small periodic volumes this is the actual physical situation, since then $A_i$ is a physical degree of freedom on the same footing as $A_4$ (spacelike Polyakov loops are also observables). Then (11) should be added to (10) for fields fulfilling the restriction $A_i<< 2\pi /L$ (L is the linear spatial dimension of the system). These terms would be in complete formal analogy with the terms representing $A_4$ in (10).
However, the expansion of the logarithms and the limit $V\rightarrow \infty$ are not interchangeable: $$\lim_{V\rightarrow\infty}\sum_{n\neq 0}\int_k\ln (1+f(A_i))\neq
\sum_{m=1}^{\infty}\lim_{V\rightarrow\infty}\sum_{n\neq 0}\int_k {(-1)^m\over
m}f(A_i)^m.$$ This can be shown the cleanest way for “quasi-abelian” configurations $(A_i^a=A_i\delta_{a3}, A_4=\Phi =0)$, when the left hand side of (12) is a well-known periodic function of $A_i$ with period $2\pi/L$ [@weiss; @gross]. The difference between the two sides of (12) is exactly given by (11), therefore we are led to the prescription to subtract it from the complete result of the calculation. Only by following this careful consideration one arrives at (10). This expression is the starting point for the discussion of the renormalisation of the effective action.
The key observation is the renormalisation invariance of $gA_i$ in background gauges [@abbott]. Exploiting this circumstance one can find all field renormalisations from appropriately selected terms of the sum $U^{(0)}+U^{(1)}$. The coefficient of $g^2A_i^2\Phi_0^2/8$ determines $Z_{\Phi}$ , that of $g^2(A_i\times A_0)^2/2$ leads to $Z_{A_0}$, and finally $g^2(A_i\times A_j)^2/2$ to $Z_{A_i}=Z_{g}^{-1}$: $$\begin{aligned}
&
Z_{\Phi} = 1+{17\alpha \over 192\pi^2}g^2-{1\over 48\pi^2}\lambda
-{9+3\alpha\over 64\pi^2}g^2D_0,\nonumber \\ &
Z_{A_0}=1-{163g^2\over 288\pi^2}+{\alpha g^2\over 12\pi^2}+
{43g^2\over 96\pi^2}D_0,\nonumber \\ &
Z_{A_i}=1-{509g^2\over 1440\pi^2}-{\alpha g^2\over 12\pi^2}+{43g^2\over
96\pi^2}D_0.\end{aligned}$$ After renormalisation, these three terms are completed in view of the minimal coupling principle into the full kinetic terms of the corresponding fields, varying in space $$\begin{aligned}
&
L_{3-d}^{(kin)}={1\over 4}F_{ij}^{a}F_{ij}^a+{1\over 2}(D_i(A)A_4)^2+
{1\over 2}(D_i(A)\Phi)^{\dagger}(D_i(A)\Phi),\nonumber \\ &
D_i(A){ A_4}=\partial_i A_0+g(A_i\times A_4).\end{aligned}$$ The second step is the renormalisation of the Higgs potential. Taking into account the effect of $Z_{\Phi}$ the regularised expression goes over into $$\begin{aligned}
&
U_{Higgs}^{(reg)}={1\over 2}\Phi^{\dagger} \Phi\bigl\{({3g^2\over 16}+
{\lambda\over 12})T^2-{1\over 8\pi^2}({3g^2\over 4}+\lambda)
(\Lambda^2-4\Lambda T)\nonumber \\ &
{}~~~~~~~~~~+m^2\bigl [1+{\lambda\over 6\pi^2}-{\alpha g^2\over 12\pi^2}+
{D_0\over 8\pi^2}({9g^2\over 4}-\lambda )\bigr ]\bigr\}\nonumber \\ &
{}~~~~+{1\over 24}(\Phi^{\dagger}\Phi)^2\bigl [\lambda+({27\over 8}g^4+
{\lambda^2\over 4}-{4\alpha g^2\lambda\over 3}){1\over 8\pi^2}\nonumber \\ &
{}~~~~~~~~+{D_0\over 4\pi^2}({9g^2\lambda\over 4}-{27g^4\over 16}-
\lambda^2)\bigr ].\end{aligned}$$ It is reassuring, that the cut-off dependences of $m^2(\Lambda )$ and $\lambda (\Lambda )$ correctly reproduce the 1-loop $\beta$-functions of the SU(2) gauge+scalar theory (this is also true for $g^2(\Lambda )$ as can be seen from (13)) [@arnold1]. For this result it is essential to employ in the Higgs potential the correct renormalisation of the $\Phi$-field.
The renormalisation conditions we have applied to the temperature-independent part of the potential, were the Linde-type conditions, used also in our previous publication [@jako]: $${dU_{Higgs}(T-indep.)\over d\Phi_0}=0,~~~~~{d^2U_{Higgs}(T-indep.)
\over d\Phi_0^2}=m_H^2(T=0),
{}~~~~\Phi_0=v_0$$ ($v_0$ is the $T=0$ expectation value of the Higgs field). The details of the corresponding subtraction procedure were discussed in [@jako] for the thermal static gauge. Here we give the final result from the analysis done along the same lines, just for the ’t Hooft gauge: $$\begin{aligned}
&
U_{Higgs}^{(1-loop)}={1\over 2}\Phi^{\dagger}\Phi\bigl [m^2+({3g^2\over 16}
+{\lambda\over 12})T^2-{\Lambda T\over 2\pi^2}({9g^2\over 4}+\lambda )\bigr ]
+{\lambda\over 24}(\Phi^{\dagger}\Phi)^2\nonumber \\ &
{}~~~~~~~~-{1\over 64\pi^2}\sum_jn_j[m_j^4(\Phi )(\ln{m_j^2(v_0)\over T^2}+
{3\over 2})-2m_j^2(v_0)m_j^2(\Phi )],\end{aligned}$$ where the index j runs over all formal degrees of freedom: j=4-d transversal (T), Higgs (H), 4-d longitudinal (L), pseudo-Goldstone (G). The corresponding quantities appearing in (17) are: $$\begin{aligned}
&
n_T=9,~~~~~m_T^2={g^2\over 4}\Phi^{\dagger} \Phi,\nonumber \\ &
n_L=-3,~~~~~m_L^2={3g^2\over 4}\Phi^{\dagger}\Phi,\nonumber \\ &
n_H=1,~~~~~m_H^2=m^2+{\lambda\over 2}\Phi^{\dagger}\Phi,\nonumber \\ &
n_G=3,~~~~~m_G^2=-m^2+({3g^2\over 4}-{\lambda\over 6})\Phi^{\dagger}\Phi.\end{aligned}$$ One has to emphasize that these formal degrees of freedom are not the diagonal modes of the coupled fluctuation matrix, therefore the masses do not correspond to any actual thermal mass. Especially, $m_G^2>0$ for the range of the $\Phi$ values between 0 and $v_0$.
The renormalisation leads also to finite rescalings of the $A_4-\Phi$ interaction and of the $A_4$-potential, due to wave function renormalisations: $$\begin{aligned}
&
U^{(1-loop)}[A_4,\Phi]={17g^4\over 192\pi^2}(A_4^2)^2(1-{153g^2\over
180\pi^2}+{2\alpha g^2\over 3\pi^2})+{1\over 8}g^2A_4^2\Phi^{\dagger}\Phi
(1+{7g^2\over 10\pi^2}+{\lambda\over 6\pi^2})\nonumber \\ &
{}~~~~~~~+{1\over 2}A_4^2({5g^2\over 6}T^2+{m^2g^2\over 8\pi^2}-
{5g^2\over 2\pi^2}\Lambda T)(1-{153g^2\over 360\pi^2}+
{\alpha g^2\over 3\pi^2}).\end{aligned}$$ The renormalised Euclidean Lagrangian density is the sum of (14),(17) and (19). The linear divergences induced for the mass terms of $A_4$ and $\Phi$ are necessary for the mass renormalisations of the 3-d theory at 1-loop.
It is important to note, that the effective theory shows dependence on the gauge fixing parameter only in (19). Clearly, a 2-loop computation of the reduced potential will also give ${\cal O}(g^4)$ contributions, therefore the present expressions of the corrections in (19) cannot be considered final. Omitting these incomplete corrections, we summarize the effective theory and the relations of the 3-d and 4-d couplings in the formulae below. It is obvious that these relations do not involve the gauge fixing parameter. $$\begin{aligned}
&
L_{3-d}={1\over 4}F_{ij}^aF_{ij}^a+{1\over 2}(D_i(A)A_0)^2+{1\over 2}(D_i(A)
\Phi)^{\dagger}(D_i(A)\Phi)\nonumber \\ &
+{1\over 2}m^2(T)\Phi^{\dagger}\Phi+{\hat\lambda\over 24}(\Phi^{\dagger}\Phi)^2
+{1\over 2}m_D^2A_4^2+{17g^4\over 192\pi^2}(A_4^2)^2+{\rm 3-d~ct.-terms}\end{aligned}$$ where $$\begin{aligned}
&
m^2(T)=\hat m^2(T)+({3g^2\over 16}+{\lambda\over 12})T^2,\nonumber \\ &
\hat m^2(T)=m^2(1-{1\over 32\pi^2}[\lambda\ln{\lambda v_0^2\over 3T^2}+
(\lambda -{9g^2\over 2})\ln{3g^2v_0^2\over 4T^2}+
{27g^4\over 4\lambda}+5\lambda-{45g^2\over 4}]),\nonumber \\ &
\hat\lambda =\lambda -{3\over 8\pi^2}[{9g^4\over 16}\ln{g^2v_0^2\over 4T^2}
+{\lambda^2\over 4}\ln{\lambda v_0^2\over 3T^2}+({\lambda^2\over 12}-
{3g^2\lambda\over 4})\ln{3g^2v_0^2\over 4T^2}
+{3\over 2} ({9g^4\over 16}+{\lambda^2\over 3}-
{3g^2\lambda\over 4})],\nonumber \\ &
m_D^2={5\over 6}g^2T^2.\end{aligned}$$ 1truecm
2\. The most adequate physical test for the effective theory seems to be the analysis of the electroweak phase transition with help of the effective Higgs potential calculated from the 3-d theory in the general 3-d Lorentz-gauge. This has been discussed already by Arnold [@arnold2], therefore we can start from his formula for the effective potential adapted to the SU(2) case: $$\begin{aligned}
&
U_{eff}^{(1-loop)}(\Phi_0)={1\over 2}m^2(T)\Phi_0^2+{\hat\lambda\over 24}
\Phi_0^4-{T\over 12\pi}\bigl\{6({g^2\over 4}\Phi_0^2)^{3/2}+
3({5g^2\over 6}T^2+{1\over 4}g^2\Phi_0^2)^{3/2}\nonumber \\ &
+(m^2(T)+{\hat\lambda\over 2}
\Phi_0^2)^{3/2}
+3[{1\over 2}(m_{\chi}+|m_{\chi}|
(m_{\chi}^2-\alpha g^2\Phi_0^2)^{1/2})]^{3/2}\nonumber \\ &
+3[{1\over 2}(m_{\chi}^2-|m_{\chi}|
(m_{\chi}^2-\alpha g^2\Phi_0^2)^{1/2})]^{3/2}\bigr \}.\end{aligned}$$ In (22) the abbreviation $$m_{\chi}^{2}=m(T)^2+{\hat\lambda\over 6}\Phi_0^2$$ is introduced and $\alpha$ is now the gauge fixing parameter of the 3-d Lorentz gauge class.
It has been argued in [@arnold2] that perturbative expansion of (22) leads to a unique barrier temperature, independent of $\alpha$. We concentrate here on $T_c$ (the transition temperature) and some further physical characteristics of the transition, which will be determined numerically. Also we use the detailed relationship between 3-d and 4-d couplings, which were not taken into account in previous analyses.
$T_c$ has been determined for two characteristic values of the Higgs mass: 60 and 80 GeV. Also the order parameter discontinuity and the surface tension between coexisting phases have been evaluated (the latter in the thin wall approximation). The gauge dependence of these quantities has been tested by varying $\alpha$ in the interval (0,1). We have restricted this interval further by requiring $U_{eff}$ to be real in the interval of more direct physical interest, $\Phi\in(0,\Phi_{min}(T_c)).$ Clearly, for large enough $\alpha$ the last two terms of (22) become complex at fixed $\Phi_0$. It is also obvious that the limiting value of $\alpha$ found in this way will depend sensitively on the Higgs-mass $(\lambda)$ (c.f. (23)). For $m_H(T=0)=60$ GeV $\alpha \leq 0.3$, for $m_H(T=0)=80$ GeV $\alpha\leq 0.7$ was found to be the “upper bound” of its allowed range of variation.
In the Table we display the relevant physical quantities, which show remarkable stability, but definitely depend on $\alpha$. It is interesting to note that in the same quantities calculated from an effective potential determined in the 3-d analogue of ’t Hooft’s gauge, more important variation can observed, namely of the same order of magnitude as the difference found between 1-loop and 2-loop calculations performed in the 4-d, finite T theory [@fodor; @arnold3]. (For a criticism concerning the physical interpretation of the effective potential determined in the ’t Hooft gauge, see [@arnold2].) 1truecm 3. In conclusion, we have determined in a cut-off regularized calculation the relationship of the effective 3-d theory to those of the original 4-d system in the general ’t Hooft gauge, subject to the renormalisation conditions (16). It has been demonstrated that the evaluation of the determinant of non-static fluctuations in a complete constant background is sufficient for the full specification of the effective theory at 1-loop. The effective action proved to be independent of the gauge fixing parameter and invariant under spatial gauge transformations. The 1-loop analysis of the electroweak phase transition in the effective model has been shown to be rather insensitive to the actual choice of the gauge fixing parameter in a general Lorentz gauge.
The method presented here is of considerable calculational advantage over the direct enumeration and evaluation of Feynman diagrams at 1-loop level. Prospective further advantages will be explored in connection of 2-loop reduction of the standard model at high T, in the near future. 1truecm [**Acknowledgements**]{}
The authors are grateful to Z. Fodor, K. Kajantie, M. Laine and M. Shaposhnikov for important suggestions during the progress of this calculation. 1truecm [**Table Caption**]{}
Dependence of some physical observables of the electroweak phase transition on the parameter of the 3-d Lorentz gauge fixing $\alpha$ for two typical values of the Higgs mass ($m_H$). In subsequent columns the transition temperature ($T_c$), the order parameter discontinuity ($\Phi_c$), the mass of the magnetic gauge quanta $(m_W)$ at $T_c$ and the surface tension ($\sigma$) are shown. $v_0$ is the vacuum expectation value of the Higgs-field. 1truecm
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Current steps attributed to resonant tunneling through individual InAs quantum dots embedded in a GaAs-AlAs-GaAs tunneling device are investigated experimentally in magnetic fields up to 28 T. The steps evolve into strongly enhanced current peaks in high fields. This can be understood as a field-induced Fermi-edge singularity due to the Coulomb interaction between the tunneling electron on the quantum dot and the partly spin polarized Fermi sea in the Landau quantized three-dimensional emitter.'
address: |
$^1$Institut für Festkörperphysik, Universität Hannover, Appelstra[ß]{}e 2, D-30167 Hannover, Germany\
$^2$Institut für Theoretische Physik, Universität Hannover, Appelstra[ß]{}e 2, D-30167 Hannover, Germany\
$^3$Grenoble High Magnetic Field Laboratory, MPIF-CNRS, B.P. 166, F-38042 Grenoble Cedex 09, France\
$^4$Physikalisch-Technische Bundesanstalt Braunschweig, Bundesallee 100, D-38116 Braunschweig, Germany
author:
- 'I. Hapke-Wurst,$^1$ U. Zeitler,$^1$ H. Frahm,$^2$ A. G. M. Jansen,$^3$ R. J. Haug,$^1$ and K. Pierz$^4$'
title: ' Magnetic-field-induced singularities in spin dependent tunneling through InAs quantum dots'
---
\#1[[$\backslash$\#1]{}]{}
The interaction of the Fermi sea of a metallic system with a local potential can lead to strong singularities close to the Fermi edge. Such effects have been predicted more than thirty years ago for the X-ray absorption and emission of metals[@theoXray] and observed subsequently[@expXray]. Similar singularities as a consequence of many body effects are also known from the luminescence of quantum wells[@Lum]. Matveev and Larkin were the first to predict interaction-induced singularities in the tunneling current via a localized state[@Matveev:1992] which were measured experimentally in several resonant tunneling experiments[@Geim:1994; @Cobden:1995; @Benedict:1998] from [*two-dimensional*]{} electrodes through a zero-dimensional system.
Here we report on singularities observed in the resonant tunneling from highly doped [*three-dimensional*]{} (3D) GaAs electrodes through an InAs quantum dot (QD) embedded in an AlAs barrier. These Fermi-edge singularities (FES) show a considerable magnetic field dependence and a strong enhancement in high magnetic fields where the 3D electrons occupy the lowest Landau level in the emitter. We observe an asymmetry in the enhancement for electrons of different spins with an extremely strong FES for electrons carrying the majority spin of the emitter. The experimental observations are explained by a theoretical model taking into account the electrostatic potential experienced by the conduction electrons in the emitter due to the charged QD. We will show that the partial spin polarisation of the emitter causes extreme values of the edge exponent $\gamma$ not observed until present and going beyond the standard theory valid for $\gamma \ll 1$ [@Matveev:1992].
The active part of our samples are self-organized InAs QDs with 3-4 nm height and 10-15 nm diameter embedded in the middle of a 10 nm-thick AlAs barrier and
sandwiched between two 3D electrodes. They consist of a 15 nm undoped GaAs spacer layer and a GaAs-buffer with graded doping. A typical InAs dot is sketched in inset (a) of Fig. \[steps\], the vertical band structure across a dot is schematically shown in inset (b).
Current voltage ($I$-$V$) characteristics were measured in large area vertical diodes ($40\times 40~\mu$m$^2$) patterned on the wafer. In Fig. \[steps\] we show a part of a typical $I$-$V$-curve with several discrete steps. We have demonstrated previously that such steps can be assigned to single electron tunneling from 3D electrodes through individual InAs QDs [@Hapke:1999] consistent with other resonant tunneling experiments through self-organized InAs QDs [@tunnel].
For the positive bias voltages shown in Fig. \[steps\] the electrons tunnel from the bottom electrode into the base of an InAs QD and leave the dot via the top. The tunneling current is mainly determined by the tunneling rate through the effectively thicker barrier below the dot (single electron tunneling regime). A step in the current occurs at bias voltages where the energy level of a dot, $E_D$, coincides with the Fermi level of the emitter, $E_F$.
In the following we will concentrate on the step labeled (\*) in Fig. \[steps\]. Other steps in the same structure as well as steps observed in the $I$-$V$-characteristics of other structures show a very similar behavior.
After the step edge a slight overshoot in the tunneling current occurs consistent with other tunneling experiments through a localized impurity [@Geim:1994] or through InAs dots [@Benedict:1998]. This effect is caused by the Coulomb interaction between a localized electron on the dot and the electrons at the Fermi edge of the emitter. The decrease of the current $I(V)$ towards higher voltages $V >V_0$ follows a power law $I \propto (V-V_0)^{-\gamma}$ [@Geim:1994] ($V_0$ is the voltage at the step edge) with an edge exponent $\gamma = 0.02 \pm 0.01$.
The evolution of step (\*) in a magnetic field applied parallel to the current direction is shown in Fig. \[Babh\]a. The step develops into two separate peaks with onset voltages marked as $V_\downarrow$ and $V_\uparrow$. The Landau quantization of the emitter leads to an oscillation of $V_\downarrow$ and $V_\uparrow$ and a shift to smaller voltages as a function of magnetic field, see Fig. \[Babh\]b. This reflects the magneto-quantum-oscillation of the Fermi energy in the emitter [@Bumbel; @Main:2000]. From the period and the amplitude of the oscillation we can extract a Fermi energy (at $B = 0$) $E_0 = 13.6~$meV and a Landau level broadening $\Gamma = 1.3~$meV in the 3D emitter. The measured $E_0 = 13.6~$meV agrees well with the expected electron concentration at the barrier derived from the doping profile in the electrodes.
For $B > 6$ T only the lowest Landau level remains occupied. With a level broadening $\Gamma = 1.3~$meV the Fermi level $E_F$ for 15 T $<$ B $<$ 30 T is within less than $2~$meV pinned to the bottom of the lowest Landau band, $E_L = \hbar \omega_c/2$. As a consequence the onset voltage shifts to lower values as $\alpha e\Delta V \approx -\hbar \omega_c/2$ with $\alpha = 0.34$. The diamagnetic shift of the energy level in the dot can be neglected compared to this shift of the Fermi energy in the emitter. For the dot investigated in [@Hapke:1999] with $r_0 = 3.7$ nm the diamagnetic shift at 30 T is $\Delta E_D = 3.5$ meV negligible compared to $E_L = 26~$meV.
The two distinct steps with onset voltages $V_\downarrow$ and $V_\uparrow$ originate from the spin-splitting of the energy level $E_D$ in the dot. Their distance $\Delta V_p$ is given by the Zeeman splitting $\Delta E_z = g_D \mu_B B = \alpha e \Delta V_p$ with an energy-to-voltage conversion factor $\alpha = 0.34$ [@explain-alpha]. As shown in Fig. \[Babh\]c $\Delta V_p$ is indeed linear in B, with a Landé factor $g_D = 0.8$ in agreement with other experiments on InAs dots [@Thornton:1998].
For low magnetic fields ($B \le 9~$T in our case, see graph for $B = 9~T$ in Fig. \[Babh\]a) the size of the steps is very similar for both spins and about half of the size at zero field. Also the slight overshoot in the current as the signature of a Fermi edge singularity is similar for both spin orientations and comparable to the zero field case with an edge exponent $\gamma < 0.05$ for all magnetic fields $B < 10~$T.
The form of the current steps changes drastically in high magnetic fields where only the lowest Landau level of the emitter remains occupied. In particular, the second current step at higher voltage evolves into a strongly enhanced peak with a peak current of one order of magnitude higher compared to the zero-field case.
Following [@Thornton:1998] we assume that $g_D$ is positive whereas the Landé factor in bulk GaAs is negative. This assumption is verified by the fact that the energetically lower lying state (first peak in Fig. \[Tabh\]) is thermally occupied at higher temperatures and can therefore be identified with the minority spin in the emitter.
The strongly enhanced current peak at higher energies is due to tunneling through the spin state corresponding to the majority spin (spin up) in the emitter. The resulting spin configuration is scetched in the inset of Fig. \[Babh\]a and will also be confirmed below by our theoretical results.
The shape of this current peak can be described by a steep ascent and a more moderate decrease of the current towards higher voltages. Down to temperatures $T<100$ mK the steepness of the ascent is only limited by thermal broadening. The decrease of the current for $V >V_0$ is again described with the characteristic behavior for a Fermi-edge singularity, $I \propto (V-V_0)^{-\gamma}$, where $V_0$ here is the voltage at the maximum peak current.
However, along with the drastic increase of the peak current the edge exponent $\gamma$ increases dramatically reaching a value $\gamma > 0.5$ for the highest fields.
A different way to visualize the signature of a FES is a temperature dependent experiment. As an example we have plotted the $I$-$V$-curve at $B=22$ T for different temperatures in Fig. \[Tabh\]. As shown in the inset the peak maximum $I_0$ for the spin-up electrons decreases according to a power law $I_0 \propto T^{-\gamma}$ with an edge exponent $\gamma = 0.43 \pm 0.05$. Such a strong temperature dependence is characteristic for a FES and allows us to exclude that pure density of states effects in the 3D emitter are responsible for the current peaks in high magnetic fields.
As shown in Fig. \[Tabh\] an edge exponent $\gamma = 0.43$ also fits within experimental accuracy the observed decrease of the current for $V>V_0$.
It is not possible to extract the edge exponent for the minority spin directly from temperature dependent experiments. At high magnetic fields the observed increase of the current with increasing temperature is mainly caused by an additional thermal population of the minority spin in the emitter. The general form of the curve is merely affected by temperature. Therefore, the edge exponent can only be gained from fitting the shape of the current peaks.
A compilation of the edge exponents $\gamma$ for various magnetic fields and both spin orientations is shown in Fig. \[gammas\].
For the data related to the majority spin two independent methods were used to extract $\gamma$. For the minority spin only fitting of the shape of the $I$-$V$-curves was used.
For a theoretical description of these effects we consider a 3D electron gas in the half space $z<0$. In a sufficiently strong magnetic field $B||\hat{z}$ all electrons are in the lowest Landau level. This defines a set of one-dimensional channels with momentum $\hbar k$ perpendicular to the boundary. This situation is different from the cases considered for scattering off point defects as in Refs. [@theoXray; @Matveev:1992] or for a 2D electron gas where the current is carried by edge states [@BaMa95]. The single particle wave functions in channel $m\ge0$ are $\psi_m(\rho,\phi) \sin kz$ with $\psi_m(\rho,\phi) \propto \rho^m \exp(-im\phi-\rho^2/4\ell_0^2)$. In the experiments the magnetic length $\ell_0=\sqrt{\hbar/eB}$ ($\ell_0 = 5.6$ nm at 20 T) is comparable to the lateral size of the QD $2 r_0 \approx 7$ nm. Hence the effect of the electrostatic potential of a charged dot on the electrons in a given channel of the emitter decreases rapidly with $m$, and the observed FES are mainly due to tunneling of electrons from the $m=0$ channel into the dot. Following [@theoXray; @Matveev:1992] tunneling processes of spin $\sigma$ electrons from the $m=0$ state in the emitter give rise to a FES with edge exponent $$\gamma_\sigma = -\frac{2}{\pi}\delta_0(k_{F\sigma})
- \frac{1}{\pi^2}\sum_{m}\sum_{\tau=\uparrow,\downarrow}
\left(\delta_m(k_{F\tau})\right)^2
\label{expo}$$ where $\delta_m(k)$ is the Fermi phase shift experienced by the electrons in the $m$-th channel due to the potential of the quantum dot [@contact]. From (\[expo\]) the observed field dependence of the edge exponents is a consequence of the variation of the Fermi momenta for spin-$\sigma$ electrons with magnetic field *and* the field dependence of the effective potential in the one-dimensional channels. The former can be computed from the one-dimensional density of states (DOS) of the lowest Landau band $$D(E,B) = \frac{e\sqrt{m^{*}}}{(2\pi\hbar)^2}\,B \left(
d(\epsilon_\uparrow) +d(\epsilon_\downarrow) \right)\ .$$ Here $\epsilon_\sigma = E-(\hbar\omega_c\pm g^*\mu_BB)/2$ is the energy of electrons with spin-$\sigma$ measured from the bottom of the Landau band. $g^* \approx -0.33$ [@Pfeffer:1985] is the effective Landé factor of the electrons in the emitter. The DOS for the spin-subbands is $d(\epsilon) = \sqrt{2}
{\mathrm{Re}} (\epsilon+ i\Gamma)^{-{1/2}}$. Without broadening, $\Gamma=0$, one has $k_{F\sigma} = \pi^2n \ell_0^2 (1\pm b^3)$ where $n$ is the 3D density of electrons and $b$ is the magnetic field measured in units of the field necessary for complete spin polarization of the 3D emitter. Using a Fermi energy $E_0=13.6$ meV and neglecting level broadening we find that only the lowest Landau level (*both spin states!*) is occupied for $B_1 >5.2$ T. Including level broadening changes $B_1$ to a slightly higher value. With the known field dependence of the Fermi energy in the quantum limit we can calculate the field for total spin polarisation $$B_{pol} = \left( \frac{16}{9 \xi}\right) ^{1/3} \frac{m^* E_0}{\hbar e}
\simeq 43~\mbox{T}$$ with $g^*= -0.33$ [@Pfeffer:1985] and $m^*=0.067\,m_0$. $\xi = \frac{1}{2} |g^*| m^*/m_0$ is the ratio between spin splitting and Landau level splitting. To make contact to the experimental observations we have to specify the interaction of the screened charge on the QD and the conduction band electrons. A Thomas-Fermi calculation gives $U(\rho,z) = (2e^2 \exp(\kappa z)/\kappa) (d/(\rho^2 +d^2)^{(3/2)})$ [@Matveev:1992]. Here $d=5\,\mathrm{nm}$ is the width of the insulating layer and $\kappa^{-1}=7\,\mathrm{nm}$ is the Debye radius. The effective potential seen by electrons in channel $m$ is $V_m\exp(\kappa z)/\kappa$ with $V_m = 2e^2d \int d\rho^2
|\psi_m(\rho,\phi)|^2/(\rho^2 +d^2)^{(3/2)}$. For large $\kappa$ we obtain for the phase shift in the $m=0$ channel $\delta_0(k) \approx -v_0f(B)k/\kappa$ where $$f(B) = \left(\frac{d}{\ell_0}\right)^2
\left\{ 1 - \sqrt{\pi\over2}\, {d\over\ell_0} {\rm
e}^{d^2\over2\ell_0^2}
\mathrm{erfc}\left({d\over\sqrt{2}\ell_0}\right)
\right\}\$$ and $v_0 \sim (m^*e^2/\hbar^2\kappa) (\kappa d)^{-2}$ up to a numerical factor. Similarly we obtain the integrated effect of the channels $m>0$ in (\[expo\]). In Fig. 4 the resulting exponents $\gamma_\sigma$ obtained for $\sigma=\uparrow,\downarrow$ are shown for $v_0= 6.75$ and a broadening $\Gamma= 0$ and $\Gamma = 1.3$ meV, respectively. The value used for $\Gamma$ reflects its realistic experimental value. $v_0$ is the only fit parameter.
Already the simple model with no level broadening ($\Gamma = 0$) is in good agreement with the experimentally measured edge exponents for both spin directions, especially in high magnetic fields where possible admixtures of higher Landau levels play a minor rule. Including level broadening leads to a less dramatic spin polarisation in the emitter and as a consequence smears out the field dependence of $\gamma$ for the minority spin. The basic features, however, remain unchanged. In particular, the edge exponent for the minority spin retains moderate values for high magnetic fields, whereas the edge exponent related to the majority spin shows a strong field dependence with very high values in high magnetic fields.
In conclusion we have evaluated experimental data concerning magnetic-field-induced FES in resonant tunneling experiments through InAs QDs. We have shown that the interaction between a localized charge and the electrons in the Landau quantized emitter leads to dramatic Fermi phase shifts if only the lowest Landau level in the 3D emitter is occupied. This results in edge exponents $\gamma > 0.5$ which were measured and described theoretically.
We would like to thank H. Marx for sample growing, P. König for experimental support and F. J. Ahlers for valuable discussions. Part of this work has been supported by the TMR Programme of the European Union under contract no. ERBFMGECT950077. We acknowledge partial support from the Deutsche Forschungsgemeinschaft under Grants HA 1826/5-1 and Fr 737/3.
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The energy-to-voltage conversion factor $\alpha$ is derived from the temperature dependence of the width $\delta V$ of a current step caused by the thermal smearing of the Fermi-edge in the emitter. Only a part $\alpha$ of the total voltage applied drops between the emitter and the dot, the rest of the voltage drop occurs inside the electrodes and between dot and collector. A. S. G. Thornton, T. Ihn, P. C. Main, L. Eaves and M. Henini, Appl. Phys. Lett. [**73**]{}, 354 (1998).
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In fact, one has to consider the change of this phase shift due to the increase in charge on the QD here. For the contact interaction considered here this is not essential.
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The electronic Landé factor in the lowest Landau level in GaAs increases linearly with magnetic field from $g^*=-0.44$ at $B=0$ to $g^*=-0.29$ at $B=30$ T. Our theory uses a constant $g^*$, for the best possible connection with the experiment we have chosen to use its value at $B=22$ T.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We establish a Fredholm criterion for an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over Carleson curves with logarithmic whirl points.'
address: ' Universidade do Minho, Centro de Matemática, Escola de Ciências, Campus de Gualtar, 4710-057 Braga, Portugal'
author:
- 'A. Yu. Karlovich'
title: ALGEBRAS OF SINGULAR INTEGRAL OPERATORS ON NAKANO SPACES WITH KHVEDELIDZE WEIGHTS OVER CARLESON CURVES WITH LOGARITHMIC WHIRL POINTS
---
Introduction
============
Fredholm theory of one-dimensional singular integral operators (SIOs) with piecewise continuous ($PC$) coefficients on weighted Lebesgue spaces was constructed by Gohberg and Krupnik [@GK92] and [@GK70; @GK71] in the beginning of 70s in the case of Khvedelidze weights and piecewise Lyapunov curves (see also the monographs [@CG81; @KS01; @LS87; @MP86]). Simonenko and Chin Ngok Min [@SCNM86] suggested another approach to the study of the Banach algebra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces with Khvedelidze weights over piecewise Lyapunov curves. This approach is based on Simonenko’s local principle [@Simonenko65]. In 1992 Spitkovsky [@Spitkovsky92] made a next significant step: he proved a Fredholm criterion for an individual SIO with $PC$ coefficients on Lebesgue spaces with Muckenhoupt weights over Lyapunov curves. Finally, Böttcher and Yu. Karlovich extended Spitkovsky’s result to the case of arbitrary Carleson curves and Banach algebras of SIOs with $PC$ coefficients. After their work the Fredholm theory of SIOs with $PC$ coefficients is available in the maximal generality (that, is, when the Cauchy singular integral operator $S$ is bounded on weighted Lebesgue spaces). We recommend the nice paper [@BK01] for a first reading about this topic and the book [@BK97] for a complete and self-contained exposition.
It is quite natural to consider the same problems in other, more general, spaces of measurable functions on which the operator $S$ is bounded. Good candidates for this role are rearrangement-invariant spaces (that is, spaces with the property that norms of equimeasurable functions are equal). These spaces have nice interpolation properties and boundedness results can be extracted from known results for Lebesgue spaces applying interpolation theorems. The author extended (some parts of) the Böttcher-Yu. Karlovich Fredholm theory of SIOs with $PC$ coefficients to the case of rearrangement-invariant spaces with Muckenhoupt weights [@K98; @K02]. Notice that necessary conditions for the Fredholmness of an individual singular integral operator with $PC$ coefficients are obtained in [@K03] for weighted reflexive Banach function spaces on which the operator $S$ is bounded.
Nakano spaces $L^{p(\cdot)}$ (generalized Lebesgue spaces with variable exponent) are a nontrivial example of Banach function spaces which are not rearrangement-invariant, in general. Many results about the behavior of some classical operators on these spaces have important applications to fluid dynamics (see [@DR03] and the references therein). Kokilashvili and Samko [@KS-GMJ] proved that the operator $S$ is bounded on weighted Nakano spaces for the case of nice curves, nice weights, and nice (but variable!) exponents. They also extended the Gohberg-Krupnik Fredholm criterion for an individual SIO with $PC$ coefficients to this situation [@KS-Proc] (see also [@Samko05]). The author [@K05] has found a Fredholm criterion and a formula for the index of an arbitrary operator in the Banach algebra of SIOs with $PC$ coefficients on Nakano spaces with Khvedelidze weights over either Lyapunov curves or Radon curves without cusps.
Very recently Kokilashvili and Samko [@KS-Memoirs] (see also [@Kokilashvili05 Theorem 7.1]) have proved a boundedness criterion for the Cauchy singular integral operator $S$ on Nakano spaces with Khvedelidze weights over arbitrary Carleson curves. Combining this boundedness result with the machinery developed in [@K03], we are able to prove a Fredholm criterion for an individual SIO on a Nakano space with a Khvedelidze weight over a Carleson curve satisfying a “logarithmic whirl condition" (see [@BK95], [@BK97 Ch. 1]) at each point. Further, we extend this result to the case of Banach algebras of SIOs with $PC$ coefficients, using the approach developed in [@BK95; @BK97; @K03; @K05].
The paper is organized as follows. In Section \[sect:preliminaries\] we define weighted Nakano spaces and discuss the boundedness of the operator $S$ on these spaces. Section \[sect:individual\] contains a Fredholm criterion for an individual SIO with $PC$ coefficients on weighted Nakano spaces. The proof of this result is based on the local principle of Simonenko type and factorization technique. In Section \[sect:tools\] we formulate the Allan-Douglas local principle and the two projections theorem. The results of Section \[sect:tools\] are the main tools allowing us to construct a symbol calculus for the Banach algebra of SIOs with $PC$ coefficients acting on a Nakano space with a Khvedelidze weight over a Carleson curve with logarithmic whirl points in Section \[sect:symbol\].
Preliminaries {#sect:preliminaries}
=============
Weighted Nakano spaces $L_w^{p(\cdot)}$
---------------------------------------
Function spaces $L^{p(\cdot)}$ of Lebesgue type with variable exponent $p$ were studied for the first time by Orlicz [@Orlicz31] in 1931, but notice that another kind of Banach spaces is called after him. Inspired by the successful theory of Orlicz spaces, Nakano defined in the late forties [@Nakano50; @Nakano51] so-called *modular spaces*. He considered the space $L^{p(\cdot)}$ as an example of modular spaces. In 1959, Musielak and Orlicz [@MO59] extended the definition of modular spaces by Nakano. Actually, that paper was the starting point for the theory of Musielak-Orlicz spaces (generalized Orlicz spaces generated by Young functions with a parameter), see [@Musielak83].
Let $\Gamma$ be a Jordan (i.e., homeomorphic to a circle) rectifiable curve. We equip $\Gamma$ with the Lebesgue length measure $|d\tau|$ and the counter-clockwise orientation. Let $p:\Gamma\to(1,\infty)$ be a measurable function. Consider the convex modular (see [@Musielak83 Ch. 1] for definitions and properties) $$m(f,p):=\int_\Gamma|f(\tau)|^{p(\tau)}|d\tau|.$$ Denote by $L^{p(\cdot)}$ the set of all measurable complex-valued functions $f$ on $\Gamma$ such that $m(\lambda f,p)<\infty$ for some $\lambda=\lambda(f)>0$. This set becomes a Banach space when equipped with the *Luxemburg-Nakano norm* $$\|f\|_{L^{p(\cdot)}}:=\inf\big\{\lambda>0: \ m(f/\lambda,p)\le 1\big\}$$ (see, e.g., [@Musielak83 Ch. 2]). Thus, the spaces $L^{p(\cdot)}$ are a special case of Musielak-Orlicz spaces. Sometimes the spaces $L^{p(\cdot)}$ are referred to as Nakano spaces. We will follow this tradition. Clearly, if $p(\cdot)=p$ is constant, then the Nakano space $L^{p(\cdot)}$ is isometrically isomorphic to the Lebesgue space $L^p$. Therefore, sometimes the spaces $L^{p(\cdot)}$ are called generalized Lebesgue spaces with variable exponent or, simply, variable $L^p$ spaces.
We shall assume that $$\label{eq:reflexivity}
1<{\rm ess}\inf_{\!\!\!\!\!\!\!\!t\in\Gamma} p(t),
\quad
{\rm ess}\sup_{\!\!\!\!\!\!\!\!\!t\in\Gamma} p(t)<\infty.$$ In this case the conjugate exponent $$q(t):=\frac{p(t)}{p(t)-1}
\quad (t\in\Gamma)$$ has the same property.
A nonnegative measurable function $w$ on the curve $\Gamma$ is referred to as a [*weight*]{} if $0<w(t)<\infty$ almost everywhere on $\Gamma$. The [*weighted Nakano space*]{} is defined by $$L_w^{p(\cdot)}=
\big\{f\mbox{ is measurable on }\Gamma\mbox{ and }fw\in L^{p(\cdot)}\big\}.$$ The norm in $L_w^{p(\cdot)}$ is defined by $\|f\|_{L_w^{p(\cdot)}}=\|fw\|_{L^{p(\cdot)}}$.
Carleson curves
---------------
A rectifiable Jordan curve $\Gamma$ is said to be a [*Carleson*]{} (or [*Ahlfors-David regular*]{}) [*curve*]{} if $$\sup_{t\in\Gamma}\sup_{R>0}\frac{|\Gamma(t,R)|}{R}<\infty,$$ where $\Gamma(t,R):=\{\tau\in\Gamma:|\tau-t|<R\}$ for $R>0$ and $|\Omega|$ denotes the measure of a measurable set $\Omega\subset\Gamma$. We can write $$\tau-t=|\tau-t|e^{i\arg(\tau-t)}
\quad\mbox{for}\quad\tau\in\Gamma\setminus\{t\},$$ and the argument can be chosen so that it is continuous on $\Gamma\setminus\{t\}$. Seifullaev [@Seif80] showed that for an arbitrary Carleson curve the estimate $\arg(\tau-t)=O(-\log|\tau-t|)$ as $\tau\to t$ holds for every $t\in\Gamma$. A simpler proof of this result can be found in [@BK97 Theorem 1.10]. One says that a Carleson curve $\Gamma$ satisfies the *logarithmic whirl condition* at $t\in\Gamma$ if $$\label{eq:spiralic}
\arg(\tau-t)=-\delta(t)\log|\tau-t|+O(1)\quad (\tau\to t)$$ with some $\delta(t)\in{\mathbb{R}}$. Notice that all piecewise smooth curves satisfy this condition at each point and, moreover, $\delta(t)\equiv 0$. For more information along these lines, see [@BK95], [@BK97 Ch. 1], [@BK01].
The Cauchy singular integral operator
-------------------------------------
The *Cauchy singular integral* of $f\in L^1$ is defined by $$(Sf)(t):=\lim_{R\to 0}\frac{1}{\pi i}\int_{\Gamma\setminus\Gamma(t,R)}
\frac{f(\tau)}{\tau-t}d\tau
\quad (t\in\Gamma).$$ Not so much is known about the boundedness of the Cauchy singular integral operator $S$ on weighted Nakano spaces $L_w^{p(\cdot)}$ for general curves, general weights, and general exponents $p(\cdot)$. [From]{} [@K03 Theorem 6.1] we immediately get the following.
Let $\Gamma$ be a rectifiable Jordan curve, let $w:\Gamma\to[0,\infty]$ be a weight, and let $p:\Gamma\to(1,\infty)$ be a measurable function satisfying . If the Cauchy singular integral generates a bounded operator $S$ on the weighted Nakano space $L_w^{p(\cdot)}$, then $$\label{eq:Ap}
\sup_{t\in\Gamma}\sup_{R>0}\frac{1}{R}
\|w\chi_{\Gamma(t,R)}\|_{L^{p(\cdot)}}
\|\chi_{\Gamma(t,R)}/w\|_{L^{q(\cdot)}}<\infty.$$
[From]{} the Hölder inequality for Nakano spaces (see, e.g., [@Musielak83] or [@KR91]) and we deduce that if $S$ is bounded on $L_w^{p(\cdot)}$, then $\Gamma$ is necessarily a Carleson curve. If the exponent $p(\cdot)=p\in(1,\infty)$ is constant, then is simply the famous Muckenhoupt condition $A_p$. It is well known that for classical Lebesgue spaces $L^p$ this condition is not only necessary, but also sufficient for the boundedness of the Cauchy singular integral operator $S$. A detailed proof of this result can be found in [@BK97 Theorem 4.15].
Let $N\in{\mathbb{N}}$. Consider now a power weight $$\label{eq:power}
\varrho(t):=\prod_{k=1}^N|t-\tau_k|^{\lambda_k},
\quad
\tau_k\in\Gamma,
\quad
k\in\{1,\dots,N\},$$ where all $\lambda_k$ are real numbers. Introduce the class ${\mathcal{P}}$ of exponents $p:\Gamma\to(1,\infty)$ satisfying and $$\label{eq:Dini-Lipschitz}
|p(\tau)-p(t)|\le\frac{A}{-\log |\tau-t|}$$ for some $A\in(0,\infty)$ and all $\tau,t\in\Gamma$ such that $|\tau-t|<1/2$.
The following criterion for the boundedness of the Cauchy singular integral operator on Nakano spaces with power weights has been recently proved by Kokilashvili and Samko [@KS-Memoirs] (see also [@Kokilashvili05 Theorem 7.1]).
\[th:KS\] Let $\Gamma$ be a Carleson Jordan curve, let $\varrho$ be a power weight of the form , and let $p\in{\mathcal{P}}$. The Cauchy singular integral operator $S$ is bounded on the weighted Nakano space $L_\varrho^{p(\cdot)}$ if and only if $$\label{eq:Khvedelidze}
0<\frac{1}{p(\tau_k)}+\lambda_k<1
\quad\mbox{for all}\quad
k\in\{1,\dots,N\}.$$
For weighted Lebesgue spaces over Lyapunov curves the above theorem was proved by Khvedelidze [@Khvedelidze56] (see also the proof in [@GK92; @Khvedelidze75; @MP86]). Therefore the weights of the form are often called *Khvedelidze weights*. We shall follow this tradition.
Notice that if $p$ is constant and $\Gamma$ is a Carleson curve, then is equivalent to the fact that $\varrho$ is a Muckenhoupt weight (see, e.g., [@BK97 Chapter 2]). Analogously one can prove that if the exponent $p$ belong to the class ${\mathcal{P}}$ and the curve $\Gamma$ is Carleson, then the power weight satisfies the condition if and only if is fulfilled. The proof of this fact is based on certain estimates for the norms of power functions in Nakano spaces with exponents in the class ${\mathcal{P}}$ (see also [@K03 Lemmas 5.7 and 5.8] and [@KS-GMJ]).
Singular integral operators with $PC$ coefficients {#sect:individual}
==================================================
The local principle of Simonenko type
-------------------------------------
Let $I$ be the identity operator on $L_\varrho^{p(\cdot)}$. Under the conditions of Theorem \[th:KS\], the operators $$P:=(I+S)/2,
\quad
Q:=(I-S)/2$$ are bounded projections on $L_\varrho^{p(\cdot)}$ (see [@K03 Lemma 6.4]). Let $L^\infty$ denote the space of all measurable essentially bounded functions on $\Gamma$. The operators of the form $aP+Q$ with $a \in L^\infty$ are called [*singular integral operators*]{} (SIOs). Two functions $a,b\in L^\infty$ are said to be locally equivalent at a point $t\in\Gamma$ if $$\inf\big\{\|(a-b)c\|_\infty\ :\ c\in C,\ c(t)=1\big\}=0.$$
\[th:local\_principle\] Suppose the conditions of Theorem [\[th:KS\]]{} are satisfied and $a\in L^\infty$. Suppose for each $t\in\Gamma$ there is a function $a_t\in L^\infty$ which is locally equivalent to $a$ at $t$. If the operators $a_tP+Q$ are Fredholm on $L_\varrho^{p(\cdot)}$ for all $t\in\Gamma$, then $aP+Q$ is Fredholm on $L_\varrho^{p(\cdot)}$.
For weighted Lebesgue spaces this theorem is known as Simonenko’s local principle [@Simonenko65]. It follows from [@K03 Theorem 6.13].
Simonenko’s factorization theorem
---------------------------------
The curve $\Gamma$ divides the complex plane $\mathbb{C}$ into the bounded simply connected domain $D^+$ and the unbounded domain $D^-$. Without loss of generality we assume that $0\in D^+$. We say that a function $a\in L^\infty$ admits a *Wiener-Hopf factorization on* $L_\varrho^{p(\cdot)}$ if $1/a\in L^\infty$ and $a$ can be written in the form $$\label{eq:WH}
a(t)=a_-(t)t^\kappa a_+(t)
\quad\mbox{a.e. on}\ \Gamma,$$ where $\kappa\in{\mathbb{Z}}$, and the factors $a_\pm$ enjoy the following properties:
1. $a_-\in QL_\varrho^{p(\cdot)}\stackrel{\cdot}{+}\mathbb{C}, \quad
1/a_-\in QL_{1/\varrho}^{q(\cdot)}\stackrel{\cdot}{+}\mathbb{C},
\quad a_+\in PL_{1/\varrho}^{q(\cdot)},\quad
1/a_+\in PL_\varrho^{p(\cdot)}$,
2. the operator $(1/a_+)Sa_+I$ is bounded on $L_\varrho^{p(\cdot)}$.
One can prove that the number $\kappa$ is uniquely determined.
\[th:factorization\] Suppose the conditions of Theorem [\[th:KS\]]{} are satisfied. A function $a\in L^\infty$ admits a Wiener-Hopf factorization [(\[eq:WH\])]{} on $L_\varrho^{p(\cdot)}$ if and only if the operator $aP+Q$ is Fredholm on $L_\varrho^{p(\cdot)}$. If $aP+Q$ is Fredholm, then its index is equal to $-\kappa$.
This theorem goes back to Simonenko [@Simonenko64; @Simonenko68]. For more about this topic we refer to [@BK97 Section 6.12], [@BS90 Section 5.5], [@GK92 Section 8.3] and also to [@CG81; @LS87] in the case of weighted Lebesgue spaces. Theorem \[th:factorization\] follows from [@K03 Theorem 6.14].
Fredholm criterion for singular integral operators with $PC$ coefficients
-------------------------------------------------------------------------
We denote by $PC$ the Banach algebra of all piecewise continuous functions on $\Gamma$: a function $a\in L^\infty$ belongs to $PC$ if and only if the finite one-sided limits $$a(t\pm 0):=\lim_{\tau\to t\pm 0}a(\tau)$$ exist for every $t\in\Gamma$.
\[th:criterion\] Let $\Gamma$ be a Carleson Jordan curve satisfying with $\delta(t)\in{\mathbb{R}}$ for every $t\in\Gamma$. Suppose $p\in{\mathcal{P}}$ and $\varrho$ is a power weight of the form which satisfies . The operator $aP+Q$, where $a\in PC$, is Fredholm on the weighted Nakano space $L^{p(\cdot)}_\varrho$ if and only if $a(t\pm 0)\ne 0$ and $$-\frac{1}{2\pi}\arg\frac{a(t-0)}{a(t+0)}
+
\frac{\delta(t)}{2\pi}\log\left|\frac{a(t-0)}{a(t+0)}\right|
+
\frac{1}{p(t)}+\lambda(t)\notin{\mathbb{Z}}\label{eq:Fredholm}$$ for all $t\in\Gamma$, where $$\lambda(t):=\left\{
\begin{array}{lcl}
\lambda_k, &\mbox{if} & t=\tau_k, \quad k\in\{1,\dots,N\},\\
0, &\mbox{if} & t\notin\Gamma\setminus\{\tau_1,\dots,\tau_N\}.
\end{array}
\right.$$
The [*necessity*]{} part follows from [@K03 Theorem 8.1] because the Böttcher-Yu. Karlovich indicator functions $\alpha_t$ and $\beta_t$ in that theorem for a Khvedelidze weight $\varrho$ and a Carleson curve $\Gamma$ satisfying the logarithmic whirl condition at $t\in\Gamma$ are calculated by $$\alpha_t(x)=\beta_t(x)=\lambda(t)+\delta(t)x\quad \mbox{for}\quad x\in{\mathbb{R}}$$ (see [@BK97 Ch. 3] or [@K02 Lemma 3.9]).
[*Sufficiency.*]{} If $aP+Q$ is Fredholm, then, by [@K03 Theorem 6.11], $a(t\pm 0)\ne 0$ for all $t\in\Gamma$.
Fix $t\in\Gamma$. For the function $a$ we construct a “canonical” function $g_{t,\gamma}$ which is locally equivalent to $a$ at the point $t\in\Gamma$. The interior and the exterior of the unit circle can be conformally mapped onto $D^+$ and $D^-$ of $\Gamma$, respectively, so that the point $1$ is mapped to $t$, and the points $0\in D^+$ and $\infty\in D^-$ remain fixed. Let $\Lambda_0$ and $\Lambda_\infty$ denote the images of $[0,1]$ and $[1,\infty)\cup\{\infty\}$ under this map. The curve $\Lambda_0\cup\Lambda_\infty$ joins $0$ to $\infty$ and meets $\Gamma$ at exactly one point, namely $t$. Let $\arg z$ be a continuous branch of argument in $\mathbb{C}\setminus(\Lambda_0\cup\Lambda_\infty)$. For $\gamma\in\mathbb{C}$, define the function $z^\gamma:=|z|^\gamma e^{i\gamma\arg z}$, where $z\in\mathbb{C}\setminus(\Lambda_0\cup\Lambda_\infty)$. Clearly, $z^\gamma$ is an analytic function in $\mathbb{C}\setminus(\Lambda_0\cup\Lambda_\infty)$. The restriction of $z^\gamma$ to $\Gamma\setminus\{t\}$ will be denoted by $g_{t,\gamma}$. Obviously, $g_{t,\gamma}$ is continuous and nonzero on $\Gamma\setminus\{t\}$. Since $a(t\pm 0)\ne 0$, we can define $\gamma_t=\gamma\in\mathbb{C}$ by the formulas $$\operatorname{Re}\gamma_t:=\frac{1}{2\pi}\arg\frac{a(t-0)}{a(t+0)},
\quad
\operatorname{Im}\gamma_t:=-\frac{1}{2\pi}\log\left|\frac{a(t-0)}{a(t+0)}\right|,$$ where we can take any value of $\arg(a(t-0)/a(t+0))$, which implies that any two choices of $\operatorname{Re}\gamma_t$ differ by an integer only. Clearly, there is a constant $c_t\in\mathbb{C}\setminus\{0\}$ such that $a(t\pm 0)=c_tg_{t,\gamma_t}(t\pm 0)$, which means that $a$ is locally equivalent to $c_tg_{t,\gamma_t}$ at the point $t\in\Gamma$. [From]{} it follows that there exists an $m_t\in{\mathbb{Z}}$ such that $$0<m_t-\operatorname{Re}\gamma_t-
\delta(t)\operatorname{Im}\gamma_t+\frac{1}{p(t)}+\lambda(t)<1.$$ By Theorem \[th:KS\], the operator $S$ is bounded on $L_{\widetilde{\varrho}}^{p(\cdot)}$, where $$\widetilde{\varrho}(\tau)=
|\tau-t|^{m_t-\operatorname{Re}
\gamma_t-\delta(t)\operatorname{Im}\gamma_t}\varrho(\tau)$$ for $\tau\in\Gamma$. In view of the logarithmic whirl condition we have $$\begin{aligned}
|(\tau-t)^{m_t-\gamma_t}|
&=&
|\tau-t|^{m_t-\operatorname{Re}\gamma_t}
e^{\operatorname{Im}\gamma_t\arg(\tau-t)}
\\
&=&
|\tau-t|^{m_t-\operatorname{Re}\gamma_t}
e^{-\operatorname{Im}\gamma_t(\delta(t)\log|\tau-t|+O(1))}
\\
&=&
|\tau-t|^{m_t-\operatorname{Re}\gamma_t-\delta(t)\operatorname{Im}\gamma_t}
e^{-\operatorname{Im}\gamma_t O(1)}\end{aligned}$$ as $\tau\to t$. Therefore the operator $\varphi_{t,m_t-\gamma_t}S\varphi_{t,\gamma_t-m_t}I$, where $$\varphi_{t,m_t-\gamma_t}(\tau)=|(\tau-t)^{m_t-\gamma_t}|,$$ is bounded on $L_\varrho^{p(\cdot)}$. Then, by [@K03 Lemma 7.1], the function $g_{t,\gamma_t}$ admits a Wiener-Hopf factorization on $L_\varrho^{p(\cdot)}$. Due to Theorem \[th:factorization\], the operator $g_{t,\gamma_t}P+Q$ is Fredholm. Then the operator $c_tg_{t,\gamma_t}P+Q$ is Fredholm, too. Since the function $c_tg_{t,\gamma_t}$ is locally equivalent to the function $a$ at every point $t\in\Gamma$, in view of Theorem \[th:local\_principle\], the operator $aP+Q$ is Fredholm on $L_\varrho^{p(\cdot)}$.
Double logarithmic spirals
--------------------------
Given $z_1,z_2\in\mathbb{C}$, $\delta\in{\mathbb{R}}$, and $r\in(0,1)$, put $$\mathcal{S}(z_1,z_2;\delta,r) := \{z_1,z_2\} \cup
\Big\{
z\in\mathbb{C}\setminus\{z_1,z_2\}:
\arg\frac{z-z_1}{z-z_2}-
\delta\log\left|\frac{z-z_1}{z-z_2}\right|\in2\pi (r+{\mathbb{Z}})
\Big\}.$$ The set $\mathcal{S}(z_1,z_2;\delta,r)$ is a double logarithmic spiral whirling about the points $z_1$ and $z_2$. It degenerates to a familiar Widom-Gohberg-Krupnik circular arc whenever $\delta=0$ (see [@BK97; @GK92]).
Fix $t\in\Gamma$ and consider a function $\chi_t\in PC$ which is continuous on $\Gamma\setminus\{t\}$ and satisfies $\chi_t(t-0)=0$ and $\chi_t(t+0)=1$.
[From]{} Theorem \[th:criterion\] we get the following.
\[co:important\] Let $\Gamma$ be a Carleson Jordan curve satisfying with $\delta(t)\in{\mathbb{R}}$ for every $t\in\Gamma$. Suppose $p\in{\mathcal{P}}$ and $\varrho$ is a power weight of the form which satisfies . Then $$\big\{\lambda\in\mathbb{C}:(\chi_t-\lambda)P+Q
\mbox{ is not Fredholm on }L_\varrho^{p(\cdot)}\big\}
=\mathcal{S}\big(0,1;\delta(t),1/p(t)+\lambda(t)\big).$$
Tools for the construction of the symbol calculus {#sect:tools}
=================================================
The Allan-Douglas local principle
---------------------------------
Let $B$ be a Banach algebra with identity. A subalgebra $Z$ of $B$ is said to be a central subalgebra if $zb=bz$ for all $z\in Z$ and all $b\in B$.
\[th:AllanDouglas\] [(see [@BS90 Theorem 1.34(a)]).]{} Let $B$ be a Banach algebra with unit $e$ and let $Z$ be closed central subalgebra of $B$ containing $e$. Let $M(Z)$ be the maximal ideal space of $Z$, and for $\omega\in M(Z)$, let $J_\omega$ refer to the smallest closed two-sided ideal of $B$ containing the ideal $\omega$. Then an element $b$ is invertible in $B$ if and only if $b+J_\omega$ is invertible in the quotient algebra $B/J_\omega$ for all $\omega\in M(Z)$.
The two projections theorem
---------------------------
The following two projections theorem was obtained by Finck, Roch, Silbermann [@FRS93] and Gohberg, Krupnik [@GK93].
\[th:2proj\] Let $F$ be a Banach algebra with identity $e$, let ${\mathcal{C}}$ be a Banach subalgebra of $F$ which contains $e$ and is isomorphic to ${\mathbb{C}}^{n \times n}$, and let $p$ and $q$ be two projections in $F$ such that $cp=pc$ and $cq=qc$ for all $c \in {\mathcal{C}}$. Let $W={\mathrm{alg}}({\mathcal{C}},p,q)$ be the smallest closed subalgebra of $F$ containing ${\mathcal{C}},p,q$. Put $$x=pqp+(e-p)(e-q)(e-p),$$ denote by $\mathrm{sp}\,x$ the spectrum of $x$ in $F$, and suppose the points $0$ and $1$ are not isolated points of $\mathrm{sp}\,x$. Then
1. for each $\mu \in \mathrm{sp}\,x$ the map $\sigma_{\mu}$ of ${\mathcal{C}}\cup \{p,q\}$ into the algebra ${\mathbb{C}}^{2n\times 2n}$ of all complex $2n\times 2n$ matrices defined by $$\label{eq:2proj1}
\sigma_{\mu}c=\left(
\begin{array}{cc}
c & 0\\
0 & c
\end{array}
\right),
\quad
\sigma_{\mu}p=\left(
\begin{array}{cc}
E & 0\\
0 & 0
\end{array}
\right),$$ $$\label{eq:2proj2}
\sigma_{\mu}q=\left(
\begin{array}{cc}
\mu E & \sqrt{\mu(1-\mu)}E \\
\sqrt{\mu(1-\mu)}E & (1-\mu)E
\end{array}
\right),$$ where $c\in {\mathcal{C}}, E$ denotes the $n \times n$ unit matrix and $\sqrt{\mu(1-\mu)}$ denotes any complex number whose square is $\mu(1-\mu)$, extends to a Banach algebra homomorphism $$\sigma_{\mu}: W \to {\mathbb{C}}^{2n \times 2n};$$
2. every element $a$ of the algebra $W$ is invertible in the algebra $F$ if and only if $$\det \sigma_{\mu} a \neq 0 \quad\mbox{for all}\quad \mu \in \mathrm{sp}\,x;$$
3. the algebra $W$ is inverse closed in $F$ if and only if the spectrum of $x$ in $W$ coincides with the spectrum of $x$ in $F$.
A further generalization of the above result to the case of $N$ projections is contained in [@BK97].
Algebra of singular integral operators with $PC$ coefficients {#sect:symbol}
=============================================================
The ideal of compact operators
------------------------------
In this section we will suppose that $\Gamma$ is a Carleson curve satisfying with $\delta(t)\in{\mathbb{R}}$ for every $t\in\Gamma$, $p\in{\mathcal{P}}$, and $\varrho$ is a Khvedelidze weight of the form which satisfies . Let $X_n:=[L_\varrho^{p(\cdot)}]_n$ be the direct sum of $n$ copies of weighted Nakano spaces $X:=L_\varrho^{p(\cdot)}$, let ${\mathcal{B}}:={\mathcal{B}}(X_n)$ be the Banach algebra of all bounded linear operators on $X_n$, and let ${\mathcal{K}}:={\mathcal{K}}(X_n)$ be the closed two-sided ideal of all compact operators on $X_n$. We denote by $C^{n\times n}$ (resp. $PC^{n\times n}$) the collection of all continuous (resp. piecewise continuous) $n\times n$ matrix functions, that is, matrix-valued functions with entries in $C$ (resp. $PC$). Put $I^{(n)}:={\mathrm{diag}}\{I,\dots, I\}$ and $S^{(n)}:={\mathrm{diag}}\{S,\dots,S\}$. Our aim is to get a Fredholm criterion for an operator $$A\in{\mathcal{U}}:={\mathrm{alg}}(PC^{n\times n},S^{(n)}),$$ the smallest Banach subalgebra of ${\mathcal{B}}$ which contains all operators of multiplication by matrix-valued functions in $PC^{n\times n}$ and the operator $S^{(n)}$.
\[le:compact\] The ideal ${\mathcal{K}}$ is contained in the algebra ${\mathrm{alg}}(C^{n\times n},S^{(n)})$, the smallest closed subalgebra of ${\mathcal{B}}$ which contains the operators of multiplication by continuous matrix-valued functions and the operator $S^{(n)}$.
The proof of this statement is standard and can be developed as in [@K96 Lemma 9.1] or [@K05 Lemma 5.1].
Operators of local type
-----------------------
We shall denote by ${\mathcal{B}}^\pi$ the Calkin algebra ${\mathcal{B}}/{\mathcal{K}}$ and by $A^\pi$ the coset $A+{\mathcal{K}}$ for any operator $A\in{\mathcal{B}}$. An operator $A\in{\mathcal{B}}$ is said to be of [*local type*]{} if $AcI^{(n)}-cA$ is compact for all $c\in C$, where $cI^{(n)}$ denotes the operator of multiplication by the diagonal matrix-valued function ${\mathrm{diag}}\{c,\dots,c\}$. This notion goes back to Simonenko [@Simonenko65] (see also the presentation of Simonenko’s local theory in his joint monograph with Chin Ngok Min [@SCNM86]). It easy to see that the set ${\mathcal{L}}$ of all operators of local type is a closed subalgebra of ${\mathcal{B}}$.
\[pr:OLT\]
1. We have ${\mathcal{K}}\subset{\mathcal{U}}\subset{\mathcal{L}}$.
2. An operator $A\in{\mathcal{L}}$ is Fredholm if and only if the coset $A^\pi$ is invertible in the quotient algebra ${\mathcal{L}}^\pi:={\mathcal{L}}/{\mathcal{K}}$.
\(a) The embedding ${\mathcal{K}}\subset{\mathcal{U}}$ follows from Lemma \[le:compact\], the embedding ${\mathcal{U}}\subset{\mathcal{L}}$ follows from the fact that $cS-ScI$ is a compact operator on $L_\varrho^{p(\cdot)}$ for $c\in C$ (see, e.g., [@K03 Lemma 6.5]).
\(b) The proof of this fact is straightforward.
Localization
------------
[From]{} Proposition \[pr:OLT\](a) we deduce that the quotient algebras ${\mathcal{U}}^\pi:={\mathcal{U}}/{\mathcal{K}}$ and ${\mathcal{L}}^\pi:={\mathcal{L}}/{\mathcal{K}}$ are well defined. We shall study the invertibility of an element $A^\pi$ of ${\mathcal{U}}^\pi$ in the larger algebra ${\mathcal{L}}^\pi$ by using a localization techniques (more precisely, Theorem \[th:AllanDouglas\]). To this end, consider $${\mathcal{Z}}^\pi:=\big\{(cI^{(n)})^\pi:c\in C\big\}.$$ [From]{} the definition of ${\mathcal{L}}$ it follows that ${\mathcal{Z}}^\pi$ is a central subalgebra of ${\mathcal{L}}^\pi$. The maximal ideal space $M({\mathcal{Z}}^\pi)$ of ${\mathcal{Z}}^\pi$ may be identified with the curve $\Gamma$ via the Gelfand map ${\mathcal{G}}$ given by $${\mathcal{G}}:{\mathcal{Z}}^\pi\to C,
\quad
\big({\mathcal{G}}(cI^{(n)})^\pi\big)(t)=c(t)
\quad (t\in\Gamma).$$ In accordance with Theorem \[th:AllanDouglas\], for every $t\in\Gamma$ we define ${\mathcal{J}}_t\subset{\mathcal{L}}^\pi$ as the smallest closed two-sided ideal of ${\mathcal{L}}^\pi$ containing the set $$\big\{(cI^{(n)})^\pi\ :\ c\in C,\ c(t)=0\big\}.$$
Consider a function $\chi_t\in PC$ which is continuous on $\Gamma\setminus\{t\}$ and satisfies $\chi_t(t-0)=0$ and $\chi_t(t+0)=1$. For $a\in PC^{n\times n}$ define the function $a_t\in PC^{n\times n}$ by $$\label{eq:at}
a_t:=a(t-0)(1-\chi_t)+a(t+0)\chi_t.$$ Clearly $(aI^{(n)})^\pi-(a_tI^{(n)})^\pi\in{\mathcal{J}}_t$. Hence, for any operator $A\in{\mathcal{U}}$, the coset $A^\pi+{\mathcal{J}}_t$ belongs to the smallest closed subalgebra ${\mathcal{W}}_t$ of ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ containing the cosets $$\label{eq:projections}
p:=\big((I^{(n)}+S^{(n)})/2\big)^\pi+{\mathcal{J}}_t,
\
q:=(\chi_tI^{(n)})^\pi+{\mathcal{J}}_t,$$ where $\chi_tI^{(n)}$ denotes the operator of multiplication by the diagonal matrix-valued function ${\mathrm{diag}}\{\chi_t,\dots,\chi_t\}$ and the algebra $$\label{eq:algebra}
{\mathcal{C}}:=\big\{(cI^{(n)})^\pi+{\mathcal{J}}_t\ : \ c\in\mathbb{C}^{n\times n}\big\}.$$ The latter algebra is obviously isomorphic to $\mathbb{C}^{n\times n}$, so ${\mathcal{C}}$ and $\mathbb{C}^{n\times n}$ can be identified with each other.
The spectrum of $pqp+(e-p)(e-q)(e-p)$
-------------------------------------
Since $P^2=P$ on $L_\varrho^{p(\cdot)}$ (see, e.g., [@K03 Lemma 6.4]) and $\chi_t^2-\chi_t\in C$, $(\chi_t^2-\chi_t)(t)=0$, it is easy to see that $$\label{eq:2proj-conditions}
p^2=p,
\quad
q^2=q,
\quad
pc=cp,
\quad
qc=cq$$ for every $c\in{\mathcal{C}}$, where $p,q$ and ${\mathcal{C}}$ are given by and . To apply Theorem \[th:2proj\] to the algebras $F={\mathcal{L}}^\pi/{\mathcal{J}}_t$ and $W={\mathcal{W}}_t={\mathrm{alg}}({\mathcal{C}},p,q)$, we have to identify the spectrum of $$pqp+(e-p)(e-q)(e-p)
=\big(P^{(n)}\chi_tP^{(n)}+Q^{(n)}(1-\chi_t)Q^{(n)}\big)^\pi+{\mathcal{J}}_t
\label{eq:element}$$ in the algebra $F={\mathcal{L}}^\pi/{\mathcal{J}}_\tau$; here $P^{(n)}:=(I^{(n)}+S^{(n)})/2$ and $Q^{(n)}:=(I^{(n)}-S^{(n)})/2$.
\[le:spectrum\] Let $\chi_t\in PC$ be a continuous function on $\Gamma\setminus\{t\}$ such that $$\chi_t(t-0)=0,
\quad
\chi_t(\tau+0)=1,$$ and $$\chi_t(\Gamma\setminus\{t\})\cap\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))=\emptyset.$$ Then the spectrum of in ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ coincides with $\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))$.
Once we have Corollary \[co:important\] at hand, the proof of this lemma can be developed by a literal repetition of the proof of [@K96 Lemma 9.4].
Symbol calculus
---------------
Now we are in a position to prove the main result of this paper.
\[th:symbol\] Define the “double logarithmic spirals bundle” $${\mathcal{M}}:=
\bigcup\limits_{t\in\Gamma} \Big(\{t\} \times
\mathcal{S}\big(0,1;\delta(t),1/p(t)+\lambda(t)\big) \Big).$$
1. For each point $(t,\mu)\in{\mathcal{M}}$, the map $$\sigma_{t,\mu} \: : \:
\{S^{(n)}\}\cup\{aI^{(n)}\: :\:
a\in PC^{n\times n}\} \to \mathbb{C}^{2n\times 2n}$$ given by $$\sigma_{t,\mu}(S^{(n)})
=
\left(
\begin{array}{ll}
E & O\\
O & -E
\end{array}
\right),
\
\sigma_{t,\mu}(aI^{(n)})
=
\left(
\begin{array}{ll}
a_{11}(t,\mu) & a_{12}(t,\mu)\\
a_{21}(t,\mu) & a_{22}(t,\mu)
\end{array}
\right),$$ where $$\begin{aligned}
a_{11}(t,\mu)
&:=&
a(t+0)\mu + a(t-0)(1-\mu),\\
a_{12}(t,\mu)
&=&
a_{21}(t,\mu)
:=
(a(t+0)-a(t-0)) \sqrt{\mu(1-\mu)},
\\
a_{22}(t,\mu)
&:=&
a(t+0)(1-\mu) + a(t-0)\mu,\end{aligned}$$ and $O$ and $E$ are the zero and identity $n\times n$ matrices, respectively, extends to a Banach algebra homomorphism $$\sigma_{t,\mu} :{\mathcal{U}}\to\mathbb{C}^{2n\times 2n}$$ with the property that $\sigma_{t,\mu}(K)$ is the zero matrix for every compact operator $K$ on $X_n$;
2. an operator $A\in{\mathcal{U}}$ is Fredholm on $X_n$ if and only if $$\det\sigma_{t,\mu} (A)\neq 0
\quad\mbox{for all}\quad (t,\mu)\in{\mathcal{M}};$$
3. the quotient algebra ${\mathcal{U}}^\pi$ is inverse closed in the Calkin algebra ${\mathcal{B}}^\pi$, that is, if a coset $A^\pi\in{\mathcal{U}}^\pi$ is invertible in ${\mathcal{B}}^\pi$, then $(A^\pi)^{-1}\in{\mathcal{U}}^\pi$.
The idea of the proof of this theorem is borrowed from [@BK97] and is based on the Allan-Douglas local principle (Theorem \[th:AllanDouglas\]) and the two projections theorem (Theorem \[th:2proj\]).
Fix $t\in\Gamma$ and choose a function $\chi_t\in PC$ such that $\chi_t$ is continuous on $\Gamma\setminus\{t\}$, $\chi_t(t-0)=0$, $\chi_t(t+0)=1$, and $\chi_t(\Gamma\setminus\{t\})\cap\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))=\emptyset$. [From]{} and Lemma \[le:spectrum\] we deduce that the algebras ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ and ${\mathcal{W}}_t={\mathrm{alg}}({\mathcal{C}},p,q)$, where $p,q$ and ${\mathcal{C}}$ are given by and , respectively, satisfy all the conditions of the two projections theorem (Theorem \[th:2proj\]).
\(a) In view of Theorem \[th:2proj\](a), for every $\mu\in\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))$, the map $\sigma_\mu:\mathbb{C}^{n\times n}\cup\{p,q\}\to\mathbb{C}^{2n\times 2n}$ given by – extends to a Banach algebra homomorphism $\sigma_\mu:{\mathcal{W}}_t\to\mathbb{C}^{2n\times 2n}$. Then the map $$\sigma_{t,\mu}=\sigma_\mu\circ\pi_t:{\mathcal{U}}\to\mathbb{C}^{2n\times 2n},$$ where $\pi_t:{\mathcal{U}}\to{\mathcal{W}}_t={\mathcal{U}}^\pi/{\mathcal{J}}_t$ is acting by the rule $A\mapsto A^\pi+{\mathcal{J}}_t$, is a well defined Banach algebra homomorphism and $$\sigma_{t,\mu}(S^{(n)})=2\sigma_\mu p-\sigma_\mu e=
\left(\begin{array}{cc}E & O \\O & -E\end{array}\right).$$ If $a\in PC^{n\times n}$, then in view of and $(aI^{(n)})^\pi-(a_tI^{(n)})^\pi\in{\mathcal{J}}_t$ it follows that $$\begin{aligned}
\sigma_{t,\mu}(aI^{(n)})
&=&
\sigma_{t,\mu}(a_tI^{(n)})
=
\sigma_\mu(a(t-0))\sigma_\mu(e-q)+\sigma_\mu(a(t+0))\sigma_\mu q
\\
&=&
\left(\begin{array}{cc} a_{11}(t,\mu) & a_{12}(t,\mu)\\
a_{21}(t,\mu) & a_{22}(t,\mu)\end{array}\right).\end{aligned}$$ [From]{} Proposition \[pr:OLT\](a) it follows that $\pi_t(K)=K^\pi+{\mathcal{J}}_t={\mathcal{J}}_t$ for every $K\in{\mathcal{K}}$ and every $t\in\Gamma$. Hence, $$\sigma_{t,\mu}(K)=\sigma_\mu(0)=
\left(\begin{array}{cc} O & O\\ O & O\end{array}\right).$$ Part (a) is proved.
\(b) [From]{} Proposition \[pr:OLT\] it follows that the Fredholmness of $A\in{\mathcal{U}}$ is equivalent to the invertibility of $A^\pi\in{\mathcal{L}}^\pi$. By Theorem \[th:AllanDouglas\], the former is equivalent to the invertibility of $\pi_t(A)=A^\pi+{\mathcal{J}}_t$ in ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ for every $t\in\Gamma$. By Theorem \[th:2proj\](b), this is equivalent to $$\begin{aligned}
\label{eq:symbol1}
\det\sigma_{t,\mu}(A)=\det\sigma_\mu\pi_t(A)\ne 0
\mbox{ for all }(t,\mu)\in{\mathcal{M}}.
&&\end{aligned}$$ Part (b) is proved.
\(c) Since $\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))$ does not separate the complex plane $\mathbb{C}$, it follows that the spectra of in the algebras ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ and ${\mathcal{W}}_t={\mathcal{U}}^\pi/{\mathcal{J}}_t$ coincide, so we can apply Theorem \[th:2proj\](c). If $A^\pi$, where $A\in{\mathcal{U}}$, is invertible in ${\mathcal{B}}^\pi$, then holds. Consequently, by Theorem \[th:2proj\](b), (c), $\pi_t(A)=A^\pi+{\mathcal{J}}_t$ is invertible in ${\mathcal{W}}_t={\mathcal{U}}^\pi/{\mathcal{J}}_t$ for every $t\in\Gamma$. Applying Theorem \[th:AllanDouglas\] to ${\mathcal{U}}^\pi$, its central subalgebra ${\mathcal{Z}}^\pi$, and the ideals ${\mathcal{J}}_t$, we obtain that $A^\pi$ is invertible in ${\mathcal{U}}^\pi$, that is, ${\mathcal{U}}^\pi$ is inverse closed in the Calkin algebra ${\mathcal{B}}^\pi$.
Note that the approach to the study of Banach algebras of SIOs based on the Allan-Douglas local principle and the two projections theorem is nowadays standard. It was successfully applied in many situations (see, e.g., [@BK95; @BK97; @FRS93; @K96; @K98; @K02; @K05]). However, it does not allow to get formulas for the index of an arbitrary operator in the Banach algebra of SIOs with $PC$ coefficients. These formulas can be obtained similarly to the classical situation considered by Gohberg and Krupnik [@GK70; @GK71] (see also [@BK97 Ch. 10]). For reflexive Orlicz spaces over Carleson curves with logarithmic whirl points this was done by the author [@K98-index]. In the case of Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points the index formulas are almost the same as in [@K98-index]. It is only necessary to replace the both Boyd indices $\alpha_M$ and $\beta_M$ of an Orlicz space $L^M$ by the numbers $1/p(t)+\lambda(t)$ in corresponding formulas.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Prior research has shown that autocorrelation and variance in voltage measurements tend to increase as power systems approach instability. This paper seeks to identify the conditions under which these statistical indicators provide reliable early warning of instability in power systems. First, the paper derives and validates a semi-analytical method for quickly calculating the expected variance and autocorrelation of all voltages and currents in an arbitrary power system model. Building on this approach, the paper describes the conditions under which filtering can be used to detect these signs in the presence of measurement noise. Finally, several experiments show which types of measurements are good indicators of proximity to instability for particular types of state changes. For example, increased variance in voltages can reliably indicate the location of increased stress, while growth of autocorrelation in certain line currents is a reliable indicator of system-wide instability.'
author:
- 'Goodarz Ghanavati, *Student Member, IEEE*, Paul D. H. Hines, *Senior Member, IEEE*, Taras I. Lakoba[^1]'
bibliography:
- 'Pow\_sys.bib'
nocite:
- '[@avalos2009equivalency; @grijalva2012individual]'
- '[@Browne2008]'
- '[@Anderson:1983]'
- '[@haesen2009probabilistic; @perninge2010risk; @bu2012probabilistic; @munoz2013affine]'
- '[@*]'
title: Identifying Useful Statistical Indicators of Proximity to Instability in Stochastic Power Systems
---
Power system stability, phasor measurement units, time series analysis, stochastic processes, principal component analysis, autocorrelation, critical slowing down.
Introduction
============
To make optimal use of constrained infrastructure, power systems frequently operate near their stability limits. Bifurcation theory provides a framework for understanding these instabilities [@alvarado1994computation]–[@perninge2010risk] and has motivated the development of new methods for online stability monitoring [@haque2003line]–[@kamwa2011robust].
This existing work has largely focused around deterministic power system models. However, real power systems are constantly influenced by stochastic perturbations in load and (increasingly) variable renewable generation. Because random fluctuations can substantially change the stability properties of a system [@wangfokker], several have proposed the use of stochastic approaches to stability analysis (e.g., [@nwankpa1992stochastic]–[@huang2013quasi]).
Indeed, outside of the power systems literature, there is growing evidence that complex systems show statistical early warning signs as they approach instability [@scheffer2009early; @Scheffer:2012]. This phenomenon, known as critical slowing down (CSD) [@mori1963relaxation], is the tendency of a dynamical system to return to equilibrium more slowly in response to perturbations as it approaches a critical bifurcation. Increasing autocorrelation and variance in measurements, two common signs of signal proximity to critical transitions in a variety of dynamical systems [@scheffer2009early]. However, not all measurements show these signs early enough to provide warning with sufficient time to take mitigating actions [@boerlijst2013catastrophic]. Understanding which variables provide useful early warning of instability is necessary for the practical application of these concepts. Doing so requires a detailed knowledge of how autocorrelation and variance change as a system’s state changes.
A few papers have studied the properties of variance and autocorrelation as indicators of instability in power systems. Reference [@Cotilla2012] showed, using simulations, that variance and autocorrelation of bus voltages increase before bifurcation. Reference [@podolsky2013] derives the autocorrelation function of a power system’s state vector near a saddle-node bifurcation and uses the result to estimate the collapse probability for power systems. In [@Dhople2013], a framework is proposed to study the impact of stochastic power injections on power system dynamics by computing the moments of the states. In [@ghanavati2013understanding], the authors showed that for some state variables, increases in autocorrelation and variance appear only when a power system is very close to the indicating that CSD does not always provide useful early warning of instability. Reference [@yuan2014stochastic] calculates the variance of state variables to analyze the impact of wind turbine mechanical power input fluctuations on small-signal stability.
The goal of this paper is to present a general method for estimating the autocorrelation and variance of state variables from a power system model and to use the results to determine which variables in a power system provide useful early warning of critical transitions in the presence of measurement noise. To this end, Sec. \[sec:Analytical\] presents a semi-analytical method for calculating the variance and autocorrelation of algebraic and differential variables. This method enables the fast calculation of voltage and current statistics for many potential operating scenarios in large power systems, and unlike the method in [@podolsky2013], is not limited to the immediate vicinity of a bifurcation. Sec. \[sec:Useful-early-signs\] illustrates the method using the 39-bus test case and shows that some variables are better indicators of proximity to instability than others. Sec. \[sec:Detectability\] extends the analysis to systems with measurement noise and presents a method for detecting CSD in the presence of measurement noise. Sec. \[sec:Stressed-Area\] uses this approach to identify stressed areas in a power network. Finally, our conclusions are presented in Sec. \[sec:Conclusions\].
Calculation of Autocorrelation and Variance in Multimachine Power Systems \[sec:Analytical\]
============================================================================================
This section presents a semi-analytical method for the fast calculation of variance $\left(\sigma^{2}\right)$ and autocorrelation $\left(R\left(\Delta t\right)\right)$ of bus voltage magnitudes and line currents in power system. Fluctuations of load and generation are well known sources of stochasticity in power systems. While this section models only randomness in load, the method can be readily extended to the case of stochasticity in power injections.
System Model\[sub:System-Model\]
--------------------------------
Adding stochastic load to the set of general differential-algebraic equations (DAE) that model a power system gives: $$\begin{aligned}
\dot{\underline{x}} & = & f\left(\underline{x},\underline{y}\right)\label{eq:diff}\\
0 & = & g\left(\underline{x},\underline{y},\underline{u}\right)\label{eq:alg}\end{aligned}$$ where $f,g$ represent differential and algebraic equations, $\underline{x},\underline{y}$ are vectors of differential and algebraic variables (generator rotor angles, bus voltage magnitudes, etc.), and $\underline{u}$ is the vector of load fluctuations. The algebraic equations consist of nodal power flow equations and static equations for components such as generator, exciter, and turbine governor. The differential equations describe the dynamic behavior of the equipment. In this paper, for modeling load fluctuations, we take an approach similar to [@perninge2010risk], [@hauer2007] and assume that load fluctuations $\underline{u}$ follow the Ornstein–Uhlenbeck process: $$\dot{\underline{u}}=-E\underline{u}+\underline{\xi}\label{eq:load_corr}$$ where $E$ is a diagonal matrix whose diagonal entries equal $t_{\textnormal{{corr}}}^{-1}$, where $t_{\textnormal{{corr}}}$ is the correlation time of the load fluctuations, and $\underline{{\xi}}$ is a vector of independent Gaussian random variables: $$\begin{aligned}
\textnormal{{E}}\left[\underline{\xi}\left(t\right)\right] & = & 0\label{eq:xi1}\\
\textnormal{\textnormal{{E}}}\left[\xi_{i}\left(t\right)\xi_{j}\left(s\right)\right] & = & \delta_{ij}\sigma_{\xi}^{2}\delta_{I}(t-s)\label{eq:xi2}\end{aligned}$$ where $t,s$ are two arbitrary times, $\delta_{ij}$ is the Kronecker delta function, $\sigma_{\xi}^{2}$ is the intensity of noise and $\delta_{I}$ represents the unit impulse (delta) function. Equations (\[eq:diff\])–(\[eq:load\_corr\]) form the set of SDAEs that models a power system with stochastic load.
We also consider the frequency-dependence of loads, which can measurably impact the statistics of voltage magnitudes [@ghanavati2013understanding]. Loads are thus modeled as follows [@berg1973power; @Milano2008]: $$\begin{aligned}
\Delta\omega & = & \frac{1}{2\pi f_{n}}\frac{d\left(\theta-\theta^{0}\right)}{dt}\label{eq:freq1}\\
P & = & P^{0}\left(1+\Delta\omega\right)^{\beta_{P}}\label{eq:freq2}\\
Q & = & Q^{0}\left(1+\Delta\omega\right)^{\beta_{Q}}\label{eq:freq3}\end{aligned}$$ where $\Delta\omega$ is the frequency deviation at the load bus, $\theta^{0},P^{0},Q^{0}$ are the baseline voltage angle, active and reactive power of each load, $\beta_{P},\beta_{Q}$ are exponents that determine the level of frequency dependence, $f_{n}$ is the nominal frequency and $\theta$ is the bus voltage angle.
Using this model, we studied the New England 39-bus test case [@pai1989energy]. As load increases, a Hopf bifurcation occurs just before the nose of the PV curve (see [@lerm2003multiparameter; @rosehart1999bifurcation]).
Solution Method\[sub:Solution-Method\]
--------------------------------------
Linearizing (\[eq:alg\]) gives the following: $$\Delta\underline{y}=\left[\begin{array}{cc}
-g_{y}^{-1}g_{x} & -g_{y}^{-1}g_{u}\end{array}\right]\left[\begin{array}{c}
\Delta\underline{x}\\
\Delta\underline{u}
\end{array}\right]\label{eq:Y-XU}$$ where $g_{x},g_{y},g_{u}$ are the Jacobian matrices of $g$ with respect to $\underline{x},\underline{y},\underline{u}$. Linearizing (\[eq:diff\]) and (\[eq:load\_corr\]) and eliminating $\Delta\underline{y}$ via (\[eq:Y-XU\]) gives the following: $$\begin{aligned}
\left[\begin{array}{c}
\Delta\underline{\dot{x}}\\
\Delta\underline{\dot{u}}
\end{array}\right] & = & \left[\begin{array}{cc}
A_{s} & -f_{y}g_{y}^{-1}g_{u}\\
0 & -E
\end{array}\right]\left[\begin{array}{c}
\Delta\underline{x}\\
\Delta\underline{u}
\end{array}\right]+\label{eq:SDE-linear}\\
& & \left[\begin{array}{c}
0\\
\textnormal{{\ensuremath{I_{n}}}}
\end{array}\right]\underline{\xi}\nonumber \end{aligned}$$ where $f_{x},f_{y}$ are the Jacobian matrices of $f$ with respect to $\underline{x},\underline{y}$ and $A_{s}=f_{x}-f_{y}g_{y}^{-1}g_{x}$; $\textnormal{{\ensuremath{I_{n}}}}$ is an identity matrix, with $n$ being the length of $\underline{u}$. If we let $\underline{z}=\left[\begin{array}{cc}
\Delta\underline{x} & \Delta\underline{u}\end{array}\right]^{T}$, (\[eq:SDE-linear\]) can be re-written in the standard form: $$\underline{\dot{z}}=A\underline{z}+B\underline{\xi}\label{eq:Ornsterin-Uhl}$$
The covariance matrix of $\underline{z}$ ($\sigma_{\underline{z}}$) satisfies the Lyapunov equation [@gardiner2012handbook]: $$A\sigma_{\underline{z}}+\sigma_{\underline{z}}A^{T}=-BB^{T}\label{eq:cov_mat}$$ which can be solved efficiently in $O\left(n^{3}\right)$ operations using MATLAB’s `lyap` function. To stress the difference between the solution from (\[eq:cov\_mat\]) and the results of direct numerical simulation of (\[eq:diff\])–(\[eq:load\_corr\]), we will refer to the former solution as semi-analytical.
The stationary autocorrelation matrix can be computed given $\sigma_{\underline{z}}$ and an equation from [@gardiner2012handbook]: $$\textnormal{{E}}\left[\underline{z}\left(t\right)\underline{z}^{T}\left(s\right)\right]=\exp\left[-A\left|\Delta t\right|\right]\sigma_{\underline{z}}\label{eq:corr_z}$$ where $\Delta t=t-s$. From (\[eq:cov\_mat\]) and (\[eq:corr\_z\]) the normalized autocorrelation function of $z_{i}$ can be calculated: $$R_{z_{i}}\left(\Delta t\right)=\textnormal{E}\left[z_{i}\left(t\right)z_{i}^{T}\left(s\right)\right]/\sigma_{z_{i}}^{2}\label{eq:norm_corr_z}$$ The covariance matrix of the algebraic variables, $\sigma_{\underline{\Delta y}}$, is found from (\[eq:Y-XU\]) and (\[eq:cov\_mat\]): $$\sigma_{\Delta\underline{{y}}}=K\sigma_{\underline{{z}}}K^{T}\label{eq:cov_y}$$ where $K$ is the matrix from (\[eq:Y-XU\]). Similarly, the autocorrelation function of $\Delta\underline{y}(t)$ is: $$\textnormal{E}\left[\Delta\underline{y}\left(t\right)\Delta\underline{y}^{T}\left(s\right)\right]=K\cdot\textnormal{{E}}\left[\underline{z}\left(t\right)\underline{z}^{T}\left(s\right)\right]K^{T}\label{eq:corr_y}$$ Finally, the covariance and autocorrelation matrices for voltage magnitudes are a subset of the matrices from (\[eq:cov\_y\]) and (\[eq:corr\_y\]).
Fluctuation-induced deviations of the current magnitudes, $\Delta I_{ik}$, in a line between buses $i$ and $k$ can be found by linearizing the following: $$I_{ik}=Y_{ii}V_{i}e^{j\left(\phi_{ik}-\phi_{ik}+\theta_{i}-\theta_{k}\right)}+Y_{ik}V_{k}\label{eq:line_cur}$$ where $I_{ik}$ is the magnitude of the current of the line between buses $i,k$; $V_{i},\theta_{i}$ are the voltage magnitude and angle of bus $i$; $Y_{ii},\phi_{ii}$ and $Y_{ik},\phi_{ik}$ are magnitudes and angles of the diagonal and off-diagonal elements of the $Y_{BUS}$ matrix. By linearizing (\[eq:line\_cur\]) one can find $\Delta\underline{I}$ from $\Delta y$ and then compute the covariance and autocorrelation matrices of $\Delta\underline{I}$ from equations similar to (\[eq:cov\_y\]) and (\[eq:corr\_y\]).
Comparing the semi-analytical method with the numerical solution shows that the former is significantly more time-efficient. For the numerical simulations in this paper, we solved (\[eq:diff\])–(\[eq:load\_corr\]) using the trapezoidal DAE solver in the Power System Analysis Toolbox (PSAT) [@milano2005open]. To find numerical values for $\sigma^{2}$ and $R(\Delta t)$ we ran 100 240s simulations, with an integration step size of 0.01s, and then computed the statistics. For the 39-bus case with 140 variables, solving for $\sigma_{z}^{2}$ using the semi-analytical method took approximately 0.08s, whereas calculating the variances using numerical simulations took about 24 hours.
Useful early warning signs: voltage magnitudes and line currents\[sec:Useful-early-signs\]
==========================================================================================
This section applies the method in Sec. \[sec:Analytical\] to calculate the autocorrelation and variance of voltages and currents in the 39-bus test case. These results are subsequently used to identify particular locations and variables in which the statistical early-warning signs are most clearly observable.
Autocorrelation and Variance of Voltages \[sub:Autoco-Var-Voltages\]
--------------------------------------------------------------------
Using the methods described in Sec. \[sec:Analytical\], we calculated $\sigma^{2},\, R\left(\Delta t\right)$ of bus voltage magnitudes in the 39-bus test case both semi-analytically and numerically using PSAT. In order to see how these statistics change as the system state moves toward the bifurcation, we increased all loads uniformly, multiplying each load by the same factor. For the correlation time and intensity of noise we used: $t_{corr}=1\textnormal{{s}}$ and $\sigma_{u}^{2}=10^{-4}$ pu. The values of $\beta_{P},\beta_{Q}$ in (\[eq:freq2\]), (\[eq:freq3\]) were chosen randomly from within $\left[2,3\right]$ and $\left[1,2\right]$, respectively [@berg1973power]. For all results in this paper, we chose the autocorrelation time lag $\Delta t=0.2\textnormal{{s}}$
Fig. \[fig:arvar\_vmag\_39bus\] shows several typical, illustrative examples of how $\sigma^{2},\, R\left(\Delta t\right)$ of bus voltage magnitudes depend on load level in the 39-bus case. These results show that, as anticipated from CSD theory, both $\sigma^{2}$ and $R\left(\Delta t\right)$ of voltage magnitudes increase as the system approaches the bifurcation. However, not all of these signs appear sufficiently early to detect the bifurcation and take mitigating actions. For example, $\sigma_{\Delta V}^{2}$ in buses 7, 14, and 26 exhibits a conspicuous increase when the load level is 10–15$\%$ below the bifurcation. These variables are good early warning signs (EWS) of the impending bifurcation. In contrast, $\sigma_{\Delta V}^{2}$ in buses 20 and 36 is not a useful warning sign as its increase occurs too close to the bifurcation. The situation with autocorrelation is reversed, as shown in the second panel of Fig. \[fig:arvar\_vmag\_39bus\].
![\[fig:arvar\_vmag\_39bus\]Variance and autocorrelation of voltage magnitudes for five buses in the 39-bus test case versus load level. Load level is the ratio of the system loads to their nominal values. b denotes the bifurcation point. The bus number associated with each curve is shown next to it. Here and everywhere below the autocorrelation time lag $t-s=0.2\textnormal{{s}}$.](compare39){width="1\columnwidth" height="0.2\textheight"}
By examining $\sigma^{2}$ and $R\left(\Delta t\right)$ for all buses in our test system, we have concluded that, as Fig. \[fig:arvar\_vmag\_39bus\] illustrates, good EWS occur in two different types of buses. We found that $\sigma^{2}$ is a good EWS for load buses, whereas $R\left(\Delta t\right)$ is a good EWS at buses that are close to generators with low inertia. In addition, we found that $\sigma_{\Delta V}^{2}$ at generator buses is much smaller than at load buses, largely due to generator voltage control systems. $R_{\Delta V}\left(\Delta t\right)$
Autocorrelation and Variance of Line Currents \[sub:Autoco-Var-Currents\]
-------------------------------------------------------------------------
The fact that autocorrelation of voltages is not uniformly useful as an EWS prompted us to look at other variables, particularly currents, that might be more useful indicators. Results for $\sigma^{2}$ and $R\left(\Delta t\right)$ of currents, shown in Fig. \[fig:Varar\_Il946\], suggest that while $\sigma_{\Delta I}^{2}$ of almost all lines increase measurably with the increase of the load level, increased $R_{\Delta I}(\Delta t)$ is clearly observable only in some of the lines, such as line $\left[6\,\,31\right]$. As was the case with voltages, the common characteristic of lines that show clear increases in $R_{\Delta I}(\Delta t)$ is that they are connected to a generator with low or moderate inertia. The explanation for this appears to be that increased $R_{\Delta I}\left(\Delta t\right)$ is closely tied to the way that generators respond to perturbations as the system approaches bifurcation. Increases in $R_{\Delta I}(\Delta t)$ are not clearly observable in lines that are close to load centers, such as line $\left[4\,\,14\right]$ in Fig. \[fig:Varar\_Il946\].
![\[fig:Varar\_Il946\]Variance and autocorrelation of current of two lines. The numbers in brackets are bus numbers at two ends of the lines. ](compare39_Il){width="1\columnwidth" height="0.2\textheight"}
Examining changes in $\sigma^{2},\, R\left(\Delta t\right)$ of several state variables showed that only magnitudes of voltages and line currents signal the proximity to the bifurcation well under certain conditions mentioned above. Other variables such as voltage angle, current angle, generator rotor angle and generator speed did not show measurable or clear monotonically increasing patterns in $\sigma^{2},\, R\left(\Delta t\right)$ that can indicate proximity to a bifurcation.
Detectability after measurement noise\[sec:Detectability\]
==========================================================
This section examines the detectability of increases in $\sigma^{2}$ and $R\left(\Delta t\right)$ of voltages and currents given the presence of measurement noise. In addition, we present a method for reducing the impact of measurement noise using a band-pass filter.
Impact of Measurement Noise on Variance and Autocorrelation\[sub:Results-Meas\_Noise\]
--------------------------------------------------------------------------------------
Clearly, measurement noise will adversely impact the observability of increases in $\sigma^{2},\, R\left(\Delta t\right)$ of voltages and currents. In order to model this impact, we assumed that measurement noise at each bus is normally distributed with a standard deviation that is proportional to the steady-state mean voltage for this load level: $\sigma_{\eta}=0.01\left\langle V\right\rangle $. As a result the measured variance, $\sigma_{\Delta V_{m}}^{2}$, of a bus voltage increases to: $$\sigma_{\Delta V_{m}}^{2}=\sigma_{\Delta V}^{2}+\sigma_{\eta}^{2}\label{eq:meas_volt}$$ where $\sigma_{\Delta V}^{2}$ is the variance before adding measurement noise.
Applying this method, Fig. \[fig:arvar\_Vmag\_39bus\_nos\] shows $\sigma^{2}$ and $R\left(\Delta t\right)$ for the voltage magnitudes of the same five buses used in Sec. \[sub:Autoco-Var-Voltages\], but after adding measurement noise. The results show that measurement noise causes the increases in $\sigma_{\Delta V_{m}}^{2}$ to occur only close to the bifurcation, except for bus 36. In fact, $\sigma_{\Delta V_{m}}^{2}$ decreases for most buses, until close to the bifurcation. Also, because of the 1% measurement noise, $\sigma_{\eta}^{2}>\sigma_{\Delta V}^{2}$ until close to the bifurcation for most buses. For bus 36, which is a generator bus, $\sigma^{2}$ is almost constant since $\left\langle V\right\rangle $ (and as a result of $\sigma_{\eta}^{2}$) is held constant by the exciter; $\sigma_{\eta}^{2}\gg\sigma_{\Delta V}^{2}$ for generator buses.
Fig. \[fig:arvar\_Vmag\_39bus\_nos\] also shows that $R_{\Delta V_{m}}\left(\Delta t\right)$ increases significantly near the bifurcation for buses 7, 14 and, to a lesser extent, for bus 26. Appendix \[meas\_nos\_artefact\] demonstrates $R_{\Delta V_{m}}\left(\Delta t\right)$is largely an artifact of adding measurement noise: it is primarily due to increases in $\sigma^{2}$ rather than that of $R\left(\Delta t\right)$. Autocorrelation of $\Delta V_{m}$ is almost zero for buses 20, 36 since the uncorrelated measurement noise dominates the voltage of buses near generators.
![\[fig:arvar\_Vmag\_39bus\_nos\]Variances and autocorrelations of voltage magnitudes of five buses in the 39-bus test case versus load level, accounting for measurement noise.](compare39_nos){width="1\columnwidth" height="0.22\textheight"}
Thus, measurement noise essentially washes out the useful EWS that we reported in Sec. \[sub:Autoco-Var-Voltages\]. In addition, there is another issue impacting the detectability of EWS, which we discuss in the next subsection.
Spread of Statistics\[sub:Spread-of-Statistics\]
------------------------------------------------
One important point regarding the detection of increased $\sigma^{2}$ and $R\left(\Delta t\right)$ is that the measured statistics of a *sample* of a variable’s measurement data (which an operator can observe in finite time) are different from the mean statistical properties of that variable over infinitely many measurements. Although the mean of these statistics typically grows as the system approaches a bifurcation, the variance (spread) of these statistics that results from finite sample sizes can cause difficulty in estimating the distance to the bifurcation.
quantify the detectability of an increase in $\sigma^{2}\,\textnormal{{or}}\, R\left(\Delta t\right)$, we introduce an index $q_{95/80}$ (see Fig. \[fig:q9580\_idx\]):
$$q_{95/80}=\int_{a}^{\infty}f_{X\left(80\%\right)}dx+\int_{-\infty}^{a}f_{X\left(95\%\right)}dx\label{eq:q9580}$$
where $X$ is the statistic of interest ($\sigma^{2}\,\textnormal{{or}}\, R\left(\Delta t\right)$), $f_{X\left(80\%\right)}\,\textnormal{{and}}\, f_{X\left(95\%\right)}$ are the probability density functions (pdfs) of $X$ for load levels of $80\%\,\textnormal{{and}}\,95\%$ of the bifurcation, and $a$ is the point where the two distributions intersect. This measure ranges from 0 to 1, where $0$ suggests that there is no overlap between the two distributions, such that detectability is unimpeded by the statistic’s spread, while $q_{95/80}=1$ means that the two distributions completely overlap—i.e. the statistic does not increase. When the statistic has a decreasing trend, we declare $q_{95/80}=NA$. $q_{95/80}$ roughly corresponds to the probability of being able to correctly distinguish between the measured statistics at 80% and 95% load levels.
![\[fig:q9580\_idx\]The left panel shows the empirical pdfs of $X$, which can be $\sigma^{2}\,\textnormal{{or}}\, R\left(\Delta t\right)$ of measurements for two load levels. Measure $q_{95/80}$ is equal to the sum of the hatched areas. The dash-dot line shows the mean of $X$ versus load level. The right panel shows an alternative view of the pdfs. ](q9580_idx){width="1\columnwidth" height="0.2\textheight"}
Filtering Measurement Noise\[sub:Band-pass-Filtering\]
-------------------------------------------------------
In this section, we explore the use of a band-pass filter to reduce the impact of e power spectral density (PSD) of voltages and currents (see Fig. \[fig:PSD\_Il46\]) shows that the power of the system noise (i.e., voltage or current magnitude variations in response to load fluctuations) is concentrated mostly in its low frequencies. This appears to be typical for Hopf and saddle-node bifurcations in power systems. On the other hand, in order to detect CSD, it is necessary to remove slow trends that result not from CSD but from other factors, such as gradual changes in the system’s operating point [@dakos2008slowing]. By experimentation, we found that a band-pass filter with a pass-band of [\[]{}0.1, 2[\]]{} Hz reduces the impact of measurement noise in this system optimally. The rationale for these bounds can be seen from Fig. \[fig:PSD\_Il46\], which shows the PSD of a typical current magnitude in our system. We use this filter for all “filtered” results reported subsequently.
![\[fig:PSD\_Il46\]Power spectral density of the current of line [\[]{}6 31[\]]{} for several load levels listed in the legend. Bifurcation is at load level=2.12.](PSD_Il46){width="1\columnwidth" height="0.2\textheight"}
Fig. \[fig:varar\_V736\_filt\] shows $\sigma_{\Delta V}^{2},\, R_{\Delta V}\left(\Delta t\right)$ of buses 7, 36 after filtering measurement noise. Comparing Fig. \[fig:varar\_V736\_filt\] with Fig. \[fig:arvar\_Vmag\_39bus\_nos\] shows that using the band-pass filter significantly improves the detectability of increases in $\sigma_{\Delta V_{7}}^{2}$, which is close to load centers, but is not effective for bus 36, which is connected to a generator. The reason is that, even with filtering, it is still necessary that $\sigma^{2}$ without measurement noise be sufficiently large so that measurement noise does not dominate it. $\sigma^{2}$ of measurement noise after filtering will approximately be: $$\sigma_{\eta f}^{2}=\sigma_{\eta}^{2}\cdot\nicefrac{\left(f_{H}-f_{L}\right)}{\left(\nicefrac{f_{s}}{2}\right)}\label{eq:var_nos_filt}$$ where $\sigma_{\eta f}^{2}$ is the variance of measurement noise after filtering; $f_{H},f_{L}$ are upper and lower cut-off frequencies of the filter; and $f_{s}$ is the sampling frequency of measurements. Assuming $\sigma_{\eta}^{2}=1e-4\,\textnormal{{and}}\, f_{s}=60\textnormal{{Hz}}$, we get $\sigma_{\eta f}^{2}=6.3\times10^{-6}$. From Fig. \[fig:arvar\_vmag\_39bus\], one can see that only $\sigma_{\Delta V}^{2}$ of the load buses exceeds this value near the bifurcation.
Fig. \[fig:varar\_V736\_filt\] also shows that after filtering out measurement noise, the increase in $R_{\Delta V_{7}}\left(\Delta t\right)$ is detectable near the bifurcation. However, as mentioned in Sec. \[sub:Results-Meas\_Noise\], increases in $R(\Delta t)$ primarily result from increases in $\sigma_{\Delta V}^{2}$, and thus do not provide additional information regarding the proximity of the system to the bifurcation. Since $\sigma_{\eta}^{2}\gg\sigma_{\Delta V}^{2}$ for generator buses, $R_{\Delta V_{36}}\left(\Delta t\right)$ also does not increase measurably as the system approaches the bifurcation, even after filtering.
![\[fig:varar\_V736\_filt\]Variance and autocorrelation of voltage magnitude of buses 7,36 versus the load level after filtering the measurement noise. In this and subsequent figures, the lines show the mean and the discrete symbols $\left(*,\triangle\right)$ represent 5th, 25th, 75th, 95th percentiles of values of $\sigma^{2},\, R\left(\Delta t\right)$ for 100 realizations at each load level. The vertical dash-dot lines show $Load\, level=80\%b,\,95\%b$.](V736_filt){width="1\columnwidth" height="0.2\textheight"}
Similar to the case without measurement noise, $R\left(\Delta t\right)$ of line currents close to generators increase more clearly than that of lines near load centers. Fig. \[fig:varar\_Il469\_filt\] shows $\sigma^{2},\, R\left(\Delta t\right)$ of currents of lines $\left[6\,\,31\right]$ and $\left[4\,\,14\right]$ after filtering the noise.
In general, filtering noise from line currents is easier than from voltages since the ratio of $\sigma^{2}$ of the system noise (defined above) to $\sigma^{2}$ of measurement noise is larger for currents.
![\[fig:varar\_Il469\_filt\]Variance and autocorrelation of currents of lines $\left[6\,\,31\right]$, $\left[4\,\,14\right]$ after filtering the measurement noise.](Il469_filt){width="1\columnwidth" height="0.2\textheight"}
Fig. \[fig:varv\_grid\] shows the index $q_{95/80}$ for $\sigma_{\Delta V}^{2}$ across the 39-bus test case after filtering measurement noise. The results in Fig. \[fig:varv\_grid\] illustrate our earlier statement that $\sigma_{\Delta V}^{2}$ of buses near load centers are good EWS of the bifurcation while $\sigma_{\Delta V}^{2}$ of generator buses are not.
![\[fig:varv\_grid\]Index $q_{95/80}$ for $\sigma_{\Delta V}^{2}$ of bus voltages across the 39-bus test case. Here, and in Fig. \[fig:arI\_grid\], each rectangle represents the index $q_{95/80}$ for $\sigma_{\Delta V}^{2}$ of the bus next to it. In order to illustrate the results more clearly, we show $q_{95/80}=0.3$ for measurements with $q_{95/80}>0.3$, because quantities with this spread become indistinguishable.](gridloc_color_varv){width="1\columnwidth" height="0.2\textheight"}
Fig. \[fig:arI\_grid\] shows the index $q_{95/80}$ for $R_{\Delta I}\left(\Delta t\right)$ of lines across the 39-bus test case after filtering the measurement noise. The results in Fig. \[fig:arI\_grid\] show that $R_{\Delta I}\left(\Delta t\right)$ of the lines near generators provide good EWS of the bifurcation while $R_{\Delta I}\left(\Delta t\right)$ of the rest of the lines do not provide useful EWS.
![\[fig:arI\_grid\]Index $q_{95/80}$ for $R_{\Delta I}\left(\Delta t\right)$ of lines across the 39-bus test case. Each rectangle represents index $q_{95/80}$ for $R_{\Delta I}\left(\Delta t\right)$ of the line next to it.](gridloc_color_arI){width="1\columnwidth" height="0.2\textheight"}
Detecting Locations of Increased Stress\[sec:Stressed-Area\]
============================================================
This section examines the potential to use statistical properties of measurements to detect the locations of increased stress in a power system.
Transmission line tripping\[sub:Transmission-line-tripping\]
------------------------------------------------------------
In the first scenario, we disconnected lines between buses 4, 14 and buses 4, 5 in order to increase stress in the area close to bus 4. For this experiment, the load level was held constant at 1.45 times the nominal. We calculated the ratio of $\sigma_{\Delta V}^{2}$ and $\sigma_{\Delta I}^{2}$ for the stressed case to the variances at the normal operating condition $\left(\textnormal{{Ratio}}\left(\sigma^{2}\right)\right)$. We also calculated the difference between $R_{\Delta V}\left(\Delta t\right)$ and $R_{\Delta I}\left(\Delta t\right)$ for the two cases $\left(\textnormal{{Diff}}\left(R\left(\Delta t\right)\right)\right)$. Values of $\textnormal{{Ratio}}\left(\sigma^{2}\right),\,\textnormal{{Diff}}\left(R\left(\Delta t\right)\right)$ that are sufficiently larger than 1 or 0 indicate significant increase in $\sigma^{2}$ or $R\left(\Delta t\right)$, respectively. Fig. \[fig:VarV\_Ratio\_lines\](a) shows $\textnormal{{Ratio}}\left(\sigma_{\Delta V}^{2}\right)$ after adding measurement noise and filtering. The five bus voltages shown have the highest mean of $\textnormal{{Ratio}}\left(\sigma^{2}\right)$ among all buses. The figure shows that the voltage of the buses near bus 4 have the largest $\textnormal{{Ratio}}\left(\sigma_{\Delta V}^{2}\right)$ among the system buses. As with voltages, $\sigma_{\Delta I}^{2}$ close to bus 4 showed more growth than $\sigma_{\Delta I}^{2}$ in the rest of the system. These results suggest that larger increases in $\sigma_{\Delta V}^{2}$ and $\sigma_{\Delta I}^{2}$ in one area of the system, relative to the rest of the system, can indicate that this area is stressed.
Our results from Sec. \[sec:Detectability\] identified certain lines whose autocorrelation of currents can be good EWS of bifurcation. We now comment on what behavior these autocorrelations exhibit in this experiment. It turns out that not all of these autocorrelations show a measurable increase; the five lines whose currents’ autocorrelations show the largest increases are shown in Fig. \[fig:VarV\_Ratio\_lines\](b). While it is not possible to pinpoint the location of the disturbance based only on these statistical characteristics, it is possible to tell, based on the statistics, that the disturbance has occurred in a certain area of the network. This knowledge would reinforce the information obtained from monitoring variances of voltages and currents. As explained in Sec. \[sub:Results-Meas\_Noise\], $R_{\Delta V}\left(\Delta t\right)$ does not provide useful information regarding which areas in the grid are most stressed.
![\[fig:VarV\_Ratio\_lines\]Panel (a) shows $\textnormal{{Ratio}}\left(\sigma_{\Delta V}^{2}\right)$ after disconnecting the two lines connected to bus 4. The mean of the $\textnormal{{Ratio}}\left(\sigma_{\Delta V}^{2}\right)$ for the 5 buses that show the highest increases in variance, as well as the 5th, 25th, 75th, 95th percentiles of their values, are shown. Panel (b) shows $\textnormal{{Diff}}\left(R_{\Delta I}\left(\Delta t\right)\right)$ for 5 lines that exhibit the largest increases in $R_{\Delta I}\left(\Delta t\right)$. The results are shown after filtering of measurement noise.](vararVIl_ratio_diff_lines){width="1\columnwidth" height="0.2\textheight"}
Capacitor tripping\[sub:Capacitor-tripping\]
--------------------------------------------
$\sigma^{2}$ and $R\left(\Delta t\right)$, (at least partially) indicate the location of stress in the network, but the mean volta do not change enough to be good indicators. This example was designed to test the hypothesis that $\sigma^{2}$ and $R\left(\Delta t\right)$ can provide information that is not readily available from the mean values.
under-load tap changing (ULTC) transformer We also transferred the load of bus 15 to bus 40. Fig. \[CPF\] shows the P-V curve of bus 40 for three cases. In Case A, the system is in normal operating condition. In Case B, a 3-MVAR capacitor at bus 40 is disconnected and in Case C, the tap changer changes the tap from 1 to 1.1 in order to return the voltage to the normal operating range ($\left[0.95\,\,1.05\right]$ pu). Fig. \[CPF\] shows that the disconnection of the capacitor reduces the stability margin significantly, which manifests itself in lower voltage at bus 40. However, the increase in the ULTC’s tap ratio to 1.1 returns the voltage to a value close to its normal level.
![\[CPF\]PV curve for the three cases described in Sec. \[sub:Capacitor-tripping\]. The vertical line corresponds to the base load level.](CPF_Cap40_11){width="1\columnwidth" height="0.2\textheight"}
Fig. \[fig:arvaril\_Cap\](a) shows $\textnormal{{Ratio}}\left(\sigma_{\Delta I}^{2}\right)=\nicefrac{{\sigma_{\Delta I,case\, C}^{2}}}{\sigma_{\Delta I,case\, A}^{2}}$ for five lines, after filtering the measurement noise. These five line currents show the largest increase in $\sigma_{\Delta I}^{2}$ among all lines. The first three highest $\textnormal{{Ratio}}\left(\sigma_{\Delta I}^{2}\right)$ occur in lines that are close to the stressed area. However, some of the lines that are close to that area do not show significant or any increase in $\sigma_{\Delta I}^{2}$. For example, $\sigma_{\Delta I}^{2}$ of line $\left[\begin{array}{cc}
14 & 15\end{array}\right]$ decreases. Nevertheless, considering lines with the highest growth in $\sigma_{\Delta I}^{2}$ can clearly be helpful in identifying the location of the area of the system under excessive stress. As was the case for line currents, the results show that buses that exhibit the largest increases in $\sigma_{\Delta V}^{2}$ are close to the stressed area. Fig. \[fig:arvaril\_Cap\](b) shows $\textnormal{{Diff}}\left(R_{\Delta I}\left(\Delta t\right)\right)=R_{\Delta I,,case\, C}\left(\Delta t\right)-R_{\Delta I,,case\, A}\left(\Delta t\right)$ for 5 lines. The positive values indicate the increase in $R_{\Delta I}\left(\Delta t\right)$. The results in Fig. \[fig:arvaril\_Cap\](b) show that lines that exhibit the largest increase in $R_{\Delta I}\left(\Delta t\right)$ are close to the stressed area.
![\[fig:arvaril\_Cap\]Panel (a) shows $\nicefrac{{\sigma_{\Delta I,case\, C}^{2}}}{\sigma_{\Delta I,case\, A}^{2}}$ for 5 lines that exhibit the largest increase in $\sigma_{\Delta I}^{2}$ among all lines. Panel (b) shows $R_{\Delta I,,case\, C}\left(\Delta t\right)-R_{\Delta I,,case\, A}\left(\Delta t\right)$ for 5 lines that exhibit the largest increase in $R_{\Delta I}\left(\Delta t\right)$.](arvarIl_Cap){width="1\columnwidth" height="0.2\textheight"}
Discussion
----------
The results presented in this section show that comparing $\sigma_{\Delta V}^{2}$ and $\sigma_{\Delta I}^{2}$ for a stressed operating condition with their variances for the normal operating condition can be useful in detecting stressed areas of a power system. The reason for this is that the variances of voltage and current magnitudes show larger increases near the stressed area of a power system, compared to variances in the rest of the system. The results also show that $R_{\Delta I}\left(\Delta t\right)$ can be helpful in detecting the stressed area’s approximate location, although it may not be helpful in pinpointing the exact location of the stress. Autocorrelation of bus voltages were not found to be useful for pinpointing the stressed location for the reason explained in Appendix \[meas\_nos\_artefact\].
Conclusions\[sec:Conclusions\]
==============================
This paper investigates the use of statistical signals (autocorrelation and variance) in time-series data, such as what is produced from synchronized phasor measurement systems, as indicators of stability in a power system.
First, we derived a semi-analytical method for quickly computing the expected autocorrelation and variance for any voltage or current in a dynamic power system model. Computing the statistics in this way was shown to be orders of magnitude faster than obtaining the same result by simulation, and allows one to quickly identify locations and variables that are reliable indicators of proximity to instability. Using this method, we showed that the variance of voltage magnitudes near load centers, the autocorrelation of line currents near generators, and the variance of almost all line currents increased measurably as the 39-bus test case approached bifurcation. We found that these trends persist, even in the presence of measurement noise, provided that the data are band-pass filtered. Finally, the paper provides results suggesting that the statistics of voltage and current data can be helpful in identifying not only whether a system is seeing increased stress, but also the location of the stress.
Together, these results suggest that, under certain conditions, these easily measured statistical quantities in synchrophasor data can be useful indicators of stability**
\[meas\_nos\_artefact\]
=======================
The equation for $R_{\Delta V_{m}}\left(\Delta t\right)$ before band-pass filtering is: $$\nicefrac{\textnormal{E}\left[\Delta V_{m}\left(t\right)\Delta V_{m}\left(s\right)\right]}{\sigma_{\Delta V_{m}}^{2}}=\nicefrac{\textnormal{E}\left[\Delta V\left(t\right)\Delta V\left(s\right)\right]}{\left(\sigma_{\Delta V}^{2}+\sigma_{\eta}^{2}\right)}\label{eq:corr_vm}$$ If $\sigma_{\Delta V}^{2}\ll\sigma_{\eta}^{2}$, $R_{\Delta V_{m}}\left(\Delta t\right)$ will be almost zero. This is the case for generator buses or buses close to generators such as buses $20,36$. However, if $\sigma_{\Delta V}^{2}$ increases such that $\sigma_{\Delta V}^{2}\sim\sigma_{\eta}^{2}$ and $R_{\Delta V}\left(\Delta t\right)$ is sufficiently larger than $0$ $\left(>0.2\right)$, then $R_{\Delta V_{m}}\left(\Delta t\right)$ will rise significantly with load level, in part because of increase in $R_{\Delta V}\left(\Delta t\right)$ and in part because of increase in $\sigma_{\Delta V}^{2}$. This happens for buses close to load centers such as $7,14$. Comparing $R\left(\Delta t\right)$ of voltage of buses $7,14$ in Fig. \[fig:arvar\_vmag\_39bus\] with those in Fig. \[fig:arvar\_Vmag\_39bus\_nos\] shows that these quantities increase significantly after adding measurement noise while their increase without measurement noise is much smaller. This shows that the increase in $R_{\Delta V_{m}}\left(\Delta t\right)$ for load buses is more due to the increase in $\sigma_{\Delta V}^{2}$ than due to the increase in $R_{\Delta V}\left(\Delta t\right)$.
Author biographies {#author-biographies .unnumbered}
==================
[Goodarz Ghanavati]{} (S‘11) received the B.S. and M.S. degrees in Electrical Engineering from Amirkabir University of Technology, Tehran, Iran in 2005 and 2008, respectively. Currently, he is pursuing the Ph.D. degree in Electrical Engineering at University of Vermont. His research interests include power system dynamics, PMU applications and smart grid.
[Paul D. H. Hines]{} (S‘96,M‘07,SM‘14) received the Ph.D. in Engineering and Public Policy from Carnegie Mellon University in 2007 and M.S. (2001) and B.S. (1997) degrees in Electrical Engineering from the University of Washington and Seattle Pacific University, respectively. He is currently an Associate Professor in the School of Engineering, with a secondary appointment in the Dept. of Computer Science, at the University of Vermont, and a member of the adjunct research faculty of the Carnegie Mellon Electricity Industry Center. Formerly he worked on various electricity industry projects at the U.S. National Energy Technology Laboratory, the US Federal Energy Regulatory Commission, Alstom ESCA, and Black and Veatch. He currently serves as the chair of the Green Mountain Section of the IEEE, as the vice-chair of the IEEE PES Working Group on Cascading Failure, and as an Associate Editor for the IEEE Transactions on Smart Grid.
[Taras I. Lakoba]{} received the Diploma in physics from Moscow State University, Moscow, Russia, in 1989, and the Ph.D. degree in applied mathematics from Clarkson University, Potsdam, NY, in 1996. His research interests include the effect of noise and nonlinearity in fiber-optic communication systems, stability of numerical methods, and perturbation techniques.
[^1]: This work was supported by the US DOE, award \#DE-OE0000447, and by the US NSF, award \#ECCS-1254549.
G. Ghanavati, P. D. H. Hines, T. I. Lakoba are with the College of Engineering and Mathematical Sciences, University of Vermont, Burlington, VT (e-mail: gghanava@uvm.edu; paul.hines@uvm.edu; tlakoba@uvm.edu).
| {
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abstract: 'We describe a practical procedure for extracting the spatial structure and the growth rates of slow eigenmodes of a spatially extended system, using a unique experimental capability both to impose and to perturb desired initial states. The procedure is used to construct experimentally the spectrum of linear modes near the secondary instability boundary in Rayleigh-Bénard convection. This technique suggests an approach to experimental characterization of more complex dynamical states such as periodic orbits or spatiotemporal chaos.'
author:
- Kapilanjan Krishan
- Andreas Handel
- 'Roman O. Grigoriev'
- 'Michael F. Schatz'
title: Modal Extraction in Spatially Extended Systems
---
Numerous nonlinear nonequilibrium systems in nature and in technology exhibit complex behavior in both space and time ; understanding and characterizing such behavior (spatiotemporal chaos) is a key unsolved problem in nonlinear science [@cross]. Many such systems are modelled by partial differential equations; hence, in principle, their dynamics takes place in an infinite dimensional phase space. However, dissipation often acts to confine these systems’ asymptotic behavior to finite-dimensional subspaces known as invariant manifolds [@manneville]. Knowledge of the invariant manifolds provides a wealth of dynamical information; thus, devising methodologies to determine invariant manifolds from experimental data would significantly advance understanding of spatiotemporal chaos.
In this Letter, we describe experiments in Rayleigh-Bénard convection where several slow eigenmodes and their growth rates associated with instability of roll states are extracted quantitatively. Rayleigh-Bénard convection (RBC) serves well as a model spatially extended system; in particular, the spiral defect chaos (SDC) state in RBC is considered an outstanding example of spatiotemporal chaos. In SDC the spatial structure is primarily composed of curved but locally parallel rolls, punctuated by defects (Fig. \[eps:sdc\]) [@morris; @egolf1]. The recurrent formation and drift of defects in SDC is believed to play a key role in driving spatiotemporal chaos; moreover, many aspects of defect nucleation in SDC are related to defect formation observed at the onset of instability in patterns of straight, parallel rolls in RBC [@busse]. We obtain experimentally a low-dimensional description of the modes responsible for the nucleation of one important class of defects (dislocations), by first imposing reproducibly a linearly stable, straight roll state (stable fixed point) near instability onset. This state is subsequently subjected to a set of distinct, well-controlled perturbations, each of which initiates a relaxational trajectory from the disturbed state to the (same) fixed point. An ensemble of such trajectories is used to construct a suitable basis for describing the embedding space by means of a modified Karhunen-Loeve decomposition. The dynamical evolution of small disturbances is then characterized by computing both finite-time Lyapunov exponents and the spatial structure of the associated eigenmodes (a similar approach was carried out numerically by [*Egolf et al.*]{} [@egolf2]). This capability is an important step toward developing a systematic way of characterizing and, perhaps, controlling, spatiotemporally chaotic states like SDC where localized “pivotal” events like defect formation play a central role in driving complex behavior.
![\[eps:sdc\] Shadowgraph visualization reveals spontaneous defect nucleation in the spiral defect chaos state of Rayleigh-Benard convection. Two convection rolls are compressed together (higher contrast region in left image). (b.) A short time later (right image), one of the rolls pinches off and two dislocations form.](sdc){width="8cm"}
The convection experiments are performed with gaseous CO$_2$ at a pressure of 3.2 MPa. A 0.697$\pm$0.06 mm-thick gas layer is contained in a 27 mm square cell, which is confined laterally by filter paper. The layer is bounded on top by a sapphire window and on the bottom by a sheet of 1 mm-thick glass neutral density filter(NDF). The neutral density filter is bonded to a heated metal plate with heat sink compound. The temperature of the sapphire window held constant at 21.3 $^{\circ}$C by water cooling. The temperature difference between the top and bottom plates $\Delta T$ is held fixed at 5.50 $\pm$ 0.01 $^{\circ}$C by computer control of a thin film heater attached to the bottom metal plate. These conditions correspond to a dimensionless bifurcation parameter $\epsilon$=$(\Delta T - \Delta T_c)/\Delta T_c=0.41$, where $\Delta
T_c$ is the temperature difference at the onset of convection. The vertical thermal diffusion time, computed to be 2.1 s at onset, represents the characteristic timescale for the system.
We use laser heating to alter the convective patterns that occur spontaneously. A focused beam from an Ar-ion laser is directed through the sapphire window at a spot on the NDF. Absorption of the laser light by the NDF increases the local temperature of the bottom boundary and hence that of the gas, thereby inducing locally a convective upflow. The convection pattern may be modifed, either locally or globally, by rastering the hot spot created by the laser beam. The beam is steered using two galvanometric mirrors rotating about axes orthogonal to each other under computer control. The intensity of the beam is modulated using an acousto-optic modulator. This technique of optical actuation is used to impose convection patterns with desired properties, to perturb these convection patterns and to change the boundary conditions. Similar approaches for manipulating convective flows were explored earlier using a high intensity lamp and masks [@whitehead] in RBC and a rastered infrared laser in Bénard-Marangoni convection [@denis].
[![\[eps:rd\] Experimental images illustrate the flow response to two different perturbations applied, in turn, to the same state of straight convection rolls. Each image represents the difference between the perturbed and unperturbed convection states and therefore, each image highlights the effect of a given perturbation on the flow. In the two cases shown, the localized perturbation is applied directly on a region of either downflow (left image) or upflow (right image). In all cases, the disturbance created by the perturbation decays away and the flow returns to the original unperturbed state.](rd1_new_stable "fig:"){width="4cm"}]{} [![\[eps:rd\] Experimental images illustrate the flow response to two different perturbations applied, in turn, to the same state of straight convection rolls. Each image represents the difference between the perturbed and unperturbed convection states and therefore, each image highlights the effect of a given perturbation on the flow. In the two cases shown, the localized perturbation is applied directly on a region of either downflow (left image) or upflow (right image). In all cases, the disturbance created by the perturbation decays away and the flow returns to the original unperturbed state.](rd2_new_stable "fig:"){width="4cm"}]{}
The experiments begin by using laser heating to impose a well-specified basic state of stable straight rolls. The basic state is typically arranged to be near the onset of instability by imposing a sufficiently large pattern wavenumber such that at fixed $\epsilon$ the system’s parameters are near the skew-varicose stability boundary [@busse]. In this regime, the modes responsible for the instability are weakly damped and, therefore, can be easily excited.
The linear stability of the basic state is probed by applying brief pulses of spatially localized laser heating. For stable patterns, all small disturbances eventually relax. To excite all modes governing the disturbance evolution, we apply a set of localized perturbations consistent with symmetries of the (ideal) straight roll pattern – continuous translation symmetry in the direction along the rolls and discrete translation symmetry in the perpendicular direction plus the reflection symmetry in both directions. Therefore, localized perturbations applied across half a wavelength of the pattern form a “basis” for all such perturbations – any other localized perturbation at a different spatial location is related by symmetry. Localized perturbations are produced in the experiment by aiming the laser beam to create a “hot spot” whose location is stepped from the center of a (cold) downflow region to the center of an adjacent (hot) upflow region in different experimental runs. The perturbations typically last approximately 5 s and have a lateral extent of approximately 0.1 mm, which is less than 10 % of the pattern wavelength.
The evolution of the perturbed convective flow is monitored by shadowgraph visualization. A digital camera with a low-pass filter (to filter out the reflections from the Ar-ion laser) is used to capture a sequence of $256\times 256$ pixel images recorded with 12 bits of intensity resolution at a rate of 41 images per second. A background image of the unperturbed flow is subtracted from each data image; such sequences of difference images comprise the time series representing the evolution of the perturbation (Fig \[eps:rd\]).
The total power for each (difference) image in a time series is obtained from 2-D spatial Fourier transforms. The resulting time series of total power shows a strong transient excursion (corresponding to the initial response of the convective flow to a localized perturbation by laser heating) followed by exponential decay as the system relaxes back to the stable state of straight convection rolls. We restrict further analysis to the region of exponential decay, which typically represents about $3.5$ seconds of data for each applied perturbation.
The dimensionality of the raw data is too high to permit direct analysis, so each difference image is first windowed (to avoid aliasing effects) and Fourier filtered by discarding the Fourier modes outside a $31\times 31$ window centered at the zero frequency. The discarded high-frequency modes are strongly damped and contain less than 1% of the total power. The basis of $31^2$ Fourier modes still contains redundant information, so we further reduce the dimensionality of the embedding space by projecting the disturbance trajectories onto the “optimal” basis constructed using a variation of the Karhunen-Loeve (KL) decomposition [@holmes; @sirovich]. The correlation matrix $C$ is computed using the Fourier filtered time series ${\bf x}^s(t)$, $$C=\sum_{s,t}({\bf x}^s(t)-\langle{\bf x}^s(t)\rangle_t)
({\bf x}^s(t)-\langle{\bf x}^s(t)\rangle_t)^\dagger,$$ where the index $s$ labels different initial conditions and the origin of time $t=0$ corresponds to the time when the perturbation applied by the laser is within the linear neighbourhood of the statioary state. The angle brackets with the subscript $t$ indicate a time average. The eigenvectors of $C$ are the KL basis vectors. It is worth noting that the average performed to compute $C$ represents an ensemble average over different initial conditions (obtained by applying different perturbations); this is distinctly different from the standard implementation of KL decomposition where statistical time averages are typically employed.
[![\[eps:kl\_new\_stable\] The first four Karhunen-Loeve eigenvectors are shown for a perturbed roll state near the skew-varicose boundary of Rayleigh-Bénard convection. The eigenvectors are ordered by their eigenvalues (largest to smallest), which are propotional to the amount of power contained in the corresponding eigenvector.](klm1 "fig:"){width="4cm"}]{} [![\[eps:kl\_new\_stable\] The first four Karhunen-Loeve eigenvectors are shown for a perturbed roll state near the skew-varicose boundary of Rayleigh-Bénard convection. The eigenvectors are ordered by their eigenvalues (largest to smallest), which are propotional to the amount of power contained in the corresponding eigenvector.](klm2 "fig:"){width="4cm"}]{}\
[![\[eps:kl\_new\_stable\] The first four Karhunen-Loeve eigenvectors are shown for a perturbed roll state near the skew-varicose boundary of Rayleigh-Bénard convection. The eigenvectors are ordered by their eigenvalues (largest to smallest), which are propotional to the amount of power contained in the corresponding eigenvector.](klm3 "fig:"){width="4cm"}]{} [![\[eps:kl\_new\_stable\] The first four Karhunen-Loeve eigenvectors are shown for a perturbed roll state near the skew-varicose boundary of Rayleigh-Bénard convection. The eigenvectors are ordered by their eigenvalues (largest to smallest), which are propotional to the amount of power contained in the corresponding eigenvector.](klm4 "fig:"){width="4cm"}]{}\
The spatial structures of the first four KL vectors are shown in Fig. \[eps:kl\_new\_stable\]. We find that the first 24 basis vectors capture over 90% of the total power, so an embedding space spanned by these vectors represents well the relaxational dynamics about the straight roll pattern. In our convection experiments, the KL eigenvectors show two distinct length scales. The first two dominant vectors are spatially localized, while the remaining vectors are spatially extended. This is consistent with earlier work as suggested in [@egolf1].
More quantitative information can be obtained by finding the eigenmodes of the system, excited by the perturbation, and their growth rates. These can be extracted from a nonlinear least squares fit with the cost function $$E_n=\sum_{i,s,t}\left[{\bf x}^s_i(t)-\left({\bf x}^s_i(\infty)
+\sum_{k=1}^n A^s_k{\bf m}^k_i e^{\lambda_k t}\right)\right]^2,$$ where ${\bf x}^s_i(t)$ is a projection of the perturbation at time $t$ in the time series $s$ onto the $i$th KL basis vector. In the fit ${\bf m}^k$ and $\lambda_k$ are the $k$th eigenmode and its growth rate and $A^s_k$ is the initial amplitude of the $k$th eigenmode excited in the experimental time series $s$. The fixed points ${\bf x}^s(\infty)$ are chosen to be different for the differing time series in the ensemble to account for a slow drift in the parameters and we assume that only $n$ eigenmodes are excited.
The results for an ensemble of time series corresponding to seven point perturbations applied across a wavelength of the pattern with $n=6$ are shown in Figs. \[eps:ps\_new\_stable\]-\[eps:gr\_new\_stable\]. (With seven different initial conditions we cannot hope to distinguish more than seven different modes). In particular, Fig. \[eps:ps\_new\_stable\] shows the projection of the experimental time series and the least squares fit on the plane spanned by the first two KL basis vectors. Such extraction of the linear manifold in experiments on spatially extended systems without the knowledge of the dynamical equations of the system aids in the application of techniques that are well developed for low dimensional systems. The manifolds of fixed points and periodic orbits are of particular interest in chaotic systems.
![\[eps:ps\_new\_stable\] A two-dimensional projection of the experimental time series (symbols) and the least squares fits (continuous curves). The time series have been shifted such that the fixed point is at the origin. ](timeser){width="7.5cm"}
The extracted growth rates $\lambda_k$ are shown in Fig. \[eps:gr\_new\_stable\]. Not surprisingly, since the pattern is stable the growth rates are negative. The leading eigenmode (see Fig. \[eps:modes\_new\_stable\]) is spatially extended and shows a diagonal structure characteristic of the skew-varicose instability in an unbounded system. This is also expected as the pattern is near the skew-varicose instability boundary. The second eigenmode is spatially localized and has no analog in spatially unbounded systems. The subsequent modes are again spatially delocalized and likely correspond to the Goldstone modes of the unbounded system (e.g., overall translation of the pattern) which are made weakly stable due to confinement by the lateral boundaries of the convection cell.
If the system is brought across the stability boundary, one of the modes is expected to become unstable (without significant change in its spatial structure), thereby determining further (nonlinear) evolution of the system towards a state with a pair of dislocation defects. We would also expect the spatially localized eigenmodes (like the second one in Fig. \[eps:modes\_new\_stable\]) to preserve their structure if the base state is smoothly distorted (as it would be, e.g., in the SDC state shown in Fig. \[eps:sdc\]), indicating the same type of a spatially localized instability. Our further experimental studies will aim to confirm these expectations.
![\[eps:gr\_new\_stable\] The growth rates of the six dominant eigenmodes and the error bars extracted from the least squares fit. The growth rates have been non-dimensionalized by the vertical thermal diffusion time.](gr_new_stable){width="8cm"}
[![\[eps:modes\_new\_stable\] Four dominant eigenmodes extracted from the least squares fit.](eim1 "fig:"){width="4cm"}]{} [![\[eps:modes\_new\_stable\] Four dominant eigenmodes extracted from the least squares fit.](eim2 "fig:"){width="4cm"}]{}\
[![\[eps:modes\_new\_stable\] Four dominant eigenmodes extracted from the least squares fit.](eim3 "fig:"){width="4cm"}]{} [![\[eps:modes\_new\_stable\] Four dominant eigenmodes extracted from the least squares fit.](eim4 "fig:"){width="4cm"}]{}\
Defects represent a type of “coherent structure” in spiral defect chaos. Previous efforts have used coherent structures to characterize spatiotemporally chaotic extended systems in both models [@sirovich] and experiments [@wolf]; the use of coherent structures to parametrize the invariant manifold was pioneered by Holmes [*et al.*]{} [@holmes] in the context of turbulence. In practice coherent structures are usually extracted using the Karhunen-Loéve (or proper orthogonal) decomposition of time series of system states, which picks out the [*statistically*]{} important patterns. This prior work has met with only limited success – indeed, it is unclear whether statistically important patterns are [*dynamically*]{} important. An alternative approach has been proposed by Christiansen [*et al.*]{} [@christiansen], who suggested instead to use the recurrent patterns corresponding to the low-period unstable periodic orbits (UPO) of the system, which are dynamically more important. Our work sets the stage for attempting the more ambitious task of extraction of UPOs and their stability properties from experimental data.
Summing up, we have developed an experimental technique which allows extraction of quantitative information describing the dynamics and stability of a pattern forming system near a fixed point. This technique should be applicable to a broad class of patterns, including unstable fixed points, periodic orbits and segments of chaotic trajectories. Moreover, we expect that a similar approach could be applied to other pattern forming systems, convective or otherwise, as long as a method of spatially distibuted actuation of their state can be devised.
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} |
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abstract: 'Frenkel and Reshetikhin [@Fre] introduced $q$-characters to study finite dimensional representations of the quantum affine algebra ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$. In the simply laced case Nakajima [@Naa][@Nab] defined deformations of $q$-characters called $q,t$-characters. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In this article we propose an algebraic general (non necessarily simply laced) new approach to $q,t$-characters motivated by the deformed screening operators [@Her01]. The $t$-deformations are naturally deduced from the structure of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$: the parameter $t$ is analog to the central charge $c\in{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$. The $q,t$-characters lead to the construction of a quantization of the Grothendieck ring and to general analogues of Kazhdan-Lusztig polynomials in the same spirit as Nakajima did for the simply laced case.'
address: 'David Hernandez: École Normale Supérieure - DMA, 45, Rue d’Ulm F-75230 PARIS, Cedex 05 FRANCE'
author:
- David Hernandez
title: 'Algebraic Approach to $q,t$-Characters'
---
Introduction
============
We suppose $q\in{\ensuremath{\mathbb{C}}}^*$ is not a root of unity. In the case of a semi-simple Lie algebra ${\mathfrak{g}}$, the structure of the Grothendieck ring $\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))$ of finite dimensional representations of the quantum algebra ${\mathcal{U}}_q({\mathfrak{g}})$ is well understood. It is analogous to the classical case $q=1$. In particular we have ring isomorphisms: $$\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))\simeq \text{Rep}({\mathfrak{g}})\simeq {\mathbb{Z}}[\Lambda]^W\simeq {\mathbb{Z}}[T_1,...,T_n]$$ deduced from the injective homomorphism of characters $\chi$: $$\chi(V)=\underset{\lambda\in\Lambda}{\sum}\text{dim}(V_{\lambda})\lambda$$ where $V_{\lambda}$ are weight spaces of a representation $V$ and $\Lambda$ is the weight lattice.
For the general case of Kac-Moody algebras the picture is less clear. In the affine case ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$, Frenkel and Reshetikhin [@Fre] introduced an injective ring homomorphism of $q$-characters: $$\chi_q:\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\rightarrow {\mathbb{Z}}[Y_{i,a}^{\pm}]_{1\leq i\leq n,a\in{\ensuremath{\mathbb{C}}}^*}={\mathcal{Y}}$$
The homomorphism $\chi_q$ allows to describe the ring $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\simeq{\mathbb{Z}}[X_{i,a}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$, where the $X_{i,a}$ are fundamental representations. It particular $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ is commutative.
The morphism of $q$-characters has a symmetry property analogous to the classical action of the Weyl group $\text{Im}(\chi)={\mathbb{Z}}[\Lambda]^W$: Frenkel and Reshetikhin defined $n$ screening operators $S_i$ such that $\text{Im}(\chi_q)=\underset{i\in I}{\bigcap}\text{Ker}(S_i)$ (the result was proved by Frenkel and Mukhin for the general case in [@Fre2]).
In the simply laced case Nakajima introduced $t$-analogues of $q$-characters ([@Naa], [@Nab]): it is a ${\mathbb{Z}}[t^{\pm}]$-linear map $$\chi_{q,t}:\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]\rightarrow{\mathcal{Y}}_t={\mathbb{Z}}[Y_{i,a}^{\pm},t^{\pm}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$$ which is a deformation of $\chi_q$ and multiplicative in a certain sense. A combinatorial axiomatic definition of $q,t$-characters is given. But the existence is non-trivial and is proved with the geometric theory of quiver varieties which holds only in the simply laced case.
In [@Her01] we introduced $t$-analogues of screening operators $S_{i,t}$ such that in the simply laced case: $$\underset{i\in I}{\bigcap}\text{Ker}(S_{i,t})=\text{Im}(\chi_{q,t})$$ It is a first step in the algebraic approach to $q,t$-characters proposed in this article: we define and construct $q,t$-characters in the general (non necessarily simply laced) case. The motivation of the construction appears in the non-commutative structure of the Cartan subalgebra ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$, the study of screening currents and of deformed screening operators.
As an application we construct a deformed algebra structure and an involution of the Grothendieck ring, and analogues of Kazhdan-Lusztig polynomials in the general case in the same spirit as Nakajima did for the simply laced case. In particular this article proves a conjecture that Nakajima made for the simply laced case (remark 3.10 in [@Nab]): there exists a purely combinatorial proof of the existence of $q,t$-characters.
This article is organized as follows: after some backgrounds in section \[back\], we define a deformed non-commutative algebra structure on ${\mathcal{Y}}_t={\mathbb{Z}}[Y_{i,a}^{\pm},t^{\pm}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$ (section \[defoal\]): it is naturally deduced from the relations of ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ (theorem \[dessus\]) by using the quantization in the direction of the central element $c$. In particular in the simply laced case it can be used to construct the deformed multiplication of Nakajima [@Nab] (proposition \[form\]) and of Varagnolo-Vasserot [@Vas] (section \[varva\]).
This picture allows us to introduce the deformed screening operators of [@Her01] as commutators of Frenkel-Reshetikhin’s screening currents of [@Freb] (section \[scr\]). In [@Her01] we gave explicitly the kernel of each deformed screening operator (theorem \[her\]).
In analogy to the classic case where $\text{Im}(\chi_q)=\underset{i\in I}{\bigcap}\text{Ker}(S_i)$, we have to describe the intersection of the kernels of deformed screening operators. We introduce a completion of this intersection (section \[complesection\]) and give its structure in proposition \[thth\]. It is easy to see that it is not too big (lemma \[leasto\]); but the point is to prove that it contains enough elements: it is the main result of our construction in theorem \[con\] which is crucial for us. It is proved by induction on the rank $n$ of ${\mathfrak{g}}$.
We define a $t$-deformed algorithm (section \[defialgo\]) analog to the Frenkel-Mukhin’s algorithm [@Fre2] to construct $q,t$-characters in the completion of ${\mathcal{Y}}_t$. An algorithm was also used by Nakajima in the simply laced case in order to compute the $q,t$-characters for some examples ([@Naa]) assuming they exist (which was geometrically proved). Our aim is different : we do not know [*a priori*]{} the existence in the general case. That is why we have to show the algorithm is well defined, never fails (lemma \[nfail\]) and gives a convenient element (lemma \[conv\]).
This construction gives $q,t$-characters for fundamental representations; we deduce from them the injective morphism of $q,t$-characters $\chi_{q,t}$ (definition \[mqt\]). We study the properties of $\chi_{q,t}$ (theorem \[axiomes\]). Some of them are generalization of the axioms that Nakajima defined in the simply laced case ([@Nab]); in particular we have constructed the morphism of [@Nab].
We have some applications: the morphism gives a deformation of the Grothendieck ring because the image of $\chi_{q,t}$ is a subalgebra for the deformed multiplication (section \[quanta\]). Moreover we define an antimultiplicative involution of the deformed Grothendieck ring (section \[invo\]); the construction of this involution is motivated by the new point view adopted in this paper : it is just replacing $c$ by $-c$ in ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$. In particular we define constructively analogues of Kazhdan-Lusztig polynomials and a canonical basis (theorem \[expol\]) motivated by the introduction of [@Nab]. We compute explicitly the polynomials for some examples.
In section \[quest\] we raise some questions : we conjecture that the coefficients of $q,t$-characters are in ${\ensuremath{\mathbb{N}}}[t^{\pm}]\subset{\mathbb{Z}}[t^{\pm}]$. In the $ADE$-case it a result of Nakajima; we give an alternative elementary proof for the $A$-cases in section \[acase\]. The cases $G_2, B_2, C_2$ are also checked in section \[fin\]. The cases $F_4, B_n, C_n$ ($n\leq 10$) have been checked on a computer.
We also conjecture that the generalized analogues to Kazhdan-Lusztig polynomials give at $t=1$ the multiplicity of simple modules in standard modules. We propose some generalizations and further applications which will be studied elsewhere.
In the appendix (section \[fin\]) we give explicit computations of $q,t$-characters for semi-simple Lie algebras of rank 2. They are used in the proof of theorem \[con\].
For convenience of the reader we give at the end of this article an index of notations defined in the main body of the text.
[**Acknowledgments.**]{} The author would like to thank M. Rosso for encouragements and precious comments on a previous version of this paper, I. B. Frenkel for having encouraged him in this direction, E. Frenkel for encouragements, useful discussions and references, E. Vasserot for very interesting explanations about [@Vas], O. Schiffmann for valuable comments and his kind hospitality in Yale university, and T. Schedler for help on programming.
Background {#back}
==========
Cartan matrix {#recalu}
-------------
A generalized Cartan matrix of rank $n$ is a matrix $C=(C_{i,j})_{1\leq i,j\leq n}$\[carmat\] such that $C_{i,j}\in{\mathbb{Z}}$ and: $$C_{i,i}=2$$ $$i\neq j\Rightarrow C_{i,j}\leq 0$$ $$C_{i,j}=0\Leftrightarrow C_{j,i}=0$$ Let $I = \{1,...,n\}$.
We say that $C$ is symmetrizable if there is a matrix $D=\text{diag}(r_1,...,r_n)$ ($r_i\in{\ensuremath{\mathbb{N}}}^*$) such that $B=DC$\[symcar\] is symmetric.
Let $q\in{\ensuremath{\mathbb{C}}}^*$ be the parameter of quantization. In the following we suppose it is not a root of unity. $z$ is an indeterminate.\[qz\]
If $C$ is symmetrizable, let $q_i=q^{r_i}$, $z_i=z^{r_i}$ and $C(z)=(C(z)_{i,j})_{1\leq i,j\leq n}$ the matrix with coefficients in ${\mathbb{Z}}[z^{\pm}]$ such that: $$C(z)_{i,j}=[C_{i,j}]_z\text{ if $i\neq j$}$$ $$C(z)_{i,i}=[C_{i,i}]_{z_i}=z_i+z_i^{-1}$$ where for $l\in{\mathbb{Z}}$ we use the notation: $$[l]_z=\frac{z^l-z^{-l}}{z-z^{-1}}\text{ ($=z^{-l+1}+z^{-l+3}+...+z^{l-1}$ for $l\geq 1$)}$$ In particular, the coefficients of $C(z)$ are symmetric Laurent polynomials (invariant under $z\mapsto z^{-1}$). We define the diagonal matrix $D_{i,j}(z)=\delta_{i,j}[r_i]_z$ and the matrix $B(z)=D(z)C(z)$.
In the following we suppose that $C$ is of finite type, in particular $\text{det}(C)\neq 0$. In this case $C$ is symmetrizable; if $C$ is indecomposable there is a unique choice of $r_i\in{\ensuremath{\mathbb{N}}}^*$ such that $r_1\wedge...\wedge r_n=1$. We have $B_{i,j}(z)=[B_{i,j}]_z$ and $B(z)$ is symmetric. See [@bou] or [@Kac] for a classification of those finite Cartan matrices.
We say that $C$ is simply-laced if $r_1=...=r_n=1$. In this case $C$ is symmetric, $C(z)=B(z)$ is symmetric. In the classification those matrices are of type $ADE$.
Denote by $\mathfrak{U}\subset{\ensuremath{\mathbb{Q}}}(z)$\[mathu\] the subgroup ${\mathbb{Z}}$-linearly spanned by the $\frac{P(z)}{Q(z^{-1})}$ such that $P(z)\in{\mathbb{Z}}[z^{\pm}]$, $Q(z)\in{\mathbb{Z}}[z]$, the zeros of $Q(z)$ are roots of unity and $Q(0)=1$. It is a subring of ${\ensuremath{\mathbb{Q}}}(z)$, and for $R(z)\in\mathfrak{U},m\in{\mathbb{Z}}$ we have $R(q^m)\in\mathfrak{U}$ and $R(q^m)\in{\ensuremath{\mathbb{C}}}$ makes sense.
It follows from lemma 1.1 of [@Fre2] that $C(z)$ has inverse $\tilde{C}(z)$\[invcar\] with coefficients of the form $R(z)\in\mathfrak{U}$.
Finite quantum algebras
-----------------------
We refer to [@Ro] for the definition of the finite quantum algebra ${\mathcal{U}}_q({\mathfrak{g}})$ associated to a finite Cartan matrix, the definition and properties of the type $1$-representations of ${\mathcal{U}}_q({\mathfrak{g}})$, the Grothendieck ring $\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))$ and the injective ring morphism of characters $\chi:\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))\rightarrow {\mathbb{Z}}[y_i^{\pm}]$.
Quantum affine algebras
-----------------------
The quantum affine algebra associated to a finite Cartan matrix $C$ is the ${\ensuremath{\mathbb{C}}}$-algebra ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$\[qadefi\] defined (Drinfeld new realization) by generators $x_{i,m}^{\pm}$ ($i\in I$, $m\in{\mathbb{Z}}$), $k_i^{\pm}$ ($i\in I$), $h_{i,m}$ ($i\in I$, $m\in{\mathbb{Z}}^*$), central elements $c^{\pm\frac{1}{2}}$, and relations: $$k_ik_j=k_jk_i$$ $$k_ih_{j,m}=h_{j,m}k_i$$ $$k_ix_{j,m}^{\pm}k_i^{-1}=q^{\pm B_{ij}}x_{j,m}^{\pm}$$ $$[h_{i,m},x_{j,m'}^{\pm}]=\pm \frac{1}{m}[mB_{ij}]_qc^{\mp\frac{\mid m\mid}{2}} x_{j,m+m'}^{\pm}$$ $$x_{i,m+1}^{\pm}x_{j,m'}^{\pm}-q^{\pm B_{ij}}x_{j,m'}^{\pm}x_{i,m+1}^{\pm}=q^{\pm B_{ij}}x_{i,m}^{\pm}x_{j,m'+1}^{\pm}-x_{j,m'+1}^{\pm}x_{i,m}^{\pm}$$ $$[h_{i,m},h_{j,m'}]=\delta_{m,-m'}\frac{1}{m}[mB_{ij}]_q\frac{c^m-c^{-m}}{q-q^{-1}}$$ $$[x_{i,m}^+,x_{j,m'}^-]= \delta_{ij}\frac{c^{\frac{m-m'}{2}}\phi^+_{i,m+m'}-c^{-\frac{m-m'}{2}}\phi^-_{i,m+m'}}{q_i-q_i^{-1}}$$ $$\underset{\pi\in \Sigma_s}{\sum}\underset{k=0..s}{\sum}(-1)^k\begin{bmatrix}s\\k\end{bmatrix}_{q_i}x_{i,m_{\pi(1)}}^{\pm}...x_{i,m_{\pi(k)}}^{\pm}x_{j,m'}^{\pm}x_{i,m_{\pi(k+1)}}^{\pm}...x_{i,m_{\pi(s)}}^{\pm}=0$$ where the last relation holds for all $i\neq j$, $s=1-C_{ij}$, all sequences of integers $m_1,...,m_s$. $\Sigma_s$ is the symmetric group on $s$ letters. For $i\in I$ and $m\in{\mathbb{Z}}$, $\phi_{i,m}^{\pm}\in {\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ is determined by the formal power series in ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})[[u]]$ (resp. in ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})[[u^{-1}]]$): $$\underset{m=0..\infty}{\sum}\phi_{i,\pm m}^{\pm}u^{\pm m}=k_i^{\pm}\text{exp}(\pm(q-q^{-1})\underset{m'=1..\infty}{\sum}h_{i,\pm m'}u^{\pm m'})$$ and $\phi_{i,m}^+=0$ for $m<0$, $\phi_{i,m}^-=0$ for $m>0$.
One has an embedding ${\mathcal{U}}_q({\mathfrak{g}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ and a Hopf algebra structure on ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ (see [@Fre] for example).
The Cartan algebra ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})\subset{\mathcal{U}}_q(\hat{{\mathfrak{g}}})$\[qhdefi\] is the ${\ensuremath{\mathbb{C}}}$-subalgebra of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ generated by the $h_{i,m},c^{\pm}$ ($i\in I, m\in{\mathbb{Z}}-\{0\}$).
Finite dimensional representations of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$
-----------------------------------------------------------------------------
A finite dimensional representation $V$ of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ is called of type $1$ if $c$ acts as $\text{Id}$ and $V$ is of type $1$ as a representation of ${\mathcal{U}}_q({\mathfrak{g}})$. Denote by $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ the Grothendieck ring of finite dimensional representations of type $1$.
The operators $\{\phi^{\pm}_{i,\pm m},i\in I, m\in{\mathbb{Z}}\}$ commute on $V$. So we have a pseudo-weight space decomposition: $$V=\underset{\gamma\in {\ensuremath{\mathbb{C}}}^{I\times {\mathbb{Z}}}\times{\ensuremath{\mathbb{C}}}^{I\times{\mathbb{Z}}}}{\bigoplus} V_{\gamma}$$ where for $\gamma=(\gamma^+,\gamma^-)$, $V_{\gamma}$ is a simultaneous generalized eigenspace: $$V_{\gamma}=\{x\in V/\exists p\in{\ensuremath{\mathbb{N}}},\forall i\in\{1,...,n\},\forall m\in{\mathbb{Z}},(\phi_{i,m}^{\pm}-\gamma_{i,m}^{\pm})^p.x=0\}$$ The $\gamma_{i,m}^{\pm}$ are called pseudo-eigen values of $V$.
([**Chari, Pressley**]{} [@Cha],[@Cha2]) Every simple representation $V\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ is a highest weight representation $V$, that is to say there is $v_0\in V$ (highest weight vector) $\gamma_{i,m}^{\pm}\in{\ensuremath{\mathbb{C}}}$ (highest weight) such that: $$V={\mathcal{U}}_q(\hat{{\mathfrak{g}}}).v_0\text{ , }c^{\frac{1}{2}}.v_0=v_0$$ $$\forall i\in I,m\in{\mathbb{Z}},x_{i,m}^+.v_0=0\text{ , }\text{ , }\phi_{i,m}^{\pm}.v_0=\gamma_{i,m}^{\pm}v_0$$ Moreover we have an $I$-uplet $(P_i(u))_{i\in I}$ of (Drinfeld-)polynomials such that $P_i(0)=1$ and: $$\gamma_i^{\pm}(u)=\underset{m\in{\ensuremath{\mathbb{N}}}}{\sum}\gamma_{i,\pm m}^{\pm}u^{\pm}=q_i^{\deg(P_i)}\frac{P_i(uq_i^{-1})}{P_i(uq_i)}\in{\ensuremath{\mathbb{C}}}[[u^{\pm}]]$$ and $(P_i)_{i\in I}$ parameterizes simple modules in $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$.
([**Frenkel, Reshetikhin**]{} [@Fre]) The eigenvalues $\gamma_i(u)^{\pm}\in{\ensuremath{\mathbb{C}}}[[u]]$ of a representation $V\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ have the form: $$\gamma_i^{\pm}(u)=q_i^{deg(Q_i)-deg(R_i)}\frac{Q_i(uq_i^{-1})R_i(uq_i)}{Q_i(uq_i)R_i(uq_i^{-1})}$$ where $Q_i(u),R_i(u)\in{\ensuremath{\mathbb{C}}}[u]$ and $Q_i(0)=R_i(0)=1$.
Note that the polynomials $Q_i,R_i$ are uniquely defined by $\gamma$. Denote by $Q_{\gamma,i}$, $R_{\gamma,i}$ the polynomials associated to $\gamma$.
q-characters {#qcar}
------------
Let ${\mathcal{Y}}$ be the commutative ring ${\mathcal{Y}}={\mathbb{Z}}[Y_{i,a}^{\pm}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$.
For $V\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ a representation, the $q$-character $\chi_q(V)$\[chiqdefi\] of $V$ is: $$\chi_q(V)=\underset{\gamma}{\sum}\text{dim}(V_{\gamma})\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\prod}Y_{i,a}^{\lambda_{\gamma,i,a}-\mu_{\gamma,i,a}}\in {\mathcal{Y}}$$ where for $\gamma\in{\ensuremath{\mathbb{C}}}^{I\times{\mathbb{Z}}}\times{\ensuremath{\mathbb{C}}}^{I\times{\mathbb{Z}}}$, $i\in I$, $a\in{\ensuremath{\mathbb{C}}}^*$ the $\lambda_{\gamma,i,a},\mu_{\gamma,i,a}\in{\mathbb{Z}}$ are defined by: $$Q_{\gamma, i}(z)=\underset{a\in{\ensuremath{\mathbb{C}}}^*}{\prod}(1-za)^{\lambda_{\gamma,i,a}}\text{ , }R_{\gamma, i}(z)=\underset{a\in{\ensuremath{\mathbb{C}}}^*}{\prod}(1-za)^{\mu_{\gamma,i,a}}$$
([**Frenkel, Reshetikhin**]{} [@Fre]) The map $$\chi_q:\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))\rightarrow {\mathcal{Y}}$$ is an injective ring homomorphism and the following diagram is commutative: $$\begin{array}{rcccl}
\text{Rep} ({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))&\stackrel{\chi_q}{\longrightarrow}&{\mathbb{Z}}[Y^{\pm}_{i,a}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}\\
\downarrow res&&\downarrow\beta\\
\text{Rep}({\mathcal{U}}_q({\mathfrak{g}}))&\stackrel{\chi}{\longrightarrow}&{\mathbb{Z}}[y^{\pm}_i]_{i\in I}\\\end{array}$$ where $\beta$ is the ring homomorphism such that $\beta(Y_{i,a})=y_i$ ($i\in I,a\in{\ensuremath{\mathbb{C}}}^*$).
For $m\in{\mathcal{Y}}$ of the form $m=\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\prod}Y_{i,a}^{u_{i,a}(m)}$ ($u_{i,a}(m)\geq 0$), denote $V_m\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ the simple module with Drinfeld polynomials $P_i(u)=\underset{a\in{\ensuremath{\mathbb{C}}}^*}{\prod}(1-ua)^{u_{i,a}(m)}$. In particular for $i\in I,a\in{\ensuremath{\mathbb{C}}}^*$ denote $V_{i,a}=V_{Y_{i,a}}$ and $X_{i,a}=\chi_q(V_{i,a})$. The simple modules $V_{i,a}$ are called fundamental representations.
Denote by $M_m\in\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ the module $M_m=\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\bigotimes}V_{i,a}^{\otimes u_{i,a}(m)}$. It is called a standard module and his $q$-character is $\underset{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}{\prod}X_{i,a}^{u_{i,a}(m)}$.
([**Frenkel, Reshetikhin**]{} [@Fre]) The ring $\text{Rep}({\mathcal{U}}_q(\hat{{\mathfrak{g}}}))$ is commutative and isomorphic to ${\mathbb{Z}}[X_{i,a}]_{i\in I,a\in{\ensuremath{\mathbb{C}}}^*}$.
([**Frenkel, Mukhin**]{} [@Fre2])\[aidafm\] For $i\in I,a\in{\ensuremath{\mathbb{C}}}^*$, we have $X_{i,a}\in{\mathbb{Z}}[Y_{j,aq^l}^{\pm}]_{j\in I,l\geq 0}$.
In particular for $a\in{\ensuremath{\mathbb{C}}}^*$ we have an injective ring homomorphism: $$\chi_q^a:\text{Rep}_a={\mathbb{Z}}[X_{i,aq^l}]_{i\in I,l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}_a={\mathbb{Z}}[Y_{i,aq^l}^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$$ For $a,b\in{\ensuremath{\mathbb{C}}}^*$ denote $\alpha_{b,a}:\text{Rep}_a\rightarrow\text{Rep}_b$ and $\beta_{b,a}:{\mathcal{Y}}_a\rightarrow{\mathcal{Y}}_b$ the canonical ring homomorphism.
We have a commutative diagram: $$\begin{array}{rcccl}
\text{Rep}_a&\stackrel{\chi_q^a}{\longrightarrow}&{\mathcal{Y}}_a\\
\alpha_{b,a}\downarrow &&\downarrow\beta_{b,a}\\
\text{Rep}_b&\stackrel{\chi_q^b}{\longrightarrow}&{\mathcal{Y}}_b\\\end{array}$$
This result is a consequence of theorem \[simme\] (or see [@Fre], [@Fre2]). In particular it suffices to study $\chi_q^1$. In the following denote $\text{Rep}=\text{Rep}_1$, $X_{i,l}=X_{i,q^l}$\[xil\], ${\mathcal{Y}}={\mathcal{Y}}_1$ and $\chi_q=\chi_q^1:\text{Rep}\rightarrow {\mathcal{Y}}$.\[rep\]
Twisted polynomial algebras related to quantum affine algebras {#defoal}
==============================================================
The aim of this section is to define the $t$-deformed algebra ${\mathcal{Y}}_t$ and to describe its structure (theorem \[dessus\]). We define the Heisenberg algebra $\mathcal{H}$, the subalgebra ${\mathcal{Y}}_u\subset \mathcal{H}[[h]]$ and eventually ${\mathcal{Y}}_t$ as a quotient of ${\mathcal{Y}}_u$.
Heisenberg algebras related to quantum affine algebras
------------------------------------------------------
### The Heisenberg algebra $\mathcal{H}$
$\mathcal{H}$\[zq\] is the ${\ensuremath{\mathbb{C}}}$-algebra defined by generators $a_i[m]$\[aim\] ($i\in I, m\in{\mathbb{Z}}-\{0\}$), central elements $c_r$\[cr\] ($r>0$) and relations ($i,j\in I,m,r\in{\mathbb{Z}}-\{0\}$): $$[a_i[m],a_j[r]]=\delta_{m,-r}(q^m-q^{-m})B_{i,j}(q^m)c_{|m|}$$
This definition is motivated by the structure of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$: in $\mathcal{H}$ the $c_r$ are algebraically independent, but we have a surjective homomorphism from $\mathcal{H}$ to ${\mathcal{U}}_q(\hat{{\mathfrak{h}}})$ such that $a_i[m]\mapsto (q-q^{-1})h_{i,m}$ and $c_r\mapsto\frac{c^{r}-c^{-r}}{r}$.
### Properties of $\mathcal{H}$
For $j\in I,m\in{\mathbb{Z}}$ we set\[yim\]: $$y_j[m]=\underset{i\in I}{\sum} \tilde{C}_{i,j}(q^m)a_i[m]\in \mathcal{H}$$
\[calczq\] We have the Lie brackets in $\mathcal{H}$ ($i,j\in I,m,r\in{\mathbb{Z}}$): $$[a_i[m],y_j[r]]=(q^{mr_i}-q^{-r_im})\delta_{m,-r}\delta_{i,j}c_{|m|}$$ $$[y_i[m],y_j[r]]=\delta_{m,-r}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})c_{|m|}$$
[[*Proof:*]{}]{}We compute in $\mathcal{H}$: $$[a_i[m],y_j[r]]
=[a_i[m],\underset{k\in I}{\sum} \tilde{C}_{k,j}(q^r)a_k[r]]
=\delta_{m,-r}c_{|m|}\underset{k\in I}{\sum}\tilde{C}_{k,j}(q^{-m})[r_i]_{q^m}C_{i,k}(q^m)(q^m-q^{-m})$$ $$=\delta_{i,j}\delta_{m,-r}(q^{mr_i}-q^{-mr_i})c_{|m|}$$ $$[y_i[m],y_j[r]]
=[\underset{k\in I}{\sum} \tilde{C}_{k,i}(q^m)a_k[m],y_j[r]]
=\delta_{m,-r}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})c_{|m|}$$
Let $\pi_+$\[piplus\] and $\pi_-$ be the ${\ensuremath{\mathbb{C}}}$-algebra endomorphisms of $\mathcal{H}$ such that ($i\in I$, $m>0$, $r<0$): $$\pi_+(a_i[m])=a_i[m]\text{ , }\pi_+(a_i[r])=0\text{ , }\pi_+(c_m)=0$$ $$\pi_-(a_i[m])=0\text{ , }\pi_-(a_i[r])=a_i[r]\text{ , }\pi_-(c_m)=0$$ They are well-defined because the relations are preserved. We set $\mathcal{H}^+=\text{Im}(\pi_+)\subset \mathcal{H}$\[zqplus\] and $\mathcal{H}^-=\text{Im}(\pi_-)\subset \mathcal{H}$.
Note that $\mathcal{H}^+$ (resp. $\mathcal{H}^-$) is the subalgebra of $\mathcal{H}$ generated by the $a_i[m]$, $i\in I,m>0$ (resp. $m<0$). So $\mathcal{H}^+$ and $\mathcal{H}^-$ are commutative algebras, and: $$\mathcal{H}^+\simeq \mathcal{H}^-\simeq {\ensuremath{\mathbb{C}}}[a_i[m]]_{i\in I,m>0}$$
We say that $m\in \mathcal{H}$ is a $\mathcal{H}$-monomial if it is a product of the generators $a_i[m],c_r$.
There is a unique ${\ensuremath{\mathbb{C}}}$-linear endomorphism $::$ of $\mathcal{H}$ such that for all $\mathcal{H}$-monomials $m$ we have: $$:m:=\pi_+(m)\pi_-(m)$$
In particular there is a vector space triangular decomposition $\mathcal{H}\simeq \mathcal{H}^+\otimes {\ensuremath{\mathbb{C}}}[c_r]_{r>0}\otimes \mathcal{H}^-$.
[[*Proof:*]{}]{}The $\mathcal{H}$-monomials span the ${\ensuremath{\mathbb{C}}}$-vector space $\mathcal{H}$, so the map is unique. But there are non trivial linear combinations between them because of the relations of $\mathcal{H}$: it suffices to show that for $m_1$, $m_2$ $\mathcal{H}$-monomials the definition of $::$ is compatible with the relations ($i,j\in I$, $l,k\in{\mathbb{Z}}-\{0\}$): $$m_1a_i[k]a_j[l]m_2-m_1a_j[l]a_i[k]m_2=\delta_{k,-l}(q^k-q^{-k})B_{i,j}(q^k)m_1c_{|k|}m_2$$ As $\mathcal{H}^+$ and $\mathcal{H}^-$ are commutative, we have: $$\pi_+(m_1a_i[k]a_j[l]m_2)\pi_-(m_1a_i[k]a_j[l]m_2)=\pi_+(m_1a_j[l]a_i[k]m_2)\pi_-(m_1a_j[l]a_i[k]m_2)$$ and we can conclude because $\pi_+(m_1c_{|k|}m_2)=\pi_-(m_1c_{|k|}m_2)=0$.
The deformed algebra ${\mathcal{Y}}_u$
--------------------------------------
### Construction of ${\mathcal{Y}}_u$
Consider the ${\ensuremath{\mathbb{C}}}$-algebra $\mathcal{H}_h=\mathcal{H}[[h]]$\[zqh\]. The application $\text{exp}$ is well-defined on the subalgebra $h\mathcal{H}_h$: $$\text{exp}:h\mathcal{H}_h\rightarrow \mathcal{H}_h$$ For $l\in{\mathbb{Z}}$, $i\in I$, introduce $\tilde{A}_{i,l},\tilde{Y}_{i,l}\in \mathcal{H}_h$\[tail\] such that: $$\tilde{A}_{i,l}=\text{exp}(\underset{m>0}{\sum}h^m a_i[m]q^{lm})\text{exp}(\underset{m>0}{\sum}h^m a_i[-m]q^{-lm})$$ $$\tilde{Y}_{i,l}=\text{exp}(\underset{m>0}{\sum}h^m y_{i}[m]q^{lm})\text{exp}(\underset{m>0}{\sum}h^m y_i[-m]q^{-lm})$$ Note that $\tilde{A}_{i,l}$ and $\tilde{Y}_{i,l}$ are invertible in $\mathcal{H}_h$ and that: $$\tilde{A}_{i,l}^{-1}=\text{exp}(-\underset{m>0}{\sum}h^m a_i[-m]q^{-lm})\text{exp}(-\underset{m>0}{\sum}h^m a_i[m]q^{lm})$$ $$\tilde{Y}_{i,l}^{-1}=\text{exp}(-\underset{m>0}{\sum}h^m y_i[-m]q^{-lm})\text{exp}(-\underset{m>0}{\sum}h^m y_{i}[m]q^{lm})$$ Recall the definition $\mathfrak{U}\subset{\ensuremath{\mathbb{Q}}}(z)$ of section \[recalu\]. For $R\in \mathfrak{U}$, introduce $t_{R}\in \mathcal{H}_h$\[tr\]: $$t_R=\text{exp}(\underset{m>0}{\sum}h^{2m}R(q^m)c_m)$$
\[yu\] ${\mathcal{Y}}_u$ is the ${\mathbb{Z}}$-subalgebra of $\mathcal{H}_h$ generated by the $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm},t_R$ ($i\in I,l\in{\mathbb{Z}},R\in\mathfrak{U}$).
In this section we give properties of ${\mathcal{Y}}_u$ and subalgebras of ${\mathcal{Y}}_u$ which will be useful in section \[studyyt\].
### Relations in ${\mathcal{Y}}_u$ {#defij}
\[relu\] We have the following relations in ${\mathcal{Y}}_u$ ($i,j\in I$ $l,k\in{\mathbb{Z}}$): $$\label{ay}\tilde{A}_{i,l}\tilde{Y}_{j,k}\tilde{A}_{i,l}^{-1}\tilde{Y}_{j,k}^{-1}
=t_{\delta_{i,j}(z^{-r_i}-z^{r_i})(-z^{(l-k)}+z^{(k-l)})}$$ $$\label{yy}\tilde{Y}_{i,l}\tilde{Y}_{j,k}\tilde{Y}_{i,l}^{-1}\tilde{Y}_{j,k}^{-1}
=t_{\tilde{C}_{j,i}(z)(z^{r_j}-z^{-r_j})(-z^{(l-k)}+z^{(k-l)})}$$ $$\label{aa}\tilde{A}_{i,l}\tilde{A}_{j,k}\tilde{A}_{i,l}^{-1}\tilde{A}_{j,k}^{-1}=t_{B_{i,j}(z)(z^{-1}-z)(-z^{(l-k)}+z^{(k-l)})}$$
[[*Proof:*]{}]{}For $A,B\in h\mathcal{H}_h$ such that $[A,B]\in h{\ensuremath{\mathbb{C}}}[c_r]_{r>0}$, we have: $$\text{exp}(A)\text{exp}(B)=\text{exp}(B)\text{exp}(A)\text{exp}([A,B])$$ So we can compute (see lemma \[calczq\]):
$\tilde{A}_{i,l}\tilde{A}_{j,k}
\\=\text{exp}(\underset{m>0}{\sum}h^m a_{i}[m]q^{lm})(\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^ma_{j}[m]q^{km}))\text{exp}(\underset{m>0}{\sum}h^ma_j[-m]q^{-km})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}B_{i,j}(q^m)(q^{-m}-q^{m})q^{m(k-l)}c_m)
\\\text{exp}(\underset{m>0}{\sum}h^m a_{i}[m]q^{lm})\text{exp}(\underset{m>0}{\sum}h^m a_{j}[m]q^{km})\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m a_j[-m]q^{-km})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}B_{i,j}(q^m)(q^{-m}-q^{m})(-q^{m(l-k)}+q^{m(k-l)})c_m)\tilde{A}_{j,k}\tilde{A}_{i,l}$
$\tilde{A}_{i,l}\tilde{Y}_{j,k}
\\=\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{lm})(\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{km}))\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-km})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}\delta_{i,j}(q^{-mr_i}-q^{mr_i})q^{m(k-l)}c_m)\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{ml})
\\\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{mk})\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-ml})\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-mk})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}\delta_{i,j}(q^{-mr_i}-q^{mr_i})(-q^{m(l-k)}+q^{m(k-l)})c_m)\tilde{Y}_{j,k}\tilde{A}_{i,l}$
$\tilde{Y}_{i,l}\tilde{Y}_{j,k}
\\=\text{exp}(\underset{m>0}{\sum}h^my_i[m]q^{ml})(\text{exp}(\underset{m>0}{\sum}h^my_i[-m]q^{-ml})\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{mk}))\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-mk})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}q^{m(k-l)}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})c_m)\text{exp}(\underset{m>0}{\sum}h^my_{i}[m]q^{ml})
\\\text{exp}(\underset{m>0}{\sum}h^my_j[m]q^{mk})\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-ml})\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-mk})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}\tilde{C}_{j,i}(q^m)(q^{mr_j}-q^{-mr_j})(-q^{m(l-k)}+q^{m(k-l)})c_m)\tilde{Y}_{j,k}\tilde{Y}_{i,l}$
### Commutative subalgebras of $\mathcal{H}_h$ {#defipip}
The ${\ensuremath{\mathbb{C}}}$-algebra endomorphisms $\pi_+,\pi_-$ of $\mathcal{H}$ are naturally extended to ${\ensuremath{\mathbb{C}}}$-algebra endomorphisms of $\mathcal{H}_h$. As ${\mathcal{Y}}_u\subset \mathcal{H}_h$, we have by restriction the ${\mathbb{Z}}$-algebra morphisms $\pi_{\pm}:{\mathcal{Y}}_u\rightarrow \mathcal{H}_h$.
Introduce ${\mathcal{Y}}=\pi_+({\mathcal{Y}}_u)\subset \mathcal{H}^+[[h]]$\[y\]. In this section \[defipip\] we study ${\mathcal{Y}}$. In particular we will see in proposition \[circ\] that the notation ${\mathcal{Y}}$ is consistent with the notation of section \[qcar\].
For $i\in I,l\in{\mathbb{Z}}$, denote\[ail\]: $$Y_{i,l}^{\pm}=\pi_+(\tilde{Y}_{i,l}^{\pm})=\text{exp}(\pm\underset{m>0}{\sum}h^m y_{i}[m]q^{lm})$$ $$A_{i,l}^{\pm}=\pi_+(\tilde{A}_{i,l}^{\pm})=\text{exp}(\pm\underset{m>0}{\sum}h^m a_i[m]q^{lm})$$
\[gen\] For $i\in I,l\in{\mathbb{Z}}$, we have: $$A_{i,l}=Y_{i,l-r_i}Y_{i,l+r_i}(\underset{j/C_{j,i}=-1}{\prod}Y_{j,l}^{-1})(\underset{j/C_{j,i}=-2}{\prod}Y_{j,l+1}^{-1}Y_{j,l-1}^{-1})(\underset{j/C_{j,i}=-3}{\prod}Y_{j,l+2}^{-1}Y_{j,l}^{-1}Y_{j,l-2}^{-1})$$ In particular ${\mathcal{Y}}$ is generated by the $Y_{i,l}^{\pm}$ ($i\in I,l\in{\mathbb{Z}}$).
[[*Proof:*]{}]{}
We have $a_i[m]=\underset{j\in I}{\sum}C_{j,i}(q^m)y_j[m]$, and: $$\pi_+(\tilde{A}_{i,l})=\text{exp}(\underset{m>0}{\sum}h^m a_i[m]q^{lm})=\underset{j\in I}{\prod}\text{exp}(\underset{m>0}{\sum}h^m C_{j,i}(q^m)y_j[m]q^{lm})$$ As $C_{i,i}(q)=q^{r_i}+q^{-r_i}$, we have: $$\text{exp}(\underset{m>0}{\sum}h^m C_{i,i}(q^m)y_i[m]q^{lm})=\text{exp}(\underset{m>0}{\sum}h^m y_i[m]q^{(l-r_i)m})\text{exp}(\underset{m>0}{\sum}h^m y_i[m]q^{(l+r_i)m})=Y_{i,l-r_i}Y_{i,l+r_i}$$ If $C_{j,i}<0$, we have $C_{j,i}(q)=-\underset{k=C_{j,i}+1, C_{j,i}+3...-C_{j,i}-1}{\sum}q^{k}$ and: $$\text{exp}(-\underset{m>0}{\sum}h^m C_{j,i}(q^m)y_j[m]q^{lm})=\underset{k=C_{j,i}+1, C_{j,i}+3...-C_{j,i}-1}{\prod}\text{exp}(-\underset{m>0}{\sum}h^m y_j[m]q^{(l+k)m})$$ As ${\mathcal{Y}}_u$ is generated by the $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm},t_R$ we get the last point.
Note that the formula of lemma \[gen\] already appeared in [@Fre].
We need a general technical lemma to describe ${\mathcal{Y}}$:
\[indgene\] Let $J=\{1,...,r\}$ and let $\Lambda$ be the polynomial commutative algebra\
$\Lambda={\ensuremath{\mathbb{C}}}[\lambda_{j,m}]_{j\in J,m\geq 0}$. For $R=(R_1,...,R_r)\in\mathfrak{U}^{r}$, consider: $$\Lambda_R=\text{exp}(\underset{j\in J,m>0}{\sum}h^mR_j(q^m)\lambda_{j,m})\in\Lambda[[h]]$$ Then the $(\Lambda_R)_{R\in\mathfrak{U}^r}$ are ${\ensuremath{\mathbb{C}}}$-linearly independent. In particular the $\Lambda_{j,l}=\Lambda_{(0,...,0,z^l,0,...,0)}$ ($j\in J$, $l\in{\mathbb{Z}}$) are ${\ensuremath{\mathbb{C}}}$-algebraically independent.
[[*Proof:*]{}]{}Suppose we have a linear combination ($\mu_R\in{\ensuremath{\mathbb{C}}}$, only a finite number of $\mu_R\neq 0$): $$\underset{R\in\mathfrak{U}^r}{\sum}\mu_R\Lambda_R=0$$ The coefficients of $h^L$ in $\Lambda_R$ are of the form $R_{j_1}(q^{l_1})^{L_1}R_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{L_N}\lambda_{j_1,l_1}^{L_1}\lambda_{j_2,l_2}^{L_2}...\lambda_{j_N,l_N}^{L_N}$ where $l_1L_1+...+l_NL_N=L$. So for $N\geq 0$, $j_1,...,j_N\in J$, $l_1,...,l_N>0$, $L_1,...,L_N\geq 0$ we have: $$\underset{R\in\mathfrak{U}^r}{\sum}\mu_RR_{j_1}(q^{l_1})^{L_1}R_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{L_N}=0$$ If we fix $L_2,...,L_{N}$, we have for all $L_1=l\geq 0$: $$\underset{\alpha_1\in{\ensuremath{\mathbb{C}}}}{\sum}\alpha_1^l\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1}{\sum}\mu_RR_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{L_N}=0$$ We get a Van der Monde system which is invertible, so for all $\alpha_1\in{\ensuremath{\mathbb{C}}}$: $$\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1}{\sum}\mu_RR_{j_2}(q^{l_2})^{L_2}...R_{j_N}(q^{l_N})^{l_N}=0$$ By induction we get for $r'\leq N$ and all $\alpha_1,...,\alpha_{r'}\in{\ensuremath{\mathbb{C}}}$: $$\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1,...,R_{j_{r'}}(q^{l_{r'}})=\alpha_{r'}}{\sum}\mu_{R}R_{j_{r'+1}}(q^{l_{r'+1}})^{L_{r'+1}}...R_{j_N}(q^{l_N})^{L_N}=0$$ And so for $r'=N$: $$\underset{R\in\mathfrak{U}^r/R_{j_1}(q^{l_1})=\alpha_1,...,R_{j_{N}}(q^{l_{N}})=\alpha_{N}}{\sum}\mu_{R}=0$$ Let be $S\geq 0$ such that for all $\mu_R,\mu_{R'}\neq 0$, $j\in J$ we have $R_j-R_j'=0$ or $R_j-R_j'$ has at most $S-1$ roots. We set $N=Sr$ and $((j_1,l_1),...,(j_S,l_S))=((1,1),(1,2),...,(1,S),(2,1),...,(2,S),(3,1),...,(r,S))$. We get for all $\alpha_{j,l}\in{\ensuremath{\mathbb{C}}}$ ($j\in J,1\leq l\leq S$): $$\underset{R\in\mathfrak{U}^r/\forall j\in J,1\leq l\leq S, R_{j}(q^{l})=\alpha_{j,l}}{\sum}\mu_{R}=0$$ It suffices to show that there is at most one term is this sum. But consider $P,Q\in\mathfrak{U}$ such that for all $1\leq l\leq S$, $P(q^l)=P'(q^l)$. As $q$ is not a root of unity the $q^l$ are different and $P-P'$ has $S$ roots, so is $0$.
For the last assertion, we can write a monomial $\underset{j\in J,l\in{\mathbb{Z}}}{\prod}\Lambda_{j,l}^{u_{j,l}}=\Lambda_{\underset{l\in{\mathbb{Z}}}{\sum}u_{1,l}z^l,...,\underset{l\in{\mathbb{Z}}}{\sum}u_{r,l}z^l}$. In particular there is no trivial linear combination between those monomials.
It follows from lemma \[gen\] and lemma \[indgene\]:
\[circ\] The $Y_{i,l}\in{\mathcal{Y}}$ are ${\mathbb{Z}}$-algebraically independent and generate the ${\mathbb{Z}}$-algebra ${\mathcal{Y}}$. In particular, ${\mathcal{Y}}$ is the commutative polynomial algebra ${\mathbb{Z}}[Y_{i,l}^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$.
The $A_{i,l}^{-1}\in{\mathcal{Y}}$ are ${\mathbb{Z}}$-algebraically independent. In particular the subalgebra of ${\mathcal{Y}}$ generated by the $A_{i,l}^{-1}$ is the commutative polynomial algebra ${\mathbb{Z}}[A_{i,l}^{-1}]_{i\in I,l\in{\mathbb{Z}}}$.
### Generators of ${\mathcal{Y}}_u$ {#ptpt}
The ${\ensuremath{\mathbb{C}}}$-linear endomorphism $::$ of $\mathcal{H}$ is naturally extended to a ${\ensuremath{\mathbb{C}}}$-linear endomorphism of $\mathcal{H}_h$. As ${\mathcal{Y}}_u\subset \mathcal{H}_h$, we have by restriction a ${\mathbb{Z}}$-linear morphism $::$ from ${\mathcal{Y}}_u$ to $\mathcal{H}_h$.
We say that $m\in{\mathcal{Y}}_u$ is a ${\mathcal{Y}}_u$-monomial if it is a product of generators $\tilde{A}_{i,l}^{\pm},\tilde{Y}_{i,l}^{\pm},t_R$.
In the following, for a product of non commuting terms, denote $\overset{\rightarrow}{\underset{s=1..S}{\prod}}U_s=U_1U_2...U_S$.
\[rel\] The algebra ${\mathcal{Y}}_u$ is generated by the $\tilde{Y}_{i,l}^{\pm},t_R$ ($i\in I,l\in{\mathbb{Z}},R\in\mathfrak{U}$).
[[*Proof:*]{}]{}Let be $i\in I$, $l\in{\mathbb{Z}}$. It follows from proposition \[circ\] that $\pi_+(\tilde{A}_{i,l})$ is of the form $\pi_+(\tilde{A}_{i,l})=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{u_{i,l}}$ and that $:m:=:\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}\tilde{Y}_{i,l}^{u_{i,l}}:$. So it suffices to show that for $m$ a ${\mathcal{Y}}_u$-monomial, there is a unique $R_m\in \mathfrak{U}$ such that $m=t_{R_m}:m:$. Let us write $m=t_R\overset{\rightarrow}{\underset{s=1.. S}{\prod}}U_s$ where $U_s\in\{\tilde{A}_{i,l}^{\pm},\tilde{Y}_{i,l}^{\pm}\}_{i\in I,l\in{\mathbb{Z}}}$ are generators. Then: $$:m:=(\underset{s=1..S}{\prod}\pi_+(U_s))(\underset{s=1..S}{\prod}\pi_-(U_s))$$ And we can conclude because it follows from the proof of lemma \[relu\] that for $1\leq s,s'\leq S$, there is $R_{s,s'}\in\mathfrak{U}$ such that $\pi_+(U_s)\pi_-(U_{s'})=t_{R_{s,s'}}\pi_-(U_{s'})\pi_+(U_s)$.
In particular it follows from this proof that $:{\mathcal{Y}}_u:\subset {\mathcal{Y}}_u$.
The deformed algebra ${\mathcal{Y}}_t$ {#studyyt}
--------------------------------------
### Construction of ${\mathcal{Y}}_t$ {#defipire}
Denote by ${\mathbb{Z}}((z^{-1}))$ the ring of series of the form $P=\underset{r\leq R_P}{\sum}P_rz^r$ where $R_P\in{\mathbb{Z}}$ and the coefficients $P_r\in{\mathbb{Z}}$. Recall the definition $\mathfrak{U}$ of section \[recalu\]. We have an embedding $\mathfrak{U}\subset{\mathbb{Z}}((z^{-1}))$ by expanding $\frac{1}{Q(z^{-1})}$ in ${\mathbb{Z}}[[z^{-1}]]$ for $Q(z)\in{\mathbb{Z}}[z]$ such that $Q(0)=1$. So we can introduce maps:\[pir\] $$\pi_r:\mathfrak{U}\rightarrow {\mathbb{Z}}\text{ , }P=\underset{k\leq R_P}{\sum}P_k z^k\mapsto P_r$$ Note that we could have consider the expansion in ${\mathbb{Z}}((z))$ and that the maps $\pi_r$ are not independent of our choice.
We define ${\mathcal{Y}}_t$\[tyt\] (resp. $\mathcal{H}_t$)\[zqt\] as the algebra quotient of ${\mathcal{Y}}_u$ (resp. $\mathcal{H}_h$) by relations: $$t_R=t_{R'}\text{ if $\pi_0(R)=\pi_0(R')$}$$
We keep the notations $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm}$ for their image in ${\mathcal{Y}}_t$. Denote by $t$ the image of $t_1=\text{exp}(\underset{m>0}{\sum}h^{2m}c_m)$ in ${\mathcal{Y}}_t$. As $\pi_0$ is additive, the image of $t_R$ in ${\mathcal{Y}}_t$ is $t^{\pi_o(R)}$\[t\]. In particular ${\mathcal{Y}}_t$ is generated by the $\tilde{Y}_{i,l}^{\pm},\tilde{A}_{i,l}^{\pm},t^{\pm}$.
As the defining relations of $\mathcal{H}_t$ involve only the $c_l$ and $\pi_+(c_l)=\pi_-(c_l)=0$, the algebra endomorphisms $\pi_+,\pi_-$ of $\mathcal{H}_t$ are well-defined. So we can define\[zqtplus\] $\mathcal{H}_t^+,\mathcal{H}_t^-,{\mathcal{Y}}_t^+,{\mathcal{Y}}_t^-$\[tytplus\] in the same way as in section \[defipip\] and $::$ a ${\ensuremath{\mathbb{C}}}$-linear endomorphism of $\mathcal{H}_t$ as in section \[ptpt\]. The ${\mathbb{Z}}[t^{\pm}]$-subalgebra ${\mathcal{Y}}_t\subset \mathcal{H}_t$ verifies $:{\mathcal{Y}}_t:\subset{\mathcal{Y}}_t$ (proof of lemma \[rel\]). We have ${\mathcal{Y}}_t^+\simeq{\mathcal{Y}}$.
We say that $m\in{\mathcal{Y}}_t$ (resp. $m\in{\mathcal{Y}}$) is a ${\mathcal{Y}}_t$-monomial (resp. a ${\mathcal{Y}}$-monomial) if it is a product of the generators $\tilde{Y}_{i,m}^{\pm},t^{\pm}$ (resp. $Y_{i,m}^{\pm}$).
### Structure of ${\mathcal{Y}}_t$ {#dessusdeux}
The following theorem gives the structure of ${\mathcal{Y}}_t$:
\[dessus\] The algebra ${\mathcal{Y}}_t$ is defined by generators $\tilde{Y}_{i,l}^{\pm}$ $(i\in I,l\in{\mathbb{Z}})$, central elements $t^{\pm}$ and relations ($i,j\in I, k,l\in{\mathbb{Z}}$): $$\tilde{Y}_{i,l}\tilde{Y}_{j,k}=t^{\gamma(i,l,j,k)}\tilde{Y}_{j,k}\tilde{Y}_{i,l}$$ where $\gamma: (I\times{\mathbb{Z}})^2\rightarrow{\mathbb{Z}}$ is given by (recall the maps $\pi_r$ of section \[defipire\])\[gamma\]: $$\gamma(i,l,j,k)=\underset{r\in{\mathbb{Z}}}{\sum}\pi_r(\tilde{C}_{j,i}(z))(-\delta_{l-k,-r_j-r}-\delta_{l-k,r-r_j}+\delta_{l-k,r_j-r}+\delta_{l-k,r_j+r})$$
[[*Proof:*]{}]{}As the image of $t_R$ in ${\mathcal{Y}}_t$ is $t^{\pi_o(R)}$, we can deduce the relations from lemma \[relu\]. For example formula \[yy\] (p. ) gives: $$\tilde{Y}_{i,l}\tilde{Y}_{j,k}\tilde{Y}_{i,l}^{-1}\tilde{Y}_{j,k}^{-1}=t^{\pi_0((\tilde{C}_{j,i}(z)(z^{r_j}-z^{-r_j})(-z^{(l-k)}+z^{(k-l)}))}$$ where: $$\pi_0(\tilde{C}_{j,i}(z)(z^{r_j}-z^{-r_j})(-z^{(l-k)}+z^{(k-l)}))$$ $$=\underset{r\in{\mathbb{Z}}}{\sum}\pi_r(\tilde{C}_{j,i}(z))(\delta_{r_j+r+k-l,0}+\delta_{-r_j+r+l-k,0}-\delta_{r_j+r+l-k,0}-\delta_{-r_j+r+k-l,0})=\gamma(i,l,j,k)$$ It follows from lemma \[gen\] that ${\mathcal{Y}}_t$ is generated by the $\tilde{Y}_{i,l}^{\pm},t^{\pm}$.
It follows from lemma \[indgene\] that the $t_R\in{\mathcal{Y}}_u$ ($R\in\mathfrak{U}$) are ${\mathbb{Z}}$-linearly independent. So the ${\mathbb{Z}}$-algebra ${\mathbb{Z}}[t_R]_{R\in\mathfrak{U}}$ is defined by generators $(t_R)_{R\in\mathfrak{U}}$ and relations $t_{R+R'}=t_Rt_{R'}$ for $R,R'\in\mathfrak{U}$. In particular the image of ${\mathbb{Z}}[t_R]_{R\in\mathfrak{U}}$ in ${\mathcal{Y}}_t$ is ${\mathbb{Z}}[t^{\pm}]$.
Let $A$ be the classes of ${\mathcal{Y}}_t$-monomials modulo $t^{{\mathbb{Z}}}$. So we have: $$\underset{m\in A}{\sum}{\mathbb{Z}}[t^{\pm}].m={\mathcal{Y}}_t$$ We prove the sum is direct: suppose we have a linear combination $\underset{m\in A}{\sum}\lambda_m(t)m=0$ where $\lambda_m(t)\in{\mathbb{Z}}[t^{\pm}]$. We saw in proposition \[circ\] that ${\mathcal{Y}}\simeq {\mathbb{Z}}[{Y}_{i,l}^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$. So $\lambda_m(1)=0$ and $\lambda_m(t)=(t-1)\lambda_m^{(1)}(t)$ where $\lambda_m^{(1)}(t)\in{\mathbb{Z}}[t^{\pm}]$. In particular $\underset{m\in A}{\sum}\lambda_m(t)^{(1)}(t)m=0$ and we get by induction $\lambda_m(t)\in (t-1)^r{\mathbb{Z}}[t^{\pm}]$ for all $r\geq 0$. This is possible if and only if all $\lambda_m(t)=0$.
In the same way using the last assertion of proposition \[circ\], we have:
\[yenga\] The sub ${\mathbb{Z}}[t^{\pm}]$-algebra of ${\mathcal{Y}}_t$ generated by the $\tilde{A}_{i,l}^{-1}$ is defined by generators $\tilde{A}_{i,l}^{-1},t^{\pm}$ $(i\in I,l\in{\mathbb{Z}})$ and relations\[alpha\]: $$\tilde{A}_{i,l}^{-1}\tilde{A}_{j,k}^{-1}=t^{\alpha(i,l,j,k)}\tilde{A}_{j,k}^{-1}\tilde{A}_{i,l}^{-1}$$ where $\alpha: (I\times{\mathbb{Z}})^2\rightarrow{\mathbb{Z}}$ is given by: $$\alpha(i,l,i,k)=2(-\delta_{l-k,2r_i}+\delta_{l-k,-2r_i})$$ $$\alpha(i,l,j,k)=2\underset{r=C_{i,j}+1,C_{i,j}+3,...,-C_{i,j}-1}{\sum}(-\delta_{l-k,-r_i+r}+\delta_{l-k,r_i+r})\text{ (if $i\neq j$)}$$
Moreover we have the following relations in ${\mathcal{Y}}_t$: $$\tilde{A}_{i,l}\tilde{Y}_{j,k}=t^{\beta(i,l,j,k)}\tilde{Y}_{j,k}\tilde{A}_{i,l}$$ where $\beta: (I\times{\mathbb{Z}})^2\rightarrow{\mathbb{Z}}$ is given by\[beta\]: $$\beta(i,l,j,k)=2\delta_{i,j}(-\delta_{l-k,r_i}+\delta_{l-k,-r_i})$$
Notations and properties related to monomials
---------------------------------------------
In this section we study some technical properties of the ${\mathcal{Y}}$-monomials and the ${\mathcal{Y}}_t$-monomials which will be used in the following.
### Basis
Denote by $A$\[a\] the set of ${\mathcal{Y}}$-monomials. It is a ${\mathbb{Z}}$-basis of ${\mathcal{Y}}$ (proposition \[circ\]). Let us define an analog ${\mathbb{Z}}[t^{\pm}]$-basis of ${\mathcal{Y}}_t$: denote $A'$\[ap\] the set of ${\mathcal{Y}}_t$-monomials of the form $m=:m:$. It follows from theorem \[dessus\] that: $${\mathcal{Y}}_t=\underset{m\in A'}{\bigoplus}{\mathbb{Z}}[t^{\pm}]m$$ The map $\pi:A'\rightarrow A$ defined by $\pi(m)=\pi_+(m)$\[pi\] is a bijection. In the following we identify $A$ and $A'$. In particular we have an embedding ${\mathcal{Y}}\subset{\mathcal{Y}}_t$ and an isomorphism of ${\mathbb{Z}}[t^{\pm}]$-modules ${\mathcal{Y}}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]\simeq{\mathcal{Y}}_t$. Note that it depends on the choice of the ${\mathbb{Z}}[t^{\pm}]$-basis of ${\mathcal{Y}}_t$.
We say that $\chi_1\in{\mathcal{Y}}_t$ has the same monomials as $\chi_2\in {\mathcal{Y}}$ if in the decompositions $\chi_1=\underset{m\in A}{\sum}\lambda_m(t)m$, $\chi_2=\underset{m\in A}{\sum}\mu_mm$ we have $\lambda_m(t)=0\Leftrightarrow\mu_m=0$.
### The notation $u_{i,l}$ {#notuil}
For $m$ a ${\mathcal{Y}}$-monomial we set $u_{i,l}(m)\in{\mathbb{Z}}$\[uil\] such that $m=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}{Y}_{i,l}^{u_{i,l}(m)}$ and $u_i(m)=\underset{l\in{\mathbb{Z}}}{\sum}u_{i,l}(m)$\[ui\]. For $m$ a ${\mathcal{Y}}_t$-monomial, we set $u_{i,l}(m)=u_{i,l}(\pi_+(m))$ and $u_i(m)=u_i(\pi_+(m))$. Note that $u_{i,l}$ is invariant by multiplication by $t$ and compatible with the identification of $A$ and $A'$.
Note that section \[dessusdeux\] implies that for $i\in I, l\in {\mathbb{Z}}$ and $m$ a ${\mathcal{Y}}_t$-monomial we have: $$\tilde{A}_{i,l}m=t^{-2u_{i,l-r_i}(m)+2u_{i,l+r_i}(m)}m\tilde{A}_{i,l}$$
Denote by $B_i\subset A$ the set of $i$-dominant ${\mathcal{Y}}$-monomials, that is to say $m\in B_i$\[bi\] if $\forall l\in{\mathbb{Z}}$, $u_{i,l}(m)\geq 0$. For $J\subset I$ denote $B_J=\underset{i\in J}{\bigcap}B_i$\[bj\] the set of $J$-dominant ${\mathcal{Y}}$-monomials. In particular, $B=B_I$ is the set of dominant ${\mathcal{Y}}$-monomials.\[b\]
We recall we can define a partial ordering on $A$ by putting $m\leq m'$ if there is a ${\mathcal{Y}}$-monomial $M$ which is a product of $A_{i,l}^{\pm}$ ($i\in I,l\in{\mathbb{Z}}$) such that $m=Mm'$ (see for example [@Her01]). A maximal (resp. lowest, higher...) weight ${\mathcal{Y}}$-monomial is a maximal (resp. minimal, higher...) element of $A$ for this ordering. We deduce from $\pi_+$ a partial ordering on the ${\mathcal{Y}}_t$-monomials.
Following [@Fre2], a ${\mathcal{Y}}$-monomial $m$ is said to be right negative if the factors $Y_{j,l}$ appearing in $m$, for which $l$ is maximal, have negative powers. A product of right negative ${\mathcal{Y}}$-monomials is right negative. It follows from lemma \[gen\] that the $A_{i,l}^{-1}$ are right negative. A ${\mathcal{Y}}_t$-monomial is said to be right negative if $\pi_+(m)$ is right negative.
### Some technical properties
\[genea\] Let $(i_1,l_1),...,(i_K,l_K)$ be in $(I\times{\mathbb{Z}})^K$. For $U\geq 0$, the set of the $m=\underset{k=1...K}{\prod}A_{i_k,l_k}^{-v_{i_k,l_k}(m)}$ ($v_{i_k,l_k}(m)\geq 0$) such that $\underset{i\in I,k\in{\mathbb{Z}}}{\text{min}}u_{i,k}(m)\geq -U$ is finite.
[[*Proof:*]{}]{}Suppose it is not the case: let be $(m_p)_{p\geq 0}$ such that $\underset{i\in I,k\in{\mathbb{Z}}}{\text{min}}u_{i,k}(m_p)\geq -U$ but\
$\underset{k=1...K}{\sum}v_{i_k,l_k}(m_p)\underset{p\rightarrow\infty}{\rightarrow}+\infty$. So there is at least one $k$ such that $v_{i_k,l_k}(m_p)\underset{p\rightarrow\infty}{\rightarrow}+\infty$. Denote by $\mathfrak{R}$ the set of such $k$. Among those $k\in\mathfrak{R}$, such that $l_k$ is maximal suppose that $r_{i_k}$ is maximal (recall the definition of $r_i$ in section \[recalu\]). In particular, we have $u_{i_k,l_k+r_{i_k}}(m_p)=-v_{i_k,l_k}(m_p)+f(p)$ where $f(p)$ depends only of the $v_{i_{k'},l_{k'}}(m_p)$, $k'\notin\mathfrak{R}$. In particular, $f(p)$ is bounded and $u_{i_k,l_k+r_{i_k}}(m_p)\underset{p\rightarrow\infty}{\rightarrow}-\infty$.
\[fini\] For $M\in B$, $K\geq 0$ the set of ${\mathcal{Y}}$-monomials $\{MA_{i_1,l_1}^{-1}...A_{i_R,l_R}^{-1}/R\geq 0,l_1,...,l_R\geq K\}\cap B$ is finite.
[[*Proof:*]{}]{}Let us write $M=Y_{i_1,l_1}...Y_{i_R,l_R}$ such that $l_1=\underset{r=1...R}{\text{min}}l_r$, $l_R=\underset{r=1...R}{\text{max}}l_r$ and consider $m$ in the set. It is of the form $m=MM'$ where $M'=\underset{i\in I,l\geq K}{\prod}A_{i,l}^{-v_{i,l}}$ ($v_{i,l}\geq 0$). Let $L=\text{max}\{l\in{\mathbb{Z}}/\exists i\in I, u_{i,l}(M')<0\}$. $M'$ is right negative so for all $i\in I$, $l>L\Rightarrow v_{i,l}=0$. But $m$ is dominant, so $L\leq l_R$. In particular $M'=\underset{i\in I,K\leq l\leq l_R}{\prod}A_{i,l}^{-v_{i,l}}$. It suffices to prove that the $v_{i,l}(m_r)$ are bounded under the condition $m$ dominant. This follows from lemma \[genea\].
Presentations of deformed algebras
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Our construction of ${\mathcal{Y}}_t$ using $\mathcal{H}_h$ (section \[studyyt\]) is a “concrete” presentation of the deformed structure. Let us look at another approach: in this section we define two bicharacters $\mathcal{N},\mathcal{N}_t$ related to basis of ${\mathcal{Y}}_t$. All the information of the multiplication of ${\mathcal{Y}}_t$ is contained is those bicharacters because we can construct a deformed $*$ multiplication on the “abstract” ${\mathbb{Z}}[t^{\pm}]$-module ${\mathcal{Y}}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]$ by putting for $m_1,m_2\in A$ ${\mathcal{Y}}$-monomials: $$m_1*m_2=t^{\mathcal{N}(m_1,m_2)-\mathcal{N}(m_2,m_1)}m_2*m_1$$ or $$m_1*m_2=t^{\mathcal{N}_t(m_1,m_2)-\mathcal{N}_t(m_2,m_1)}m_2*m_1$$ Those presentations appeared earlier in the literature [@Nab], [@Vas] for the simply laced case. In particular this section identifies our approach with those articles and gives an algebraic motivation of the deformed structures of [@Nab], [@Vas] related to the structure of ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$.
### The bicharacter $\mathcal{N}$
It follows from the proof of lemma \[rel\] that for $m$ a ${\mathcal{Y}}_t$-monomial, there is $N(m)\in{\mathbb{Z}}$\[n\] such that $m=t^{N(m)}:m:$. For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials we define $\mathcal{N}(m_1,m_2)=N(m_1m_2)-N(m_1)-N(m_2)$. We have $N(Y_{i,l})=N(A_{i,l})=0$. Note that for $\alpha,\beta\in{\mathbb{Z}}$ we have: $$N(t^{\alpha}m)=\alpha+N(m)\text{ , }\mathcal{N}(t^{\alpha}m_1,t^{\beta}m_2)=\mathcal{N}(m_1,m_2)$$ In particular the map $\mathcal{N}:A\times A\rightarrow{\mathbb{Z}}$ is well-defined and independent of the choice of a representant in $\pi_+^{-1}(A)$.
\[aaa\] For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials, we have in $\mathcal{H}_t$: $$\pi_-(m_1)\pi_+(m_2)=t^{\mathcal{N}(m_1,m_2)}\pi_+(m_2)\pi_-(m_1)$$
[[*Proof:*]{}]{}We have: $$m_1=t^{N(m_1)}\pi_+(m_1)\pi_-(m_1)\text{ , }m_2=t^{N(m_2)}\pi_+(m_2)\pi_-(m_2)$$ and so: $$m_1m_2=t^{N(m_1m_2)}\pi_+(m_1)\pi_+(m_2)\pi_-(m_1)\pi_-(m_2)=t^{N(m_1)+N(m_2)}\pi_+(m_1)\pi_-(m_1)\pi_+(m_2)\pi_-(m_2)$$
\[bibi\] The map $\mathcal{N}:A\times A\rightarrow {\mathbb{Z}}$ is a bicharacter, that is to say for $m_1,m_2,m_3\in A$, we have: $$\mathcal{N}(m_1m_2,m_3)=\mathcal{N}(m_1,m_3)+\mathcal{N}(m_2,m_3)\text{ and }\mathcal{N}(m_1,m_2m_3)=\mathcal{N}(m_1,m_2)+\mathcal{N}(m_1,m_3)$$ Moreover for $m_1,...,m_k$ ${\mathcal{Y}}_t$-monomials, we have: $$N(m_1m_2...m_k)=N(m_1)+N(m_2)+...+N(m_k)+\underset{1\leq i<j\leq k}{\sum}\mathcal{N}(m_i,m_j)$$
[[*Proof:*]{}]{}For the first point it follows from lemma \[aaa\]: $$\pi_-(m_1m_2)\pi_+(m_3)=t^{\mathcal{N}(m_1m_2,m_3)}\pi_+(m_3)\pi_-(m_1m_2)=t^{\mathcal{N}(m_2,m_3)}\pi_-(m_1)\pi_+(m_3)\pi_-(m_2)$$ $$=t^{\mathcal{N}(m_1,m_3)+\mathcal{N}(m_2,m_3)}\pi_+(m_3)\pi_-(m_1m_2)$$ For the second point we have first: $$N(m_1m_2)=N(m_1)+N(m_2)+\mathcal{N}(m_1,m_2)$$ and by induction: $$N(m_1m_2...m_k)=N(m_1)+N(m_2...m_k)+\mathcal{N}(m_1,m_2...m_k)$$ $$=N(m_1)+N(m_2)+...+N(m_k)+\underset{1<i<j\leq k}{\sum}\mathcal{N}(m_i,m_j)+\mathcal{N}(m_1,m_2)+...+\mathcal{N}(m_1,m_k)$$
### The bicharacter $\mathcal{N}_t$ {#tildeat}
For $m$ a ${\mathcal{Y}}_t$-monomial and $l\in{\mathbb{Z}}$, denote $\pi_l(m)=\underset{j\in I}{\prod}\tilde{Y}_{j,l}^{u_{j,l}(m)}$. It is well defined because for $i,j\in I$ and $l\in{\mathbb{Z}}$ we have $\tilde{Y}_{i,l}\tilde{Y}_{j,l}=\tilde{Y}_{j,l}\tilde{Y}_{i,l}$ (theorem \[dessus\]). Moreover for $m_1,m_2$ ${\mathcal{Y}}_t$-monomials we have $\pi_l(m_1m_2)=\pi_l(m_1)\pi_l(m_2)=\underset{i\in I}{\prod}\tilde{Y}_{i,l}^{u_{i,l}(m_1)+u_{i,l}(m_2)}$.
For $m$ a ${\mathcal{Y}}_t$-monomial denote $\tilde{m}={\underset{l\in{\mathbb{Z}}}{\overset{\rightarrow}{\prod}}}\pi_l(m)$\[tm\], and $A_t$\[tat\] the set of ${\mathcal{Y}}_t$-monomials of the form $\tilde{m}$. From theorem \[dessus\] there is a unique $N_t(m)\in{\mathbb{Z}}$\[nt\] such that $m=t^{N_t(m)}\tilde{m}$, and: $${\mathcal{Y}}_t=\underset{m\in A_t}{\bigoplus}{\mathbb{Z}}[t^{\pm}]m$$ For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials we define $\mathcal{N}_t(m_1,m_2)=N_t(m_1m_2)-N_t(m_1)-N_t(m_2)$. We have $N_t(Y_{i,l})=0$. Note that for $\alpha,\beta\in{\mathbb{Z}}$ we have: $$N_t(t^{\alpha}m)=\alpha+N_t(m)\text{ , }\mathcal{N}_t(t^{\alpha}m_1,t^{\beta}m_2)=\mathcal{N}_t(m_1,m_2)$$ In particular the map $\mathcal{N}_t:A\times A\rightarrow{\mathbb{Z}}$ is well-defined and independent of the choice of $A$.
For $m_1,m_2$ ${\mathcal{Y}}_t$-monomials, we have: $$\mathcal{N}_t(m_1,m_2)=\underset{l>l'}{\sum}(\mathcal{N}(\pi_l(m_1),\pi_{l'}(m_2))-\mathcal{N}(\pi_{l'}(m_2),\pi_{l}(m_1)))$$ In particular, $\mathcal{N}_t$ is a bicharacter and for $m_1,...,m_k$ ${\mathcal{Y}}_t$-monomials, we have: $$N_t(m_1m_2...m_k)=N_t(m_1)+N_t(m_2)+...+N_t(m_k)+\underset{1\leq i<j\leq k}{\sum}\mathcal{N}_t(m_i,m_j)$$
[[*Proof:*]{}]{}For the first point, it follows from the definition that $\tilde{(m_1m_2)}=t^{\mathcal{N}_t(m_1,m_2)}\tilde{m_1}\tilde{m_2}$. But: $$\tilde{(m_1m_2)}=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\pi_l(m_1)\pi_l(m_2)\text{ , }\tilde{m_1}\tilde{m_2}=(\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\pi_l(m_1))(\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\pi_l(m_2))$$ So we have to commute $\pi_l(m_1)$ and $\pi_{l'}(m_2)$ for $l>l'$. The last assertion is proved as in lemma \[bibi\].
### Presentation related to the basis $A_t$ and identification with [@Nab]
We suppose we are in the $ADE$-case.
Let be $m_1=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}}\tilde{A}_{i,l}^{-v_{i,l}}:,m_2=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}'}\tilde{A}_{i,l}^{-v_{i,l}'}:\in{\mathcal{Y}}_t$. We set $m_1^y=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}}:$ and $m_2^y=:\underset{i\in I,l\in{\mathbb{Z}}}{\prod}\tilde{Y}_{i,l}^{y_{i,l}'}:$.
\[form\] We have $\mathcal{N}_t(m_1,m_2)=\mathcal{N}_t(m_1^y,m_2^y)+2d(m_1,m_2)$, where:\[d\] $$d(m_1,m_2)=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}u_{i,l}'+y_{i,l+1}v_{i,l}'=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}u_{i,l+1}v_{i,l}'+v_{i,l+1}y_{i,l}'$$ where $u_{i,l}=y_{i,l}-v_{i,l-1}-v_{i,l+1}+\underset{j/C_{i,j}=-1}{\sum}v_{j,l}$ and $u_{i,l}'=y_{i,l}'-v_{i,l-1}'-v_{i,l+1}'+\underset{j/C_{i,j}=-1}{\sum}v_{j,l}'$.
[[*Proof:*]{}]{}
First notice that we have ($i\in I,l\in{\mathbb{Z}}$): $$\mathcal{N}_t(Y_{i,l},A_{i,l-1}^{-1})=2\text{ , }\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l})=2\text{ , }\mathcal{N}_t(A_{i,l+1}^{-1},A_{i,l-1}^{-1})=-2$$ $$\mathcal{N}_t(Y_{i,l+1}^{-1},Y_{i,l-1}^{-1})=-2\text{ , }\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l})=2$$ For example $\mathcal{N}_t(Y_{i,l},A_{i,l-1}^{-1})=\mathcal{N}(Y_{i,l},A_{i,l-1}^{-1})-\mathcal{N}(A_{i,l-1}^{-1},Y_{i,l})=2$ because $\tilde{Y}_{i,l}\tilde{A}_{i,l-1}^{-1}=t^2\tilde{A}_{i,l-1}^{-1}\tilde{Y}_{i,l}$.
We have $\mathcal{N}_t(m_1,m_2)=A+B+C+D$ where:
$A=\mathcal{N}_t(m_1^y,m_2^y)$
$B=\underset{i,j\in I,l,k\in{\mathbb{Z}}}{\sum}y_{i,l}v_{j,k}'\mathcal{N}_t(Y_{i,l},A_{j,k}^{-1})=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}y_{i,l}v_{i,l-1}'\mathcal{N}_t(Y_{i,l},A_{i,l-1}^{-1})=2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}y_{i,l}v_{i,l-1}'$
$C=\underset{i,j\in I,l,k\in{\mathbb{Z}}}{\sum}v_{i,l}y_{j,k}'\mathcal{N}_t(A_{i,l}^{-1},Y_{j,k})=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}y_{i,l}'\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l})=2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}y_{i,l}'$
$D=\underset{i,j\in I,l,k\in{\mathbb{Z}}}{\sum}v_{i,l}v_{j,k}'\mathcal{N}_t(A_{i,l}^{-1},A_{j,k}^{-1})
\\=\underset{i\in I,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{i,l-1}'\mathcal{N}_t(A_{i,l+1}^{-1},A_{i,l-1}^{-1})+v_{i,l}v_{i,l}'\mathcal{N}_t(Y_{i,l+1}^{-1},Y_{i,l-1}^{-1})+\underset{C_{j,i}=-1,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{j,l}'\mathcal{N}_t(A_{i,l+1}^{-1},Y_{i,l})
\\=-2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}(v_{i,l+1}v_{i,l-1}'+v_{i,l}v_{i,l'})+2\underset{C_{j,i}=-1,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{j,l}'$
In particular, we have: $$B+C+D=2\underset{i\in I,l\in{\mathbb{Z}}}{\sum}(y_{i,l}v_{i,l-1}'+v_{i,l+1}y_{i,l}'-v_{i,l+1}v_{i,l-1}'-v_{i,l}v_{i,l}')+2\underset{C_{j,i}=-1,l\in{\mathbb{Z}}}{\sum}v_{i,l+1}v_{j,l}'$$
The bicharacter $d$ was introduced for the $ADE$-case by Nakajima in [@Nab] motivated by geometry. It particular this proposition \[form\] gives a new motivation for this deformed structure.
### Presentation related to the basis $A$ and identification with [@Vas] {#varva}
For $m_1,m_2\in A$, we have: $$\mathcal{N}(m_1,m_2)=\underset{i,j\in I, l,k\in{\mathbb{Z}}}{\sum}u_{i,l}(m_1)u_{j,k}(m_2)((\tilde{C}_{j,i}(z))_{r_j+l-k}-(\tilde{C}_{j,i}(z))_{-r_j+l-k})$$
[[*Proof:*]{}]{}First we can compute in ${\mathcal{Y}}_u$: $$\tilde{Y}_{i,l}\tilde{Y}_{j,k}=\text{exp}(\underset{m>0}{\sum}h^{2m}[y_i[-m],y_j[m]]q^{m(k-l)}):\tilde{Y}_{i,l}\tilde{Y}_{j,k}:=t_{\tilde{C}_{j,i}(z)z^{k-l}(z^{-r_j}-z^{r_j})}:\tilde{Y}_{i,l}\tilde{Y}_{j,k}:$$ and as $N(\tilde{Y}_{i,l})=N(\tilde{Y}_{j,k})=0$ we have $\mathcal{N}(\tilde{Y}_{i,l},\tilde{Y}_{j,k})=(\tilde{C}_{j,i}(z))_{r_j+l-k}-(\tilde{C}_{j,i}(z))_{-r_j+l-k}$.
In $sl_2$-case we have $C(z)=z+z^{-1}$ and $\tilde{C}(z)=\frac{1}{z+z^{-1}}=\underset{r\geq 0}{\sum}(-1)^rz^{-2r-1}$. So: $$\tilde{Y}_l\tilde{Y}_k=t^s:\tilde{Y}_l\tilde{Y}_k:$$ where:
$s=0$ if $l-k=1+2r$, $r\in{\mathbb{Z}}$
$s=0$ if $l-k=2r$, $r>0$
$s=2(-1)^{r+1}$ if $l-k=2r$, $r<0$
$s=-1$ if $l=k$
It is analogous to the multiplication introduced for the $ADE$-case by Varagnolo-Vasserot in [@Vas]: we suppose we are in the $ADE$-case, denote $P=\underset{i\in I}{\bigoplus}{\mathbb{Z}}\omega_i$ (resp. $Q=\underset{i\in I}{\bigoplus}{\mathbb{Z}}\alpha_i$) the weight-lattice (resp. root-lattice) and:
$\bar{}:P\otimes{\mathbb{Z}}[z^{\pm}]\rightarrow P\otimes{\mathbb{Z}}[z^{\pm}]$ is defined by $\overline{\lambda \otimes P(z)}=\lambda\otimes P(z^{-1})$.
$(,):Q\otimes {\mathbb{Z}}((z^{-1}))\times P\otimes{\mathbb{Z}}((z^{-1}))\rightarrow {\mathbb{Z}}((z^{-1}))$ is the ${\mathbb{Z}}((z^{-1}))$-bilinear form defined by $(\alpha_i,\omega_j)=\delta_{i,j}$.
$\Omega^{-1}:P\otimes{\mathbb{Z}}[z^{\pm}]\rightarrow Q\otimes{\mathbb{Z}}((z^{-1}))$ is defined by $\Omega^{-1}(\omega_i)=\underset{k\in I}{\sum}\tilde{C}_{i,k}(z)\alpha_k$.
The map $\epsilon:P\otimes{\mathbb{Z}}[z^{\pm}]\times P\otimes{\mathbb{Z}}[z^{\pm}]\rightarrow {\mathbb{Z}}$ is defined by: $$\epsilon_{\lambda,\mu}=\pi_0((z^{-1}\Omega^{-1}(\bar{\lambda})|\mu))$$ The multiplication of [@Vas] is defined by: $$Y_{i,l}Y_{j,m}=t^{2\epsilon_{z^l\omega_i,z^m\omega_j}-2\epsilon_{z^m\omega_j,z^l\omega_i}}Y_{j,m}Y_{i,l}$$ So we can compute: $$\epsilon_{z^l\omega_i,z^m\omega_j}=\pi_0((z^{-1}\Omega^{-1}(z^{-l}\omega_i)|z^m\omega_j))=\pi_0(\underset{k\in I}{\sum}(z^{-1-l}\tilde{C}_{i,k}(z)\alpha_k|z^m\omega_j))$$ $$=\pi_0(z^{m-l-1}\tilde{C}_{i,j}(z))=(\tilde{C}_{i,j}(z))_{l+1-m}$$ If we set $\epsilon'_{\lambda,\mu}=\pi_0((z\Omega^{-1}(\bar{\lambda})|\mu))$ then we have $\epsilon'_{z^l\omega_i,z^m\omega_j}=(\tilde{C}_{i,j}(z))_{l-1-m}$ and: $$\epsilon_{z^l\omega_i,z^m\omega_j}-\epsilon'_{z^l\omega_i,z^m\omega_j}=\mathcal{N}({Y}_{i,l},{Y}_{j,m})$$
Deformed screening operators {#scr}
============================
Motivated by the screening currents of [@Freb] we give in this section a “concrete” approach to deformations of screening operators. In particular the $t$-analogues of screening operators defined in [@Her01] will appear as commutators in $\mathcal{H}_h$. Let us begin with some background about classic screening operators.
Reminder: classic screening operators ([@Fre],[@Fre2]) {#screclas}
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### Classic screening operators and symmetry property of $q$-characters
Recall the definition of\
$\pi_+(\tilde{A}_{i,l}^{\pm})=A_{i,l}^{\pm}\in {\mathcal{Y}}$ and of $u_{i,l}:A\rightarrow{\mathbb{Z}}$ in section \[defoal\].
The $i^{\text{th}}$-screening operator is the ${\mathbb{Z}}$-linear map defined by:\[si\] $$S_i:{\mathcal{Y}}\rightarrow{\mathcal{Y}}_i=\frac{\underset{l\in{\mathbb{Z}}}{\bigoplus}{\mathcal{Y}}.S_{i,l}}{\underset{l\in{\mathbb{Z}}}{\sum}{\mathcal{Y}}.(S_{i,l+2r_i}-A_{i,l+r_i}.S_{i,l})}$$\[yi\] $$\forall m\in A, S_i(m)=\underset{l\in{\mathbb{Z}}}{\sum}u_{i,l}(m)S_{i,l}$$
Note that the $i^{\text{th}}$-screening operator can also be defined as the derivation such that: $$S_i(1)=0\text{ , }\forall j\in I,l\in{\mathbb{Z}},S_i(Y_{j,l})=\delta_{i,j}Y_{i,l}.S_{i,l}$$
([**Frenkel, Reshetikhin, Mukhin**]{} [@Fre],[@Fre2])\[simme\] The image of $\chi_q:{\mathbb{Z}}[X_{i,l}]_{i\in I,l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}$ is: $$\text{Im}(\chi_q)=\underset{i\in I}{\bigcap}\text{Ker}(S_i)$$
It is analogous to the classical symmetry property of $\chi$: $\text{Im}(\chi)={\mathbb{Z}}[y_i^{\pm}]_{i\in I}^W$.
### Structure of the kernel of $S_i$ {#mi}
Let $\mathfrak{K}_i=\text{Ker}(S_i)$\[ki\]. It is a ${\mathbb{Z}}$-subalgebra of ${\mathcal{Y}}$.
\[deux\]([**Frenkel, Reshetikhin, Mukhin**]{} [@Fre],[@Fre2]) The ${\mathbb{Z}}$-subalgebra $\mathfrak{K}_i$ of ${\mathcal{Y}}$ is generated by the $Y_{i,l}(1+A_{i,l+r_i}^{-1}),Y_{j,l}^{\pm}$ ($j\neq i,l\in {\mathbb{Z}}$).
For $m\in B_i$, we denote: $$E_i(m)=m\underset{l\in{\mathbb{Z}}}{\prod}(1+{{A_{i,l+r_i}}}^{-1})^{u_{i,l}(m)}\in\mathfrak{K}_i$$\[eim\] In particular:
\[el\] The ${\mathbb{Z}}$-module $\mathfrak{K}_i$ is freely generated by the $E_i(m)$ ($m\in B_i$): $$\mathfrak{K}_i=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}E_i(m)\simeq {\mathbb{Z}}^{(B_i)}$$
### Examples in the $sl_2$-case {#exlsl}
We suppose in this section that we are in the $sl_2$-case. For $m\in B$, let $L(m)=\chi_q(V_m)$ be the $q$-character of the ${\mathcal{U}}_q(\hat{sl_2})$-irreducible representation of highest weight $m$. In particular $L(m)\in\mathfrak{K}$ and $\mathfrak{K}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}L(m)$.
In [@Fre] an explicit formula for $L(m)$ is given: a $\sigma\subset{\mathbb{Z}}$ is called a $2$-segment if $\sigma$ is of the form $\sigma=\{l,l+2,...,l+2k\}$. Two $2$-segment are said to be in special position if their union is a $2$-segment that properly contains each of them. All finite subset of ${\mathbb{Z}}$ with multiplicity $(l,u_l)_{l\in{\mathbb{Z}}}$ ($u_l\geq 0$) can be broken in a unique way into a union of $2$-segments which are not in pairwise special position.
For $m\in B$ we decompose $m=\underset{j}{\prod}\underset{l\in \sigma_j}{\prod}Y_l\in B$ where the $(\sigma_j)_j$ is the decomposition of the $(l,u_l(m))_{l\in{\mathbb{Z}}}$. We have: $$L(m)=\underset{j}{\prod}L(\underset{l\in \sigma_j}{\prod}Y_l)$$ So it suffices to give the formula for a $2$-segments: $$L(Y_lY_{l+2}Y_{l+4}...Y_{l+2k})=Y_lY_{l+2}Y_{l+4}...Y_{l+2k}+Y_lY_{l+2}...Y_{l+2(k-1)}Y_{l+2(k+1)}^{-1}$$ $$+Y_lY_{l+2}...Y_{l+2(k-2)}Y_{l+2k}^{-1}Y_{l+2(k+1)}^{-1}+...+Y_{l+2}^{-1}Y_{l+2}^{-1}...Y_{l+2(k+1)}^{-1}$$ We say that $m$ is irregular if there are $j_1\neq j_2$ such that $$\sigma_{j_1}\subset \sigma_{j_2}\text{ and }\sigma_{j_1}+2\subset\sigma_{j_2}$$
([**Frenkel, Reshetikhin**]{} [@Fre])\[dominl\] There is a dominant ${\mathcal{Y}}$-monomial other than $m$ in $L(m)$ if and only if $m$ is irregular.
### Complements: another basis of $\mathfrak{K}_i$ {#compl}
Let us go back to the general case. Let ${\mathcal{Y}}_{sl_2}={\mathbb{Z}}[Y_l^{\pm}]_{l\in{\mathbb{Z}}}$ the ring ${\mathcal{Y}}$ for the $sl_2$-case. Let $i$ be in $I$ and for $0\leq k\leq r_i-1$, let $\omega_k: A\rightarrow {\mathcal{Y}}_{sl_2}$ be the map defined by: $$\omega_k(m)=\underset{l\in{\mathbb{Z}}}{\prod}Y_l^{u_{i,k+lr_i}(m)}$$ and $\nu_k:{\mathbb{Z}}[(Y_{l-1}Y_{l+1})^{-1}]_{l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}$ be the ring homomorphism such that $\nu_k((Y_{l-1}Y_{l+1})^{-1})=A_{i,k+lr_i}^{-1}$.
For $m\in B_i$, $\omega_k(m)$ is dominant in ${\mathcal{Y}}_{sl_2}$ and so we can define $L(\omega_k(m))$ (see section \[exlsl\]). We have $L(\omega_k(m))\omega_k(m)^{-1}\in{\mathbb{Z}}[(Y_{l-1}Y_{l+1})^{-1}]_{l\in{\mathbb{Z}}}$. We introduce: $$L_i(m)=m\underset{0\leq k\leq r_i-1}{\prod}\nu_k(L(\omega_k(m))\omega_k(m)^{-1})\in \mathfrak{K}_i$$\[lim\] In analogy with the corollary \[el\] the ${\mathbb{Z}}$-module $\mathfrak{K}_i$ is freely generated by the $L_i(m)$ ($m\in B_i$): $$\mathfrak{K}_i=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}L_i(m)\simeq {\mathbb{Z}}^{(B_i)}$$
Screening currents
------------------
Following [@Freb], for $i\in I,l\in{\mathbb{Z}}$, introduce $\tilde{S}_{i,l}\in \mathcal{H}_h$:\[tsil\] $$\tilde{S}_{i,l}=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q_i^{m}-q_i^{-m}}q^{lm})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q_i^{-m}-q_i^{m}}q^{-lm})$$
\[currents\] We have the following relations in $\mathcal{H}_h$: $$\tilde{A}_{i,l}\tilde{S}_{i,l-r_i}=t_{-z^{-2r_i}-1}\tilde{S}_{i,l+r_i}$$ $$\tilde{S}_{i,l}\tilde{A}_{j,k}=t_{C_{i,j}(z)(z^{(k-l)}+z^{(l-k)})}\tilde{A}_{j,k}\tilde{S}_{i,l}$$ $$\tilde{S}_{i,l}\tilde{Y}_{j,k}=t_{\delta_{i,j}(z^{(k-l)}+z^{(l-k)})}\tilde{Y}_{j,k}\tilde{S}_{i,l}$$
[[*Proof:*]{}]{}
As for lemma \[relu\] we compute in $\mathcal{H}_h$:
$\tilde{A}_{i,l}\tilde{S}_{i,l-r_i}
\\=\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{lm})(\text{exp}(\underset{m>0}{\sum}h^ma_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{m(l-r_i)})
\\\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{m(r_i-l)})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}\frac{-q^{2mr_i}+q^{-2mr_i}}{q^{mr_i}-q^{-mr_i}}q^{-mr_i}c_m)\text{exp}(\underset{m>0}{\sum}h^ma_{i}[m]q^{lm}+h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{m(l-r_i)})
\\\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{m(r_i-l)}+h^ma_i[-m]q^{-lm})
\\=t_{-z^{-2r_i}-1}\text{exp}(\underset{m>0}{\sum}h^m a_i[m](1+\frac{q^{-mr_i}}{q^{mr_i}-q^{-mr_i}})q^{lm})\text{exp}(\underset{m>0}{\sum}h^m a_i[m](1+\frac{q^{mr_i}}{q^{-mr_i}-q^{mr_i}})q^{-lm})
\\=t_{-z^{-2r_i}-1}\tilde{S}_{i,l+r_i}$
$\tilde{S}_{i,l}\tilde{A}_{j,k}
\\=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm})(\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^ma_{j}[m]q^{km}))
\\\text{exp}(\underset{m>0}{\sum}h^ma_j[-m]q^{-km})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}\frac{(q^{-mr_i}-q^{mr_i})C_{i,j}(q^m)}{q^{-mr_i}-q^{mr_i}}q^{m(k-l)}c_m)\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm})
\\\text{exp}(\underset{m>0}{\sum}h^ma_{j}[m]q^{km})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^ma_j[-m]q^{-km})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}C_{i,j}(q^m)(q^{m(k-l)}+q^{(l-k)m})c_m)\tilde{A}_{j,k}\tilde{S}_{i,l}$
Finally:
$\tilde{S}_{i,l}\tilde{Y}_{j,k}
\\=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm})(\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^my_{j}[m]q^{km}))
\\\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-km})
\\=\text{exp}(\underset{m>0}{\sum}h^{2m}\delta_{i,j}\frac{(q^{-mr_i}-q^{mr_i})}{q^{-mr_i}-q^{mr_i}}q^{m(k-l)}c_m)\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q^{mr_i}-q^{-mr_i}}q^{lm})
\\\text{exp}(\underset{m>0}{\sum}h^my_{j}[m]q^{km})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q^{-mr_i}-q^{mr_i}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^my_j[-m]q^{-km})
\\=\text{exp}(\underset{m>0}{\sum}\delta_{i,j}h^{2m}(q^{m(k-l)}+q^{(l-k)m})c_m)\tilde{Y}_{j,k}\tilde{S}_{i,l}$
Deformed bimodules
------------------
In this section we define and study a $t$-analogue ${\mathcal{Y}}_{i,t}$ of the module ${\mathcal{Y}}_i$.
For $i\in I$, let ${\mathcal{Y}}_{i,u}$\[yiu\] be the ${\mathcal{Y}}_u$ sub left-module of $\mathcal{H}_h$ generated by the $\tilde{S}_{i,l}$ ($l\in {\mathbb{Z}}$). It follows from lemma \[currents\] that $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$ generate ${\mathcal{Y}}_{i,u}$ and that it is also a subbimodule of $\mathcal{H}_h$. Denote by $\tilde{S}_{i,l}\in \mathcal{H}_t$\[tsit\] the image of $\tilde{S}_{i,l}\in \mathcal{H}_h$ in $\mathcal{H}_t$.
${\mathcal{Y}}_{i,t}$\[yit\] is the sub left-module of $\mathcal{H}_t$ generated by the $\tilde{S}_{i,l}$ ($l\in{\mathbb{Z}}$).
In particular it is to say the image of ${\mathcal{Y}}_{i,u}$ in $\mathcal{H}_t$. It follows from lemma \[currents\] that for $l\in{\mathbb{Z}}$, we have in ${\mathcal{Y}}_{i,t}$: $$\tilde{A}_{i,l}\tilde{S}_{i,l-r_i}=t^{-1}\tilde{S}_{i,l+r_i}$$ It particular ${\mathcal{Y}}_{i,t}$ is generated by the $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$.
It follows from lemma \[currents\] that for $l\in{\mathbb{Z}}$, we have: $$\tilde{S}_{i,l}.\tilde{Y}_{j,k}=t^{2\delta_{i,j}\delta_{l,k}}\tilde{Y}_{j,k}.\tilde{S}_{i,l}\text{ , }\tilde{S}_{i,l}.t=t.\tilde{S}_{i,l}$$ In particular ${\mathcal{Y}}_{i,t}$ a subbimodule of $\mathcal{H}_t$. Moreover: $$\tilde{S}_{i,l}.\tilde{A}_{i,k}=t^{2\delta_{l-k,r_i}+2\delta_{l-k,-r_i}}\tilde{A}_{i,k}.\tilde{S}_{i,l}$$ $$\tilde{S}_{i,l}.\tilde{A}_{j,k}=t^{-2\underset{r=C_{i,j}+1,C_{i,j}+3,...,-C_{i,j}-1}{\sum}\delta_{l-k,r}}\tilde{A}_{j,k}.\tilde{S}_{i,l}\text{ (if $i\neq j$)}$$
\[tcur\] The ${\mathcal{Y}}_t$ left module ${\mathcal{Y}}_{i,t}$ is freely generated by $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$: $${\mathcal{Y}}_{i,t}=\underset{-r_i\leq l <r_i}{\bigoplus}{\mathcal{Y}}_t\tilde{S}_{i,l}\simeq {\mathcal{Y}}_t^{2r_i}$$
[[*Proof:*]{}]{}We saw that $(\tilde{S}_{i,l})_{-r_i\leq l <r_i}$ generate ${\mathcal{Y}}_{i,t}$. We prove they are ${\mathcal{Y}}_t$-linearly independent:\
for $(R_1,...,R_n)\in\mathfrak{U}^n$, introduce: $$Y_{R_1,...,R_n}=\text{exp}(\underset{m>0, j\in I}{\sum}h^my_j[m]R_j(q^m))\in \mathcal{H}_t^+$$ It follows from lemma \[indgene\] that the $(Y_{R})_{R\in\mathfrak{U}^n}$ are ${\mathbb{Z}}$-linearly independent. Note that we have $\pi_+({\mathcal{Y}}_{i,t})\subset \underset{R\in \mathfrak{U}^n}{\bigoplus}{\mathbb{Z}}Y_R$ and that ${\mathcal{Y}}=\underset{R\in{\mathbb{Z}}[z^{\pm}]^n}{\bigoplus}{\mathbb{Z}}Y_R$. Suppose we have a linear combination ($\lambda_r\in{\mathcal{Y}}_t$): $$\lambda_{-r_i}\tilde{S}_{i,-r_i}+...+\lambda_{r_i-1}\tilde{S}_{i,r_i-1}=0$$ Introduce $\mu_{k,R}\in{\mathbb{Z}}$ such that: $$\pi_+(\lambda_k)=\underset{R\in{\mathbb{Z}}[z^{\pm}]^n}{\sum}\mu_{k,R}Y_R$$ and $R_{i,k}=(R_{i,k}^{1}(z),...,R_{i,k}^{n}(z))\in\mathfrak{U}^n$ such that $\pi_+(\tilde{S}_{i,k})=Y_{R_{i,k}}$. If we apply $\pi_+$ to the linear combination, we get: $$\underset{R\in{\mathbb{Z}}[z^{\pm}]^n,-r_i\leq k\leq r_i-1}{\sum}\mu_{k,R}Y_RY_{R_{i,k}}=0$$ and we have for all $R'\in \mathfrak{U}$: $$\underset{-r_i\leq k\leq r_i-1/R'-R_{i,k}\in{\mathbb{Z}}[z^{\pm}]^n}{\sum}\mu_{k,R'-R_{i,k}}=0$$ Suppose we have $-r_i\leq k_1\neq k_2\leq r_i -1$ such that $R'-R_{i,k_1},R'-R_{i,k_2}\in {\mathbb{Z}}[z^{\pm}]^n$. So $R_{i,k_1}-R_{i,k_2}\in{\mathbb{Z}}[z^{\pm}]^n$. But $a_i[m]=\underset{j\in I}{\sum}C_{j,i}(q^m)y_j[m]$, so for $j\in I$: $$C_{j,i}(z)\frac{z^{k_1}-z^{k_2}}{z^{r_i}-z^{-r_i}}=(R_{i,k_1}^j(z)-R_{i,k_2}^j(z))\in {\mathbb{Z}}[z^{\pm}]$$ In particular for $j=i$ we have $C_{i,i}(z)\frac{z^{k_1}-z^{k_2}}{z^{r_i}-z^{-r_i}}=\frac{(z^{r_i}+z^{-r_i})(z^{k_1}-z^{k_2})}{z^{r_i}-z^{-r_i}}\in {\mathbb{Z}}[z^{\pm}]$. This is impossible because $|k_1-k_2|<2r_i$. So we have only one term in the sum and all $\mu_{k,R}=0$. So $\pi_+(\lambda_k)=0$, and $\lambda_k\in (t-1){\mathcal{Y}}_t$. We have by induction for all $m>0$, $\lambda_k\in (t-1)^m{\mathcal{Y}}_t$. It is possible if and only if $\lambda_k=0$.
Denote by ${\mathcal{Y}}_i$ the ${\mathcal{Y}}$-bimodule $\pi_+({\mathcal{Y}}_{i,t})$. It is consistent with the notations of section \[screclas\].
$t$-analogues of screening operators
------------------------------------
We introduced $t$-analogues of screening operators in [@Her01]. The picture of the last section enables us to define them from a new point of view.
For $m$ a ${\mathcal{Y}}_t$-monomial, we have: $$[\tilde{S}_{i,l},m]=\tilde{S}_{i,l}m-m\tilde{S}_{i,l}=(t^{2u_{i,l}(m)}-1)m\tilde{S}_{i,l}=t^{u_{i,l}(m)}(t-t^{-1})[u_{i,l}(m)]_tm\tilde{S}_{i,l}$$ So for $\lambda\in {\mathcal{Y}}_t$ we have $[\tilde{S}_{i,l},\lambda]\in (t^2-1){\mathcal{Y}}_{i,t}$, and $[\tilde{S}_{i,l},\lambda]\neq 0$ only for a finite number of $l\in{\mathbb{Z}}$. So we can define:
The $i^{th}$ $t$-screening operator is the map $S_{i,t}:{\mathcal{Y}}_t\rightarrow {\mathcal{Y}}_{i,t}$ such that ($\lambda\in{\mathcal{Y}}_t$): $$S_{i,t}(\lambda)=\frac{1}{t^2-1}\underset{l\in{\mathbb{Z}}}{\sum}[\tilde{S}_{i,l},\lambda]\in {\mathcal{Y}}_{i,t}$$
In particular, $S_{i,t}$ is ${\mathbb{Z}}[t^{\pm}]$-linear and a derivation. It is our map of [@Her01].
For $m$ a ${\mathcal{Y}}_t$-monomial, we have $\pi_+(S_{i,t}(m))=\pi_+(t^{u_{i,l}(m)-1}[u_{i,l}(m)]_t)\pi_+(m\tilde{S}_{i,l})=u_{i,l}(m)\pi_+(m\tilde{S}_{i,l})$ and the following commutative diagram: $$\begin{array}{rcccl}
{\mathcal{Y}}_t&\stackrel{S_{i,t}}{\longrightarrow}&{\mathcal{Y}}_{i,t}\\
\pi_+\downarrow &&\downarrow&\pi_+\\
{\mathcal{Y}}&\stackrel{S_i}{\longrightarrow}&{\mathcal{Y}}_i\end{array}$$
Kernel of deformed screening operators {#kernelun}
--------------------------------------
### Structure of the kernel {#defitl}
We proved in [@Her01] a $t$-analogue of theorem \[deux\]:
\[her\] ([@Her01]) The kernel of the $i^{th}$ $t$-screening operator $S_{i,t}$ is the ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t$ generated by the $\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}),\tilde{Y}_{j,l}^{\pm}$ ($j\neq i,l\in {\mathbb{Z}}$).
[[*Proof:*]{}]{}For the first inclusion we compute: $$S_{i,t}(\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}))=\tilde{Y}_{i,l}\tilde{S}_{i,l}+t\tilde{Y}_{i,l}\tilde{A}_{i,l+r_i}^{-1}(-t^{-2})\tilde{S}_{i,l+2r_i}=\tilde{Y}_{i,l}(\tilde{S}_{i,l}-t^{-1}\tilde{A}_{i,l+r_i}^{-1}\tilde{S}_{i,l+2r_i})=0$$ For the other inclusion we refer to [@Her01].
Let $\mathfrak{K}_{i,t}=\text{Ker}(S_{i,t})$\[kit\]. It is a ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t$. In particular we have $\pi_+(\mathfrak{K}_{i,t})=\mathfrak{K}_i$ (consequence of theorem \[deux\] and \[her\]).
For $m\in B_i$ introduce: (recall that $\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}U_l$ means $...U_{-1}U_0U_1U_2...$): $$E_{i,t}(m)=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}((\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}))^{u_{i,l}(m)}\underset{j\neq i}{\prod}\tilde{Y}_{j,l}^{u_{j,l}(m)})$$\[teitm\] It is well defined because it follows from theorem \[dessus\] that for $j\neq i, l\in{\mathbb{Z}}$, $(\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1}))$ and $\tilde{Y}_{j,l}$ commute. For $m\in B_i$, the formula shows that the ${\mathcal{Y}}_t$-monomials of $E_{i,t}(m)$ are the ${\mathcal{Y}}$-monomials of $E_i(m)$ (with identification by $\pi_+$). Such elements were used in [@Nab] for the $ADE$ case.
The theorem \[her\] allows us to describe $\mathfrak{K}_{i,t}$:
\[hers\] For all $m\in B_i$, we have $E_{i,t}(m)\in\mathfrak{K}_{i,t}$. Moreover: $$\mathfrak{K}_{i,t}=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}[t^{\pm}]E_{i,t}(m)\simeq {\mathbb{Z}}[t^{\pm}]^{(B_i)}$$
[[*Proof:*]{}]{}First $E_{i,t}(m)\in\mathfrak{K}_{i,t}$ as product of elements of $\mathfrak{K}_{i,t}$. We show easily that the $E_{i,t}(m)$ are ${\mathbb{Z}}[t^{\pm}]$-linearly independent by looking at a maximal ${\mathcal{Y}}_t$-monomial in a linear combination.
Let us prove that the $E_{i,t}(m)$ are ${\mathbb{Z}}[t^{\pm}]$-generators of $\mathfrak{K}_{i,t}$: for a product $\chi$ of the algebra-generators of theorem \[her\], let us look at the highest weight ${\mathcal{Y}}_t$-monomial $m$. Then $E_{i,t}(m)$ is this product up to the order in the multiplication. But for $p=1$ or $p\geq 3$, $Y_{i,l}Y_{i,l+pr_i}$ is the unique dominant ${\mathcal{Y}}$-monomial of $E_i(Y_{i,l})E_i(Y_{i,l+pr_i})$, so: $$\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})\tilde{Y}_{i,l+pr_i}(1+t\tilde{A}_{i,l+pr_i+r_i}^{-1})\in t^{{\mathbb{Z}}}\tilde{Y}_{i,l+pr_i}(1+t\tilde{A}_{i,l+pr_i+r_i}^{-1})\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})$$ And for $p=2$: $$\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})\tilde{Y}_{i,l+2r_i}(1+t\tilde{A}_{i,l+3r_i}^{-1})-\tilde{Y}_{i,l+2r_i}(1+t\tilde{A}_{i,l+3r_i}^{-1})\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})$$ $$\in {\mathbb{Z}}[t^{\pm}]+t^{{\mathbb{Z}}}\tilde{Y}_{i,l}(1+t\tilde{A}_{i,l+r_i}^{-1})\tilde{Y}_{i,l+2r_i}(1+t\tilde{A}_{i,l+3r_i}^{-1})$$
### Elements of $\mathfrak{K}_{i,t}$ with a unique $i$-dominant ${\mathcal{Y}}_t$-monomial
\[defifprem\] For $m\in B_i$, there is a unique $F_{i,t}(m)\in\mathfrak{K}_{i,t}$\[tfitm\] such that $m$ is the unique $i$-dominant ${\mathcal{Y}}_t$-monomial of $F_{i,t}(m)$. Moreover : $$\mathfrak{K}_{i,t}=\underset{m\in B_i}{\bigoplus}F_{i,t}(m)$$
[[*Proof:*]{}]{}It follows from corollary \[hers\] that an element of $\mathfrak{K}_{i,t}$ has at least one $i$-dominant ${\mathcal{Y}}_t$-monomial. In particular we have the uniqueness of $F_{i,t}(m)$.
For the existence, let us look at the $sl_2$-case. Let $m$ be in $B$. It follows from the lemma \[fini\] that $\{MA_{i_1,l_1}^{-1}...A_{i_R,l_R}^{-1}/R\geq 0,l_1,...,l_R\geq l(M)\}\cap B$ is finite (where $l(M)=\text{min}\{l\in{\mathbb{Z}}/\exists i\in I,u_{i,l}(M)\neq 0\}$). We define on this set a total ordering compatible with the partial ordering: $m_L=m>m_{L-1}>...>m_1$. Let us prove by induction on $l$ the existence of $F_t(m_l)$. The unique dominant ${\mathcal{Y}}_t$-monomial of $E_t(m_1)$ is $m_1$ so $F_t(m_1)=E_t(m_1)$. In general let $\lambda_1(t),...,\lambda_{l-1}(t)\in{\mathbb{Z}}[t^{\pm}]$ be the coefficient of the dominant ${\mathcal{Y}}_t$-monomials $m_1,...,m_{l-1}$ in $E_t(m_l)$. We put: $$F_t(m_l)=E_t(m_l)-\underset{r=1...l-1}{\sum}\lambda_r(t)F_t(m_r)$$ Notice that this construction gives $F_t(m)\in m{\mathbb{Z}}[\tilde{A}_l^{-1},t^{\pm}]_{l\in{\mathbb{Z}}}$.
For the general case, let $i$ be in $I$ and $m$ be in $B_i$. Consider $\omega_k(m)$ as in section \[compl\]. The study of the $sl_2$-case allows us to set $\chi_k=\omega_k(m)^{-1}F_t(\omega_k(m))\in{\mathbb{Z}}[\tilde{A}_l^{-1},t^{\pm}]_l$. And using the ${\mathbb{Z}}[t^{\pm}]$-algebra homomorphism $\nu_{k,t}:{\mathbb{Z}}[\tilde{A}_l^{-1},t^{\pm}]_{l\in{\mathbb{Z}}}\rightarrow {\mathbb{Z}}[\tilde{A}_{i,l}^{-1},t^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$ defined by $\nu_{k,t}(\tilde{A}_l^{-1})=\tilde{A}_{i,k+lr_i}^{-1}$, we set (the terms of the product commute): $$F_{i,t}(m)=m\underset{0\leq k\leq r_i-1}{\prod}\nu_{k,t}(\chi_k)\in\mathfrak{K}_{i,t}$$ For the last assertion, we have $E_{i,t}(m)=\underset{l=1...L}{\sum}\lambda_l(t)F_{i,t}(m_l)$ where $m_1,...,m_L$ are the $i$-dominant ${\mathcal{Y}}_t$-monomials of $E_{i,t}(m)$ with coefficients $\lambda_1(t),...,\lambda_L(t)\in{\mathbb{Z}}[t^{\pm}]$.
In the same way there is a unique $F_i(m)\in\mathfrak{K}_i$\[fim\] such that $m$ is the unique $i$-dominant ${\mathcal{Y}}$-monomial of $F_i(m)$. Moreover $F_i(m)=\pi_+(F_{i,t}(m))$.
### Examples in the $sl_2$-case {#examples-in-the-sl_2-case}
In this section we suppose that ${\mathfrak{g}}=sl_2$ and we compute $F_t(m)=F_{1,t}(m)$ in some examples with the help of section \[exlsl\].
\[fexp\] Let $\sigma=\{l,l+2,...,l+2k\}$ be a $2$-segment and $m_{\sigma}=\tilde{Y}_l\tilde{Y}_{l+2}...\tilde{Y}_{l+2k}\in B$. Then we have the formula: $$F_t(m_{\sigma})=m_{\sigma}(1+t\tilde{A}_{l+2k+1}^{-1}+t^2\tilde{A}_{l+(2k+1)}^{-1}\tilde{A}_{l+(2k-1)}^{-1}+...+t^k\tilde{A}_{l+(2k+1)}^{-1}\tilde{A}_{l+(2k-1)}^{-1}...\tilde{A}_{l+1}^{-1})$$ If $\sigma_1,\sigma_2$ are $2$-segments not in special position, we have: $$F_t(m_{\sigma_1})F_t(m_{\sigma_2})=t^{\mathcal{N}(m_{\sigma_1},m_{\sigma_2})-\mathcal{N}(m_{\sigma_2},m_{\sigma_1})}F_t(m_{\sigma_2})F_t(m_{\sigma_1})$$ If $\sigma_1,...,\sigma_R$ are $2$-segments such that $m_{\sigma_1}...m_{\sigma_r}$ is regular, we have: $$F_t(m_{\sigma_1}...m_{\sigma_R})=F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$$
In particular if $m\in B$ verifies $\forall l\in{\mathbb{Z}}, u_l(m)\leq 1$ then it is of the form $m=m_{\sigma_1}...m_{\sigma_R}$ where the $\sigma_r$ are $2$-segments such that $\text{max}(\sigma_r)+2<\text{min}(\sigma_{r+1})$. So the lemma \[fexp\] gives an explicit formula $F_t(m)=F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$.
[[*Proof:*]{}]{}First we need some relations in ${\mathcal{Y}}_{1,t}$ : we know that for $l\in{\mathbb{Z}}$ we have $t\tilde{S}_{l-1}=\tilde{A}_l^{-1}\tilde{S}_{l+1}=t^2\tilde{S}_{l+1}\tilde{A}_l^{-1}$, so $t^{-1}\tilde{S}_{l-1}=\tilde{S}_{l+1}\tilde{A}_l^{-1}$. So we get by induction that for $r\geq 0$: $$t^{-r}\tilde{S}_{l+1-2r}=\tilde{S}_{l+1}\tilde{A}_l^{-1}\tilde{A}_{l-2}^{-1}...\tilde{A}_{l-2(r-1)}^{-1}$$ As $u_{i,l+1}(\tilde{A}_l^{-1}\tilde{A}_{l-2}^{-1}...\tilde{A}_{l-2(r-1)}^{-1})=u_{i,l+1}(\tilde{A}_l^{-1})=-1$, we get: $$t^{-r}\tilde{S}_{l+1-2r}=t^{-2}\tilde{A}_l^{-1}\tilde{A}_{l-2}^{-1}...\tilde{A}_{l-2(r-1)}^{-1}\tilde{S}_{l+1}$$ For $r'\geq 0$, by multiplying on the left by $\tilde{A}_{l+2r'}^{-1}\tilde{A}_{l+2(r'-1)}^{-1}...\tilde{A}_{l+2}^{-1}$, we get: $$t^{-r}\tilde{A}_{l+2r'}^{-1}\tilde{A}_{l+2(r'-1)}^{-1}...\tilde{A}_{l+2}^{-1}\tilde{S}_{l+1-2r}=t^{-2}\tilde{A}_{l+2r'}^{-1}\tilde{A}_{l+2(r'-1)}^{-1}...\tilde{A}_{l-2(r-1)}^{-1}\tilde{S}_{l+1}$$ If we put $r'=1+R',r=R-R',l=L-1-2R'$, we get for $0\leq R'\leq R$: $$t^{R'}\tilde{A}_{L+1}^{-1}\tilde{A}_{L-1}^{-1}...\tilde{A}_{L+1-2R'}^{-1}\tilde{S}_{L-2R}=t^{R-2}\tilde{A}_{L+1}^{-1}\tilde{A}_{L-1}^{-1}...\tilde{A}_{L+1-2R}^{-1}\tilde{S}_{L-2R'}$$ Now let be $m=\tilde{Y}_0\tilde{Y}_2...\tilde{Y}_l$ and $\chi\in{\mathcal{Y}}_t$ given by the formula in the lemma. Let us compute $\tilde{S}_t(\chi)$: $$\tilde{S}_t(\chi)=m(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_l)$$ $$+tm\tilde{A}^{-1}_{l+1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-2}-t^{-2}\tilde{S}_{l+2})$$ $$+t^2m\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-4}-t^{-2}\tilde{S}_{l}-t^{-2}\tilde{S}_{l+2})$$ $$+...$$ $$+t^lm\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}...\tilde{A}^{-1}_{1}(-t^{-2}\tilde{S}_2+...-t^{-2}\tilde{S}_{l}-t^{-2}\tilde{S}_{l+2})$$ $$=m(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_l)$$ $$+tm\tilde{A}^{-1}_{l+1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-2})-m\tilde{S}_l$$ $$+t^2m\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}(\tilde{S}_0+\tilde{S}_2+...+\tilde{S}_{l-4})-tm\tilde{A}^{-1}_{l+1}\tilde{S}_{l-2}-m\tilde{S}_{l-2}$$ $$+...$$ $$-mt^{l-1}\tilde{A}^{-1}_{l+1}\tilde{A}^{-1}_{l-1}...\tilde{A}^{-1}_{3}-...-t^{-2}\tilde{S}_{l}-m\tilde{S}_0$$ $$=0$$ So $\chi\in\mathfrak{K}_t$. But we see on the formula that $m$ is the unique dominant monomial of $\chi$. So $\chi=F_t(m)$.
For the second point, we have two cases:
if $m_{\sigma_1}m_{\sigma_2}$ is regular, it follows from lemma \[dominl\] that $L(m_{\sigma_1})L(m_{\sigma_2})=L(m_{\sigma_2})L(m_{\sigma_1})$ has no dominant monomial other than $m_{\sigma_1}m_{\sigma_2}$. But our formula shows that $F_t(m_{\sigma_1})$ (resp. $F_t(m_{\sigma_2}$)) has the same monomials than $L(m_{\sigma_1})$ (resp. $L(m_{\sigma_2})$). So $$F_t(m_{\sigma_1})F_t(m_{\sigma_2})-t^{\mathcal{N}(m_{\sigma_1},m_{\sigma_2})-\mathcal{N}(m_{\sigma_2},m_{\sigma_1})}F_t(m_{\sigma_2})F_t(m_{\sigma_1})$$ has no dominant ${\mathcal{Y}}_t$-monomial because $m_{\sigma_1}m_{\sigma_2}-t^{\mathcal{N}(m_{\sigma_1},m_{\sigma_2})-\mathcal{N}(m_{\sigma_2},m_{\sigma_1})}m_{\sigma_2}m_{\sigma_1}=0$.
if $m_{\sigma_1}m_{\sigma_2}$ is irregular, we have for example $\sigma_{j_1}\subset \sigma_{j_2}$ and $\sigma_{j_1}+2\subset\sigma_{j_2}$. Let us write $\sigma_{j_1}=\{l_1,l_1+2,...,,p_1\}$ and $\sigma_2=\{l_2,l_2+2,...,,p_2\}$. So we have $l_2\leq l_1$ and $p_1\leq p_2-2$. Let $m=m_1m_2$ be a dominant ${\mathcal{Y}}$-monomial of $L(m_{\sigma_1}m_{\sigma_2})=L(m_{\sigma_1})L(m_{\sigma_2})$ where $m_1$ (resp. $m_2$) is a ${\mathcal{Y}}$-monomial of $L(m_{\sigma_1})$ (resp. $L(m_{\sigma_2})$). If $m_2$ is not $m_{\sigma_2}$, we have $Y_{p_2}^{-1}$ in $m_2$ which can not be canceled by $m_1$. So $m=m_1m_{\sigma_2}$. Let us write $m_1=m_{\sigma_1}A_{p_1+1}^{-1}...A_{p_1+1-2r}^{-1}$. So we just have to prove: $$\tilde{A}_{p_1+1}^{-1}...\tilde{A}_{p_1+1-2r}^{-1}m_{\sigma_2}=m_{\sigma_2}\tilde{A}_{p_1+1}^{-1}...\tilde{A}_{p_1+1-2r}^{-1}$$ This follows from ($l\in{\mathbb{Z}}$): $$\tilde{A}_l^{-1}\tilde{Y}_{l-1}\tilde{Y}_{l+1}=\tilde{Y}_{l-1}\tilde{Y}_{l+1}\tilde{A}_l^{-1}$$
For the last assertion it suffices to show that $F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$ has no other dominant ${\mathcal{Y}}_t$-monomial than $m_{\sigma_1}...m_{\sigma_R}$. But $F_t(m_{\sigma_1})...F_t(m_{\sigma_R})$ has the same monomials than $L(m_{\sigma_1})...L(m_{\sigma_R})=L(m_{\sigma_1}...m_{\sigma_R})$. As $m_{\sigma_1}...m_{\sigma_R}$ is regular we get the result.
### Technical complements
Let us go back to the general case. We give some technical results which will be used in the following to compute $F_{i,t}(m)$ in some cases (see proposition \[cpfacile\] and section \[fin\]).
\[calcn\] Let $i$ be in $I$, $l\in{\mathbb{Z}}$, $M\in A$ such that $u_{i,l}(M)=1$ and $u_{i,l+2r_i}=0$. Then we have $\mathcal{N}(M,\tilde{A}_{i,l+r_i}^{-1})=-1$. In particular $\pi^{-1}(MA_{i,l+r_i}^{-1})=tM\tilde{A}_{i,l+r_i}^{-1}$.
[[*Proof:*]{}]{}We can suppose $M=:M:$ and we compute in ${\mathcal{Y}}_u$: $$M\tilde{A}_{i,l+r_i}^{-1}=\pi_+(m)\text{exp}(\underset{m>0,r\in{\mathbb{Z}},j\in I}{\sum}u_{j,r}(M)h^mq^{-rm}y_j[-m])$$ $$\text{exp}(\underset{m>0}{\sum}-h^mq^{-(l+r_i)m}a_i[-m])\text{exp}(\underset{m>0}{\sum}-h^mq^{(l+r_i)m}a_i[m])$$ $$=:M\tilde{A}_{i,l+r_i}^{-1}:\text{exp}(\underset{m>0}{\sum}h^{2m}([a_i[-m],a_i[m]]-\underset{r\in{\mathbb{Z}}}{\sum}u_{i,r}(m)[y_i[-m],a_i[m]]q^{(l+r_i-r)m}c_m)=t_R:\tilde{Y}_{i,l}\tilde{A}_{i,l+r_i}^{-1}:$$ where: $$R(z)=-(z^{2r_i}-z^{-2r_i})+\underset{r\in{\mathbb{Z}}}{\sum}u_{i,r}(M)z^{(l+r_i-r)}(z^{r_i}-z^{-r_i})$$ So: $$\mathcal{N}(\tilde{Y}_{i,l},M\tilde{A}_{i,l+r_i}^{-1})=\underset{r\in{\mathbb{Z}}}{\sum}u_{i,r}(M)(z^{2r_i+l-r}-z^{l-r})_0=-u_{i,l}(M)+u_{i,l+2r_i}(M)=-1$$
\[pidonne\] Let $m$ be in $B_i$ such that $\forall l\in{\mathbb{Z}}, u_{i,l}(m)\leq 1$ and for $1\leq r\leq 2r_i$ the set\
$\{l\in{\mathbb{Z}}/u_{i,r+2lr_i}(m)=1\}$ is a $1$-segment. Then we have $F_{i,t}(m)=\pi^{-1}(F_i(m))$.
[[*Proof:*]{}]{}Let us look at the $sl_2$-case : $m=m_1m_2=m_{\sigma_1}m_{\sigma_2}$ where $\sigma_1,\sigma_2$ are $2$-segment. So the lemma \[fexp\] gives an explicit formula for $F_t(m)$ and it follows from lemma \[calcn\] that $F_t(m)=\pi^{-1}(F(m))$.
We go back to the general case : let us write $m=m'm_1...m_{2r_i}$ where $m'=\underset{j\neq i,l\in{\mathbb{Z}}}{\prod}Y_{j,l}^{u_{j,l}(m)}$ and $m_r=\underset{l\in{\mathbb{Z}}}{\prod}Y_{i,r+2lr_i}^{u_{i,r+2lr_i}(m)}$. We have $m_r$ of the form $m_r=Y_{i,l_r}Y_{i,l_r+2r_i}...Y_{i,l_r+2n_ir_i}$. We have $F_{i,t}(m)=t^{-N(m'm_1...m_r)}m'F_{i,t}(m_1)...F_{i,t}(m_{2r_i})$. The study of the $sl_2$-case gives $F_{i,t}(m_r)=\pi^{-1}(F_i(m_r))$. It follows from lemma \[calcn\] that: $$t^{-N(m'm_1...m_r)}m'\pi^{-1}(F_i(m_1))...\pi^{-1}(F_i(m_r))=\pi^{-1}(m'F_i(m_1)...F_i(m_r))=\pi^{-1}(F_i(m))$$
Intersection of kernels of deformed screening operators {#rests}
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Motivated by theorem \[simme\] we study the structure of a completion of $\mathfrak{K}_t=\underset{i\in I}{\bigcap}\text{Ker}(S_{i,t})$ in order to construct $\chi_{q,t}$ in section \[conschi\]. Note that in the $sl_2$-case we have $\mathfrak{K}_t=\text{Ker}(S_{1,t})$ that was studied in section \[scr\].
Reminder: classic case ([@Fre], [@Fre2])
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### The elements $E(m)$ and $q$-characters
For $J\subset I$, denote the ${\mathbb{Z}}$-subalgebra $\mathfrak{K}_J=\underset{i\in J}{\bigcap}\mathfrak{K}_i\subset{\mathcal{Y}}$ and $\mathfrak{K}=\mathfrak{K}_I$.
\[least\] ([@Fre], [@Fre2]) A non zero element of $\mathfrak{K}_J$ has at least one $J$-dominant ${\mathcal{Y}}$-monomial.
[[*Proof:*]{}]{}It suffices to look at a maximal weight ${\mathcal{Y}}$-monomial $m$ of $\chi\in\mathfrak{K}_J$: for $i\in J$ we have $m\in B_i$ because $\chi\in\mathfrak{K}_i$.
([@Fre], [@Fre2]) For $i\in I$ there is a unique $E(Y_{i,0})\in \mathfrak{K}$\[ei\] such that ${Y}_{i,0}$ is the unique dominant ${\mathcal{Y}}$-monomial in $E(Y_{i,0})$.
The uniqueness follows from lemma \[least\]. For the existence we have $E(Y_{i,0})=\chi_q(V_{\omega_i}(1))$ (theorem \[simme\]).
Note that the existence of $E(Y_{i,0})\in\mathfrak{K}$ suffices to characterize $\chi_q: \text{Rep}\rightarrow \mathfrak{K}$. It is the ring homomorphism such that $\chi_q(X_{i,l})=s_l(E(Y_{i,0}))$ where $s_l:{\mathcal{Y}}\rightarrow {\mathcal{Y}}$ is given by $s_l({Y}_{j,k})={Y}_{j,k+l}$.
For $m\in B$, we defined the standard module $M_m$ in section \[back\]. We set: $$E(m)=\underset{m\in B}{\prod}s_l(E(Y_{i,0}))^{u_{i,l}(m)}=\chi_q(M_m)\in\mathfrak{K}$$\[em\] We defined the simple module $V_m$ in section \[back\]. We set $L(m)=\chi_q(V_m)\in\mathfrak{K}$\[lm\]. We have: $$\mathfrak{K}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}E(m)=\underset{m\in B}{\bigoplus}{\mathbb{Z}}L(m)\simeq {\mathbb{Z}}^{(B)}$$ For $m\in B$, we can also define a unique $F(m)\in\mathfrak{K}$\[fm\] such that $m$ is the unique dominant ${\mathcal{Y}}$-monomial which appears in $F(m)$ (see for example the proof of proposition \[defifprem\]).
### Technical complements
For $J\subset I$, let ${\mathfrak{g}}_J$ be the semi-simple Lie algebra of Cartan Matrix $(C_{i,j})_{i,j\in J}$ and ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J$ the associated quantum affine algebra with coefficient $(r_i)_{i\in J}$. In analogy with the definition of $E_i(m),L_i(m)$ using the $sl_2$-case (section \[compl\]), we define for $m\in B_J$: $E_J(m)$, $L_J(m)$, $F_J(m)\in \mathfrak{K}_J$ using ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J$. We have: $$\mathfrak{K}_J=\underset{m\in B_J}{\bigoplus}{\mathbb{Z}}E_J(m)=\underset{m\in B_J}{\bigoplus}{\mathbb{Z}}L_J(m)=\underset{m\in B_J}{\bigoplus}{\mathbb{Z}}F_J(m)\simeq {\mathbb{Z}}^{(B_J)}$$
As a direct consequence of proposition \[aidafm\] we have :
\[aidahfm\] For $m\in B$, we have $E(m)\in{\mathbb{Z}}[Y_{i,l}]_{i\in I,l\geq l(m)}$ where $l(m)=\text{min}\{l\in{\mathbb{Z}}/\exists i\in I,u_{i,l}(m)\neq 0\}$.
Completion of the deformed algebras {#complesection}
-----------------------------------
In this section we introduce completions of ${\mathcal{Y}}_t$ and of $\mathfrak{K}_{J,t}=\underset{i\in J}{\bigcap}\mathfrak{K}_{i,t}\subset {\mathcal{Y}}_t$ ($J\subset I$). We have the following motivation: we have seen $\pi_+(\mathfrak{K}_{J,t})\subset \mathfrak{K}_J$ (section \[scr\]). In order to prove an analogue of the other inclusion (theorem \[con\]) we have to introduce completions where infinite sums are allowed.
### The completion ${\mathcal{Y}}_t^{\infty}$ of ${\mathcal{Y}}_t$
Let $\overset{\infty}{A}_t$\[infat\] be the ${\mathbb{Z}}[t^{\pm}]$-module $\overset{\infty}{A}_t=\underset{m\in A}{\prod}{\mathbb{Z}}[t^{\pm}].m\simeq {\mathbb{Z}}[t^{\pm}]^A$. An element $(\lambda_m(t)m)_{m\in A}\in \overset{\infty}{A}_t$ is noted $\underset{m\in A}{\sum}\lambda_m(t)m$. We have $\underset{m\in A}{\bigoplus}{\mathbb{Z}}[t^{\pm}].m={\mathcal{Y}}_t\subset \overset{\infty}{A}_t$. The algebra structure of ${\mathcal{Y}}_t$ gives a ${\mathbb{Z}}[t^{\pm}]$-bilinear morphisms ${\mathcal{Y}}_t\otimes\overset{\infty}{A}_t\rightarrow\overset{\infty}{A}_t$ and $\overset{\infty}{A}_t\otimes{\mathcal{Y}}_t\rightarrow\overset{\infty}{A}_t$ such that $\overset{\infty}{A}_t$ is a ${\mathcal{Y}}_t$-bimodule. But the ${\mathbb{Z}}[t^{\pm}]$-algebra structure of ${\mathcal{Y}}_t$ can not be naturally extended to $\overset{\infty}{A}_t$. We define a ${\mathbb{Z}}[t^{\pm}]$-submodule ${\mathcal{Y}}_t^{\infty}$\[tytinf\] with ${\mathcal{Y}}_t\subset{\mathcal{Y}}_t^{\infty}\subset\overset{\infty}{A}_t$, for which it is the case:
Let ${\mathcal{Y}}_t^A$\[tyta\] be the ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t$ generated by the $(\tilde{A}_{i,l}^{-1})_{i\in I,l\in{\mathbb{Z}}}$. We gave in proposition \[yenga\] the structure of ${\mathcal{Y}}_t^A$. In particular we have ${\mathcal{Y}}_t^A=\underset{K\geq 0}{\bigoplus}{\mathcal{Y}}_t^{A,K}$ where for $K\geq 0$: $${\mathcal{Y}}_t^{A,K}=\underset{m=:\tilde{A}_{i_1,l_1}^{-1}...\tilde{A}_{i_K,l_K}^{-1}:}{\bigoplus}{\mathbb{Z}}[t^{\pm}].m\subset {\mathcal{Y}}_t^A$$ Note that for $K_1,K_2\geq 0$, ${\mathcal{Y}}_t^{A,K_1}{\mathcal{Y}}_t^{A,K_2}\subset{\mathcal{Y}}_t^{A,K_1+K_2}$ for the multiplication of ${\mathcal{Y}}_t$. So ${\mathcal{Y}}_t^A$ is a graded algebra if we set $\text{deg}(x)=K$ for $x\in{\mathcal{Y}}_t^{A,K}$. Denote by ${\mathcal{Y}}_t^{A,\infty}$ the completion of ${\mathcal{Y}}_t^A$ for this gradation. It is a sub-${\mathbb{Z}}[t^{\pm}]$-module of $\overset{\infty}{A}_t$.
We define ${\mathcal{Y}}_t^{\infty}$ as the sub ${\mathcal{Y}}_t$-leftmodule of $\overset{\infty}{A}_t$ generated by ${\mathcal{Y}}_t^{A,\infty}$.
In particular, we have: ${\mathcal{Y}}_t^{\infty}=\underset{M\in A}{\sum}M.{\mathcal{Y}}_t^{A,\infty}\subset \overset{\infty}{A}_t$.
There is a unique algebra structure on ${\mathcal{Y}}_t^{\infty}$ compatible with the structure of ${\mathcal{Y}}_t\subset {\mathcal{Y}}_t^{\infty}$.
[[*Proof:*]{}]{}The structure is unique because the elements of ${\mathcal{Y}}_t^{\infty}$ are infinite sums of elements of ${\mathcal{Y}}_t$. For $M\in A$, we have ${\mathcal{Y}}_t^{A,\infty}.M\subset M.{\mathcal{Y}}_t^{A,\infty}$, so ${\mathcal{Y}}_t^{\infty}$ is a sub ${\mathcal{Y}}_t$-bimodule of $\overset{\infty}{A}_t$. For $M\in A$ and $\lambda\in{\mathcal{Y}}_t^{A, \infty}$ denote $\lambda^M\in{\mathcal{Y}}_t^{A, \infty}$ such that $\lambda.M=M.\lambda^M$. We define the ${\mathbb{Z}}[t^{\pm}]$-algebra structure on ${\mathcal{Y}}_t^{\infty}$ by ($M,M'\in A,\lambda,\lambda'\in{\mathcal{Y}}_t^{A,\infty}$): $$(M.\lambda)(M'.\lambda')=MM'.(\lambda^{M'}\lambda')$$ It is well defined because for $M_1,M_2,M\in A,\lambda,\lambda_2\in{\mathcal{Y}}_t^A$ we have $M_1\lambda_1=M_2\lambda_2\Rightarrow M_1M\lambda_1^M=M_2M\lambda_2^M$.
### The completion $\mathfrak{K}_{i,t}^{\infty}$ of $\mathfrak{K}_{i,t}$
We define a completion of $\mathfrak{K}_{i,t}$ analog to the completed algebra ${\mathcal{Y}}_t^{\infty}$.
For $M\in A$, we define a ${\mathbb{Z}}[t^{\pm}]$-linear endomorphism $E_{i,t}^M:M{\mathcal{Y}}_t^{A,\infty}\rightarrow M{\mathcal{Y}}_t^{A,\infty}$\[teitmap\] such that ($m$ ${\mathcal{Y}}_t^A$-monomial): $$E_{i,t}^M(Mm)=0\text{ if $:Mm:\notin B_i$}$$ $$E_{i,t}^M(Mm)=E_{i,t}(Mm)\text{ if $:Mm:\in B_i$}$$ It is well-defined because if $m\in{\mathcal{Y}}_t^{A,K}$ and $:Mm:\in B_i$ we have $E_{i,t}(Mm)\in M\underset{K'\geq K}{\bigoplus}{\mathcal{Y}}^{A,K'}$.
We define $\mathfrak{K}_{i,t}^{\infty}=\underset{M\in A}{\sum}\text{Im}(E_{i,t}^M)\subset{\mathcal{Y}}_t^{\infty}$\[kitinf\].
For $J\subset I$, we set $\mathfrak{K}_{J,t}^{\infty}=\underset{i\in J}{\bigcap}\mathfrak{K}_{i,t}^{\infty}$ and $\mathfrak{K}_t^{\infty}=\mathfrak{K}_{I,t}^{\infty}$.
\[leasto\] A non zero element of $\mathfrak{K}_{J,t}^{\infty}$ has at least one $J$-dominant ${\mathcal{Y}}_t$-monomial.
[[*Proof:*]{}]{}Analog to the proof of lemma \[least\].
\[alginf\] For $J\subset I$, we have $\mathfrak{K}_{J,t}^{\infty}\cap {\mathcal{Y}}_t=\mathfrak{K}_{J,t}$. Moreover $\mathfrak{K}_{J,t}^{\infty}$ is a ${\mathbb{Z}}[t^{\pm}]$-subalgebra of ${\mathcal{Y}}_t^{\infty}$.
[[*Proof:*]{}]{}It suffices to prove the results for $J=\{i\}$. First for $m\in B_i$ we have $E_{i,t}(m)=E_{i,t}^m(m)\in\mathfrak{K}_{i,t}^{\infty}$ and so $\mathfrak{K}_{i,t}=\underset{m\in B_i}{\bigoplus}{\mathbb{Z}}[t^{\pm}]E_{i,t}(m)\subset \mathfrak{K}_{i,t}^{\infty}\cap{\mathcal{Y}}_t$. Now let $\chi$ be in $\mathfrak{K}_{i,t}^{\infty}$ such that $\chi$ has only a finite number of ${\mathcal{Y}}_t$-monomials. In particular it has only a finite number of $i$-dominant ${\mathcal{Y}}_t$-monomials $m_1,...,m_r$ with coefficients $\lambda_1(t),...,\lambda_r(t)$. In particular it follows from lemma \[leasto\] that $\chi=\lambda_1(t)F_{i,t}(m_1)+...+\lambda_r(t)F_{i,t}(m_r)\in\mathfrak{K}_{i,t}$ (see proposition \[defifprem\] for the definition of $F_{i,t}(m)$).
For the last assertion, consider $M_1$, $M_2\in A$ and $m_1$, $m_2$ ${\mathcal{Y}}_t^A$-monomials such that $:M_1m_1:,:M_2m_2:\in B_i$. Then $E_{i,t}(M_1m_1)E_{i,t}(M_2m_2)$ is in the the sub-algebra $\mathfrak{K}_{i,t}\subset{\mathcal{Y}}_t$ and in $\text{Im}(E_{i,t}^{M_1M_2})$.
In the same way for $t=1$ we define the ${\mathbb{Z}}$-algebra $\mathfrak{{\mathcal{Y}}}^{\infty}$ and the ${\mathbb{Z}}$-subalgebras $\mathfrak{K}_J^{\infty}\subset{\mathcal{Y}}^{\infty}$.
The surjective map $\pi_+:{\mathcal{Y}}_t\rightarrow{\mathcal{Y}}$ is naturally extended to a surjective map $\pi_+:{\mathcal{Y}}_t^{\infty}\rightarrow{\mathcal{Y}}^{\infty}$. For $i\in I$, we have $\pi_+(\mathfrak{K}_{i,t}^{\infty})=\mathfrak{K}_i^{\infty}$ and for $J\subset I$, $\pi_+(\mathfrak{K}_{J,t}^{\infty})\subset\mathfrak{K}_J^{\infty}$. The other inclusion is equivalent to theorem \[con\].
### Special submodules of ${\mathcal{Y}}_t^{\infty}$ {#infsum}
For $m\in A$, $K\geq 0$ we construct a subset $D_{m,K}\subset m\{\tilde{A}_{i_1,l_1}^{-1}...\tilde{A}_{i_K,l_K}^{-1}\}$\[dmk\] stable by the maps $E_{i,t}^{m}$ such that $\underset{K\geq 0}{\bigcup}D_{m,K}$ is countable: we say that $m'\in D_{m,K}$ if and only if there is a finite sequence $(m_0=m,m_1,...,m_R=m')$ of length $R\leq K$, such that for all $1\leq r\leq R$, there is $r'<r$, $J\subset I$ such that $m_{r'}\in B_J$ and for $r'<r''\leq r$, $m_{r''}$ is a ${\mathcal{Y}}$-monomial of $E_J(m_{r'})$ and $m_{r''}m_{r''-1}^{-1}\in\{A_{j,l}^{-1}/l\in{\mathbb{Z}},j\in J\}$.
The definition means that “there is chain of monomials of some $E_J(m'')$ from $m$ to $m'$”.
The set $D_{m,K}$ is finite. In particular, the set $D_m$ is countable.
[[*Proof:*]{}]{}Let us prove by induction on $K\geq 0$ that $D_{m,K}$ is finite: we have $D_{m,0}=\{m\}$ and: $$D_{m,K+1}\subset \underset{J\subset I, m'\in D_{m,K}\cap B_J}{\bigcup}\{\text{${\mathcal{Y}}$-monomials of }E_J(m')\}$$
\[ordrep\] For $m,m'\in A$ such that $m'\in D_{m}$ we have $D_{m'}\subseteq D_{m}$. For $M\in A$, the set $B\cap D_M$ is finite.
[[*Proof:*]{}]{}Consider $(m_0=m,m_1,...,m_R=m')$ a sequence adapted to the definition of $D_m$. Let $m''$ be in $D_{m'}$ and $(m_R=m',m_{R+1},...,m_{R'}=m'')$ a sequence adapted to the definition of $D_{m'}$. So $(m_0,m_1,...,m_{R'})$ is adapted to the definition of $D_m$, and $m''\in D_m$.
Let us look at $m\in B\bigcap D_M$: we can see by induction on the length of a sequence $(m_0=M,m_1,...,m_R=m)$ adapted to the definition of $D_M$ that $m$ is of the form $m=MM'$ where $M'=\underset{i\in I,l\geq l_1}{\prod}A_{i,l}^{-v_{i,l}}$ ($v_{i,l}\geq 0$). So the last assertion follows from lemma \[fini\].
$\tilde{D}_m$\[tdm\] is the ${\mathbb{Z}}[t^{\pm}]$-submodule of ${\mathcal{Y}}_t^{\infty}$ whose elements are of the form $(\lambda_m(t)m)_{m\in D_m}$.
For $m\in A$ introduce $m_0=m>m_1>m_2>...$ the countable set $D_m$ with a total ordering compatible with the partial ordering. For $k\geq 0$ consider an element $F_k\in \tilde{D}_{m_k}$.
Note that some infinite sums make sense in $\tilde{D}_m$: for $k\geq 0$, we have $D_{m_k}\subset \{m_k,m_{k+1},...\}$. So $m_k$ appears only in the $F_{k'}$ with $k'\leq k$ and the infinite sum $\underset{k\geq 0}{\sum}F_k$ makes sense in $\tilde{D}_m$.
Crucial result for our construction
-----------------------------------
Our construction of $q,t$-characters is based on theorem \[con\] proved in this section.
### Statement
\[tftm\] For $n\geq 1$ denote $P(n)$\[pn\] the property “for all semi-simple Lie-algebras ${\mathfrak{g}}$ of rank $\text{rk}({\mathfrak{g}})=n$, for all $m\in B$ there is a unique $F_t(m)\in\mathfrak{K}_t^{\infty}\cap \tilde{D}_m$ such that $m$ is the unique dominant ${\mathcal{Y}}_t$-monomial of $F_t(m)$.”.
\[con\] For all $n\geq 1$, the property $P(n)$ is true.
Note that for $n=1$, that is to say ${\mathfrak{g}}=sl_2$, the result follows from section \[scr\].
The uniqueness follows from lemma \[leasto\] : if $\chi_1,\chi_2\in \mathfrak{K}_t^{\infty}$ are solutions, then $\chi_1-\chi_2$ has no dominant ${\mathcal{Y}}_t$-monomial, so $\chi_1=\chi_2$.
Remark: in the simply-laced case the existence is a consequence of the geometric theory of quivers [@Naa], [@Nab], and in $A_n,D_n$-cases of algebraic explicit constructions [@Nac]. In the rest of this section \[rests\] we give an algebraic proof of this theorem in the general case.
### Outline of the proof
First we give some preliminary technical results (section \[conste\]) in which we construct $t$-analogues of the $E(m)$. Next we prove $P(n)$ by induction on $n$. Our proof has 3 steps:
Step 1 (section \[petitdeux\]): we prove $P(1)$ and $P(2)$ using a more precise property $Q(n)$ such that $Q(n)\Rightarrow P(n)$. The property $Q(n)$ has the following advantage: it can be verified by computation in elementary cases $n=1,2$.
Step 2 (section \[consp\]): we give some consequences of $P(n)$ which will be used in the proof of $P(r)$ ($r>n$): we give the structure of $\mathfrak{K}_t^{\infty}$ (proposition \[thth\]) for $\text{rk}({\mathfrak{g}})=n$ and the structure of $\mathfrak{K}_{J,t}^{\infty}$ where $J\subset I$, $|J|=n$ and $|I|>n$ (corollary \[recufond\]).
Step 3 (section \[proof\]): we prove $P(n)$ ($n \geq 3$) assuming $P(r)$, $r\leq n$ are true. We give an algorithm (section \[defialgo\]) to construct explicitly $F_t(m)$. It is called $t$-algorithm and is a $t$-analogue of Frenkel-Mukhin algorithm [@Fre2] (a deformed algorithm was also used by Nakajima in the $ADE$-case [@Naa]). As we do not know [*a priori*]{} the algorithm is well defined the general case, we have to show that it never fails (lemma \[nfail\]) and gives a convenient element (lemma \[conv\]).
Preliminary: Construction of the $E_t(m)$ {#conste}
-----------------------------------------
\[copieun\] We suppose that for $i\in I$, there is $F_t(\tilde{Y}_{i,0})\in\mathfrak{K}_t^{\infty}\cap \tilde{D}_{\tilde{Y}_{i,0}}$ such that $\tilde{Y}_{i,0}$ is the unique dominant ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})$. Then:
i\) All ${\mathcal{Y}}_t$-monomials of $F_t(\tilde{Y}_{i,0})$, except the highest weight ${\mathcal{Y}}_t$-monomial, are right negative.
ii\) All ${\mathcal{Y}}_t$-monomials of $F_t(\tilde{Y}_{i,0})$ are products of $\tilde{Y}_{j,l}^{\pm}$ with $l\geq 0$.
iii\) The only ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})$ which contains a $\tilde{Y}_{j,0}^{\pm}$ ($j\in I$) is the highest weight monomial $\tilde{Y}_{i,0}$.
iv\) The $F_t(\tilde{Y}_{i,0})$ ($i\in I$) commute.
Note that (i),(ii) and (iii) appeared in [@Fre2].
[[*Proof:*]{}]{}
i\) It suffices to prove that all ${\mathcal{Y}}_t$-monomials $m_0=Y_{i,0},m_1,...$ of $D_{Y_{i,0}}$ except $Y_{i,0}$ are right negative. But $m_1$ is the monomial $Y_{i,0}A_{i,1}^{-1}$ of $E_i(Y_{i,0})$ and it is right negative. We can now prove the statement by induction: suppose that $m_r$ is a monomial of $E_J(m_{r'})$, where $m_{r'}$ is right negative. So $m_r$ is a product of $m_{r'}$ by some $A_{j,l}^{-1}$ ($l\in{\mathbb{Z}}$).Those monomials are right negative because a product of right negative monomial is right negative.
ii\) Suppose that $m\in A$ is product of $Y_{k,l}^{\pm}$ with $l\geq 0$. It follows from lemma \[aidahfm\] that all monomials of $D_{m}$ are product of $Y_{k,l}^{\pm}$ with $l\geq 0$.
iii\) All ${\mathcal{Y}}$-monomials of $D_{Y_{i,0}}$ except $\tilde{Y}_{i,0}$ are in $D_{Y_{i,0}A_{i,r_i}^{-1}}$. But $l(Y_{i,0}A_{i,r_i}^{-1})\geq 1$ and we can conclude with the help of lemma \[aidahfm\].
iv\) Let $i\neq j$ be in $I$ and look at $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})$. Suppose we have a dominant ${\mathcal{Y}}_t$-monomial $m_0=m_1m_2$ in $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})$ different from the highest weight ${\mathcal{Y}}_t$-monomial $\tilde{Y}_{i,0}\tilde{Y}_{j,0}$. We have for example $m_1\neq \tilde{Y}_{i,0}$, so $m_1$ is right negative. Let $l_1$ be the maximal $l$ such that a $\tilde{Y}_{k,l}$ appears in $m_1$. We have $u_{k,l}(m_1)<0$ and $l>0$. As $u_{k,l}(m_0)\geq 0$ we have $u_{k,l}(m_2)>0$ and $m_2\neq {Y}_{j,0}$. So $m_2$ is right negative and there is $k'\in I$ and $l'>l$ such that $u_{k',l'}(m_2)<0$. So $u_{k',l'}(m_1)>0$, contradiction. So the highest weight ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})$ is the unique dominant ${\mathcal{Y}}_t$-monomial. In the same way the highest weight ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{j,0})F_t(\tilde{Y}_{i,0})$ is the unique dominant ${\mathcal{Y}}_t$-monomial. But we have $\tilde{Y}_{i,0}\tilde{Y}_{j,0}=\tilde{Y}_{j,0}\tilde{Y}_{i,0}$, so $F_t(\tilde{Y}_{i,0})F_t(\tilde{Y}_{j,0})-F_t(\tilde{Y}_{j,0})F_t(\tilde{Y}_{i,0})\in\mathfrak{K}_t^{\infty}$ has no dominant ${\mathcal{Y}}_t$-monomial, so is equal to $0$.
Denote, for $l\in{\mathbb{Z}}$, by $s_l:{\mathcal{Y}}_t^{\infty}\rightarrow {\mathcal{Y}}_t^{\infty}$ the endomorphism of ${\mathbb{Z}}[t^{\pm}]$-algebra such that $s_l(\tilde{Y}_{j,k})=\tilde{Y}_{j,k+l}$ (it is well-defined because the defining relations of ${\mathcal{Y}}_t$ are invariant for $k\mapsto k+l$). If the hypothesis of the lemma \[copieun\] are verified, we can define for $m\in t^{{\mathbb{Z}}}B$ : $$E_t(m)=m (\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}\tilde{Y}_{i,l}^{u_{i,l}(m)})^{-1}\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{u_{i,l}(m)}\in\mathfrak{K}_t^{\infty}$$\[tetm\] because for $l\in{\mathbb{Z}}$ the product $\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{u_{i,l}(m)}$ is commutative (lemma \[copieun\]).
Step 1: Proof of $P(1)$ and $P(2)$ {#petitdeux}
----------------------------------
The aim of this section is to prove $P(1)$ and $P(2)$. First we define a more precise property $Q(n)$ such that $Q(n)\Rightarrow P(n)$.
### The property $Q(n)$ {#qn}
For $n\geq 1$ denote $Q(n)$ the property “for all semi-simple Lie-algebras ${\mathfrak{g}}$ of rank $\text{rk}({\mathfrak{g}})=n$, for all $i\in I$ there is a unique $F_t(\tilde{Y}_{i,0})\in\mathfrak{K}_t\cap \tilde{D}_{\tilde{Y}_{i,0}}$ such that $\tilde{Y}_{i,0}$ is the unique dominant ${\mathcal{Y}}_t$-monomial of $F_t(\tilde{Y}_{i,0})$. Moreover $F_t(\tilde{Y}_{i,0})$ has the same monomials as $E(Y_{i,0})$”.
The property $Q(n)$ is more precise than $P(n)$ because it asks that $F_t(\tilde{Y}_{i,0})$ has only a finite number of monomials.
For $n\geq 1$, the property $Q(n)$ implies the property $P(n)$.
[[*Proof:*]{}]{}We suppose $Q(n)$ is true. In particular the section \[conste\] enables us to construct $E_t(m)\in\mathfrak{K}_t^{\infty}$ for $m\in B$. The defining formula of $E_t(m)$ shows that it has the same monomials as $E(m)$. So $E_t(m)\in\tilde{D}_m$ and $E_t(m)\in\mathfrak{K}_t$.
Let us prove $P(n)$: let $m$ be in $B$. The uniqueness of $F_t(m)$ follows from lemma \[leasto\]. Let $m_L=m>m_{L-1}>...>m_1$ be the dominant monomials of $D_m$ with a total ordering compatible with the partial ordering (it follows from lemma \[fini\] that $D_m\cap B$ is finite). Let us prove by induction on $l$ the existence of $F_t(m_l)$. The unique dominant of $D_{m_1}$ is $m_1$ so $F_t(m_1)=E_t(m_1)\in\tilde{D}_{m_1}$. In general let $\lambda_1(t),...,\lambda_{l-1}(t)\in{\mathbb{Z}}[t^{\pm}]$ be the coefficient of the dominant ${\mathcal{Y}}_t$-monomials $m_1,...,m_{l-1}$ in $E_t(m_l)$. We put: $$F_t(m_l)=E_t(m_l)-\underset{r=1...l-1}{\sum}\lambda_r(t)F_t(m_r)$$ We see in the construction that $F_t(m)\in\tilde{D}_m$ because for $m'\in D_m$ we have $E_t(m')\in \tilde{D}_{m'}\subseteq\tilde{D}_m$ (lemma \[ordrep\]).
### Cases $n=1,n=2$ {#petit}
We need the following general technical result:
\[cpfacile\] Let $m$ be in $B$ such that all monomial $m'$ of $F(m)$ verifies : $\forall i\in I, m'\in B_i$ implies $\forall l\in{\mathbb{Z}}, u_{i,l}(m')\leq 1$ and for $1\leq r\leq 2r_i$ the set $\{l\in{\mathbb{Z}}/u_{i,r+2lr_i}(m')=1\}$ is a $1$-segment. Then $\pi^{-1}(F(m))\in{\mathcal{Y}}_t$ is in $\mathfrak{K}_t$ and has a unique dominant monomial $m$.
[[*Proof:*]{}]{}Let us write $F(m)=\underset{m'\in A}{\sum}\mu(m')m'$ ($\mu(m')\in{\mathbb{Z}}$). Let $i$ be in $I$ and consider the decomposition of $F(m)$ in $\mathfrak{K}_i$: $$F(m)=\underset{m'\in B_i}{\sum}\mu(m')F_i(m')$$ But $\mu(m')\neq 0$ implies the hypothesis of lemma \[pidonne\] is verified for $m'\in B_i$. So $\pi^{-1}(F_i(m'))=F_{i,t}(m')$. And: $$\pi^{-1}(F(m))=\underset{m'\in B_i}{\sum}\mu(m')F_{i,t}(m')\in \mathfrak{K}_{i,t}$$
For $n=1$ (section \[kernelun\]), $n=2$ (section \[fin\]), we can give explicit formula for the $E(Y_{i,0})=F(Y_{i,0})$. In particular we see that the hypothesis of proposition \[cpfacile\] are verified, so:
The properties $Q(1)$, $Q(2)$ and so $P(1)$, $P(2)$ are true.
This allow us to start our induction in the proof of theorem \[con\].
In section \[acase\] we will see other applications of proposition \[cpfacile\].
Note that the hypothesis of proposition \[cpfacile\] are not verified for fundamental monomials $m=Y_{i,0}$ in general: for example for the $D_5$-case we have in $F(Y_{2,0})$ the monomial $Y_{3,3}^2Y_{5,4}^{-1}Y_{2,4}^{-1}Y_{4,4}^{-1}$.
Step 2: consequences of the property $P(n)$ {#consp}
-------------------------------------------
Let be $n\geq 1$. We suppose in this section that $P(n)$ is proved. We give some consequences of $P(n)$ which will be used in the proof of $P(r)$ ($r>n$).
Let $\mathfrak{K}_t^{\infty,f}$\[ktinff\] be the ${\mathbb{Z}}[t^{\pm}]$-submodule of $\mathfrak{K}_t^{\infty}$ generated by elements with a finite number of dominant ${\mathcal{Y}}_t$-monomials.
\[thth\] We suppose $\text{rk}({\mathfrak{g}})=n$. We have: $$\mathfrak{K}_t^{\infty, f}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}[t^{\pm}]F_t(m)\simeq{\mathbb{Z}}[t^{\pm}]^{(B)}$$ Moreover for $M\in A$, we have: $$\mathfrak{K}_t^\infty\cap\tilde{D}_M=\underset{m\in B\cap D_M}{\bigoplus}{\mathbb{Z}}[t^{\pm}]F_t(m)\simeq{\mathbb{Z}}[t^{\pm}]^{B\cap D_M}$$
[[*Proof:*]{}]{}Let $\chi$ be in $\mathfrak{K}_t^{\infty, f}$ and $m_1,...,m_L\in B$ the dominant ${\mathcal{Y}}_t$-monomials of $\chi$ and $\lambda_1(t),...,\lambda_L(t)\in{\mathbb{Z}}[t^{\pm}]$ their coefficients. It follows from lemma \[leasto\] that $\chi=\underset{l=1...L}{\sum}\lambda_l(t)F_t(m_l)$.
Let us look at the second point: lemma \[ordrep\] shows that $m\in B\cap D_M\Rightarrow F_t(m)\in\tilde{D}_M$. In particular the inclusion $\supseteq$ is clear. For the other inclusion we prove as in the first point that $\mathfrak{K}_t^\infty\cap\tilde{D}_M=\underset{m\in B\cap D_M}{\sum}{\mathbb{Z}}[t^{\pm}]F_t(m)$. We can conclude because it follows from lemma \[fini\] that $D_M\cap B$ is finite.
We recall that have seen in section \[infsum\] that some infinite sum make sense in $\tilde{D}_M$.
\[recufond\] We suppose $\text{rk}({\mathfrak{g}})>n$ and let $J$ be a subset of $I$ such that $|J|=n$. For $m\in B_J$, there is a unique $F_{J,t}(m)\in\mathfrak{K}_{J,t}^{\infty}$ such that $m$ is the unique $J$-dominant ${\mathcal{Y}}_t$-monomial of $F_{J,t}(m)$. Moreover $F_{J,t}(m)\in\tilde{D}_{m}$.
For $M\in A$, the elements of $\mathfrak{K}_{J,t}^\infty\cap\tilde{D}_M$ are infinite sums $\underset{m\in B_J\cap D_M}{\sum}\lambda_m(t)F_{J,t}(m)$. In particular: $$\mathfrak{K}_{J,t}^\infty\cap\tilde{D}_M\simeq{\mathbb{Z}}[t^{\pm}]^{B_J\cap D_M}$$
[[*Proof:*]{}]{}The uniqueness of $F_{J,t}(m)$ follows from lemma \[leasto\]. Let us write $m=m_Jm'$ where\
$m_J=\underset{i\in J,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{u_{i,l}(m)}$. So $m_J$ is a dominant ${\mathcal{Y}}_t$-monomial of ${\mathbb{Z}}[Y_{i,l}^{\pm}]_{i\in J,l\in{\mathbb{Z}}}$. In particular the proposition \[thth\] with the algebra ${\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J$ of rank $n$ gives $m_J\chi$ where $\chi\in{\mathbb{Z}}[\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,-1},t^{\pm}]_{i\in J,l\in{\mathbb{Z}}}$ (where for $i\in I,l\in{\mathbb{Z}}$, $\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,\pm}=\beta_{I,J}(\tilde{A}_{i,l}^{\pm})$ where $\beta_{I,J}(\tilde{Y}_{i,l}^{\pm})=\delta_{i\in J}\tilde{Y}_{i,l}^{\pm}$). So we can put $F_t(m)=m\nu_{J,t}(\chi)$ where $\nu_{J,t}:{\mathbb{Z}}[\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,-1},t^{\pm}]_{i\in J,l\in{\mathbb{Z}}}\rightarrow{\mathcal{Y}}_t$ is the ring homomorphism such that $\nu_{J,t}(\tilde{A}_{i,l}^{{\mathcal{U}}_q(\hat{{\mathfrak{g}}})_J,-1})=\tilde{A}_{i,l}^{-1}$.
The last assertion is proved as in proposition \[thth\].
Step 3: $t$-algorithm and end of the proof of theorem \[con\] {#proof}
-------------------------------------------------------------
In this section we explain why the $P(r)$ ($r<n$) imply $P(n)$. In particular we define the $t$-algorithm which constructs explicitly the $F_t(m)$.
### The induction
We prove the property $P(n)$ by induction on $n\geq 1$. It follows from section \[petitdeux\] that $P(1)$ and $P(2)$ are true. Let be $n\geq 3$ and suppose that $P(r)$ is proved for $r<n$.
Let $m_+$ be in $B$ and $m_0=m_+>m_1>m_2>...$ the countable set $D_{m_+}$ with a total ordering compatible with the partial ordering.
For $J\varsubsetneq I$ and $m\in B_J$, it follows from $P(r)$ and corollary \[recufond\] that there is a unique $F_{J,t}(m)\in\tilde{D}_m\cap\mathfrak{K}_{J,t}^{\infty}$ such that $m$ is the unique $J$-dominant monomial of $F_{J,t}(m)$ and that the elements of $\tilde{D}_{m_+}\cap \mathfrak{K}_{J,t}^{\infty}$ are the infinite sums of ${\mathcal{Y}}_t^{\infty}$: $\underset{m\in D_{m_+}\cap B_J}{\sum}\lambda_m(t)F_{J,t}(m)$ where $\lambda_m(t)\in{\mathbb{Z}}[t^{\pm}]$.
If $m\in A-B_J$, denote $F_{J,t}(m)=0$.
### Definition of the $t$-algorithm {#defialgo}
For $r,r'\geq 0$ and $J\subsetneq I$ denote $[F_{J,t}(m_{r'})]_{m_r}\in{\mathbb{Z}}[t^{\pm}]$ the coefficient of $m_r$ in $F_{J,t}(m_{r'})$.
We call $t$-algorithm the following inductive definition of the sequences $(s(m_r)(t))_{r\geq 0}\in{\mathbb{Z}}[t^{\pm}]^{{\ensuremath{\mathbb{N}}}}$, $(s_J(m_r)(t))_{r\geq 0}\in{\mathbb{Z}}[t^{\pm}]^{{\ensuremath{\mathbb{N}}}}$ ($J\varsubsetneq I$)\[smrt\]: $$s(m_0)(t)=1\text{ , }s_J(m_0)(t)=0$$ and for $r\geq 1, J\subsetneq I$: $$s_J(m_r)(t)=\underset{r'<r}{\sum}(s(m_{r'})(t)-s_J(m_{r'})(t))[F_{J,t}(m_{r'})]_{m_r}$$ $$\text{ if }m_r\notin B_J, s(m_r)(t)=s_J(m_r)(t)$$ $$\text{ if }m_r\in B, s(m_r)(t)=0$$
We have to prove that the $t$-algorithm defines the sequences in a unique way. We see that if $s(m_r),s_J(m_r)$ are defined for $r\leq R$ so are $s_J(m_{R+1})$ for $J\subsetneq I$. The $s_J(m_R)$ impose the value of $s(m_{R+1})$ and by induction the uniqueness is clear. We say that the $t$-algorithm is well defined to step $R$ if there exist $s(m_{r}), s_J(m_r)$ such that the formulas of the $t$-algorithm are verified for $r\leq R$.
The $t$-algorithm is well defined to step $r$ if and only if: $$\forall J_1,J_2\varsubsetneq I, \forall r'\leq r, m_{r'}\notin B_{J_1} \text{ and }m_{r'}\notin B_{J_2}\Rightarrow s_{J_1}(m_{r'})(t)=s_{J_2}(m_{r'})(t)$$
[[*Proof:*]{}]{}If for $r'<r$ the $s(m_{r'})(t),s_J(m_{r'})(t)$ are well defined, so is $s_J(m_r)(t)$. If $m_r\in B$, $s(m_r)(t)=0$ is well defined. If $m_r\notin B$, it is well defined if and only if $\{s_J(m_r)(t)/m_r\notin B_J\}$ has one unique element.
### The $t$-algorithm never fails
If the $t$-algorithm is well defined to all steps, we say that the $t$-algorithm never fails. In this section we show that the $t$-algorithm never fails.
If the $t$-algorithm is well defined to step $r$, for $J\varsubsetneq I$ we set: $$\mu_J(m_r)(t)=s(m_r)(t)-s_J(m_r)(t)$$ $$\chi_J^r=\underset{r'\leq r}{\sum}\mu_{J}(m_{r'}(t))F_{J,t}(m_{r'})\in\mathfrak{K}_{J,t}^{\infty}$$
If the $t$-algorithm is well defined to step $r$, for $J\subset I$ we have: $$\chi_J^r\in (\underset{r'\leq r}{\sum} s(m_{r'})(t)m_{r'})+s_J(m_{r+1})(t)m_{r+1}+\underset{r'>r+1}{\sum}{\mathbb{Z}}[t^{\pm}]m_{r'}$$ For $J_1\subset J_2\subsetneq I$, we have: $$\chi_{J_2}^r=\chi_{J_1}^r+\underset{r'>r}{\sum}\lambda_{r'}(t)F_{J_1,t}(m_{r'})$$ where $\lambda_{r'}(t)\in{\mathbb{Z}}[t^{\pm}]$. In particular, if $m_{r+1}\notin B_{J_1}$, we have $s_{J_1,t}(m_{r+1})=s_{J_2,t}(m_{r+1})$.
[[*Proof:*]{}]{}For $r'\leq r$ let us compute the coefficient $(\chi_J^r)_{m_{r'}}\in{\mathbb{Z}}[t^{\pm}]$ of $m_{r'}$ in $\chi_J^r$: $$(\chi_J^r)_{m_{r'}}=\underset{r''\leq r'}{\sum}(s(m_{r''})(t)-s_J(m_{r''})(t))[F_{J,t}(m_{r''})]_{m_{r'}}$$ $$=(s(m_{r'})(t)-s_J(m_{r'})(t))[F_{J,t}(m_{r'})]_{m_{r'}}+\underset{r''<r'}{\sum}(s(m_{r''})(t)-s_J(m_{r''})(t))[F_{J,t}(m_{r''})]_{m_{r'}}$$ $$=(s(m_{r'})(t)-s_J(m_{r'})(t))+s_J(m_{r'})(t)=s(m_{r'})(t)$$ Let us compute the coefficient $(\chi_J^r)_{m_{r+1}}\in{\mathbb{Z}}[t^{\pm}]$ of $m_{r+1}$ in $\chi_J^r$: $$(\chi_J^r)_{m_{r+1}}=\underset{r''<r+1}{\sum}(s(m_{r''})(t)-s_J(m_{r''})(t))[F_{J,t}(m_{r''})]_{m_{r+1}}=s_J(m_{r+1})$$ For the second point let $J_1\subset J_2\subsetneq I$. We have $\chi_{J_2}^r\in \mathfrak{K}_{J_1,t}^{\infty}\cap\tilde{D}_{m+}$ and it follows from $P(|J_1|)$ and corollary \[recufond\] (or section \[petit\] if $|J_1|\leq 2$) that we can introduce $\lambda_{m_{r'}}(t)\in{\mathbb{Z}}[t^{\pm}]$ such that : $$\chi_{J_2}^r=\underset{r'\geq 0}{\sum}\lambda_{m_{r'}}(t)F_{J_1,t}(m_{r'})$$ We show by induction on $r'$ that for $r'\leq r$, $m_{r'}\in B_{J_1}\Rightarrow \lambda_{m_{r'}}(t)=\mu_{J_1}(m_{r'})(t)$. First we have $\lambda_{m_0}(t)=(\chi_{J_2}^r)_{m_0}=s(m_0)(t)=1=\mu_{J_1}(m_0)$. For $r'\leq r$: $$s(m_{r'})(t)=\lambda_{m_{r'}}(t)+\underset{r''<r'}{\sum}\lambda_{m_{r''}}(t)[F_{J_1,t}(m_{r''})]_{m_{r'}}$$ $$\lambda_{m_{r'}}(t)=s(m_{r'})(t)-\underset{r''<r'}{\sum}\mu_{J_1}(m_{r'})(t)[F_{J_1,t}(m_{r''})]_{m_{r'}}=s(m_{r'})(t)-s_{J_1}(m_{r'})(t)=\mu_{J_1}(m_{r'})(t)$$ For the last assertion if $m_{r+1}\notin B_{J_1}$, the coefficient of $m_{r+1}$ in $\underset{r'>r}{\sum}{\mathbb{Z}}[t^{\pm}]F_{J_1,t}(m_{r'})$ is 0, and $(\chi_{J_2}^r)_{m_{r+1}}=(\chi_{J_1}^r)_{m_{r+1}}$. It follows from the first point that $s_{J_1,t}(m_{r+1})=s_{J_2,t}(m_{r+1})$.
\[nfail\] The $t$-algorithm never fails.
[[*Proof:*]{}]{}Suppose the sequence is well defined until the step $r-1$ and let $J_1,J_2\varsubsetneq I$ such that $m_r\notin B_{J_1}$ and $m_r\notin B_{J_2}$. Let $i$ be in $J_1$, $j$ in $J_2$ such that $m_r\notin B_i$ and $m_r\notin B_j$. Consider $J=\{i,j\}\varsubsetneq I$. The $\chi_J^{r-1},\chi_i^{r-1},\chi_j^{r-1}\in{\mathcal{Y}}_t$ have the same coefficient $s(m_{r'})(t)$ on $m_{r'}$ for $r'\leq r-1$. Moreover: $$s_i(m_r)(t)=(\chi_i^{r-1})_{m_r}\text{ , }s_j(m_r)(t)=(\chi_j^{r-1})_{m_r}\text{ , }s_J(m_r)(t)=(\chi_J^{r-1})_{m_r}$$ But $m_r\notin B_J$, so: $$\chi_J^{r-1}=\underset{r'\leq r-1}{\sum}\mu_i(m_{r'})(t)F_{i,t}(m_{r'})+\underset{r'\geq r+1}{\sum}\lambda_{m_{r'}}(t)F_{i,t}(m_{r'})$$ So $(\chi_J^{r-1})_{m_r}=(\chi_i^{r-1})_{m_r}$ and we have $s_i(m_r)(t)=s_J(m_r)(t)$. In the same way we have $s_i(m_r)(t)=s_{J_1}(m_r)(t)$, $s_j(m_r)(t)=s_J(m_r)(t)$ and $s_j(m_r)(t)=s_{J_2}(m_r)(t)$. So we can conclude $s_{J_1}(m_r)(t)=s_{J_2}(m_r)(t)$.
### Proof of $P(n)$
It follows from lemma \[nfail\] that $\chi=\underset{r\geq 0}{\sum}s(m_r)(t)m_r\in{\mathcal{Y}}_t^{\infty}$ is well defined.
\[conv\] We have $\chi\in \mathfrak{K}_t^{\infty}\bigcap\tilde{D}_{m_+}$. Moreover the only dominant ${\mathcal{Y}}_t$-monomial in $\chi$ is $m_0=m_+$.
[[*Proof:*]{}]{}The defining formula of $\chi$ gives $\chi\in\tilde{D}_{m_+}$. Let $i$ be in $I$ and: $$\chi_i=\underset{r\geq 0}{\sum}\mu_i(m_r)(t)F_{i,t}(m_r)\in\mathfrak{K}_{i,t}^{\infty}$$ Let us compute for $r\geq 0$ the coefficient of $m_r$ in $\chi-\chi_i$: $$(\chi-\chi_i)_{m_r}=s(m_r)(t)-\underset{r'\leq r}{\sum}\mu_i(m_{r'})(t)[F_{i,t}(m_{r'})]_{m_r}$$ $$=s(m_r)(t)-s_i(m_r)(t)-\mu_i(m_r)(t)[F_{i,t}(m_{r})]_{m_r}=(s(m_r)(t)-s_i(m_r)(t))(1-[F_{i,t}(m_{r})]_{m_r})$$ We have two cases:
if $m_r\in B_i$, we have $1-[F_{i,t}(m_{r})]_{m_r}=0$.
if $m_r\notin B_i$, we have $s(m_r)(t)-s_i(m_r)(t)=0$.
So $\chi=\chi_i\in\mathfrak{K}_{i,t}^{\infty}$, and $\chi\in \mathfrak{K}_t^{\infty}$.
The last assertion follows from the definition of the algorithm: for $r>0$, $m_r\in B\Rightarrow s(m_r)(t)=0$.
This lemma implies:
For $n\geq 3$, if the $P(r)$ ($r<n$) are true, then $P(n)$ is true.
In particular the theorem \[con\] is proved by induction on $n$.
Morphism of $q,t$-characters and applications {#conschi}
=============================================
Morphism of $q,t$-characters {#aida}
----------------------------
### Definition of the morphism
We set $\text{Rep}_t=\text{Rep}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t^{\pm}]={\mathbb{Z}}[X_{i,l},t^{\pm}]_{i\in I,l\in{\mathbb{Z}}}$\[rept\]. We say that $M\in\text{Rep}_t$ is a $\text{Rep}_t$-monomial if it is of the form $M=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}X_{i,l}^{x_{i,l}}$ ($x_{i,l}\geq 0$). In this case denote $x_{i,l}(M)=x_{i,l}$. Recall the definition of the $E_t(m)$ (section \[conste\]).
\[mqt\] The morphism of $q,t$-characters is the ${\mathbb{Z}}[t^{\pm}]$-linear map $\chi_{q,t}:\text{Rep}_t\rightarrow {\mathcal{Y}}_t^{\infty}$\[chiqt\] such that ($u_{i,l}\geq 0$): $$\chi_{q,t}(\underset{i\in I,l\in{\mathbb{Z}}}{\prod}X_{i,l}^{u_{i,l}})=E_t(\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{u_{i,l}})$$
### Properties of $\chi_{q,t}$
\[axiomes\] We have $\pi_+(\text{Im}(\chi_{q,t}))\subset{\mathcal{Y}}$ and the following diagram is commutative: $$\begin{array}{rcccl}
\text{Rep}&\stackrel{\chi_{q,t}}{\longrightarrow}&\text{Im}(\chi_{q,t})\\
\text{id}\downarrow &&\downarrow&\pi_+\\
\text{Rep}&\stackrel{\chi_q}{\longrightarrow}&{\mathcal{Y}}\end{array}$$ In particular the map $\chi_{q,t}$ is injective. The ${\mathbb{Z}}[t^{\pm}]$-linear map $\chi_{q,t}:\text{Rep}_t\rightarrow{\mathcal{Y}}_t^{\infty}$ is characterized by the three following properties:
1\) For a $\text{Rep}_t$-monomial $M$ define $m=\pi^{-1}(\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{x_{i,l}(M)})\in A$ and $\tilde{m}\in A_t$ as in section \[tildeat\]. Then we have : $$\chi_{q,t}(M)=\tilde{m}+\underset{m'<m}{\sum}a_{m'}(t)m'\text{ (where $a_{m'}(t)\in{\mathbb{Z}}[t^{\pm}]$)}$$
2\) The image of $\text{Im}(\chi_{q,t})$ is contained in $\mathfrak{K}_t^{\infty}$.
3\) Let $M_1,M_2$ be $\text{Rep}_t$-monomials such that $\text{max}\{l/\underset{i\in I}{\sum}x_{i,l}(M_1)>0\}\leq \text{min}\{l/\underset{i\in I}{\sum}x_{i,l}(M_2)>0\}$. We have : $$\chi_{q,t}(M_1M_2)=\chi_{q,t}(M_1)\chi_{q,t}(M_2)$$
Note that the properties $1,2,3$ are generalizations of the defining axioms introduced by Nakajima in [@Nab] for the $ADE$-case; in particular in the $ADE$-case $\chi_{q,t}$ is the morphism of $q,t$-characters constructed in [@Nab].
[[*Proof:*]{}]{}$\pi_+(\text{Im}(\chi_{q,t}))\subset{\mathcal{Y}}$ means that only a finite number of ${\mathcal{Y}}_t$-monomials of $E_t(m)$ have coefficient $\lambda(t)\notin (t-1){\mathbb{Z}}[t^{\pm}]$. As $F_t(\tilde{Y}_{i,0})$ has no dominant ${\mathcal{Y}}_t$-monomial other than $\tilde{Y}_{i,0}$, we have the same property for $\pi_+(F_t(\tilde{Y}_{i,0}))\in\mathfrak{K}^{\infty}$ and $\pi_+(F_t(\tilde{Y}_{i,0}))=E(Y_{i,0})\in {\mathcal{Y}}$. As ${\mathcal{Y}}$ is a subalgebra of ${\mathcal{Y}}^{\infty}$ we get $\pi_+(E_t(m))\in{\mathcal{Y}}$ with the help of the defining formula.
The diagram is commutative because $\pi_+\circ s_l=s_l\circ \pi_+$ and $\pi_+(F_t(\tilde{Y}_{i,0}))=E(Y_{i,0})$. It is proved by Frenkel, Reshetikhin in [@Fre] that $\chi_q$ is injective, so $\chi_{q,t}$ is injective.
Let us show that $\chi_{q,t}$ verifies the three properties:
1\) By definition we have $\chi_{q,t}(M)=E_t(m)$. But $s_l(F_t(\tilde{Y}_{i,0}))=F_t(\tilde{Y}_{i,l})\in\tilde{D}(\tilde{Y}_{i,l})$. In particular $s_l(F_t(\tilde{Y}_{i,0}))$ is of the form $\tilde{Y}_{i,l}+\underset{m'<Y_{i,l}}{\sum}\lambda_{m'}(t)m'$ and we get the property for $E_t(m)$ by multiplication.
2\) We have $s_l(F_t(\tilde{Y}_{i,0}))=E_t(\tilde{Y}_{i,l})\in\mathfrak{K}_t^{\infty}$ and $\mathfrak{K}_t^{\infty}$ is a subalgebra of ${\mathcal{Y}}_t^{\infty}$, so $\text{Im}(\chi_{q,t})\subset\mathfrak{K}_t^{\infty}$.
3\) If we set $L=\text{max}\{l/\underset{i\in I}{\sum}x_{i,l}(M_1)>0\}$, $m_1=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{x_{i,l}(M_1)}$, $m_2=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}Y_{i,l}^{x_{i,l}(M_2)}$, we have: $$E_t(m_1)=\overset{\rightarrow}{\underset{l\leq L}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{x_{i,l}(M_1)}\text{ , }E_t(m_2)=\overset{\rightarrow}{\underset{l\geq L}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{x_{i,l}(M_2)}$$ and in particular: $$E_t(m_1m_2)=E_t(m_1)E_t(m_2)$$ Finally let $f:\text{Rep}_t\rightarrow{\mathcal{Y}}_t^{\infty}$ be a ${\mathbb{Z}}[t^{\pm}]$-linear homomorphism which verifies properties 1,2,3. We saw that the only element of $\mathfrak{K}_t^{\infty}$ with highest weight monomial $\tilde{Y}_{i,l}$ is $s_l(F_t(\tilde{Y}_{i,0}))$. In particular we have $f(X_{i,l})=E_t(Y_{i,l})$. Using property 3, we get for $M\in\text{Rep}_t$ a monomial : $$f(M)=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}f(X_{i,l})^{u_{i,l}(m)}=\overset{\rightarrow}{\underset{l\in{\mathbb{Z}}}{\prod}}\underset{i\in I}{\prod}s_l(F_t(\tilde{Y}_{i,0}))^{u_{i,l}(m)}=\chi_{q,t}(M)$$
Quantization of the Grothendieck Ring {#quanta}
-------------------------------------
In this section we see that $\chi_{q,t}$ allows us to define a deformed algebra structure on $\text{Rep}_t$ generalizing the quantization of [@Nab]. The point is to show that $\text{Im}(\chi_{q,t})$ is a subalgebra of $\mathfrak{K}_t^{\infty}$.
### Generators of $\mathfrak{K}_t^{\infty,f}$
Recall the definition of $\mathfrak{K}_t^{\infty,f}$ in section \[consp\]. For $m\in B$, all monomials of $E_t(m)$ are in $\{mA_{i_1,l_1}^{-1}...A_{i_K,l_K}^{-1}/k\geq 0,l_k\geq L\}$ where $L=\text{min}\{l\in{\mathbb{Z}},\exists i\in I,u_{i,l}(m)>0\}$. So it follows from lemma \[fini\] that $E_t(m)\in\mathfrak{K}_t^{\infty}$ has only a finite number of dominant ${\mathcal{Y}}_t$-monomials, that is to say $E_t(m)\in\mathfrak{K}_t^{\infty,f}$.
\[diago\] The ${\mathbb{Z}}[t^{\pm}]$-module $\mathfrak{K}_t^{\infty,f}$ is freely generated by the $E_t(m)$: $$\mathfrak{K}_t^{\infty, f}=\underset{m\in B}{\bigoplus}{\mathbb{Z}}[t^{\pm}]E_t(m)\simeq{\mathbb{Z}}[t^{\pm}]^{(B)}$$
[[*Proof:*]{}]{}The $E_t(m)$ are ${\mathbb{Z}}[t^{\pm}]$-linearly independent and we saw $E_t(m)\in\mathfrak{K}_t^{\infty, f}$. It suffices to prove that the $E_t(m)$ generate the $F_t(m)$: let us look at $m_0\in B$ and consider $L=\text{min}\{l\in{\mathbb{Z}},\exists i\in I,u_{i,l}(m_0)>0\}$. In the proof of lemma \[fini\] we saw there is only a finite dominant monomials in $\{m_0A_{i_1,l_1}^{-v_{i_1,l_1}}...A_{i_R,l_R}^{-v_{i_R,l_R}}/R\geq 0,i_r\in I,l_r\geq L\}$. Let $m_0>m_1>...>m_D\in B$ be those monomials with a total ordering compatible with the partial ordering. In particular, for $0\leq d\leq D$ the dominant monomials of $E_t(m_d)$ are in $\{m_d,m_{d+1},...,m_D\}$. So there are elements $(\lambda_{d,d'}(t))_{0\leq d,d'\leq D}$ of ${\mathbb{Z}}[t^{\pm}]$ such that: $$E_t(m_d)=\underset{d\leq d'\leq D}{\sum}\lambda_{d,d'}(t)F_t(m_{d'})$$ We have $\lambda_{d,d'}(t)=0$ if $d'<d$ and $\lambda_{d,d}(t)=1$. We have a triangular system with $1$ on the diagonal, so it is invertible in ${\mathbb{Z}}[t^{\pm}]$.
### Construction of the quantization
$\mathfrak{K}_t^{\infty,f}$ is a subalgebra of $\mathfrak{K}_t^{\infty}$.
[[*Proof:*]{}]{}It suffices to prove that for $m_1,m_2\in B$, $E_t(m_1)E_t(m_2)$ has only a finite number of dominant ${\mathcal{Y}}_t$-monomials. But $E_t(m_1)E_t(m_2)$ has the same monomials as $E_t(m_1m_2)$.
It follows from proposition \[diago\] that $\chi_{q,t}$ is a ${\mathbb{Z}}[t^{\pm}]$-linear isomorphism between $\text{Rep}_t$ and $\mathfrak{K}_t^{\infty,f}$. So we can define:
The associative deformed ${\mathbb{Z}}[t^{\pm}]$-algebra structure on $\text{Rep}_t$ is defined by: $$\forall \lambda_1,\lambda_2\in\text{Rep}_t,\lambda_1*\lambda_2=\chi_{q,t}^{-1}(\chi_{q,t}(\lambda_1)\chi_{q,t}(\lambda_2))$$\[star\]
### Examples: $sl_2$-case
We make explicit computation of the deformed multiplication in the $sl_2$-case:
In the $sl_2$-case, the deformed algebra structure on $\text{Rep}_t={\mathbb{Z}}[X_l,t^{\pm}]_{l\in{\mathbb{Z}}}$ is given by: $$X_{l_1}*X_{l_2}*...*X_{l_m}=X_{l_1}X_{l_2}...X_{l_m}\text{ if $l_1\leq l_2\leq ...\leq l_m$}$$ $$X_{l}*X_{l'}=t^{\gamma}X_{l}X_{l'}=t^{\gamma}X_{l'}*X_l\text{ if $l>l'$ and $l\neq l'+2$}$$ $$X_{l}*X_{l-2}=t^{-2}X_{l}X_{l-2}+t^{\gamma}(1-t^{-2})=t^{-2}X_{l-2}*X_{l}+(1-t^{-2})$$ where $\gamma\in{\mathbb{Z}}$ is defined by $\tilde{Y}_l\tilde{Y}_{l'}=t^{\gamma}\tilde{Y}_{l'}\tilde{Y}_l$.
[[*Proof:*]{}]{}For $l\in{\mathbb{Z}}$ we have the $q,t$-character of the fundamental representation $X_l$: $$\chi_{q,t}(X_l)=\tilde{Y}_l+\tilde{Y}_{l+2}^{-1}=\tilde{Y}_l(1+t\tilde{A}_{l+1}^{-1})$$ The first point of the proposition follows immediately from the definition of $\chi_{q,t}$. For example, for $l,l'\in{\mathbb{Z}}$ we have: $$\chi_{q,t}(X_lX_{l'})=\chi_{q,t}(X_{\text{min}(l,l')})\chi_{q,t}(X_{\text{max}(l,l')})$$ In particular if $l\leq l'$, we have $X_l*X_{l'}=X_lX_{l'}$. Suppose now that $l>l'$ and introduce $\gamma\in{\mathbb{Z}}$ such that $\tilde{Y}_l\tilde{Y}_{l'}=t^{\gamma}\tilde{Y}_{l'}\tilde{Y}_l$. We have: $$\chi_{q,t}(X_l)\chi_{q,t}(X_l')=\tilde{Y}_l(1+t\tilde{A}_{l+1}^{-1})\tilde{Y}_{l'}(1+t\tilde{A}_{l'+1}^{-1})$$ $$=t^{\gamma}\tilde{Y}_{l'}\tilde{Y}_l+t^{\gamma +1}\tilde{Y}_{l'}\tilde{Y}_l\tilde{A}_{l+1}^{-1}+t^{\gamma +1+2\delta_{l,l'+2}}\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_l+t^{\gamma+ 2}\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_{l}\tilde{A}_{l+1}^{-1}$$ $$=t^{\gamma}\chi_{q,t}(X_{l'}X_l)+t^{\gamma +1}(t^{2\delta_{l,l'+2}}-1)\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_l$$ If $l\neq l'+2$ we get $X_l*X_{l'}=t^{\gamma}X_{l'}*X_l$. If $l=l'+2$, we have: $$\tilde{Y}_{l'}\tilde{A}_{l'+1}^{-1}\tilde{Y}_{l'+2}=t^{-1}\tilde{Y}_{l'+2}^{-1}\tilde{Y}_{l'+2}=t^{-1}$$ But $t^2\tilde{Y}_l\tilde{Y}_{l-2}=\tilde{Y}_{l-2}\tilde{Y}_l$, so $X_l*X_{l-2}=t^{-2}X_{l-2}*X_l+t^{-2}(t^2-1)$.
Note that $\gamma$ were computed in section \[varva\].
We see that the new ${\mathbb{Z}}[t^{\pm}]$-algebra structure is not commutative and not even twisted polynomial.
An involution of the Grothendieck ring {#invo}
--------------------------------------
In this section we construct an antimultiplicative involution of the Grothendieck ring $\text{Rep}_t$. The construction is motivated by the point view adopted in this article : it is just replacing $c_{|l|}$ by $-c_{|l|}$. In the $ADE$-case such an involution were introduced Nakajima [@Nab] with different motivations.
### An antihomomorphism of $\mathcal{H}$
There is a unique ${\ensuremath{\mathbb{C}}}$-linear isomorphism of $\mathcal{H}$ which is antimultiplicative and such that: $$\overline{c_m}=-c_m\text{ , }\overline{a_i[r]}=a_i[r]\text{ ($m>0,i\in I,r\in{\mathbb{Z}}-\{0\}$)}$$ Moreover it is an involution.
[[*Proof:*]{}]{}It suffices to show it is compatible with the defining relations of $\mathcal{H}$ ($i,j\in I,m,r\in{\mathbb{Z}}-\{0\}$): $$\overline{[a_i[m],a_j[r]]}=\overline{a_i[m]a_j[r]}-\overline{a_j[r]a_i[m]}=-[a_i[m],a_j[r]]$$ $$\overline{\delta_{m,-r}(q^m-q^{-m})B_{i,j}(q^m)c_{|m|}}=-\delta_{m,-r}(q^m-q^{-m})B_{i,j}(q^m)c_{|m|}$$ For the last assertion, we have $\overline{\overline{c_m}}=c_m$ and $\overline{\overline{a_i[r]}}=a_i[r]$, and an algebra morphism which fixes the generators is the identity.
It can be naturally extended to an antimultiplicative ${\ensuremath{\mathbb{C}}}$-isomorphism of $\mathcal{H}_h$.
\[stbl\] The ${\mathbb{Z}}$-subalgebra ${\mathcal{Y}}_u\subset \mathcal{H}_h$ verifies $\overline{{\mathcal{Y}}_u}\subset{\mathcal{Y}}_u$.
[[*Proof:*]{}]{}It suffices to check on the generators of ${\mathcal{Y}}_u$ ($R\in\mathfrak{U},i\in I,l\in{\mathbb{Z}}$): $$\overline{t_R}=\text{exp}(\underset{m>0}{\sum}h^{2m}R(q^m)(-c_m))=t_{-R}$$ $$\overline{\tilde{Y}_{i,l}}=\text{exp}(\underset{m>0}{\sum}h^m y_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m y_i[m]q^{lm})$$ $$=\text{exp}(\underset{m>0}{\sum}h^{2m}[y_i[-m],y_i[m]])\tilde{Y}_{i,l}
=t_{-\tilde{C}_{i,i}(q)(q_i-q_i^{-1})}\tilde{Y}_{i,l}\in{\mathcal{Y}}_u$$ $$\overline{\tilde{Y}_{i,l}}^{-1}=(\overline{\tilde{Y}_{i,l}})^{-1}=t_{\tilde{C}_{i,i}(q)(q_i-q_i^{-1})}\tilde{Y}_{i,l}^{-1}\in{\mathcal{Y}}_u$$
### Involution of ${\mathcal{Y}}_t$
As for $R,R'\in\mathfrak{U}$, we have $\pi_0(R)=\pi_0(R')\Leftrightarrow\pi_0(-R)=\pi_0(-R')$, the involution of ${\mathcal{Y}}_u$ (resp. of $\mathcal{H}_h$) is compatible with the defining relations of ${\mathcal{Y}}_t$ (resp. $\mathcal{H}_t$). We get a ${\mathbb{Z}}$-linear involution of ${\mathcal{Y}}_t$ (resp. of $\mathcal{H}_t$). For $\lambda,\lambda'\in{\mathcal{Y}}_t,\alpha\in{\mathbb{Z}}$, we have: $$\overline{\lambda.\lambda'}=\overline{\lambda'}.\overline{\lambda}\text{ , }\overline{t^{\alpha}\lambda}=t^{-\alpha}\overline{\lambda}$$
Note that in ${\mathcal{Y}}_u$ for $i\in I,l\in{\mathbb{Z}}$: $$\overline{\tilde{A}_{i,l}}=\text{exp}(\underset{m>0}{\sum}h^m a_i[-m]q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m a_i[m]q^{lm})$$ $$=\text{exp}(\underset{m>0}{\sum}h^{2m}[a_i[-m],a_i[m]]c_m)\tilde{A}_{i,l}=t_{(-q_i^2+q_i^{-2})}\tilde{A}_{i,l}$$ So in ${\mathcal{Y}}_t$ we have $\overline{\tilde{A}_{i,l}}=\tilde{A}_{i,l}$ and $\overline{\tilde{A}_{i,l}^{-1}}=\tilde{A}_{i,l}^{-1}$.
### The involution of deformed bimodules
For $i\in I$, the ${\mathcal{Y}}_{i,u}\subset \mathcal{H}_h$ verifies $\overline{{\mathcal{Y}}_{i,u}}\subset{\mathcal{Y}}_{i,u}$.
[[*Proof:*]{}]{}First we compute for $i\in I,l\in{\mathbb{Z}}$: $$\overline{\tilde{S}_{i,l}}=\text{exp}(\underset{m>0}{\sum}h^m\frac{a_i[-m]}{q_i^{-m}-q_i^{m}}q^{-lm})\text{exp}(\underset{m>0}{\sum}h^m\frac{a_{i}[m]}{q_i^{m}-q_i^{-m}}q^{lm})$$ $$=\text{exp}(\underset{m>0}{\sum}h^{2m}\frac{[a_i[-m],a_i[m]]}{-(q_i^{-m}-q_i^{m})^2}c_m)\tilde{S}_{i,l}=t_{\frac{q_i+q_i^{-1}}{q_i-q_i^{-1}}}\tilde{S}_{i,l}\in{\mathcal{Y}}_{i,u}$$ Now for $\lambda\in{\mathcal{Y}}_u$, we have $\overline{\lambda.\tilde{S}_{i,l}}=t_{\frac{q_i+q_i^{-1}}{q_i-q_i^{-1}}}\tilde{S}_{i,l}\overline{\lambda}$. But it is in ${\mathcal{Y}}_{i,u}$ because $\overline{\lambda}\in{\mathcal{Y}}_u$ (lemma \[stbl\]) and ${\mathcal{Y}}_{i,u}$ is a ${\mathcal{Y}}_u$-subbimodule of $\mathcal{H}_h$ (lemma \[currents\]).
In $\mathcal{H}_t$ we have $\overline{\tilde{S}_{i,l}}=t\tilde{S}_{i,l}$ because $\pi_0(\frac{q_i+q_i^{-1}}{q_i-q_i^{-1}})=1$. As said before we get a ${\mathbb{Z}}$-linear involution of ${\mathcal{Y}}_{i,t}$ such that: $$\overline{\lambda \tilde{S}_{i,l}}=t\tilde{S}_{i,l}\overline{\lambda}$$
We introduced such an involution in [@Her01]. With this new point of view, the compatibility with the relation $\tilde{A}_{i,l-r_i}\tilde{S}_{i,l}=t^{-1}\tilde{S}_{i,l+r_i}$ is a direct consequence of lemma \[currents\] and needs no computation; for example: $$\overline{\tilde{A}_{i,l-r_i}\tilde{S}_{i,l}}=t\tilde{S}_{i,l}\tilde{A}_{i,l-r_i}=t^3\tilde{A}_{i,l-r_i}\tilde{S}_{i,l}=t^2\tilde{S}_{i,l+r_i}$$ $$\overline{t^{-1}\tilde{S}_{i,l+r_i}}=t\overline{\tilde{S}_{i,l+r_i}}=t^2\tilde{S}_{i,l+r_i}$$
### The induced involution of $\text{Rep}_t$
For $i\in I$, the subalgebra $\mathfrak{K}_{i,t}\subset{\mathcal{Y}}_t$ verifies $\overline{\mathfrak{K}_{i,t}}\subset\mathfrak{K}_{i,t}$.
[[*Proof:*]{}]{}Suppose $\lambda\in\mathfrak{K}_{i,t}$, that is to say $S_{i,t}(\lambda)=0$. So $\overline{(t^2-1)S_{i,t}(\lambda)}=0$ and: $$\underset{l\in{\mathbb{Z}}}{\sum}(\overline{\tilde{S}_{i,l}\lambda}-\overline{\lambda\tilde{S}_{i,l}})=0\Rightarrow t\underset{l\in{\mathbb{Z}}}{\sum}(\overline{\lambda}\tilde{S}_{i,l}-\tilde{S}_{i,l}\overline{\lambda})=0$$ So $t(1-t^2)S_{i,t}(\overline{\lambda})=0$ and $\overline{\lambda}\in\mathfrak{K}_{i,t}$.
Note that $\chi\in{\mathcal{Y}}_t$ has the same monomials as $\overline{\chi}$, that is to say if $\chi=\underset{m\in A}{\sum}\lambda(t)m$ and $\overline{\chi}=\underset{m\in A}{\sum}\mu(t)m$, we have $\lambda(t)\neq 0\Leftrightarrow\mu(t)\neq 0$. In particular we can naturally extend our involution to an antimultiplicative involution on ${\mathcal{Y}}_t^{\infty}$. Moreover we have $\overline{\mathfrak{K}_t^{\infty}}\subset \mathfrak{K}_t^{\infty}$ and $\overline{\mathfrak{K}_t^{\infty,f}}=\overline{\text{Im}(\chi_{q,t})}\subset \text{Im}(\chi_{q,t})$. So we can define:
The ${\mathbb{Z}}$-linear involution of $\text{Rep}_t$ is defined by: $$\forall \lambda\in\text{Rep}_t\text{ , }\overline{\lambda}=\chi_{q,t}^{-1}(\overline{\chi_{q,t}(\lambda)})$$
Analogues of Kazhdan-Lusztig polynomials
----------------------------------------
In this section we define analogues of Kazhdan-Lusztig polynomials (see [@kalu]) with the help of the antimultiplicative involution of section \[invo\] in the same spirit Nakajima did for the $ADE$-case [@Nab]. Let us begin we some technical properties of the action of the involution on monomials.
### Invariance of monomials {#invmon}
We recall that the ${\mathcal{Y}}_t^A$-monomials are products of the $\tilde{A}_{i,l}^{-1}$ ($i\in I, l\in{\mathbb{Z}}$).
For $M$ a ${\mathcal{Y}}_t$-monomial and $m$ a ${\mathcal{Y}}_t^A$-monomial there is a unique $\alpha(M,m)\in{\mathbb{Z}}$\[alpham\] such that $\overline{t^{\alpha(M,m)}Mm}=t^{\alpha(M,m)}\overline{M}m$.
[[*Proof:*]{}]{}Let $\beta\in{\mathbb{Z}}$ such that $\overline{m}=t^{\beta}m$. We have $\overline{Mm}=\overline{m}\overline{M}=t^{\beta+\gamma}\overline{M}m$ where $\gamma\in 2{\mathbb{Z}}$ (section \[notuil\]). So it suffices to prove that $\beta\in 2{\mathbb{Z}}$.
Let us compute $\beta$. Let $\pi_+(m)=\underset{i\in I,l\in{\mathbb{Z}}}{\prod}A_{i,l}^{-v_{i,l}}$. In ${\mathcal{Y}}_u$ we have $\pi_+(m)\pi_-(m)=t_R\pi_-(m)\pi_+(m)$ where $\pi_0(R)=\beta$ and: $$R(q)=\underset{i,j\in I,r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}v_{j,r'}\underset{l>0}{\sum}q^{lr-lr'}\frac{[a_i[l],a_j[-l]]}{c_l}$$ where for $l>0$ we set $\frac{[a_i[l],a_j[-l]]}{c_l}=B_{i,j}(q^l)(q^{l}-q^{-l})\in{\mathbb{Z}}[q^{\pm}]$ which is antisymmetric. For $i=j$, we have the term: $$\underset{r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}v_{i,r'}\underset{l>0}{\sum}q^{lr-lr'}\frac{[a_i[l],a_i[-l]]}{c_l}$$ $$=\underset{l>0}{\sum}(\underset{\{r,r'\}\subset{\mathbb{Z}}, r\neq r'}{\sum}v_{i,r}(m)v_{i,r'}(m)(q^{l(r-r')}+q^{l(r'-r)})+\underset{r\in{\mathbb{Z}}}{\sum}v_{i,r}(m)^2)\frac{[a_i[l],a_i[-l]]}{c_l}$$ It is antisymmetric, so it has no term in $q^0$. So $\pi_0(R)=\pi_0(R')$ where $R'$ is the sum of the contributions for $i\neq j$: $$\underset{r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}(m)v_{j,r'}(m)\underset{l>0}{\sum}q^{lr-lr'}(\frac{[a_i[l],a_j[-l]]}{c_l}+\frac{[a_j[l],a_i[-l]]}{c_l})$$ $$=2\underset{r,r'\in{\mathbb{Z}}}{\sum}v_{i,r}(m)v_{j,r'}(m)\underset{l>0}{\sum}q^{lr-lr'}\frac{[a_i[l],a_j[-l]]}{c_l}$$ In particular $\pi_0(R')\in 2{\mathbb{Z}}$.
For $M$ a ${\mathcal{Y}}_t$-monomial denote $A^{\text{inv}}_M=\{t^{\alpha(m,M)}Mm/\text{$m$ ${\mathcal{Y}}_t^A$-monomial}\}$\[ainv\]. In particular for $m'\in A^{\text{inv}}_M$ we have $\overline{m'}{m'}^{-1}=\overline{M}M^{-1}$.
### The polynomials
For $M$ a ${\mathcal{Y}}_t$-monomial, denote $B^{\text{inv}}_M=t^{{\mathbb{Z}}}B\cap A^{\text{inv}}_M\label{binv}$.
\[expol\] For $m\in t^{{\mathbb{Z}}}B$ there is a unique $L_t(m)\in\mathfrak{K}_t^{\infty}$\[tltm\] such that: $$\overline{L_t(m)}=(\overline{m}m^{-1})L_t(m)$$ $$E_t(m)=L_t(m)+\underset{m'<m, m' \in B^{\text{inv}}_m}{\sum}P_{m',m}(t)L_t(m')$$ where $P_{m',m}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$.
Those polynomials $P_{m',m}(t)$ are called analogues to Kazhdan-Lusztig polynomials and the $L_t(m)$ ($m\in B$) for a canonical basis of $\mathfrak{K}_t^{f, \infty}$. Such polynomials were introduced by Nakajima [@Nab] for the $ADE$-case.
[[*Proof:*]{}]{}First consider $\overline{F_t(m)}$: it is in $\mathfrak{K}_t^{\infty}$ and has only one dominant ${\mathcal{Y}}_t$-monomial $\overline{m}$, so $\overline{F_t(m)}=\overline{m}m^{-1}F_t(m)$.
Let be $m=m_L>m_{L-1}>...>m_0$ the finite set $t^{{\mathbb{Z}}}D(m)\cap B^{\text{inv}}_m$ (see lemma \[ordrep\]) with a total ordering compatible with the partial ordering. Note that it follows from section \[invmon\] that for $L\geq l\geq 0$, we have $\overline{m_l}m_l^{-1}=\overline{m}m^{-1}$.
We have $E_t(m_0)=F_t(m_0)$ and so $\overline{E_t(m_0)}=\overline{m_0}m_0^{-1}E_t(m_0)$. As $B_{m_0}^{\text{inv}}=\{m_0\}$, we have $L_t(m_0)=E_t(m_0)$. We suppose by induction that the $L_t(m_l)$ ($L-1\geq l\geq 0$) are uniquely and well defined. In particular $m_l$ is of highest weight in $L_t(m_l)$, $\overline{L_t(m_l)}=\overline{m_l}m_l^{-1}L_t(m_l)=\overline{m}m^{-1}L_t(m_l)$, and we can write: $$\tilde{D}_t(m_L)\cap \mathfrak{K}_t^{\infty}={\mathbb{Z}}[t^{\pm}]F_t(m_L)\oplus\underset{0\leq l\leq L-1}{\bigoplus}{\mathbb{Z}}[t^{\pm}]L_t(m_l)$$ In particular consider $\alpha_{l,L}(t)\in{\mathbb{Z}}[t^{\pm}]$ such that: $$E_t(m)=F_t(m)+\underset{l<L}{\sum}\alpha_{l,L}(t)L_t(m_l)$$ We want $L_t(m)$ of the form : $$L_t(m)=F_t(m)+\underset{l<L}{\sum}\beta_{l,L}(t)L_t(m_l)$$ The condition $\overline{L_t(m)}=\overline{m}m^{-1}mL_t(m)$ means that the $\beta_{l,L}(t)$ are symmetric. The condition $P_{m',m}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$ means $\alpha_{l,L}(t)-\beta_{l,L}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$. So it suffices to prove that those two conditions uniquely define the $\beta_{l,L}(t)$: let us write $\alpha_{l,L}(t)=\alpha_{l,L}^+(t)+\alpha_{l,L}^0(t)+\alpha_{l,L}^-(t)$ (resp. $\beta_{l,L}(t)=\beta_{l,L}^+(t)+\beta_{l,L}^0(t)+\beta_{l,L}^-(t)$) where $\alpha_{l,L}^{\pm}(t)\in t^{\pm}{\mathbb{Z}}[t^{\pm}]$ and $\alpha_{l,L}^0(t)\in{\mathbb{Z}}$ (resp. for $\beta$). The condition $\alpha_{l,L}(t)-\beta_{l,L}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$ means $\beta_{l,L}^0(t)=\alpha_{l,L}^0(t)$ and $\beta_{l,L}^-(t)=\alpha_{l,L}^-(t)$. The symmetry of $\beta_{l,L}(t)$ means $\beta_{l,L}^+(t)=\beta_{l,L}^-(t^{-1})=\alpha_{l,L}^-(t^{-1})$.
### Examples for ${\mathfrak{g}}=sl_2$ {#exltcalc}
In this section we suppose that ${\mathfrak{g}}=sl_2$.
\[calcex\] Let $m\in t^{{\mathbb{Z}}}B$ such that $\forall l\in{\mathbb{Z}}, u_l(m)\leq 1$. Then $L_t(m)=F_t(m)$. Moreover: $$E_t(m)=L_t(m)+\underset{m'<m/m'\in B_m^{\text{inv}}}{\sum}t^{-R(m')}L_t(m')$$ where $R(m')\geq 1$ is given by $\pi_+(m'm^{-1})=A_{i_1,l_1}^{-1}...A_{i_R,l_R}^{-1}$. In particular for $m'\in B^{\text{inv}}_m$ such that $m'<m$ we have $P_{m',m}(t)=t^{-R(m')}$.
[[*Proof:*]{}]{}Note that a dominant monomial $m'<m$ verifies $\forall l\in{\mathbb{Z}}, u_l(m')\leq 1$ and appears in $E_t(m)$. We know that $\tilde{D}_m\cap\mathfrak{K}_t=\underset{m'\in t^{{\mathbb{Z}}}D_m\cap B^{\text{inv}}_m}{\bigoplus}{\mathbb{Z}}[t^{\pm}]F_t(m')$. We can introduce $P_{m',m}(t)\in{\mathbb{Z}}[t^{\pm}]$ such that: $$E_t(m)=F_t(m)+\underset{m'\in t^{{\mathbb{Z}}}D_m\cap B^{\text{inv}}-\{m\} }{\sum}P_{m',m}(t)F_t(m')$$ So by induction it suffices to show that $P_{m',m}(t)\in t^{-1}{\mathbb{Z}}[t^{-1}]$.
$P_{m',m}(t)$ is the coefficient of $m'$ in $E_t(m)$. A dominant ${\mathcal{Y}}_t$-monomial $M$ which appears in $E_t(m)$ is of the form: $$M=m(m_1...m_{R+1})^{-1}m_1t\tilde{A}_{l_1}^{-1}m_2t\tilde{A}_{l_2}^{-1}m_3...t\tilde{A}_{l_R}^{-1}m_{R+1}$$ where $l_1<...<l_R\in{\mathbb{Z}}$ verify $\{l_r+2,l_r-2\}\cap\{l_1,...,l_{r-1},l_{r+1},...,l_R\}$ is empty, $u_{l_r-1}(m)=u_{l_r+1}(m)=1$ and we have set $m_r=\underset{l_{r-1}<l\leq l_r}{\overset{\rightarrow}{\prod}}\tilde{Y}_l^{u_l(m)}$. Such a monomial appears one time in $E_t(m)$. In particular $P_{m',m}(t)=t^{\alpha}$ where $\alpha\in{\mathbb{Z}}$ is given by $M=t^{\alpha}m'$ that is to say $\overline{M}M^{-1}=t^{-2\alpha}{m'}^{-1}m'=t^{-2\alpha}m^{-1}m$. So we compute:
$\overline{M}M^{-1}=t^{-2R}\overline{m_{R+1}}\tilde{A}_{l_R}^{-1}\overline{m_R}...\tilde{A}_{l_1}^{-1}\overline{m_1}(\overline{m}_1^{-1}...\overline{m}_{R+1}^{-1})\overline{m}m_{R+1}^{-1}\tilde{A}_{l_R}m_R^{-1}...\tilde{A}_{l_1}m_1^{-1}(m_1...m_{R+1})m^{-1}
\\=t^{-2R}t^{4R}\tilde{A}_{l_R}^{-1}...\tilde{A}_{l_1}^{-1}\overline{m}\tilde{A}_{l_R}...\tilde{A}_{l_1}m^{-1}
\\=t^{2R}\tilde{A}_{l_R}^{-1}...\tilde{A}_{l_1}^{-1}\tilde{A}_{l_R}...\tilde{A}_{l_1}\overline{m}m^{-1}=t^{2R}\overline{m}m^{-1}$
Let us look at another example $m=\tilde{Y}_0^2\tilde{Y}_2$. We have: $$E_t(m)=L_t(m)+t^{-2}L_t(m')$$ where $m'=t\tilde{Y}_0^2\tilde{Y}_2\tilde{A}_1^{-1}\in B_{m}^{\text{inv}}$ and: $$L_t(m)=F_t(\tilde{Y}_0)F_t(\tilde{Y}_0\tilde{Y}_2)=\tilde{Y}_0(1+t\tilde{A}_1^{-1})\tilde{Y}_0\tilde{Y}_2(1+t\tilde{A}_3^{-1}(1+t\tilde{A}_1^{-1}))$$ $$L_t(m')=F_t(m')=t\tilde{Y}_0^2\tilde{Y}_2\tilde{A}_1^{-1}(1+t\tilde{A}_1^{-1})$$ Indeed the dominant monomials appearing in $E_t(m)$ are $m$ and $\tilde{Y}_0t\tilde{A}_1^{-1}\tilde{Y}_0\tilde{Y}_2+\tilde{Y}_0^2t\tilde{A}_1^{-1}\tilde{Y}_2=(1+t^{-2})m'$.
In particular: $P_{m',m}(t)=t^{-2}$.
### Example in non-simply laced case {#exnons}
We suppose that $C=\begin{pmatrix}2 & -2\\-1 & 2\end{pmatrix}$ and $m=\tilde{Y}_{2,0}\tilde{Y}_{1,5}$. The formulas for $E_t(\tilde{Y}_{2,0})$ and $E_t(\tilde{Y}_{1,5})$ are given is section \[fin\]. We have: $$E_t(m)=L_t(m)+t^{-1}L_t(m')$$ where $m'=t\tilde{Y}_{2,0}\tilde{Y}_{1,5}\tilde{A}_{2,2}^{-1}\tilde{A}_{1,4}^{-1}\in B_m^{\text{inv}}$ and: $$L_t(m')=F_t(m')=t\tilde{Y}_{2,0}\tilde{Y}_{1,5}\tilde{A}_{2,2}^{-1}\tilde{A}_{1,4}^{-1}(1+t\tilde{A}_{1,2}^{-1}(1+t\tilde{A}_{2,4}^{-1}(1+t\tilde{A}_{1,6}^{-1})))$$ Indeed the dominant monomials appearing in $E_t(m)$ are $m=\tilde{Y}_{2,0}\tilde{Y}_{1,5}$ and $\tilde{Y}_{2,0}t\tilde{A}_{2,2}^{-1}t\tilde{A}_{1,4}^{-1}\tilde{Y}_{1,5}=t^{-1}m'$.
In particular $P_{m',m}(t)=t^{-1}$.
Questions and conjectures {#quest}
=========================
Positivity of coefficients {#acase}
--------------------------
\[cpaconj\] If ${\mathfrak{g}}$ is of type $A_n$ ($n\geq 1$), the coefficients of $\chi_{q,t}(Y_{i,0})$ are in ${\ensuremath{\mathbb{N}}}[t^{\pm}]$.
[[*Proof:*]{}]{}We show that for all $i\in I$ the hypothesis of proposition \[cpfacile\] for $m=Y_{i,0}$ are verified; in particular the property $Q$ of section \[qn\] will be verified.
Let $i$ be in $I$. For $j\in I$, let us write $E(Y_{i,0})=\underset{m\in B_j}{\sum}\lambda_j(m)E_j(m)\in\mathfrak{K}_j$ where $\lambda_j(m)\in{\mathbb{Z}}$. Let $D$ be the set $D=\{\text{monomials of $E_j(m)$ }/j\in I,m\in B_j,\lambda_j(m)\neq 0\}$. It suffices to prove that for $j\in I$, $m\in B_j\cap D\Rightarrow u_j(m)\leq 1$ (because proposition \[cpfacile\] implies that for all $i\in I$, $F_t(\tilde{Y}_{i,0})=\pi^{-1}(E(Y_{i,0}))$).
As $E(Y_{i,0})=F(Y_{i,0})$, $Y_{i,0}$ is the unique dominant ${\mathcal{Y}}$-monomial in $E(Y_{i,0})$. So for a monomial $m\in D$ there is a finite sequence $\{m_0=Y_{i,0},m_1,...,m_R=m\}$ such that for all $1\leq r\leq R$, there is $r'<r$ and $j\in I$ such that $m_{r'}\in B_j$ and for $r'<r''\leq r$, $m_{r''}$ is a monomial of $E_j(m_{r'})$ and $m_{r''}m_{r''-1}^{-1}\in\{A_{j,l}^{-1}/l\in{\mathbb{Z}}\}$. Such a sequence is said to be adapted to $m$. Suppose there is $j\in I$ and $m\in B_j\cap D$ such that $u_j(m)\geq 2$. So there is $m'\leq m$ in $D\cap B_j$ such that $u_j(m)=2$. So we can consider $m_0\in D$ such that there is $j_0\in I$, $m_0\in B_{j_0}$, $u_{j_0}(m)\geq 2$ and for all $m'<m_0$ in $D$ we have $\forall j\in I,m'\in B_j\Rightarrow u_j(m')\leq 1$. Let us write: $$m_0=Y_{j_0,q^{l}}Y_{j_0,q^{m}}\underset{j\neq j_0}{\prod}m_0^{(j)}$$ where for $j\neq j_0$, $m_0^{(j)}=\underset{l\in{\mathbb{Z}}}{\prod}Y_{j,l}^{u_{j,l}(m_0)}$. In a finite sequence adapted to $m_0$, a term $Y_{j_0,q^{l}}$ or $Y_{j_0,q^{m}}$ must come from a $E_{j_0+1}(m_1)$ or a $E_{j_0-1}(m_1)$. So for example we have $m_1<m_0$ in $D$ of the form $m_1=Y_{j_0,q^{m}}Y_{j_0+1,q^{l-1}}\underset{j\neq j_0,j_0+1}{\prod}m_1^{(j)}$. In all cases we get a monomial $m_1<m_0$ in $D$ of the form: $$m_1=Y_{j_1,q^{m_1}}Y_{j_1+1,q^{l_1}}\underset{j\neq j_1,j_1+1}{\prod}m_1^{(j)}$$ But the term $Y_{j_1+1,q^{l-1}}$ can not come from a $E_{j_1}(m_2)$ because we would have $u_{j_1}(m_2)\geq 2$. So we have $m_2<m_1$ in $D$ of the form: $$m_2=Y_{j_2,q^{m_2}}Y_{j_2+2,q^{l_2}}\underset{j\neq j_2,j_2+1,j_2+2}{\prod}m_2^{(j)}$$ This term must come from a $E_{j_2-1}, E_{j_2+3}$. By induction, we get $m_N<m_0$ in $D$ of the form : $$m_N=Y_{1,q^{m_N}}Y_{n,q^{l_N}}\underset{j\neq 1,..,n}{\prod}m_N^{(j)}=Y_{1,q^{m-N}}Y_{n,q^{l_N}}$$ It is a dominant monomial of $D\subset D_{Y_{i,0}}$ which is not $Y_{i,0}$. It is impossible (proof of lemma \[copieun\]).
An analog result is also geometrically proved by Nakajima for the $ADE$-case in [@Nab] (it is also algebraically for $AD$-cases proved in [@Nac]). Those results and the explicit formulas in $n=1,2$-cases (see section \[fin\]) suggest:
\[conun\] The coefficients of $F_t(\tilde{Y}_{i,0})=\chi_{q,t}(Y_{i,0})$ are in ${\ensuremath{\mathbb{N}}}[t^{\pm}]$.
In particular for $m\in B$, the coefficients of $E_t(m)$ would be in ${\ensuremath{\mathbb{N}}}[t^{\pm}]$; moreover $\chi_{q,t}(Y_{i,0})$ and $\chi_q(Y_{i,0})$ would have the same monomials, the $t$-algorithm would stop and $\text{Im}(\chi_{q,t})\subset{\mathcal{Y}}_t$.
At the time he wrote this paper the author does not know a general proof of the conjecture. However a case by case investigation seems possible: the cases $G_2, B_2, C_2$ are checked in section \[fin\] and the cases $F_4, B_n, C_n$ ($n\leq 10$) have been checked on a computer. So a combinatorial proof for series $B_n, C_n$ ($n\geq 2$) analog to the proof of proposition \[cpaconj\] would complete the picture.
Decomposition in irreducible modules
------------------------------------
The proposition \[calcex\] suggests:
\[condeux\] For $m\in B$ we have $\pi_+(L_t(m))=L(m)$.
In the $ADE$-case the conjecture \[condeux\] is proved by Nakajima with the help of geometry ([@Nab]). In particular this conjecture implies that the coefficients of $\pi_+(L_t(m))$ are non negative. It gives a way to compute explicitly the decomposition of a standard module in irreducible modules, because the conjecture \[condeux\] implies: $$E(m)=L(m)+\underset{m'<m}{\sum}P_{m',m}(1)L(m')$$ In particular we would have $P_{m',m}(1)\geq 0$.
In section \[exltcalc\] we have studied some examples:
-In proposition \[calcex\] for ${\mathfrak{g}}=sl_2$ and $m\in B$ such that $\forall l\in{\mathbb{Z}}, u_l(m)\leq 1$: we have $\pi_+(L_t(m))=F(m)=L(m)$ and: $$E(m)=\underset{m'\in B/m'\leq m}{\sum}L(m')$$ -For ${\mathfrak{g}}=sl_2$ and $m=\tilde{Y}_0^2\tilde{Y}_2$: we have $\pi_+(L_t(m))=F(Y_0)F(Y_0Y_2)=L(m)$ and: $$E(Y_0^2Y_2)=L(Y_0^2Y_2)+L(Y_0)$$ Note that $L(Y_0^2Y_2)$ has two dominant monomials $Y_0^2Y_2$ and $Y_0$ because $Y_0^2Y_2$ is irregular (lemma \[dominl\]).
-For $C=B_2$ and $m=\tilde{Y}_{2,0}\tilde{Y}_{1,5}$. The $\pi_+(L_t(\tilde{Y}_{2,0}\tilde{Y}_{1,5}))$ has non negative coefficients and the conjecture implies $E(Y_{2,0}Y_{1,5})=L(Y_{2,0}Y_{1,5})+L(Y_{1,1})$.
Further applications and generalizations
----------------------------------------
We hope to address the following questions in the future:
### Iterated deformed screening operators
Our presentation of deformed screening operators as commutators leads to the definition of iterated deformed screening operators. For example in order 2 we set: $$\tilde{S}_{j,i,t}(m)=[\underset{l\in{\mathbb{Z}}}{\sum}\tilde{S}_{j,l},S_{i,t}(m)]$$
### Possible generalizations
Some generalizations of the approach used in this article will be studied:
a\) the theory of $q$-characters at roots of unity ([@Fre3]) suggests a generalization to the case $q^N=1$.
b\) in this article we decided to work with ${\mathcal{Y}}_t$ which is a quotient of ${\mathcal{Y}}_u$. The same construction with ${\mathcal{Y}}_u$ will give characters with an infinity of parameters of deformation $t_r=\text{exp}(\underset{l>0}{\sum}h^{2l}q^{lr}c_l)$ ($r\in{\mathbb{Z}}$).
c\) our construction is independent of representation theory and could be established for other generalized Cartan matrices (in particular for twisted affine cases).
Appendix {#fin}
========
There are 5 types of semi-simple Lie algebra of rank 2: $A_1\times A_1$, $A_2$, $C_2$, $B_2$, $G_2$ (see for example [@Kac]). In each case we give the formula for $E(1),E(2)\in\mathfrak{K}$ and we see that the hypothesis of proposition \[cpfacile\] is verified. In particular we have $E_t(\tilde{Y}_{1,0})=\pi^{-1}(E(1)),E_t(\tilde{Y}_{2,0})=\pi^{-1}(E(2))\in\mathfrak{K}_t$.
Following [@Fre], we represent the $E(1),E(2)\in\mathfrak{K}$ as a $I\times{\mathbb{Z}}$-oriented colored tree. For $\chi\in\mathfrak{K}$ the tree $\Gamma_{\chi}$ is defined as follows: the set of vertices is the set of ${\mathcal{Y}}$-monomials of $\chi$. We draw an arrow of color $(i,l)$ from $m_1$ to $m_2$ if $m_2=A_{i,l}^{-1}m_1$ and if in the decomposition $\chi=\underset{m\in B_i}{\sum}\mu_m L_i(m)$ there is $M\in B_i$ such that $\mu_M\neq 0$ and $m_1,m_2$ appear in $L_i(M)$.
Then we give a formula for $E_t(\tilde{Y}_{1,0}),E_t(\tilde{Y}_{2,0})$ and we write it in $\mathfrak{K}_{1,t}$ and in $\mathfrak{K}_{2,t}$.
$A_1\times A_1$-case
--------------------
The Cartan matrix is $C=\begin{pmatrix}2 & 0\\0 & 2 \end{pmatrix}$ and $r_1=r_2=1$ (note that in this case the computations keep unchanged for all $r_1,r_2$).
$$\xymatrix{Y_{1,0} \ar[d]^{1,1}
\\Y_{1,2}^{-1}}\text{ and }\xymatrix{Y_{2,0} \ar[d]^{2,1}
\\Y_{2,2}^{-1}}$$
$$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1})=\tilde{Y}_{1,0}(1+t\tilde{A}_{1,1}^{-1})\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0}+\tilde{Y}_{1,2}^{-1}\in\mathfrak{K}_{2,t}$$ $$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0}+Y_{2,2}^{-1})=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,1}^{-1})\in\mathfrak{K}_{2,t}$$ $$=\tilde{Y}_{2,0}+\tilde{Y}_{2,2}^{-1}\in\mathfrak{K}_{1,t}$$
$A_2$-case
----------
The Cartan matrix is $C=\begin{pmatrix}2 & -1\\-1 & 2 \end{pmatrix}$. It is symmetric, $r_1=r_2=1$:
$$\xymatrix{Y_{1,0} \ar[d]^{1,1}
\\Y_{1,2}^{-1}Y_{2,1}\ar[d]^{2,2}
\\Y_{2,3}^{-1}}
\text{ and }
\xymatrix{Y_{2,0} \ar[d]^{2,1}
\\Y_{2,2}^{-1}Y_{1,1}\ar[d]^{1,2}
\\Y_{1,3}^{-1}}$$
$$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1}Y_{2,1}+Y_{2,3}^{-1})=\tilde{Y}_{1,0}(1+t\tilde{A}_{1,1}^{-1})+\tilde{Y}_{2,3}^{-1}\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0}+:\tilde{Y}_{1,2}^{-1}\tilde{Y}_{2,1}:(1+t\tilde{A}_{2,2}^{-1})\in\mathfrak{K}_{2,t}$$ $$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0}+Y_{2,2}^{-1}Y_{1,1}+Y_{1,3}^{-1})=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,1}^{-1})+\tilde{Y}_{1,3}^{-1}\in\mathfrak{K}_{2,t}$$ $$=\tilde{Y}_{2,0}+:\tilde{Y}_{2,2}^{-1}\tilde{Y}_{1,1}:(1+t\tilde{A}_{1,2}^{-1})\in\mathfrak{K}_{1,t}$$
$C_2,B_2$-case
--------------
The two cases are dual so it suffices to compute for the Cartan matrix $C=\begin{pmatrix}2 & -2\\-1 & 2 \end{pmatrix}$ and $r_1=1$, $r_2=2$.
$$\xymatrix{Y_{1,0} \ar[d]^{1,1}
\\Y_{1,2}^{-1}Y_{2,1}\ar[d]^{2,3}
\\Y_{2,5}^{-1}Y_{1,4}\ar[d]^{1,5}
\\Y_{1,6}^{-1}}
\text{ and }
\xymatrix{Y_{2,0}\ar[d]^{2,2}
\\Y_{2,4}^{-1}Y_{1,1}Y_{1,3}\ar[d]^{1,4}
\\Y_{1,1}Y_{1,5}^{-1}\ar[d]^{1,2}
\\Y_{1,3}^{-1}Y_{1,5}^{-1}Y_{2,2}\ar[d]^{2,4}
\\Y_{2,6}^{-1}}$$
$$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1}Y_{2,1}+Y_{2,5}^{-1}Y_{1,4}+Y_{1,6}^{-1})$$ $$=\tilde{Y}_{1,0}(1+t\tilde{A}_{1,1}^{-1})+:\tilde{Y}_{2,5}^{-1}\tilde{Y}_{1,4}:(1+t\tilde{A}_{1,5}^{-1})\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0}+:\tilde{Y}_{1,2}^{-1}\tilde{Y}_{2,1}:(1+t\tilde{A}_{2,3}^{-1})+\tilde{Y}_{1,6}^{-1}\in\mathfrak{K}_{2,t}$$
$$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0}+Y_{2,4}^{-1}Y_{1,1}Y_{1,3}+Y_{1,1}Y_{1,5}^{-1}+Y_{1,3}^{-1}Y_{1,5}^{-1}Y_{2,2}+Y_{2,6}^{-1})$$ $$=\tilde{Y}_{2,0}+:\tilde{Y}_{2,4}^{-1}\tilde{Y}_{1,1}\tilde{Y}_{1,3}:(1+t\tilde{A}_{1,4}^{-1}+t^2\tilde{A}_{1,4}^{-1}\tilde{A}_{1,2}^{-1})+\tilde{Y}_{2,6}^{-1}\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,2}^{-1})+:\tilde{Y}_{1,1}\tilde{Y}_{1,5}^{-1}:+:\tilde{Y}_{1,3}^{-1}\tilde{Y}_{1,5}^{-1}\tilde{Y}_{2,2}:(1+t\tilde{A}_{2,4}^{-1})\in\mathfrak{K}_{2,t}$$
$G_2$-case
----------
The Cartan matrix is $C=\begin{pmatrix}2 & -3\\-1 & 2 \end{pmatrix}$ and $r_1=1$, $r_2=3$.
### First fundamental representation
$$\xymatrix{Y_{1,0} \ar[d]^{1,1}
\\Y_{1,2}^{-1}Y_{2,1}\ar[d]^{2,4}
\\Y_{2,7}^{-1}Y_{1,4}Y_{1,6}\ar[d]^{1,7}
\\Y_{1,4}Y_{1,8}^{-1}\ar[d]^{1,5}
\\Y_{1,6}^{-1}Y_{1,8}^{-1}Y_{2,5}\ar[d]^{2,8}
\\Y_{2,11}^{-1}Y_{1,10}\ar[d]^{1,11}
\\Y_{1,12}^{-1}}$$
$$E_t(\tilde{Y}_{1,0})=\pi^{-1}(Y_{1,0}+Y_{1,2}^{-1}Y_{2,1}+Y_{2,7}^{-1}Y_{1,4}Y_{1,6}+Y_{1,4}Y_{1,6}+Y_{1,6}^{-1}Y_{1,8}^{-1}Y_{2,5}+Y_{2,11}^{-1}Y_{1,10}+Y_{1,12}^{-1})$$ $$=\tilde{Y}_{1,0}(1+\tilde{A}_{1,1}^{-1})
+:\tilde{Y}_{2,7}^{-1}\tilde{Y}_{1,4}\tilde{Y}_{1,6}:(1+t\tilde{A}_{1,7}^{-1}+t^2\tilde{A}_{1,7}^{-1}\tilde{A}_{1,5}^{-1})
+:\tilde{Y}_{2,11}^{-1}\tilde{Y}_{1,10}:(1+t\tilde{A}_{1,11}^{-1})\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{1,0}
+:\tilde{Y}_{1,2}^{-1}\tilde{Y}_{2,1}:(1+t\tilde{A}_{2,4}^{-1})
+:\tilde{Y}_{1,4}\tilde{Y}_{1,6}:
+:\tilde{Y}_{1,6}^{-1}\tilde{Y}_{1,8}^{-1}\tilde{Y}_{2,5}:(1+t\tilde{A}_{2,8}^{-1})+:\tilde{Y}_{1,12}^{-1}:\in\mathfrak{K}_{2,t}$$
### Second fundamental representation
$$\xymatrix{&Y_{2,0} \ar[d]^{2,3}&
\\&Y_{2,6}^{-1}Y_{1,5}Y_{1,3}Y_{1,1}\ar[d]^{1,6}&
\\&Y_{1,7}^{-1}Y_{1,3}Y_{1,1}\ar[d]^{1,4}&
\\&Y_{2,4}Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,1}\ar[ld]^{1,2}\ar[rd]^{2,7}&
\\Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,3}^{-1}Y_{2,4}Y_{2,2}\ar[d]^{2,5}\ar[rd]^{2,7}&&Y_{2,10}^{-1}Y_{1,9}Y_{1,1}\ar[ld]^{1,2}\ar[d]^{1,10}
\\Y_{2,4}Y_{2,8}^{-1}\ar[d]^{2,7}&Y_{2,2}Y_{2,10}^{-1}Y_{1,9}Y_{1,3}^{-1}\ar[ld]^{2,5}\ar[rd]^{1,10}&Y_{1,11}^{-1}Y_{1,1}\ar[d]^{1,2}
\\Y_{2,8}^{-1}Y_{2,10}^{-1}Y_{1,9}Y_{1,7}Y_{1,5}\ar[rd]^{1,10}&&Y_{2,2}Y_{1,11}^{-1}Y_{1,3}^{-1}\ar[ld]^{2,5}
\\&Y_{2,8}^{-1}Y_{1,11}^{-1}Y_{1,7}Y_{1,5}\ar[d]^{1,8}&
\\&Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,5}\ar[d]^{1,6}&
\\&Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,7}^{-1}Y_{2,6}\ar[d]^{2,9}&
\\&Y_{2,12}^{-1}&}$$
$$E_t(\tilde{Y}_{2,0})=\pi^{-1}(Y_{2,0}
+Y_{2,6}^{-1}Y_{1,5}Y_{1,3}Y_{1,1}
+Y_{1,7}^{-1}Y_{1,3}Y_{1,1}0
+Y_{2,4}Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,1}
+Y_{1,7}^{-1}Y_{1,5}^{-1}Y_{1,3}^{-1}Y_{2,4}Y_{2,2}$$ $$+Y_{2,10}^{-1}Y_{1,9}Y_{1,1}
+Y_{2,4}Y_{2,8}^{-1}+Y_{2,2}Y_{2,10}^{-1}Y_{1,9}Y_{1,3}^{-1}
+Y_{1,11}^{-1}Y_{1,1}
+Y_{2,8}^{-1}Y_{2,10}^{-1}Y_{1,9}Y_{1,7}Y_{1,5}
+Y_{2,2}Y_{1,11}^{-1}Y_{1,3}^{-1}$$ $$+Y_{2,8}^{-1}Y_{1,11}^{-1}Y_{1,7}Y_{1,5}
+Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,5}+Y_{1,11}^{-1}Y_{1,9}^{-1}Y_{1,7}^{-1}Y_{2,6}+Y_{2,12}^{-1})$$ We use the following relations to write $E_t(\tilde{Y}_{2,0})$ in $\mathfrak{K}_{1,t}$ and in $\mathfrak{K}_{2,t}$: $\tilde{A}_{1,2}\tilde{A}_{2,7}=\tilde{A}_{2,7}\tilde{A}_{1,2}$, $\tilde{A}_{2,5}\tilde{A}_{2,7}=\tilde{A}_{2,7}\tilde{A}_{2,5}$, $\tilde{A}_{1,2}\tilde{A}_{1,10}=\tilde{A}_{1,10}\tilde{A}_{1,2}$, $\tilde{A}_{2,5}\tilde{A}_{1,10}=\tilde{A}_{1,10}\tilde{A}_{2,5}$. $$E_t(\tilde{Y}_{2,0})=\tilde{Y}_{2,0}
+:\tilde{Y}_{2,6}^{-1}\tilde{Y}_{1,5}\tilde{Y}_{1,3}\tilde{Y}_{1,1}:(1+t\tilde{A}_{1,6}^{-1}(1+t\tilde{A}_{1,4}^{-1}(1+t\tilde{A}_{1,2}^{-1})))
+:\tilde{Y}_{2,10}^{-1}\tilde{Y}_{1,9}\tilde{Y}_{1,1}:(1+t\tilde{A}_{1,2}^{-1})(1+t\tilde{A}_{1,10}^{-1})$$ $$+:\tilde{Y}_{2,4}\tilde{Y}_{2,8}^{-1}:
+:\tilde{Y}_{2,8}^{-1}\tilde{Y}_{2,10}^{-1}\tilde{Y}_{1,9}\tilde{Y}_{1,7}\tilde{Y}_{1,5}:(1+t\tilde{A}_{1,10}^{-1}(1+t\tilde{A}_{1,8}^{-1}(1+t\tilde{A}_{1,6}^{-1})))
+\tilde{Y}_{2,12}^{-1}\in\mathfrak{K}_{1,t}$$ $$=\tilde{Y}_{2,0}(1+t\tilde{A}_{2,3}^{-1})
+:\tilde{Y}_{1,7}^{-1}\tilde{Y}_{1,3}\tilde{Y}_{1,1}:
+:\tilde{Y}_{2,4}\tilde{Y}_{1,7}^{-1}\tilde{Y}_{1,5}^{-1}\tilde{Y}_{1,1}:(1+t\tilde{A}_{2,7}^{-1})$$ $$+:\tilde{Y}_{1,7}^{-1}\tilde{Y}_{1,5}^{-1}\tilde{Y}_{1,3}^{-1}\tilde{Y}_{2,4}\tilde{Y}_{2,2}:(1+t\tilde{A}_{2,7}^{-1})(1+t\tilde{A}_{2,5}^{-1})
+:\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,1}:
+:\tilde{Y}_{2,2}\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,3}^{-1}:(1+t\tilde{A}_{2,5}^{-1})$$ $$+:\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,9}^{-1}\tilde{Y}_{1,5}:
+:\tilde{Y}_{1,11}^{-1}\tilde{Y}_{1,9}^{-1}\tilde{Y}_{1,7}^{-1}\tilde{Y}_{2,6}:(1+t\tilde{A}_{2,9}^{-1})\in\mathfrak{K}_{2,t}$$
Notations {#notations .unnumbered}
=========
------------------------------------------------------------------------------------- -------------------------------------------------------- ---
$A$ set of ${\mathcal{Y}}$-monomials p
$A_t$ set of ${\mathcal{Y}}_t$-monomials p
$A_m^{\text{inv}}, B_m^{\text{inv}}$ set of ${\mathcal{Y}}_t$-monomials p
$\overset{\infty}{A}_t$ product module p
$\alpha$ map $(I\times {\mathbb{Z}})^2\rightarrow {\mathbb{Z}}$ p
$\alpha(m)$ character p
$a_i[m]$ element of $\mathcal{H}$ p
$\tilde{A}_{i,l}, \tilde{A}_{i,l}^{-1}$ elements of ${\mathcal{Y}}_u$ or ${\mathcal{Y}}_t$ p
$A_{i,l}, A_{i,l}^{-1}$ elements of ${\mathcal{Y}}$ p
$B$ a set of ${\mathcal{Y}}$-monomials p
$B_i$, $B_J$ a set of ${\mathcal{Y}}$-monomials p
$(B_{i,j})$ symmetrized
Cartan matrix p
$\beta$ map $(I\times {\mathbb{Z}})^2\rightarrow {\mathbb{Z}}$ p
$(C_{i,j})$ Cartan matrix p
$(\tilde{C}_{i,j})$ inverse of $C$ p
$c_r$ central element of $\mathcal{H}$ p
$d$ bicharacter p
$D_{m,K}, D_m$ set of monomials p
$\tilde{D}_m$ submodule of ${\mathcal{Y}}_t^{\infty}$ p
$E_i(m)$ element of $\mathfrak{K}_i$ p
$E_{i,t}(m)$ element of $\mathfrak{K}_{i,t}$ p
$E_{i,t}^m$ map p
$E(m)$ element of $\mathfrak{K}$ p
$E_t(m)$ element of $\mathfrak{K}_{t}^{\infty}$ p
$F_i(m)$ element of $\mathfrak{K}_i$ p
$F_{i,t}(m)$ element of $\mathfrak{K}_{i,t}$ p
$F(m)$ element of $\mathfrak{K}$ p
$F_t(m)$ element of $\mathfrak{K}_{t}^{\infty}$ p
$\gamma$ map $(I\times {\mathbb{Z}})^2\rightarrow {\mathbb{Z}}$ p
$\mathcal{H}$ Heisenberg algebra p
$\mathcal{H}^+, \mathcal{H}^-$ subalgebras of $\mathcal{H}$ p
$\mathcal{H}_h$ formal series in $\mathcal{H}$ p
$\mathcal{H}_t$ quotient of $\mathcal{H}_h$ p
$\mathcal{H}_t^+, \mathcal{H}_t^-$ subalgebras of $\mathcal{H}_t$ p
$\mathfrak{K}_i, \mathfrak{K}_J, \mathfrak{K}$ subrings of ${\mathcal{Y}}$ p
$\mathfrak{K}_{i,t}, \mathfrak{K}_{J,t}, \mathfrak{K}_t$ subrings of ${\mathcal{Y}}_t$ p
$\mathfrak{K}_{i,t}^{\infty}, \mathfrak{K}_{J,t}^{\infty}, \mathfrak{K}_t^{\infty}$ subrings of ${\mathcal{Y}}_t^{\infty}$ p
$\chi_q$ morphism
of $q$-characters p
$\chi_{q,t}$ morphism
of $q,t$-characters p
$L_i(m)$ element of $\mathfrak{K}_i$ p
$L_t(m)$ element of $\mathfrak{K}_t^{\infty}$ p
$:m:$ monomial in $A$ p
$\tilde{m}$ monomial in $A_t$ p
$N,N_t,\mathcal{N},\mathcal{N}_t$ characters, bicharacters p
$P(n)$ property of $n\in{\ensuremath{\mathbb{N}}}$ p
------------------------------------------------------------------------------------- -------------------------------------------------------- ---
--------------------------------------------------------- ---------------------------------------------------------------- ---
$\pi$ map p
$\pi_r$ map to ${\mathbb{Z}}$ p
$\pi_+,\pi_-$ endomorphisms of
$\mathcal{H}_h$, $\mathcal{H}_t$ p
$q$ complex number p
$Q(n)$ property of $n\in{\ensuremath{\mathbb{N}}}$ p
$\text{Rep}$ Grothendieck ring p
$\text{Rep}_t$ deformed
Grothendieck ring p
$s(m_r)_J,s(m_r)$ sequences of ${\mathbb{Z}}[t^{\pm}]$ p
$S_i$ screening operator p
$\tilde{S}_{i,l}$ screening current p
$S_{i,t}$ $t$-screening operator p
$t$ central element of ${\mathcal{Y}}_t$ p
$t_R$ central element of ${\mathcal{Y}}_u$ p
$u_{i,l}$ multiplicity of $Y_{i,l}$ p
$u_i$ sum of the $u_{i,l}$ p
$\mathfrak{U}$ subring of ${\ensuremath{\mathbb{Q}}}(q)$ p
${\mathcal{U}}_q(\hat{{\mathfrak{g}}})$ quantum
affine algebra p
${\mathcal{U}}_q(\hat{{\mathfrak{h}}})$ Cartan algebra p
$X_{i,l}$ element of $\text{Rep}$ p
$y_i[m]$ element of $\mathcal{H}$ p
$Y_{i,l}, Y_{i,l}^{-1}$ elements of ${\mathcal{Y}}$ p
$\tilde{Y}_{i,l},\tilde{Y}_{i,l}^{-1}$ elements of ${\mathcal{Y}}_u$ or ${\mathcal{Y}}_t$ p
${\mathcal{Y}}$ subalgebra of $\mathcal{H}_h$ p
${\mathcal{Y}}_t$ quotient of ${\mathcal{Y}}_u$ p
${\mathcal{Y}}_t^+, {\mathcal{Y}}_t^-$ subalgebras of $\mathcal{H}_t$ p
${\mathcal{Y}}_u$ subalgebra of $\mathcal{H}_h$ p
${\mathcal{Y}}_{i,t}$ ${\mathcal{Y}}_t$-module p
${\mathcal{Y}}_{i,u}$ ${\mathcal{Y}}_u$-module p
${\mathcal{Y}}_t^{\infty}, {\mathcal{Y}}_t^{A, \infty}$ submodules of $\overset{\infty}{A}_t$ p
${\mathcal{Y}}_t^A, {\mathcal{Y}}_t^{A,K}$ submodules of ${\mathcal{Y}}_t$ p
$z$ indeterminate p
$::$ endomorphism of
$\mathcal{H}, \mathcal{H}_h, {\mathcal{Y}}_u, {\mathcal{Y}}_t$ p
$*$ deformed
multiplication p
--------------------------------------------------------- ---------------------------------------------------------------- ---
[99]{}
, [*Groupes et algèbres de Lie*]{}
[Chapitres IV-VI, Hermann (1968)]{}
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[in Representations of groups (Banff, AB, 1994),59-78, CMS Conf. Proc, 16, Amer. Math. Soc., Providence, RI (1995)]{}
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[Cambridge University Press, Cambridge (1994)]{}
, [*Deformations of $W$-algebras associated to simple Lie algebras*]{}
[Comm. Math. Phys. 197, no. 1, 1–32 (1998)]{}
, [*The $q$-Characters of Representations of Quantum Affine Algebras and Deformations of $W$-Algebras*]{}
[Recent Developments in Quantum Affine Algebras and related topics, Cont. Math., vol. 248, pp 163-205 (1999)]{}
, [*Combinatorics of $q$-Characters of Finite-Dimensional Representations of Quantum Affine Algebras*]{}
[Comm. Math. Phy., vol 216, no. 1, pp 23-57 (2001)]{}
, [*The $q$-characters at roots of unity*]{}
[Adv. Math. 171, no. 1, 139–167 (2002)]{}
, [*t-analogues des opérateurs d’écrantage associés aux q-caractères*]{}
[Int. Math. Res. Not., vol. 2003, no. 8, pp 451-475 (2003)]{}
, [*Infinite dimensional Lie algebras*]{}
[3rd Edition, Cambridge University Press (1990)]{}
, [*Representations of Coxeter Groups and Hecke Algebras*]{}
[Inventiones math. 53, pp. 165-184 (1979)]{}
, [*$t$-Analogue of the $q$-Characters of Finite Dimensional Representations of Quantum Affine Algebras*]{}
[“Physics and Combinatorics”, Proc. Nagoya 2000 International Workshop, World Scientific, pp 181-212 (2001)]{}
, [*Quiver Varieties and $t$-Analogs of $q$-Characters of Quantum Affine Algebras*]{}
[Preprint arXiv:math.QA/0105173]{}
, [*t-analogs of q-characters of quantum affine algebras of type $A_n$, $D_n$*]{}
[Preprint arXiv:math.QA/0204184]{}
, [*Représentations des groupes quantiques*]{}
[Séminaire Bourbaki exp. no. 744, Astérisque 201-203, 443-83, SMF (1992)]{}
, [*Perverse Sheaves and Quantum Grothendieck Rings*]{}
[Preprint arXiv:math.QA/0103182]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose a list of conditions that consistency with thermodynamics imposes on linear and nonlinear generalizations of standard unitary quantum mechanics that assume a set of true quantum states without the restriction $\rho^2=\rho$ even for strictly isolated systems and that are to be considered in experimental tests of the existence of intrinsic (spontaneous) decoherence at the microscopic level.'
author:
- Gian Paolo Beretta
title: |
Nonlinear extensions of Schrödinger–vonNeumann quantum dynamics:\
a list of conditions for compatibility with thermodynamics
---
Understanding and predicting ’decoherence’ is important in future applications involving nanometric devices, fast switching times, clock synchronization, superdense coding, quantum computation, teleportation, quantum cryptography, etc. where entanglement structure and dynamics play a key role [@decoherence]. In the last three decades it has also been central in exploring possible limitations to the validity of standard unitary quantum mechanics (QM), by studying a variety of linear and nonlinear extensions that have been advocated by several authors on a variety of conceptual grounds [@extensions]. It has been suggested [@Domokos] that long-baseline neutrino oscillation experiments may provide means of testing the existence of spontaneous decoherence at the microscopic level and the validity of linear and nonlinear extensions of the Schrödinger–vonNeumann equation of motion of QM, thus prompting a renewed interest on such extensions [@Domokos; @Czachor; @Gheorghiu].
The aim of this Letter is to list the main conditions that must be imposed and checked on linear and nonlinear extensions of QM which assume an augmented set of true quantum states described by state operators $\rho$ without the restriction $\rho^2=\rho$. The reasoning and framework proposed here should provide useful guidance also to current efforts to define general measures of entanglement [@Yukalov].
The conditions proposed here form a very restrictive set. Yet, at least one possible extension has been proved to satisfy them all[@Beretta], with mathematics that has been partially rediscovered recently by researchers in different contexts and fields [@Gheorghiu; @Englman; @Aerts].
[**1. Causality. Forward and backward in time.\
**]{}\[causality\] We consider the set ${{\mathcal P}}$ of all linear, hermitian, nonnegative-definite, unit-trace operators $\rho$ (without the restriction $\rho^2=\rho$) on the standard QM Hilbert space ${{\mathcal H}}$ associated with a strictly isolated system[@isolated]. Every solution of the equation of motion, i.e., every trajectory $u(t,\rho)$ passing at time $t=0$ through state $\rho$ in ${{\mathcal P}}$, should lie entirely in ${{\mathcal P}}$ for all times $t$, $-\infty<t<+\infty$. This strong causality condition is nontrivial and demanding both from the conceptual and the technical mathematical points of view.
[**2. Conservation of energy and other invariants.**]{}\[energy\] The value of the energy functional $e(\rho)={{\rm Tr}}(\rho H)$, where $H$ is the standard QM Hamiltonian operator associated with the isolated system \[$H\ne H(t)$\], must remain invariant along every trajectory. If ${{\mathcal H}}$ is the Fock space of an isolated system consisting of $M$ types of elementary constituents (e.g., atoms and molecules if chemical and nuclear reactions are inhibited; or atomic nuclei and electrons for modelling chemical reactions) each with a number operator $N_i$ ($[H,N_i]=0$ and $[N_i,N_j]=0$), then also the value of each number-of-constituents functional $n_i(\rho)={{\rm Tr}}(\rho N_i)$ must remain invariant along every trajectory. Depending on the type of system, there may be other time-invariant functionals, e.g., the total momentum components $p_j (\rho)={{\rm Tr}}(\rho P_j)$, with $j=x,y,z$, for a free particle (in which case Galileian invariance must also be verified, for $[H,P_j]=0$ and $[P_i,P_j]=0$). In what follows, we denote by $g_i(\rho)={{\rm Tr}}(\rho G_i)$ the set of non-Hamiltonian time-invariant functionals, if any, with $[H,G_i]=0$ and $[G_i,G_j]=0$ (clearly, $H$ and the $G_i$’s have a common eigenbasis that we denote by $\{|\psi_\ell\rangle\}$).
[**3. Standard QM unitary evolution of $\rho^2=\rho$ states.\
**]{}\[standardQM\] Unitary time evolution of the states of QM according to the Schrödinger equation of motion must be compatible with the more general dynamical law. These trajectories, passing through any state $\rho$ such that $\rho^2=\rho$ and entirely contained in the state domain of QM, must be solutions also of the extended dynamical law. Because the states of QM are extreme points of the state domain ${{\mathcal P}}$, the trajectories of QM must be boundary solutions (limit cycles) of the extended dynamical law.
In general, any extended dynamical equation may be written in the form $$\begin{aligned}
\label{eqofmotion}
&&{\displaystyle {{\frac{\displaystyle {\rm d}\rho}{\displaystyle {\rm d}t}}} = - \frac{i}{\hbar}}[H,\rho] +D_M \\&&{\rm with}\ D_M={{\bf \hat D}}_M(\rho,H,G_i,\dots) \ ,\end{aligned}$$ where operator $D_M$ represents the [*dissipative*]{} part of the equation of motion and may depend linearly and/or nonlinearly (through superoperator ${{\bf \hat D}}_M$) on the state operator $\rho$, on the Hamiltonian $H$, on the linear operators $G_i$ associated with the other time invariants (if any), as well as on the structure and the number $M$ of elementary constituents of the system. Like the Schrödinger–vonNeumann term, also the dissipative term should not be responsible for rates of change of any of the invariant functionals ${{\rm Tr}}(\rho)$, $e(\rho) $, $g_i(\rho) $ and, therefore, $${{\rm Tr}}D_M=0\qquad{{\rm Tr}}D_M H=0\qquad{{\rm Tr}}D_M G_i=0\ .$$
If the complete dynamics preserves the feature of uniqueness of solutions throughout the state domain ${{\mathcal P}}$, then pure states can only evolve according to the Schrödinger equation of motion and, therefore, ${{\bf \hat D}}_M(\rho,H,G_i,\dots)=0$ when $\rho^2=\rho$. This feature that may be responsible for hiding the presence of deviations from QM in experiments where the isolated system is prepared in a pure state. It also implies that no trajectory can enter or leave the state domain of QM. Thus, by continuity, there must be trajectories that approach indefinitely these boundary solutions (of course, this can only happen backward in time, as $t\to -\infty$, for otherwise the entropy of the isolated system would decrease in forward time).
[**4. Conservation of effective Hilbert space dimensionality.**]{}
Unitary dynamics \[Eq. (\[eqofmotion\]) with $D_M=0$\] would maintain unchanged all the eigenvalues of $\rho$ and therefore cannot satisfy Condition 5 below [@unitary]. Instead, we only require that the dynamical law maintains zero the initially zero eigenvalues of $\rho$ and, therefore, conserves the cardinality of the set of zero eigenvalues, $ \dim{{\rm Ker}}(\rho)$. In other words, if the isolated system is prepared in a state that does not ’occupy’ the eigenvector $ |\psi_\ell\rangle $ of $H$ (and the $G_i$’s), i.e., if $\rho(0)|\psi_\ell\rangle =0$ (so that $|\psi_\ell\rangle $ is also an eigenvector of $\rho$ corresponding to a zero eigenvalue), then such energy eigenvector remains ’unoccupied’ at all times, i.e., $\rho(t)|\psi_\ell\rangle =0$.
This condition preserves an important feature that allows remarkable model simplifications within QM: the dynamics is fully equivalent to that of a model system with Hilbert space ${{\mathcal H}}'$ (a subspace of ${{\mathcal H}}$) defined by the linear span of all the $|\psi_\ell\rangle$’s such that $\rho(t)|\psi_\ell\rangle \ne 0$ at some time $t$ (and, hence, by our condition, at all times). The relevant operators $X'$ on ${{\mathcal H}}'$ ($\rho'$, $H'$, $G'_i$, …) are defined from the original $X$ on ${{\mathcal H}}$ ($\rho$, $H$, $G_i$, …) so that $\langle \alpha_k|X'|\alpha_\ell\rangle =\langle \alpha_k|X|\alpha_\ell\rangle$ with $|\alpha_k\rangle$ any basis of ${{\mathcal H}}'$.
It is also consistent with recent experimental tests [@exp1] that rule out, for pure states, deviations from linear and unitary dynamics and confirm that initially unoccupied eigenstates cannot spontaneously become occupied. This fact adds nontrivial experimental and conceptual difficulty to the problem of designing a fundamental test of QM, capable of ascertaining whether decoherence originates from uncontrolled interactions with the environment due to the practical impossibility of obtaining strict isolation, or else it is a more fundamental intrinsic feature of microscopic dynamics requiring an extension of QM. In the latter case, this condition will preserve within the extended theory the exact validity of all the remarkable successes of QM.
[**5. Entropy nondecrease. Irreversibility.**]{}\[irreversibility\] The principle of nondecrease of entropy [@nonextensiveS] for an isolated system must be satisfied, i.e., the rate of change of the entropy functional $-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\rho\ln\rho)$ must be nonnegative along every trajectory, $-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}[u(t,\rho)\ln u(t,\rho)]\ge
-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\rho\ln\rho)$.
[**6. Stability and uniqueness of the thermodynamic equilibrium states. Second law.**]{}\[secondLaw\] A state operator $\rho$ of the isolated system represents an [*equilibrium state*]{} if ${\rm
d}\rho/{\rm d}t=0$. For each given set $(\tilde e, {\bf \tilde g})$ of feasible values of the energy functional $e(\rho)$ and the other time-invariant functionals $g_i(\rho)$, if any, among all the equilibrium states that the dynamical law may admit there must be one and only one which is [*globally stable*]{}[@stability].
This stable equilibrium state must be that of equilibrium thermodynamics and, therefore, of the form $$\label{stableeq}
\rho_{\rm e} =\frac{ \exp[-\beta(\tilde e, {\bf \tilde g}) H+\sum_i\nu_i(\tilde e, {\bf \tilde g}) G_i]}{{{\rm Tr}}\exp[-\beta(\tilde e, {\bf \tilde g}) H+\sum_i\nu_i(\tilde e, {\bf \tilde g}) G_i] }\ ,$$ where $G_i$ are defined above. Of course, states given by Eq. (\[stableeq\]) are solutions of the constrained maximization problem
\[maxproblem\]$$\begin{aligned}
&& {\rm max}\ -{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\rho\ln\rho)\ {\rm subject\ to\ }\\
&& {{\rm Tr}}(\rho)=1,\ {{\rm Tr}}(\rho H)=\tilde e,\ {{\rm Tr}}(\rho G_i)=\tilde g_i,\ {\rm and\ } \rho \ge 0 \ .\end{aligned}$$
and reduce to the canonical equilibrium states $\rho_{\rm e} = \exp(-\beta H)/{{\rm Tr}}\exp(-\beta H)$ when $G_i=g_iI$ for all $i$’s (with $g_i$ scalars and $I$ the identity on ${{\mathcal H}}$), and to the microcanonical state $\rho_{\rm e}=I/\dim{{\mathcal H}}$ if also $H=eI$ (and $\dim{{\mathcal H}}<\infty$).
As discussed in Ref. [@JMathPhys], the entropy functional is not a Lyapunov function, even if, in a strict sense that depends on the continuity and the conditional stability of states $\rho_{\rm e}$, it does provide a criterion for the local stability of these states. In addition to this, the second law requires however that no other equilibrium state of the dynamical law be be globally stable[@stability; @unitary].
Consider the noteworthy family of states $$\rho_{\rm nd} =\frac{ B\exp[-\beta(\tilde e, {\bf \tilde g}) H +\sum_i\nu_i(\tilde e, {\bf \tilde g}) G_i ] B}{{{\rm Tr}}B\exp[-\beta(\tilde e, {\bf \tilde g}) H +\sum_i\nu_i(\tilde e, {\bf \tilde g}) G_i ] }\ ,$$ where $ B$ is any given idempotent operator $ B^2= B$. This family, which includes pure states \[${{\rm Tr}}(B)=1$\], maximizes the entropy \[Prob. (\[maxproblem\])\] subject to the additional constraint $ \rho=B\rho B$ for the given $B$. All eigenvalues of $\rho_{\rm nd}$ must remain invariant (otherwise the entropy would decrease) and the state is equilibrium if $[B,H]=0$ or otherwise it evolves unitarily (limit cycle) with $B(t)=\exp(-iHt/\hbar)B(0) \exp(iHt/\hbar)$. They have a thermal-like distribution (positive and negative temperatures) over a finite number \[${{\rm Tr}}(B)$\] of ’occupied’ eigenvectors. Because entropy cannot decrease and $-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\rho\ln\rho)$ is an $S$-function, they are conditionally locally stable equilibrium states or limit cycles [@JMathPhys]. For them not to be globally stable, as required by the second law, it suffices that the extended dynamics imply that at least one state of equal energy and other invariants (not necessarily neighboring nor with the same kernel) evolves towards higher entropy than $\rho_{\rm nd}$.
[**7. Non-interacting subsystems. Separate energy conservation.**]{}\[separateEnergyConservation\] For an isolated system composed of two distinguishable subsystems $A$ and $B$ with associated Hilbert spaces ${{\mathcal H}}^A$ and ${{\mathcal H}}^B$, so that the Hilbert space of the system is ${{\mathcal H}}={{\mathcal H}}^A{{ \otimes }}{{\mathcal H}}^B$, if the two subsystems are non-interacting, i.e., the Hamiltonian operator $H = H_A{{ \otimes }}I_B + I_A{{ \otimes }}H_B$, then the functionals ${{\rm Tr}}[(H_A{{ \otimes }}I_B)\rho]= {{\rm Tr}}_A(H_A\rho_A) $ and ${{\rm Tr}}[(I_A{{ \otimes }}H_B)\rho]= {{\rm Tr}}_B(H_B\rho_B)$ represent the energies of the two subsystems and must remain invariant along every trajectory, even if the states of $A$ and $B$ are correlated, i.e., even if $\rho\ne \rho_A{{ \otimes }}\rho_B$. Of course, $\rho_A={{\rm Tr}}_B(\rho)$, $\rho_B={{\rm Tr}}_A(\rho)$, ${{\rm Tr}}_B$ denotes the partial trace over ${{\mathcal H}}^B$ and ${{\rm Tr}}_A$ the partial trace over ${{\mathcal H}}^A$.
[**8. Independent states. Weak separability. Separate entropy nondecrease.**]{}\[separateEntropyConservation\] Two distinguishable subsystems $A$ and $B$ are in independent states if the state operator $\rho=\rho_A{{ \otimes }}\rho_B$. For any given $\rho$, let us define the idempotent operator $B$ obtained from $\rho$ by substituting in its spectral expansion each nonzero eigenvalue with unity [@operatorB] and the [*entropy operator*]{} $S =-{ k_{\rm\scriptscriptstyle B}}B\ln\rho$ (always well-defined). For independent states, $S= S_A{{ \otimes }}I_B +
I_A{{ \otimes }}S_B= -{ k_{\rm\scriptscriptstyle B}}[B_A\ln\rho_A{{ \otimes }}I_B + I_A{{ \otimes }}B_B\ln\rho_B ]$. For permanently non-interacting subsystems, every trajectory passing through a state in which the subsystems are in independent states must proceed through independent states along the entire trajectory, i.e., when two uncorrelated systems do not interact with each other, each must evolve in time independently of the other.
In addition, if at some instant of time two subsystems $A$ and $B$, not necessarily non-interacting, are in independent states, then the instantaneous rates of change of the subsystem’s entropy functionals $-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\rho_A\ln\rho_A)$ and $-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\rho_B\ln\rho_B)$ must both be nondecreasing in time.
[**9. Correlations, entanglement and locality. Strong separability.**]{}\[strongSeparability\] Two non-interacting subsystems $A$ and $B$ initially in correlated and/or entangled states (possibly due to a previous interaction that has then been turned off) should in general proceed in time towards less correlated and entangled states. In any case, in order for the dynamics not to generate locality problems, i.e., faster-than-light communication between noninteracting subsystems (even if in entangled or correlated states), entanglement and correlations must not increase in the absence of interactions. In other words, when subsystem $A$ is not interacting with subsystem $B$, it should never be possible to influence the local observables of $A$ by acting only on the interactions within $B$, such as switching on and off parameters or measurement devices within $B$.
This however does not mean that existing entanglement and/or correlations between $A$ and $B$ established by past interactions should have no influence whatsoever on the time evolution of the local observables of either $A$ or $B$. In particular, there is no physical reason to request that two different states $\rho$ and $\rho'$ such that $\rho'_A=\rho_A$ should evolve with identical local dynamics (${\rm d}\rho'_A /{\rm d}t ={\rm d}\rho_A /{\rm
d}t$) whenever $A$ does not interact with $B$, even if entanglement and/or correlations in state $\rho$ differ from those in state $\rho'$. Rather, the two local evolutions should be different until spontaneous decoherence (if any) will have fully erased memory of the entanglement and the correlations established by the past interactions now turned off. In fact, this may be a possible experimental scheme to detect spontaneous decoherence.
Compatibility with the predictions of QM about the generation of quantum entanglement between interacting subsystems that emerge through the Schrödinger-vonNeumann term $-i[H,\rho]/\hbar$ of Eq. (\[eqofmotion\]), requires that the dissipative term $D_M$ may entail (spontaneous) loss of entanglement and loss of correlations between subsystems, but should not be able to create them.
[**10. Onsager reciprocity.**]{}\[reciprocity\] First, we introduce a particularly useful representation of general nonequilibrium states [@Beretta]. Given any state $\rho$ on ${{\mathcal H}}$, we define the effective Hilbert space ${{\mathcal H}}'$ as above, and choose a set of operators $\{I',X'_1,X'_2,\dots\}$ spanning the linear space ${{\mathcal L}}_h({{\mathcal H}}')$ of linear hermitian operators on ${{\mathcal H}}'$; the corresponding state $\rho'$ on ${{\mathcal H}}'$ has no zero eigenvalues, so that $S=-{ k_{\rm\scriptscriptstyle B}}B \ln\rho$ becomes $S'=-{ k_{\rm\scriptscriptstyle B}}\ln\rho'$ on ${{\mathcal H}}'$, which can be written as $ S'=f_0 I' +{\scriptstyle \sum_j} f_j X'_j$ because it belongs to ${{\mathcal L}}_h({{\mathcal H}}')$. Thus, $$\label{anystate}
\rho' =\frac{ \exp (-\sum_j f_j X'_j/{ k_{\rm\scriptscriptstyle B}}) }{{{\rm Tr}}\exp (-\sum_j f_j X'_j/{ k_{\rm\scriptscriptstyle B}})} \ ,$$ where $f_0={ k_{\rm\scriptscriptstyle B}}\ln{{\rm Tr}}\exp(-\sum_j f_j X'_j/{ k_{\rm\scriptscriptstyle B}})$. Similarly, we can also write $S'_e=-{ k_{\rm\scriptscriptstyle B}}\ln\rho'_e =f_{0e} I' +{\scriptstyle \sum_j} f_{je} X'_j $, for the target maximum-entropy equilibrium state on ${{\mathcal H}}'$ $$\rho'_e(\rho')=\frac{\exp(-\beta H'+\sum_k \nu_k G'_k)} {{{\rm Tr}}\exp(-\beta H'+\sum_k \nu_k G'_k)} \ ,$$ where $\beta$ and $\nu_k$ are such that $ e(\rho'_e)=e(\rho')$ and $ g_i(\rho'_e)=g_i(\rho')$, so that ${{\rm Tr}}(\rho'\ln\rho'_e)= {{\rm Tr}}(\rho'_e\ln\rho'_e)$ and ${{\rm Tr}}[({\rm d}\rho'/{\rm d}t)S'_e]=0$. As a result, the following relations can be easily proved, $$\begin{aligned}
&s(\rho')- s(\rho'_e(\rho'))= f_0- f_{0e} + \sum_i (f_i- f_{ie})\, x_i (\rho') \ ,&\\
&\frac{\displaystyle \partial [s(\rho')- s(\rho'_e(\rho'))]}{ \displaystyle \partial x_i (\rho')}= f_i- f_{ie}\ ,&\label{affinity}\\
&\frac{\displaystyle {\rm d}s(\rho')}{\displaystyle {\rm d}t} = \sum_i f_i\frac{\displaystyle {\rm D} x_i(\rho')} {\displaystyle {\rm D}t}=\sum_i (f_i- f_{ie})\frac{\displaystyle {\rm D} x_i(\rho')} {\displaystyle {\rm D}t}\ ,&\\
&{{\langle\Delta S'\Delta S'\rangle}}=\sum_{ij} f_i f_j{{\langle\Delta X'_i\Delta X'_j\rangle}} \ , &\end{aligned}$$ where ${\rm D}x_i(\rho')/{\rm D}t={{\rm Tr}}(D_M X'_i)$ denotes the [*dissipative rate of change*]{} of the linear mean-value functional $x_i (\rho')={{\rm Tr}}(\rho' X_i')$, ${{\langle\Delta S'\Delta S'\rangle}}={{\rm Tr}}[\rho'(-{ k_{\rm\scriptscriptstyle B}}\ln\rho')^2] -s(\rho')^2$, ${{\langle\Delta X'_i\Delta X'_j\rangle}} = \frac{1}{2}{{\rm Tr}}[(\rho\{ X'_i, X'_j\} ]- x_i(\rho') x_j(\rho')$. When the system is in state $\rho'$, we interpret $ {{\langle\Delta X'_i\Delta X'_j\rangle}}$ as the codispersion (covariance) of simultaneous measurements of observables $X'_i$ and $X'_j$, $ {{\langle\Delta X'_i\Delta X'_i\rangle}}$ as the dispersion (or fluctuations) of observable $X'_i$ and ${{\langle\Delta S'\Delta S'\rangle}}$ the entropy fluctuations.
By Eq. (\[affinity\]), we may also interpret $f_i- f_{ie}$ as the [*generalized affinity*]{} or [*force*]{} conjugated with the mean value of the linear observable $X_i$. In order to recover Onsager’s theory, we may impose that (at least in the vicinity of state $\rho'_e$) the extended dynamics be such that the dissipative rates be linearly related to the generalized affinities through generalized-conductivity functionals, i.e., $$\frac{\displaystyle {\rm D} x_i(\rho')} {\displaystyle {\rm D}t}={ \sum_j }L_{ij}(\rho',H',G'_k,X'_\ell,\dots) (f_j- f_{je}) \ ,$$ where the $ L_{ij}$’s may be nonlinear functionals of $\rho'$ (possibly to be approximated with their values at $\rho'_e$, in its vicinity), but should form a symmetric ($\bf{H}\rightarrow -\bf{H}$, if $H'$ depends on an external magnetic field) non-negative definite matrix, so that the rate of entropy production results in a quadratic form ${ \sum_{ij} }(f_i- f_{ie}) L_{ij} (f_j- f_{je})$. Moreover, the $ L_{ij}$’s should be linearly interrelated with the matrix of codispersions $ {{\langle\Delta X'_i\Delta X'_j\rangle}}$, in order to recover also Callen’s fluctuation-dissipation theorem.
[16]{}
An account of this vaste literature can be found in the following papers and references therein: S. Weinberg, Phys. Rev. Lett. [**62**]{}, 485 (1989); A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A [**41**]{}, 3436 (1990); A.K. Ekert, Phys. Rev. Lett. [**67**]{}, 661 (1991); J.A. Holyst and L.A. Turski, Phys. Rev. A [**45**]{}, 6180 (1992); G. Vidal and R.F. Werner, Phys. Rev. A [**65**]{}, 032314 (1993); W.G. Unruh and R.M. Wald, Phys. Rev. D [**52**]{}, 2176 (1995); C.H. Bennett et al., Phys. Rev. Lett. [**76**]{}, 722 (1996); M. Grigorescu, Physica A [**256**]{}, 149 (1998); A. Miranowicz, H. Matsueda and M.R.B. Wahiddin, J. Phys. A [**33**]{}, 5159 (2000).
See, e.g., G.J. Milburn, Phys. Rev. A [**44**]{}, 5401 (1991); M. Hensel and H.J. Korsch, J. Phys. A [**25**]{}, 2043 (1992); M.R. Gallis, Phys. Rev. A [**48**]{}, 1028 (1993); A.K. Rajagopal, Phys. Rev. A [**54**]{}, 1124 (1996); B. Reznik, Phys. Rev. Lett. [**76**]{}, 1192 (1996); and references therein
G. Domokos and S. Kovesi-Domokos, J. Phys. A [**32**]{}, 4105 (1999).
M. Czachor, Phys. Rev. A [**57**]{}, 4122 (1998); M. Czachor and J. Naudts, Phys. Rev. E [**59**]{}, R2497 (1999).
S. Gheorghiu-Svirschevski, Phys. Rev. A [**63**]{}, 022105 and 054102 (2001).
See, e.g., V.I. Yukalov, Phys. Rev. Lett. [**90**]{}, 167905 (2003) and references therein.
G.P. Beretta, Found. Phys. [**17**]{}, 365 (1987). See the references therein for credit to where a general explicit form of ${{\bf \hat D}}_M$ for steepest-entropy-ascent quantum dynamics was first formulated and where the conceptual need to remove the restriction $\rho^2=\rho$ was first advocated.
R. Englman, Appendix in M. Lemanska and Z. Jaeger, Physica D [**170**]{}, 72 (2002).
D. Aerts et al., Phys. Rev. E [**67**]{}, 051926 (2003).
By [*strictly isolated*]{} we mean that the system interacts with no other systems and at some time (and, hence, at all times) is in an independent state when viewed as a subsystem of any conceivable composite system containing it.
See references \[1\] in Ref. [@Gheorghiu].
Given a $\rho$, $B=B^2= I- P_{{{\rm Ker}}(\rho)}= P_{\bot{{\rm Ker}}(\rho)}$, so that $B=B^2$, $ B\rho=\rho B=\rho$, and $B= B_A{{ \otimes }}B_B$ when $\rho= \rho_A{{ \otimes }}\rho_B$. Moreover, from $B=B^2$ follows that $B\dot B B=0$ and ${{\rm Tr}}(\rho \dot B \ln\rho)= {{\rm Tr}}(B\dot B B \rho \ln\rho)=0$, thus $\dot s(\rho)={{\rm Tr}}(\dot\rho S)-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\dot\rho)$ with $S=-{ k_{\rm\scriptscriptstyle B}}B\ln\rho$.
For a discussion on the relation between the notion of stability in thermodynamics and the mathematical concept of stability see Ref. [@JMathPhys]. The relevant definitions of local and global stability and of metastabilty are as follows.
An equilibrium state is stable in the sense required by the second law if it can be altered to a different state only by interactions that leave net effects in the state of the enviromment, i.e., alter the values of the energy and the other invariants. We call this notion [*global stability*]{}.
Lyapunov local stability is necessary for global stability, but not sufficient: we must exclude [*metastability*]{}. As a result, the concept of global stability implied by the second law is as follows. An equilibrium state $\rho_e$ is [*globally stable*]{} if for every $\eta>0$ and every $\epsilon>0$ there is a $\delta(\epsilon,\eta)>0$ such that every trajectory $u(t,\rho)$ with $\eta<d(u(0,\rho),\rho_e) <
\eta+\delta(\epsilon,\eta)$, i.e., passing at time $t = 0$ between distance $\eta$ and $\eta+\delta$ from $\rho_e$, remains within $d(u(t,\rho),\rho_e) <
\eta+\epsilon$ for every $t>0$, i.e., proceeds in time without ever exceeding the distance $\eta+\epsilon$.
The dynamical law may admit many equilibrium states that all share the same values of the invariants ${{\rm Tr}}(\rho H)$ and ${{\rm Tr}}(\rho G_i)$ and the parameters embedded in the Hilbert space ${{\mathcal H}}$ and the Hamiltonian $H$ describing the external forces (such as the size of a container), but among all these only one must globally stable, i.e., all the other equilibrium states must either be unstable or metastable.
Under a unitary (Hamiltonian) dynamical law the trajectories would be $u(t,\rho) =
U(t)\rho U^{-1}(t)$ with $U(t) = \exp(-itH/\hbar)$. The equilibrium states $\rho_e$, with $\rho_eH=H\rho_e$, would all be globally stable. Indeed, with respect to the metric $d(\rho_1,\rho_2) = {{\rm Tr}}|\rho_1-\rho_2|$, it is easy to show that every trajectory $u(t,\rho)$ would be equidistant from any given equilibrium state $\rho_e$, i.e., $d(u(t,\rho),\rho_e) = d(u(0,\rho),\rho_e)$ for all $t$ and all $\rho$. Because for each set of values of the invariant functianal these globally stable equilibrium states are in general more than one, the second-law requirement of uniqueness would be violated.
G.P. Beretta, J. Math. Phys. [**27**]{}, 305 (1986). The conjecture therein was later found proved in F. Hiai, M. Ohya, and M. Tsukada, Pacific J. Math. [**96**]{}, 99 (1981).
Some recent nonextensive quantum theories are based on other well-behaved entropy functionals such as the Daroczy-Tsallis functional. However \[see e.g., E.P. Gyftopoulos and E. Çubukçu, Phys. Rev. E [**55**]{}, 3851 (1997)\], compatibility with thermodynamics requires the Gibbs-Shannon-vonNeumann functional $s(\rho) =-{ k_{\rm\scriptscriptstyle B}}{{\rm Tr}}(\rho\ln\rho)$.
| {
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---
abstract: 'We present a comparison of Doppler-shifted H$\alpha$ line emission observed by the Global Jet Watch from freshly-launched jet ejecta at the nucleus of the Galactic microquasar SS433 with subsequent ALMA imaging at mm-wavelengths of [*the same*]{} jet ejecta. There is a remarkable similarity between the transversely-resolved synchrotron emission and the prediction of the jet trace from optical spectroscopy: this is an a priori prediction not an a posteriori fit, confirming the ballistic nature of the jet propagation. The mm-wavelength of the ALMA polarimetry is sufficiently short that the Faraday rotation is negligible and therefore that the observed ${\bf E}$-vector directions are accurately orthogonal to the projected local magnetic field. Close to the nucleus the ${\bf B}$-field vectors are perpendicular to the direction of propagation. Further out from the nucleus, the ${\bf B}$-field vectors that are coincident with the jet instead become parallel to the ridge line; this occurs at a distance where the jet bolides are expected to expand into one another. X-ray variability has also been observed at this location; this has a natural explanation if shocks from the expanding and colliding bolides cause particle acceleration. In regions distinctly separate from the jet ridge line, the fractional polarisation approaches the theoretical maximum for synchrotron emission.'
author:
- 'Katherine M. Blundell , Robert Laing, Steven Lee and Anita Richards,'
title: |
SS433’s jet trace from ALMA imaging and Global Jet Watch spectroscopy:\
evidence for post-launch particle acceleration
---
Introduction
============
Since shortly after its discovery four decades ago the prototypical Galactic microquasar SS433 has been known to eject oppositely-directed jets whose launch axis precesses with a cone angle of about 19 degrees approximately every 162 days and whose speeds average to about a quarter of the speed of light. Striking images of the emission at radio (cm) wavelengths reveal a zigzag/corkscrew structure that arises because of the above properties modulated by light-travel time effects arising from its orientation with respect to our line-of-sight [@Hjellming1981; @Stirling2002; @Blundell04; @Roberts2008; @Miller-Jones2008]. The optical spectra of this object are characterised by a strong Balmer H$\alpha$ emission line complex close to the rest-wavelength of this line, and also blue-shifted and red-shifted lines whose observed wavelengths change successively on a daily basis according to the instantaneous speed and angle of travel with respect to our line of sight. Fitted parameters to the kinematic model developed from the first few years of optical spectroscopy were presented by e.g. @Margon84 and @Eikenberry01. Hitherto the timing of optical spectroscopy and spatially resolved radio imaging has not permitted the observation of the same ejecta both at launch and after propagation. We present the first mm-wave image of SS433 from ALMA in combination with optical spectroscopy (Sec\[sec:SpeedsFromZeds\]) of the same ejecta observed during the year prior to the ALMA observations (Sec\[sec:ALMA\]). This allows us to distinguish ballistic motion post-launch from deceleration [e.g. @Stirling2004].
A long standing question is why SS433’s jet ejecta are primarily line emitting at launch yet synchrotron emitting at largest distances from the nucleus; the polarisation changes explored in Sec.\[sec:polar\] shed light on this. Inference of the magnetic-field structure in the jets is complicated by the combined effects of Faraday rotation and time-variable structure. Previous studies [@Stirling2004; @Roberts2008; @Miller-Jones2008] have been hampered by lack of resolution and frequency coverage as well as the uncertain effects of spatial- and temporal-variations in Faraday rotation. The dependence of Faraday rotation on the square of the wavelength ($\lambda$), means that wide-band observations at mm wavelengths allow the projected field direction to be determined accurately in a single observation even close to the core, where Faraday rotation measures may be large [@Roberts2008]. We present our polarimetric 230GHz results in Sec.\[sec:polar\].
Optical spectroscopy and inference from Doppler shifts {#sec:SpeedsFromZeds}
======================================================
Spectra of SS433 spanning a wavelength range of approximately 5800 to 8500 Angstroms, and having a spectral resolution of $\sim4000$ were observed in the year prior to the ALMA observations whenever this target was a nighttime object. These were carried out with the multi-longitude Global Jet Watch telescopes each of which is equipped with an Aquila spectrograph; the design and testing of these high-throughput spectrographs are described by @Lee2018. The observatories, astronomical operations, processing and calibration of the spectroscopic data streams are described in @Blundell2018. Almost all of these spectra contain a pair of so-called “moving lines” arising from the most recently launched jet bolides in SS433. The wavelengths corresponding to the centroids of the blue-shifted and red-shifted H$\alpha$ emission were converted into redshift pairs with respect to H$\alpha$ in the rest frame of SS433 according to its systemic velocity with respect to Earth [@lockman2007]. From these redshift and blueshift pairs from a given spectrum were derived the launch speed of each pair of bolides [@Blundell05 equation 2]. This avoids the approximation of constant ejection speed, which has been shown to be inaccurate from archival spectroscopy [@Blundell05; @Blundell11]. Assuming that the subsequent motion is ballistic (this assumption is discussed in Sec\[sec:ballistic\]), and adopting the standard kinematic model [@Hjellming1981], the locations they attain by the Julian Date of the mid-point of the ALMA observations (2457294.4836) are calculated, and plotted in Fig\[fig:combine\]. The assumed parameters of the kinematic model using the notation of @Eikenberry01 and @Hjellming1981 are: cone angle $\theta = 19^\circ$ (Hjellming et al. use $\psi$), inclination $i = 79^\circ$, rotation on the sky $\chi = 10^\circ$ (position angle $+100^\circ$), period $P = 162.34$day (Blundell et al., in preparation) and distance $d =
5.5$kpc [@Blundell04; @lockman2007]. The ejection phase was determined by fitting to the observed redshift pairs from JD 2457000 to JD 2457293.5. The phase $\phi = (2\pi/P)(t - t_{\rm ref}) + \phi_0$ with $\phi_0 =
-0.241$rad for a reference Julian date of $t_{\rm ref} =
2456000$. $\phi$ is used as in equation 1 of @Eikenberry01; @Hjellming1981 denote the same quantity by $\Omega(t_0-t_{\rm ref})$.
Millimetre polarimetric imaging {#sec:ALMA}
===============================
SS433 was observed using 27 ALMA antennas between 2015 September 28 21:26 and September 29 01:46 UT. Three execution blocks were run almost in sequence and under similar conditions. The precipitable water vapour column was around 1.4mm. The correlator was set up in Time Division Multiplex mode with a total bandwidth of 7.5GHz, in four 1.75-GHz spectral windows (spw) centred at 224, 226, 240 and 242GHz. Each spw was divided into 64 spectral channels and XX, YY, XY and YX correlations were recorded. The longest and shortest baselines were 2270 and 43m, sensitive to angular scales $\la 3.7$arcsec.
The quasar J1751+0939 was used as a bandpass, polarization and flux scale calibrator and J1832+0731 was used as the phase reference source on an approximately 8 min cycle. The total integration time on SS433 was $\approx$2hr. Initial data reduction followed standard ALMA scripts, executed in CASA [@Schnee2014]. The flux density of J1751+0939 during these observations was taken to be 3.7275Jy at 232.86GHz with a spectral index $\alpha = -$0.441 (defined in the sense $S(\nu) \propto \nu^{-\alpha}$) and the total flux scale uncertainty is about 10%. Polarization leakage was calibrated as described by @Nagai16. Several iterations of [clean]{} in multi-frequency synthesis mode [@Rau] followed by self-calibration were used to improve the imaging of SS433. The final iteration of amplitude and phase self-calibration was made by combining all four spectral windows using a model with two Taylor series terms. We show the zero-order Taylor series $I$ image after self-calibration, together with polarised intensity images derived from $Q$ and $U$ for the entire band (we demonstrate below that Faraday rotation is negligible for our frequency range). The off-source rms levels are 13, 11 and 12$\mu$Jybeam$^{-1}$ in $I$, $Q$ and $U$, respectively, consistent with the expectations for thermal noise alone. The restoring beam has FWHM $0.19 \times 0.16$arcsec$^2$.
The $I$ image (Fig\[fig:combine\], central panel greyscale) shows the familiar zigzag/corkscrew shape of SS433. The peak flux density at 233GHz is 86.0 mJy/beam. The in-band spectral index of the core is $-0.29 \pm 0.14$.
Comparison of time-extrapolated spectroscopy with ALMA imaging {#sec:ballistic}
==============================================================
![[*Central panel:*]{} mm-wave image made from ALMA observations in 2015 September described in Sec\[sec:ALMA\] overlaid with symbols representing the positions attained by the bolides whose speeds are measured via optical spectroscopy and assumed to move ballistically after launch. [*Upper two and lower two panels:*]{} Representative spectra from four different dates prior to the ALMA observations are shown, revealing a pair of Doppler shifted lines corresponding to emission from the oppositely moving jet bolides, recently launched and still optically radiant. These spectra are from each of four different observatories, from bottom to top, namely eastern Australia (GJW-OZ), Western Australia (GJW-WA), Chile (GJW-CL) and South Africa (GJW-SA). []{data-label="fig:combine"}](combine_redblue_axis_labels.eps){width="53.00000%"}
Fig\[fig:combine\] shows excellent agreement between the jet trace predicted by the redshift and blueshift pairs from the optical spectroscopy and the brightness distribution subsequently measured by ALMA at mm-wavelengths. There are only three free parameters in the superposition: two positional offsets to align the ejection centre with the peak of the radio emission and the rotation of the precession axis on the sky (which was taken from @Hjellming1981, not determined independently). This is the first time that it has been possible to demonstrate the superposition directly with optical spectroscopy covering the appropriate time period. We find no evidence for significant deceleration of the jet post-launch (which would be evinced by radio emission systematically lagging the optical bolides). In particular, we can rule out a deceleration of 0.02$c$ just outside the launch location as suggested[^1] by @Stirling2004: this would lead to a systematic offset of $\approx$0.3arcsec between the predicted jet trace and the ALMA brightness distribution at projected distances $\ga$1arcsec, most obviously on the East side of the source; this is not seen. More stringent constraints on deceleration can be obtained from a comparison of the jet trace predicted by optical spectroscopy with VLA radio imaging between 8GHz and 12GHz, in which the trace is detectable out to much greater distances from the nucleus than is possible in our current 230-GHz observations. We will address this comparison in a future paper, together with possible correlations of speed with launch angle.
$B$-field structure {#sec:polar}
===================
The apparently conflicting results in the literature for the relation between the projected magnetic-field direction and the underlying jet flow in SS433 can be understood by consideration of the different distances from the nucleus probed by these studies. @Stirling2004 and @Roberts2008 found a preferential alignment between the magnetic field and the jet ridge line from $\approx$0.4 – 2arcsec from the nucleus, whereas figure 8 of @Miller-Jones2008 suggests that the field is instead parallel to the ballistic velocity of the jet knots at distances larger than 2arcsec.
Our measurements of the magnetic-field direction are much less affected by Faraday rotation than those in earlier work. @Stirling2004 found a mean rotation measure (RM) of 119radm$^{-2}$ (excluding the core), which implies a position angle rotation of 0.01$^\circ$ at 230GHz. Even for the RM’s of $\approx$600radm$^{-2}$ estimated for distances within 0.4arcsec of the core [@Stirling2004; @Roberts2008], the inferred rotation is still only 0.06$^\circ$ at this ALMA band. We also find no evidence for any wavelength-dependent rotation across our observing band. In particular, the position angles measured for the individual spws at the location of the core (where we might expect maximum Faraday rotation) are consistent with the mean value for the band with an rms scatter of 1.0$^\circ$ and Fig. \[fig:paplot\] shows no systematic trend. We therefore conclude that the position angles plotted in Fig. \[fig:fracpolvectors\] are not significantly affected by Faraday rotation.
![The direction of the ${\bf B}$-field at four different wavelengths plotted as a function of $\lambda^2$. All measurements are consistent with $-17$ degrees. The solid bars represent errors due to thermal noise alone while the dotted bars include systematic errors. \[fig:paplot\]](paplot.eps){width="6.5cm"}
While the fractional polarisation of the nucleus of SS433 at 230GHz is low ($p = 0.011$), the position angle is still securely determined. The inset to Fig\[fig:fracpolvectors\] shows that within $\approx$0.35arcsec of the nucleus the orientation of the $B$-field vectors is consistent with being perpendicular to the jet ridge line (and also to line of radial ejection, which is indistinguishable from it at this distance), consistent with the tentative suggestion by @Roberts2008. At a distance of $\approx$0.35arcsec, the degree of polarisation increases to $p \approx 0.1$. Here, the field directions in both jets become parallel to the ridge line and clearly inconsistent with the direction of ballistic motion. This relative orientation persists out to at least 0.7arcsec, beyond which we cannot measure accurate position angles. Our result is consistent with that of @Stirling2004, but with higher angular resolution and lower uncertainties from Faraday rotation. For distances from the launch-point exceeding $\approx$2arcsec, Miller-Jones et al (2008) report the magnetic field of the jet as being parallel to the local velocity vector (using the value for rotation measure reported by Stirling et al 2004).
The transition in field direction at 0.35arcsec ($3 \times
10^{14}$m) may bear on the oft-debated question of whether the outflow in SS433 is best described as a succession of independent bolides or a continuous jet. It is interesting to compare this distance with the point at which the expanding bow shocks surrounding neighbouring bolides first intersect. If we assume that one bolide is ejected per day at a speed of 0.26$c$, their radial separation is $\approx$6.7 $\times 10^{12}$m. If the shock expansion speed is comparable with the expansion rate of the radio knots measured with VLBI ($\approx 0.015c$; @Jeffrey2016), then the shock fronts will indeed expand into each other and interact when the bolides have travelled $\approx 3 \times 10^{14}$m from the nucleus, roughly where the change in field direction occurs. Bolides will coalesce to form larger structures which will then cease to interact with one another, when the paths of successive bolide conglomerates are too angularly divergent. Thereafter, the magnetic field observed to be associated with the jet trace will no longer reflect the details of the bolides as launched but rather their interactions with the (magnetised) medium through which they flow. This magnetic field behaviour appears to dominate for distances from the launch-point exceeding $\approx$2arcsec.
Very highly polarised emission is observed away from the jet trace on the East side of the source (labelled A in Fig \[fig:fracpolvectors\]). Both @Roberts2008 and @Miller-Jones2008 have drawn attention to these off-ridgeline regions being significantly more polarised than the jet ridgeline itself in VLA data at 15GHz and 8GHz respectively. Our 230GHz data show significantly higher fractional polarisation values (0.6 – 0.7) in these regions. Comparison of all these data suggest the degree and direction of polarisation may change with precession period and possibly also distance from the nucleus.
The interaction of bolides into larger coalescences has been reasoned above to occur where there is a change in polarisation behaviour namely at approximately 0.35arcsec from the nucleus. We note that this coincides with the region reported by @Migliari2002 [@Migliari2005] to show distinct X-ray variability which we suggest arises from shocks formed by the coalescence. Such a mechanism would naturally give rise to the stochastic nature of the X-ray variations reported by @Migliari2002 [@Migliari2005].
![Panel (a) shows a colour scale depicting the fractional polarisation, $p$, of SS433 averaged over the ALMA observing band. Points are shown blanked (grey) wherever the total intensity $I < 5\sigma_I$. Panel (b) shows vectors whose lengths are proportional to $p$ (with the scale indicated by the labelled bar) and whose directions are along the apparent ${\bf B}$-field direction (i.e. rotated by 90$^\circ$ from the ${\bf E}$-vector direction with no correction for Faraday rotation: see text). The vectors are plotted where $I > 5\sigma_I$ and $P > 3\sigma_I$ and are superimposed on a grey-scale of total intensity, as indicated by the wedge labelled in mJy/beam. The inset shows the core with vectors plotted on an expanded scale for $I > 1$mJy/beam and $P > 3\sigma_I$. Panel (c): as (b), but with vectors superimposed on the inferred locations reached by pairs of plasma bolides (green crosses, as in Fig\[fig:combine\]). The inset again shows the core region.[]{data-label="fig:fracpolvectors"}](pol.eps){width="8.5cm"}
Conclusions
===========
Over one precession period of the jets in the Galactic microquasar SS433 is shown to be traced out at mm-wavelengths in our ALMA imaging. This shows remarkable concordance with the predicted trace of the same ejecta from Global Jet Watch spectroscopy at earlier epochs.
At mm-wavelengths the Faraday Rotation towards SS433 is negligible. By 0.35arcsec from the nucleus the ${\bf B}$-field direction has changed from being perpendicular to parallel to the jet ridge line. This occurs where bolides are expected to have expanded into one another, and where X-ray variability has been reported, consistent with the onset of particle acceleration and the change from line-emission at launch to dominant synchrotron emission further out.
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2013.1.01369.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. A great many organisations and individuals have contributed to the success of the Global Jet Watch observatories and these are listed on [www.GlobalJetWatch.net]{} but we particularly thank the University of Oxford and the Australian Astronomical Observatory.
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Rau, U. & Cornwell, T.J., 2011, , 532, 71
Roberts, D. H., Wardle, J. F. C., Lipnick, S. L., Selesnick, P. L., & Slutsky, S. 2008, , 676, 584
Schnee, S.L., Brogan, C., Espada, D., et al 2014 in Observatory Operations: Strategies, Processes and Systems, Proc. SPIE, 9149, 91490
Stirling, A.M., Jowett, F. H., Spencer, R. E., et al, 2002, , 337, 657
Stirling, A. M., Spencer, R. E., Cawthorne, T. V., & Paragi, Z. 2004, , 354, 1239
[^1]: The lower speeds suggested by @Stirling2004 are a direct consequence of their adoption of a smaller distance $d = 4.8$kpc for SS433.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose an efficient method for mapping and storage of a quantum state of propagating light in atoms. The quantum state of the light pulse is stored in two sublevels of the ground state of a macroscopic atomic ensemble by activating a synchronized Raman coupling between the light and atoms. We discuss applications of the proposal in quantum information processing and in atomic clocks operating beyond quantum limits of accuracy. The possibility of transferring the atomic state back on light via teleportation is also discussed.'
address: 'Institute of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark'
author:
- 'A. E. Kozhekin, K. M[ø]{}lmer and E. Polzik'
title: Quantum Memory for Light
---
Light is an ideal carrier of quantum information, but photons are difficult to store for a long time. In order to implement a storage device for quantum information transmitted as a light signal, it is necessary to faithfully map the quantum state of the light pulse onto a medium with low dissipation, allowing for storage of this quantum state. Depending on the particular application of the memory, the next step may be either a (delayed) measurement projecting the state onto a certain basis, or further processing of the stored quantum state, e.g., after a read-out via the teleportation process. The delayed projection measurement is relevant for the security of various quantum cryptography and bit commitment schemes [@Bras]. The teleportation read-out is relevant for full scale quantum computing.
In this Letter we propose a method that enables quantum state transfer between propagating light and atoms with an efficiency up to 100% for certain classes of quantum states. The long term storage of these quantum states is achieved by utilizing atomic ground states. In the end of the paper we propose an atom-back-to-light teleportation scheme as a read-out method for our quantum memory.
We consider the stimulated Raman absorption of propagating quantum light by a cloud of $\Lambda$ atoms. As shown in the inset of Fig.\[fig:var\], the weak quantum field and the strong classical field are both detuned from the upper intermediate atomic state(s) by $\Delta$ which is much greater than the strong field Rabi frequency $\Omega_{s}$, the width of an upper level $\gamma_{i}$ and the spectral width of the quantum light $\Gamma_{q}$. The Raman interaction “maps” the non-classical features of the quantum field onto the coherence of the lower atomic doublet, distributed over the atomic cloud.
In our analysis we eliminate the excited intermediate states, and we treat the atoms by an effective two-level approximation. We start with the quantum Maxwell-Bloch equations in the lowest order for the slowly varying operator $\hat{Q}$: $\hat{Q}=\hat{\sigma _{31}}
e^{-i(\omega_{q} - \omega_{s})t +i (k_{q}-k_{s})z}$ (it will be assumed, that $(k_{q}-k_{s}) L \ll 1$, where $L$ is the length of the atomic cloud, $z$ is the propagation direction, and $\omega_{q,s}$ and $k_{q,s}$ are frequencies and wavevectors of “quantum” and “strong” fields respectively) [@Raym81; @Raym85]
$$\begin{aligned}
&& \frac{d}{dt}\hat{Q}(z,t) =-i\kappa_{1}^{\ast} \hat{E}_{q}(z,t)
E_{s}^{\ast} (z,t) - \Gamma \hat{Q}(z,t) + \hat{F}(z,t) \label{Bloch} \\
&& \left( \frac{\partial}{\partial z} + \frac{1}{c} \frac{\partial}{\partial t}
\right) \hat{E}_{q}(z,t) = -i \kappa_{2} \hat{Q}(z,t) E_{s}(z,t)
\label{Max}\end{aligned}$$
$\Gamma$ is the dephasing rate of the $1\leftrightarrow 3$ coherence which also includes the strong field power broadening $\Gamma_{s}
\simeq \omega^{3} \hbar \kappa_{1}^{2} |E_{s}|^{2}/(3c^{3})$ due to spontaneous Raman scattering [@Raym81], $\hat{F}(z,t)$ is the associated quantum Langevin force with correlation function $\langle
\hat{F^{\ast}}(z,t) \hat{F}(z^{\prime}, t^{\prime })\rangle =2\Gamma
/n \delta (z-z^{\prime })\delta (t-t^{\prime })$, and $\kappa_{1} =
\sum_{i} \mu_{1i} \mu_{3i}/(\hbar^{2} \Delta_{i})$, $\kappa_{2} = 2\pi
n\hbar \omega \kappa_{1}/c$, where $\mu_{ji}$ are dipole moments of the atomic transitions and $n$ is the density of the atoms. A one-dimensional wave equation is sufficient to describe the spatial propagation of light in a pencil-shaped sample with a Fresnel number ${\cal F}= A / \lambda L$ near unity ($A$ is the cross-sectional area of the sample and $\lambda$ is the optical wavelength) [@Raym85].
If the strong field is not depleted in the process of quantum field absorption and if most of the atomic population stays in the initial level $1$, Eqs.(\[Bloch\]-\[Max\]) can be integrated to get
$$\begin{aligned}
\hat{Q}(z,\tau ) &=&e^{-\Gamma \tau }\hat{Q}(z,0)-e^{-\Gamma \tau
}\int_{0}^{z}dz^{\prime }\,\hat{Q}(z^{\prime },0)\sqrt{\frac{a(\tau )}
{z-z^{\prime }}}J_{1}(2\sqrt{a(\tau )(z-z^{\prime })}) \nonumber \\
&-&i\kappa _{1}\int_{0}^{\tau }d\tau ^{\prime }\,e^{-\Gamma (\tau -\tau
^{\prime })}\hat{E}_{q}(0,\tau ^{\prime })E_{s}(\tau ^{\prime })J_{0}(2
\sqrt{z(a(\tau )-a(\tau ^{\prime }))})+\int_{0}^{\tau }d\tau ^{\prime
}\,e^{-\Gamma (\tau -\tau ^{\prime })}\hat{F}(z,\tau ^{\prime }) \nonumber
\\
&-&\int_{0}^{\tau }d\tau ^{\prime }\,\int_{0}^{z}dz^{\prime }\,e^{-\Gamma
(\tau -\tau ^{\prime })}\hat{F}(z^{\prime },\tau ^{\prime })
\sqrt{\frac{a(\tau )-a(\tau ^{\prime })}{z-z^{\prime }}}
J_{1}(2\sqrt{(a(\tau )-a(\tau
^{\prime }))(z-z^{\prime })}) \label{Q} \\
\hat{E}_{q}(z,\tau ) &=&\hat{E}_{q}(0,\tau )-i\kappa _{2}E_{s}(\tau
)e^{-\Gamma \tau }\int_{0}^{z}dz^{\prime }\,\hat{Q}(z^{\prime },0)J_{0}(2
\sqrt{a(\tau )(z-z^{\prime })}) \nonumber \\
&-&\kappa _{1}^{\ast }\kappa _{2}E_{s}(\tau )\int_{0}^{\tau }d\tau ^{\prime
}\,e^{-\Gamma (\tau -\tau ^{\prime })}\hat{E}_{q}(0,\tau ^{\prime
})E_{s}^{\ast }(\tau ^{\prime })\sqrt{\frac{z}{a(\tau )-a(\tau ^{\prime })}}
J_{1}(2\sqrt{z(a(\tau )-a(\tau ^{\prime }))}) \nonumber \\
&-&i\kappa _{2}E_{s}(\tau )\int_{0}^{\tau }d\tau ^{\prime
}\,\int_{0}^{z}dz^{\prime }\,e^{-\Gamma (\tau -\tau ^{\prime })}\hat{F}
(z^{\prime },\tau ^{\prime })J_{0}(2\sqrt{(a(\tau )-a(\tau ^{\prime
}))(z-z^{\prime })}) \label{E}\end{aligned}$$
where $\tau =t-z/c$, and $a(\tau )=\kappa _{1}^{\ast }\kappa
_{2}\int_{0}^{\tau }d\tau ^{\prime \prime }\,|E_{s}(\tau ^{\prime
\prime })|^{2}$ and $\hat{Q}(z,0)$ is the initial atomic coherence.
Integrating Eq.(\[Q\]) over space we obtain the collective atomic spin operator, which is the atomic variable on which the quantum light field is mapped. $$\begin{aligned}
\hat{{\cal Q}}_{L}(\tau ) \equiv n\int_{0}^{L}\,dz\,\hat{Q}(z,\tau )
&=&ne^{-\Gamma \tau }\int_{0}^{L}\,dz^{\prime }J_{0}(2\sqrt{a(\tau
)(L-z^{\prime })})\hat{Q}(z^{\prime },0) \nonumber \\
&+&n\int_{0}^{\tau }\,d\tau ^{\prime }e^{-\Gamma (\tau -\tau ^{\prime
})}\int_{0}^{L}\,dz^{\prime }J_{0}(2\sqrt{(a(\tau )-a(\tau ^{\prime
}))(L-z^{\prime })})\hat{F}(z^{\prime },\tau ^{\prime }) \nonumber \\
&-&in\kappa _{1}\int_{0}^{\tau }\,d\tau ^{\prime }e^{-\Gamma (\tau -\tau
^{\prime })}\hat{E}_{q}(\tau ^{\prime })E_{s}(\tau ^{\prime })\sqrt{\frac{L}{
a(\tau )-a(\tau ^{\prime })}}J_{1}(2\sqrt{a(\tau )-a(\tau ^{\prime })L})
\label{Q-L}\end{aligned}$$ Eq.(\[Q-L\]) is the main result of this Letter. The first term represents the decaying memory of the initial atomic coherence in the sample, the second term is the contribution from the Langevin noise associated with the decay of the coherence, and the last term represents the contribution from the absorbed quantum light. It is thus the last term, that describes the quantum memory capability of the atomic system. Note that the strong classical field $E_{s}(\tau^{\prime})$ can be turned on and off, so that only the value of the quantum field in a certain time window is mapped onto the atomic system, where it is subsequently kept. We assume that the rate $\Gamma $ is dominated by the power broadening contribution $\Gamma_{s}$ when the classical field is turned on, and it can be quite small $\Gamma =\Gamma_{0}$ when the classical field is turned off to ensure long storage times. If the quantum field pulse $\hat{E}_{q}(\tau)$ and the overlapping classical pulse $E_{s}(\tau)$ are long enough so that $\Gamma \tau \gg 1$ the initial atomic state decays and the state determined by $\hat{E}_{q}(\tau )$ emerges instead. After the light pulses are turned off, the atomic ”memory” state decays slowly with the rate $\Gamma_{0}$.
As an example of storing a quantum feature of light in atoms let us consider storing a squeezed state, which plays an important role in quantum information with continious variables [@Brau98b]. For infinitely broadband squeezed light the quadrature operator $\hat{X}_{q}(z,\tau) = \text{Re} \hat{E}_{q} (z,\tau )$ on the entry face of the sample can be written as $\langle \hat{X}_{q}(0,\tau)
\hat{X}_{q}(0,\tau^{\prime }) \rangle = 2 \pi \hbar \omega / c \langle
X_{0}^{2}\rangle \delta (\tau -\tau^{\prime })$, where $\langle
X_{0}^{2}\rangle$ is the dimensionless light noise, $\langle
X_{0}^{2}\rangle =1$ in the case of broad band vacuum. In steady-state the variance of the atomic noise $\hat{X}=\text{Re}\hat{{\cal Q}}_{L}$ becomes $$\begin{aligned}
\langle X^{2}\rangle &=& nL \left( e^{-\alpha} \left( I_{0}(\alpha)
+I_{1}(\alpha) \right) \right) \nonumber \\
&+& nL \langle X_{0}^{2}\rangle \left( 1-e^{-\alpha} \left(
I_{0}(\alpha)+I_{1}(\alpha) \right) \right)
\label{var}\end{aligned}$$ where $\alpha=aL/\Gamma$ is the optical depth of the sample, $a=\kappa_{1}^{\ast} \kappa_{2} |E_{s}|^{2}$ and $I_{0}$ and $I_{1}$ are Bessel functions of the first kind. In the case of vacuum incident on the sample we recover the atomic vacuum noise $\langle X^{2}\rangle
=nL$, the number of atoms per unit area. The second term in (\[var\]), represents the light contribution to atomic noise, it is reduced when the light is squeezed, and in the case of ideally squeezed light $\langle X_{0}^{2}\rangle =0$ only the first term contributes to the atomic noise variance. We define the dimensionless expression in the parenthesis as a mapping efficiency for the Gaussian fields $\eta = (1-\langle X^2 \rangle / nL)/(1-\langle X_0^2 \rangle)$ (for ideally squeezed light $1-\eta$ quantifies the amount of spin squeezing). The results are plotted in Fig.\[fig:var\] (solid line) as a function of the optical depth $\alpha$. Storing squeezing in atoms with an efficiency higher than 90% requires an atomic sample with an optical depth of the order of $\gtrsim 60$. Note that by absorption of EPR beams in separate atomic samples, we may, e.g., prepare entangled atomic gases, see also [@Polz99]. If $\Gamma
\approx \Gamma_{s}$, and the decoherence is dominated by the strong field that is required for the operational memory, then $\alpha \simeq
(3/2\pi) \lambda^{2} nL$, i.e., the optical depth is the same as for a resonant narrowband field. The dependence on the optical depth arises because the more squeezed light is absorbed in the sample, the more the atoms become squeezed. If only a fraction of the light field is absorbed, the atomic spins will not only be correlated with each other but also with the field leaving the sample, and thus the squeezing will be degraded, see also [@Molm99].
Various schemes for quantum state exchange between light and atoms based on cavity QED Raman-type interactions have been proposed in the past [@Cira97; @Park99; @Zeng]. Quantum memory with a microwave cavity field as storage medium has been demonstrated in [@Mait97]. The fact, that the present proposal does not utilize high finesse cavities significantly simplifies the experimental realization. The above result can be compared with the proposal [@Kuzm97] and its experimental verification [@Hald99] for squeezing the collective spin of an optically thick sample of $V$-type excited atoms via the interaction with squeezed light. As opposed to the theoretical bound of 50% mapping efficiency found in [@Kuzm97] the present proposal offers in principle a perfect transfer of the state of light onto atoms.
A steady state analysis in frequency domain similar to that in [@Kuzm97] leads to the following expression for the spectral collective atomic spin operator $$\begin{aligned}
\tilde{{\cal Q}}_{L}(\Delta ) &=&-\frac{i n}{\kappa _{2}E_{s}}\left(
1-e^{ik(\Delta )L}\right) \tilde{E}_{q}(\Delta ) \nonumber \\
&+&\int_{0}^{L}\,dz\frac{n}{\Gamma -i\Delta }e^{ik(\Delta) (L-z)}
\tilde{F}(z,\Delta ) \label{spectra}\end{aligned}$$ where $\Delta$ is the detuning from the two-photon resonance and $k(\Delta)$ is the Lorentzian absorption profile $ik(\Delta) =
-a/(\Gamma -i\Delta )$. The atomic noise variance $\langle
X^{2}\rangle =\int d\Delta \tilde{X} (\Delta )\tilde{X}(-\Delta )$ gives the same result as Eq.(\[var\]).
The simplest approach to quantum field propagation in a medium is the model of scattering by a collection of frequency-dependent beam splitters [@Jeff]. Each beam splitter removes a small fraction of a propagating light beam and it simultaneously couples in a small fraction of vacuum into the beam. The result for the noise spectrum of the transmitted light in our model coincides with such a simplified treatment and is given by $$\langle \tilde{X}^{2}(\Delta ) \rangle = \langle \tilde{X}_{0}^{2}(\Delta)
\rangle
e^{-\frac{a\Gamma L}{\Gamma ^{2}+\Delta^{2}}} + \left( 1-e^{
-\frac{a\Gamma L}{\Gamma^{2} + \Delta ^{2}}} \right)$$ For infinite bandwidth squeezed incident light this spectrum approaches the vacuum value $1$, for the frequencies where light is strongly attenuated. The width of this noise region grows with optical depth of the system. It is within this spectral region that quantum features of the light field are transferred onto atoms.
In the case of the finite bandwidth of ideal squeezing [@Shap] $\langle \hat{X}_{q}(0,\tau)
\hat{X}_{q}(0,\tau^{\prime }) \rangle \simeq 2 \pi \hbar \omega / c
(\delta (\tau -\tau^{\prime})- \Gamma_{q} / 2e^{-\Gamma_{q} | \tau -
\tau^{\prime }|})$, calculations based on either Eq.(\[Q-L\]) or Eq.(\[spectra\]) have to be carried out numerically and the mapping efficiencies for different spectral widths of squeezing $\Gamma_{q}$ are shown in Fig.\[fig:var\]. We observe in the figure that when the entire bandwidth of squeezed light is completely absorbed in the sample, further growth of the optical depth leads only to the reduction of the spin squeezing, because the atoms which are not reached by the squeezed light are subject to the standard vacuum noise.
The macroscopic number of atoms in our atomic sample, of which most remain in the ground state, allows us to replace the sum of fermionic atomic operators by an effective bosonic operator $\hat{{\cal Q}}_{L}$ matching the bosonic operator of the light field. This restriction should be kept in mind when comparing our results to other analyses of spin-squeezing [@Wine].
A suitable experimental setup for realization of the storage of field correlations in atoms is the cold atom fountain, e.g. as used in a frequency standard. A recent paper [@Sant99] reports operation of a laser cooled cesium fountain clock in the quantum limited regime meaning that the variance $\langle X^{2}\rangle =nL$ of the collective atomic spin associated with the $F=4,m=0$ – $F=3,m=0$ two level system has been achieved. This means that the setup is suitable for the observation of squeezing of $\langle X^{2} \rangle$. The decoherence time $\Gamma_{0}$ of the order of a second reached in the atomic standard setup in principle allows quantum memory on this time scale. We thus propose to prepare atoms in the $F=3,m=0$ state (our state $1$, the level $F=4,m=0$ plays the role of our state $3$) and to illuminate them by a Raman pulse containing the squeezed vacuum and the strong field as described above. After the pulse and after some delay the atoms are interrogated in a microwave cavity where their collective spin state is analyzed to verify that the memory works.
We now wish to address the experimental requirements for our proposal. For our two-level analysis to be valid, we assume that $\Delta \gg \Gamma_{q}, \Gamma_{s}, \gamma_{i}$ and $\sigma_{R} \gg
\sigma_{\text{2-level}}$ where $\sigma_{R}=(6\pi)^{4} c^{8}
I_{\text{sat}}^{2} / (2 \Gamma_{q} S \omega^{11} \hbar^{3}
\Delta_{i}^{2})$ is the stimulated Raman cross section for the quantum field, $S=I_{s}/I_{\text{sat}}$ is the saturation parameter and $I_{\text{sat}}=\omega^{6}/ (9\pi c^{5}) \sum \mu_{1i} \mu_{3i}$ is the saturation intensity for the strong field for $1 \leftrightarrow
i$, $3 \leftrightarrow i$ transitions, $\sigma_{\text{2-level}} = 3
\lambda^{2} \gamma_{i}^{2} / (8\pi \Delta_{i}^{2})$ is the spontaneous 2-level cross section. In order to carry out the steady state solution of (\[Bloch\]-\[Max\]) we assume $\Gamma_{s} \gg
\tau_{\text{pulse}}^{-1}$ - where $\tau_{\text{pulse}}$ is the duration of the Raman pulse. Finally, the condition on the bandwidth of the quantum field $\Gamma_{q} \gg \tau_{\text{pulse}}^{-1}$ ensures that the pulse is long enough to contain all relevant correlations of the quantum state of the field. It is possible to satisfy all those conditions with the following set of parameters: $\Gamma_{q}=10^{7}$Hz, $\Delta = 10^{9}$Hz, $S>4$, $\tau_{\text{pulse}} = 10$ $m$sec. With the resonant optical depth of $20$ achievable for $5 \times 10^{5}$ atoms a mapping efficiency exceeding 80% is possible (Fig.\[fig:var\]). After the pulse is switched off the memory time $\Gamma_{0}^{-1}$ is set by the free evolution of the $F=4,m=0$ – $F=3,m=0$ system and as mentioned above it can be as long as a second.
We have analyzed the possibility to transfer (write-down) a quantum state of light onto an atomic sample. And we have suggested how to perform a delayed measurement of the quantum state. We will now briefly discuss how to map the atomic state back onto a light field by interspecies teleportation [@Maie98]. To realize an effective teleportation of an atomic collective spin onto a light beam we suggest an approach similar to teleportation of light [@Furu98; @Brau98; @Vaid94] with EPR correlated light beams and a beam-splitter type interaction between one of the beams and the atomic collective spin. Making a homodyne measurement of the light quadrature and a Ramsey measurement of the atomic spin we may employ the protocol used for light teleportation [@Furu98; @Brau98; @Vaid94] and restore the atomic state in the other light beam.
To realize the “beam-splitter” we send a short pulse ($\tau_{\text{pulse}} \Gamma \ll 1$ - so that dissipation processes do not take place) of one of the EPR beams through our atomic sample in the small optical depth regime ($\alpha \rightarrow a
\tau_{\text{pulse}} L \ll 1$). In our scheme the switching from high to small optical depth is made simply by adjusting the intensity of the coupling field $E_{s}$. In the weak coupling regime (small optical depth) the interaction between light and atoms (\[E\]) - (\[Q-L\]) can be described by a linear approximation leading to a “beam-splitter” type interaction. Introducing a new rescaled atomic operator $\hat{q} = (nL)^{-1/2} \hat{{\cal Q}}_{L}$ and the field “area” operator $\hat{\theta} = \sqrt{\lambda / 2 \pi \hbar
\tau_{\text{pulse}}} \int_{0}^{\tau_{\text{pulse} }} d \tau^{\prime}
\hat{E}_{q}(\tau^{\prime})$ we obtain:
$$\begin{aligned}
\hat{q}_{\text{out}} &=&\hat{q}_{\text{in}}-ir\hat{\theta}_{\text{in}} \\
\hat{\theta}_{\text{out}} &=&\hat{\theta}_{\text{in}}-ir\hat{q}_{\text{in}}\end{aligned}$$
The condition for such a linearization is a weak interaction, hence our “beam-splitter” is highly asymmetric, $r = \sqrt{
\alpha} = \sqrt{\sigma_{R} \tau_{\text{pulse}} L /
\Gamma_{q}} \ll 1$. Teleportation with asymmetric beam splitters is possible but it requires a higher degree of correlation in the EPR beams. A simple estimate suggests, that the residual noise in the EPR pair must be smaller than $r$. If one assumes a stronger coupling in order to approach the symmetric beam-splitter case, the field probes a component of the atomic coherence, which deviates from the uniform integral in Eq.(\[Q-L\]) due to the spatial variation of the probe light. If, for example, the probe is damped by a factor of order 2, it is reasonable to decompose the probed atomic coherence as a roughly even mixture of the uniform integral $\hat{{\cal Q}}_{L}$ and a “noise” operator which we, for simplicity may assume to be the standard vacuum noise. This noise is comparable to the “quduty” of noise [@Brau98] of a direct detection of the atomic ensemble and reconstruction of a corresponding field state (“classical teleportation”).
We are grateful to Prof. Sam Braunstein for stimulating discussions of the atomic teleportation. This research has been funded by the Danish Research Council and by the Thomas B. Thriges Center for Quantum Information. AK acknowledge support of the ESF-QIT programme.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $\Gamma$ be a crystallographic group of dimension $n,$ i.e. a discrete, cocompact subgroup of $\operatorname{Isom}(\R^n)$ = $O(n)\ltimes\R^n.$ For any $n\geq 2,$ we construct a crystallographic group with a trivial center and a trivial outer automorphism group.'
author:
- 'R. Lutowski, A. Szczepański [^1]'
title: Crystallographic groups with trivial center and outer automorphism group
---
[^2] [^3]
Introduction
============
Let $\Gamma$ be a discrete, cocompact subgroup of $O(n)\ltimes\R^n$ = $\operatorname{Isom}(\R^n)$ i.e. a crystallographic group. If $\Gamma$ is a torsion free group, then $M = \R^n/\Gamma$ is a flat manifold (that is a compact Riemannian manifold without boundary with the sectional curvature $K_x = 0$ for any $x\in M$). Moreover $\pi_{1}(M) = \Gamma.$ In 2003 R. Waldmüller found a torsion free crystallographic group $\Gamma\subset O(141)\ltimes\R^{141}$ (a flat manifold $M = \R^{141}/\Gamma$) with the following properties: $(i)$ $Z(\Gamma) = \{e\},$ $(ii)$ $\operatorname{Out}(\Gamma) = \{e\},$ where $Z(\Gamma)$ is the center of the group $\Gamma,$ and $\operatorname{Out}(\Gamma) = \operatorname{Aut}(\Gamma)/\operatorname{Inn}(\Gamma)$ denotes the group of outer automorphisms of $\Gamma$ (see [@S Appendix C] and [@wald]). Equivalently, $(i)$ means that the abelianization of $\Gamma$ is finite (the first Betti number of $M$ is equal to zero). Moreover, if both conditions $(i)$ and $(ii)$ are satisfied, then the group of affine diffeomorphisms $\operatorname{Aff}(M)$ of the manifold $M$ is trivial (see [@charlap] and [@S]). We do not know if there exist such flat manifolds in dimensions less than $141.$ For example in dimensions up to six such Bieberbach groups do not exist. In this paper we are interested in the existence of not necessarily torsion free crystallographic groups with the above properties. We shall prove that for any $n\geq 2$ there exists a crystallographic group of dimension $n$ which satisfies conditions $(i)$ and $(ii).$
The main motivation for us is the article [@BL] of M. Belolipetsky and A. Lubotzky. For any $n\geq 3$ they found an infinite family of hyperbolic compact manifolds of dimension $n$ with the following property: for every manifold $M$ from this family, $\operatorname{Out}(\pi_1(M))$ = $\{e\}.$ Since the center of the fundamental group of a compact hyperbolic manifold is trivial, the above result gives us an infinite family of groups which satisfy conditions $(i)$ and $(ii).$ The construction of the above hyperbolic examples uses the properties of simple Lie groups of $\R$-rank one and, in particular, follows from the existence of non arithmetic lattices. In our construction the most important are Bieberbach theorems, and specific properties of crystallographic groups.
Crystallographic groups with trivial center and outer automorphism group
========================================================================
In this part we shall prove our main result. Let $\Gamma$ be a torsion free crystallographic group. From Bieberbach’s theorems (see [@S Chapter 2]) we have a short exact sequence of groups $$0\to\Z^n\to\Gamma\stackrel{p}{\rightarrow} G\to 0,$$ where $\Z^n$ is a maximal abelian subgroup of $\Gamma$ and $G$ is a finite group. Moreover, let $h_{\Gamma}:G\to \operatorname{GL}(n,\Z)$ be the integral holonomy representation defined by the formula $$\forall_{g\in G} h_{\Gamma}(g)(e) = \bar{g}e\bar{g}^{-1},$$ where $\bar{g}\in\Gamma, p(\bar{g}) = g$ and $e\in\Z^n.$ Let $$N = N_{\operatorname{GL}(n,\Z)}(h_{\Gamma}(G)) = \{X\in \operatorname{GL}(n,\Z)\mid \forall_{f\in h_{\Gamma}(G)}\hskip 2mm XfX^{-1}\in h_{\Gamma}(G)\}.$$ In the case when $Z(\Gamma) = \{e\},$ we have the following commutative diagram ([@S p. 65-69]) with exact rows and columns: $$\begin{diagram}
\node{}\node{0}\arrow{s}\node{0}\arrow{s}\node{0}\arrow{s}\\
\node{0}\arrow{e}\node{\Z^n}\arrow{s}\arrow{e}\node{\Gamma}\arrow{s}\arrow{e}\node{G}\arrow{s,r}{h_\Gamma}\arrow{e}\node{0}\\
\node{0}\arrow{e}\node{Z^1(G,\Z^n)}\arrow{s}\arrow{e}\node{\operatorname{Aut}(\Gamma)}\arrow{s}\arrow{e,t}{F}\node{N_\alpha}\arrow{s}\arrow{e}\node{0}\\
\node{0}\arrow{e}\node{H^1(G,\Z^n)}\arrow{s}\arrow{e}\node{\operatorname{Out}(\Gamma)}\arrow{s}\arrow{e}\node{N_\alpha/G}\arrow{s}\arrow{e}\node{0}\\
\node{}\node{0}\node{0}\node{0}
\end{diagram}$$
Diagram 1
where $Z^1(G,\Z^n)$ is the group of 1-cocycles. Moreover $$N_{\alpha} = \{n\in N\mid n\ast\alpha =\alpha\},$$ and $\alpha\in H^2(G,\Z^n)$ is the cohomology class of the first row of the diagram. The action $\ast:N\times H^2(G,\Z^n)\to H^2(G,\Z^n)$ is defined by the formula $$n\ast [a] = [n\ast a],$$ where $n\in N, a\in Z^2(G,\Z^n),\hskip 2mm [a]$ is the cohomology class of $a$ and $$\forall_{g_1,g_2\in G} \; n\ast a(g_1,g_2) = n a(n^{-1}g_{1}n,n^{-1}g_{2}n).$$ We have the following proposition.
$\operatorname{Aut}(\Gamma)$ is a crystallographic group if and only if $\operatorname{Out}(\Gamma)$ is a finite group.
[**Proof:**]{} We start with an observation that $Z^1(G,\Z^n)$ is a free abelian group of rank $n$ which is a faithful $N_{\alpha}$ module. First, assume that $\operatorname{Aut}(\Gamma)$ is a crystallographic group with the maximal abelian subgroup $M.$ From [@charlap Proposition I.4.1], $M$ is the unique normal maximal abelian subgroup of $\operatorname{Aut}(\Gamma).$ Hence, $M = Z^1(G,\Z^n),$ and $\operatorname{Out}(\Gamma)$ is a finite group. The reverse implication is obvious. This finishes the proof of the proposition. $\Box$ Let us formulate our main result.
\[main\] For every $n\geq 2$ there exists a crystallographic group $\Gamma$ of dimension $n$ with $Z(\Gamma) = \operatorname{Out}(\Gamma) = \{e\}.$
[**Proof:**]{} We shall need the following lemma.
Let $G, H$ be finite groups and $H\subset G\subset \operatorname{GL}(n,\Z).$ If the group $N_{\operatorname{GL}(n,\Z)}(H)$ is finite, then $N_{\operatorname{GL}(n,\Z)}(G)$ is finite.
[**Proof of Lemma:**]{} From the assumption, $\operatorname{Aut}(H)$ and $\operatorname{Aut}(G)$ are finite. Moreover, we have monomorphisms: $$N_{\operatorname{GL}(n,\Z)}(H)/C_{\operatorname{GL}(n,\Z)}(H)\stackrel{\bar{\phi}}{\rightarrow}\operatorname{Aut}(H)$$ and $$N_{\operatorname{GL}(n,\Z)}(G)/C_{\operatorname{GL}(n,\Z)}(G)\stackrel{\bar{\phi}}{\rightarrow}\operatorname{Aut}(G),$$ where $\bar{\phi}$ is induced by $\phi(s)(g)=sgs^{-1},g\in G, s\in \operatorname{GL}(n,\Z).$ Since $C_{\operatorname{GL}(n,\Z)}(G)\subset C_{\operatorname{GL}(n,\Z)}(H),$ our Lemma is proved. $\Box$
If $\mid\operatorname{Out}(\Gamma)\mid < \infty,$ then $\mid\operatorname{Out}(\operatorname{Aut}(\Gamma))\mid < \infty.$
$\Box$
Assume $Z(\Gamma) = \{e\},$ then
1. $H^1(G,\Z^n)\simeq (\Q^{n}/\Z^{n})^{G} = H^{0}(G,\Q^{n}/\Z^{n});$
2. $Z^1(G,\Z^n)\simeq \{m\in\Q^n\mid\forall_{g\in G}\hskip 2mm gm-m \in \Z^n\} = A^{0}(\Gamma)$ as $N_{\alpha}$ modules;
3. $A(\Gamma) = N_{\operatorname{Aff}(\R^n)}(\Gamma) = \{a\in \operatorname{Aff}(\R^n)\mid\forall_{\gamma\in\Gamma} a\gamma a^{-1}\in\Gamma\}\simeq \operatorname{Aut}(\Gamma).$
$\Box$ We have the following modification of the Diagram 2. $$\begin{diagram}
\node{}\node{0}\arrow{s}\node{0}\arrow{s}\node{0}\arrow{s}\\
\node{0}\arrow{e}\node{\Z^n}\arrow{s}\arrow{e}\node{\Gamma}\arrow{s}\arrow{e}\node{G}\arrow{s,r}{h_\Gamma}\arrow{e}\node{0}\\
\node{0}\arrow{e}\node{A^{0}(\Gamma)}\arrow{s}\arrow{e}\node{A(\Gamma)}\arrow{s}\arrow{e,t}{F}\node{N_\alpha}\arrow{s}\arrow{e}\node{0}\\
\node{0}\arrow{e}\node{(\Q^{n}/\Z^{n})^{G}}\arrow{s}\arrow{e}\node{\operatorname{Out}(\Gamma)}\arrow{s}\arrow{e}\node{N_\alpha/G}\arrow{s}\arrow{e}\node{0}\\
\node{}\node{0}\node{0}\node{0}
\end{diagram}$$
Diagram 2
Let $\Gamma$ be a crystallographic group of rank $n$ with trivial center and holonomy group $G.$ Moreover, assume that the group $H^1(G,\Z^n) = \{e\},$ and the group $\operatorname{Out}(\Gamma)$ is finite. Inductively, put $\Gamma_{0} = \Gamma$ and $\Gamma_{i+1} =$A($\Gamma_{i}$), for $i\geq 0.$
\[seq\] $\exists N$ such that $\Gamma_{N+1} = \Gamma_{N}.$
[**Proof:**]{} We start from observations that for $i > 0,$ $\Gamma_i$ is a crystallographic group, $Z(\Gamma_i) = \{e\}$ and $M_0 = M_i,$ where $M_i = A^{0}(\Gamma_{i-1})\subset\Gamma_i$ is the maximal abelian normal subgroup (a subgroup of translations). Let $G_i =\Gamma_i/M_i.$ From definition we can consider $(G_i)$ as a nondecreasing sequence of finite subgroups of $\operatorname{GL}(n,\Z).$ From Bieberbach theorems [@S Chapter 2] and from Diagrams 1 and 2, there is only a finite number of possibilities for $G_i.$ Hence $\exists N\in\N$ such that $\forall_{i > N}\hskip 2mm G_i = G_{N}.$ This finishes the proof. $\Box$
Let $\Gamma_1 = G_1\ltimes\Z^2$ be the crystallographic group of dimension 2 with holonomy group $G_1 = D_{12},$ where $$D_{12} = \text{gen}\left\{
\left [
\begin{matrix}
0 & -1\cr
1 & -1
\end{matrix}\right ],
\left [
\begin{matrix}
-1 & 0\cr
0 & -1
\end{matrix}\right ],
\left [
\begin{matrix}
0 & 1\cr
1 & 0
\end{matrix}\right ]\right\}$$ is the dihedral group of order 12. Moreover, let $\Gamma_2 = G_2\ltimes\Z^3$ be the crystallographic group of dimension 3, with holonomy group $G_2 = S_4\times\Z_2$ generated by matrices $$\left [
\begin{matrix}
0 & 1 & 0\cr
0 & -1 & -1\cr
1 & 1 & 0
\end{matrix}
\right ],
\quad
\left [
\begin{matrix}
0 & 0 & 1\cr
0 & -1 & -1\cr
-1 & 0 & 1
\end{matrix}
\right ].$$ Here $S_4$ denotes the symmetric group on four letters. For $i = 1,2$ we have $$N_{\operatorname{GL}(n_{i},\Z)}(G_{i}) = G_{i}$$ and $$H^1(G_i,\Z^{n_i}) = 0,$$ where $n_i$ is the rank of $\Gamma_i.$ Hence $A(\Gamma_i) = \Gamma_i,$ and $\operatorname{Out}(\Gamma_i) = \{e\},$ for $i = 1,2.$
Now we are ready to finish the proof of Theorem \[main\]. The cases $n = 2,3$ are done in the above example. Assume $n\geq 4.$ Let $n = 2k+3i,$ where $i\in\{0,1\}.$ Put $\Gamma' =\Gamma_{1}^{k}\times\Gamma_{2}^{i}.$ Then $\Gamma'$ is centerless and by [@lut2 Theorem 3.4] the bottom exact sequence of the Diagram 2 looks as follows $$0\to 0\to \operatorname{Out}(\Gamma')\to S_k\to 0.$$ Hence, $\Gamma'$ satisfies the assumption of Lemma \[seq\] and the sequence $\Gamma_{0} = \Gamma',\hskip 2mm \Gamma_{i+1} =$ A($\Gamma_{i}$) stabilizes, i.e., $\exists N$ such that $\forall_{i\geq N}\hskip 3mm \Gamma_{i}$ = $\Gamma_{N}.$ Moreover, $\operatorname{Out}(\Gamma_{N}) = \{e\}$ and $Z(\Gamma_{N}) = \{e\}.$ $\Box$
[99]{} M. Belolipetsky, A. Lubotzky, *Finite groups and hyperbolic manifolds*, Invent. Math. 162, (2005) no. 3 459 - 472 L. S. Charlap, *Bieberbach Groups and Flat Manifolds*, Universitext, Springer-Verlag, New York, 1986 R. Lutowski, *On symmetry of flat manifolds*, Exp. Math. 18 (2009), no. 3, 201 - 204 A. Szczepański, *Geometry of the crystallographic groups*, Algebra and Discrete Mathematics Vol. 4, World Scientific, Shanghai 2012 R. Waldmüller, *A flat manifold with no symmetries*, Experiment. Math. 12 (2003), no. 1, 71 - 77
Institute of Mathematics\
University of Gdańsk\
ul. Wita Stwosza 57\
80-952 Gdańsk\
Poland\
E-mail: `rlutowsk@mat.ug.edu.pl`, `matas@univ.gda.pl`
[^1]: Both authors are supported by the Polish National Science Center grant 2013/09/B/ST1/04125.
[^2]: 2010 *Mathematics Subject Classification*: Primary 20H15; Secondary 57S30.
[^3]: *Key words and phrases*: Euclidean orbifolds, Crystallographic groups.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'N. G. Guseva'
- 'P. Papaderos'
- 'H. T. Meyer'
- 'Y. I. Izotov'
- 'K. J. Fricke'
date: 'Received ; Accepted'
title: 'An investigation of the luminosity-metallicity relation for a large sample of low-metallicity emission-line galaxies [^1], [^2]'
---
[We present 8.2m VLT spectroscopic observations of 28 H [[ii]{}]{} regions in 16 emission-line galaxies and 3.6m ESO telescope spectroscopic observations of 38 H [[ii]{}]{} regions in 28 emission-line galaxies. These emission-line galaxies were selected mainly from the Data Release 6 (DR6) of the Sloan Digital Sky Survey (SDSS) as metal-deficient galaxy candidates. ]{} [We collect photometric and high-quality spectroscopic data for a large uniform sample of star forming galaxies including new observations. Our aim is to study the luminosity-metallicity ($L-Z$) relation for nearby galaxies, especially at its low-metallicity end and compare it with that for higher-redshift galaxies. ]{} [Physical conditions and element abundances in the new sample are derived with the $T_{\rm e}$-method, excluding six H [[ii]{}]{} regions from the VLT observations and nearly two third of the H [[ii]{}]{} regions from the 3.6m observations. Element abundances for the latter galaxies were derived with the semiempirical strong-line method. ]{} [ From our new observations we find that the oxygen abundance in 61 out of the 66 H [[ii]{}]{} regions of our sample ranges from 12 + log O/H = 7.05 to 8.22. Our sample includes 27 new galaxies with 12 + log O/H $<$ 7.6 which qualify as extremely metal-poor star-forming galaxies (XBCDs). Among them are 10 H [[ii]{}]{} regions with 12 + log O/H $<$ 7.3. The new sample is combined with a further 93 low-metallicity galaxies with accurate oxygen abundance determinations from our previous studies, yielding in total a high-quality spectroscopic data set of 154 H [[ii]{}]{} regions. 9000 more galaxies with oxygen abundances, based mainly on the $T_{\rm e}$-method, are compiled from the SDSS. Photometric data for all galaxies of our combined sample are taken from the SDSS database while distances are from the NED. Our data set spans a range of 8 mag with respect to its absolute magnitude in SDSS $g$ (–12 $\ga M_g \ga$ –20) and nearly 2 dex in its oxygen abundance (7.0$\la$12 + log O/H$\la$8.8), allowing us to probe the $L-Z$ relation in the nearby universe down to the lowest currently studied metallicity level. The $L-Z$ relation established on the basis of the present sample is consistent with previous ones obtained for emission-line galaxies. ]{}
Introduction \[intro\]
======================
It was shown more than 20 years ago that low-luminosity dwarf galaxies have systematically lower metallicities compared to more luminous galaxies [@Lequeux1979; @Skillman1989; @RicherMcC1995]. This dependence, initially obtained for irregular galaxies, was later confirmed for galaxies of different morphological types [e.g. @Vila1992; @KobylZarit1999; @MelbourneSalzer2002; @Lee2004; @Pil2004; @Lee45mu2006].
The differences between giant and dwarf galaxies are usually attributed to different chemical evolution of galaxies with different masses [e.g. @Lequeux1979; @Tremonti2004; @Lee45mu2006; @Ellison2008; @Gavilan2009]. Thus, more efficient mechanisms seem to be at work in massive galaxies converting gas into stars and/or less efficient ones ejecting enriched matter into the galactic halo or even into the intergalactic medium. While the mass of a galaxy is one of the key physical parameters governing galaxy evolution, its determination is not easy and somewhat uncertain. Therefore, very often the luminosity, which is directly derived from observations, is used instead of the mass. In addition, some authors also use other global characteristics of a galaxy such as Hubble morphological type, rotation velocity, the gas mass fraction, surface brightness of the galaxy, to study correlations between metallicity and macroscopic properties of a galaxy [e.g. @Tremonti2004; @Pil2004].
Metallicity reflects the level of the gas astration in the galaxy. Hence, the metallicity of a galaxy depends strongly on its evolutionary state, specifically, on the fraction of the gas converted into stars. The metallicity in emission-line galaxies is defined in terms of the relative abundance of oxygen to hydrogen (usually 12 + log O/H) in the interstellar medium (ISM). Different mechanisms were considered in chemical evolution models to account for the low metallicity of dwarf galaxies, mainly 1) enriched galactic wind outflow which expells the newly synthesized heavy elements from the galaxy, resulting in slowing enrichment of the galaxy ISM; 2) inflow of metal-poor intergalactic gas and its mixing with the galaxy ISM which results in decreasing ISM metallicity, and 3) the burst character of star formation with a very low level of astration between the bursts. In principle, chemical evolution models could predict the slope and scatter of the mass-metallicity $M-Z$ (and luminosity-metallicity $L-Z$) relations over a large range in mass (luminosity) and metallicity invoking the mechanisms mentioned above.
Usually, $L-Z$ relations are based on optical observations of nearby galaxies. However, it was shown in recent studies that the near infrared (NIR) range could be more promising for such studies. @Saviane2008 collected abundances obtained by means of the temperature-sensitive method and NIR luminosities for a sample of dwarf irregular galaxies with –20 $<$ $M_H$ $<$ –13, located in nearby groups of galaxies. They obtained a tight $M-Z$ relation with a low scatter of 0.11 dex around its linear fit. @Salzer05 [see also @Vaduvescu07] noted that the NIR luminosities are more fundamental than the $B$-band ones, since they are largely free of absorption effects and are more directly related to the stellar mass of the galaxy than optical luminosities. Nevertheless, this statement is correct only for galaxies with low and moderate SF activity. In emission-line galaxies with high star formation rate (SFR), such as blue compact dwarf (BCD) galaxies, the young, low mass-to-light ($M/L$) ratio stellar component may provide up to $\sim$50% of the total $K$ band emission [@Noeske03]. Additionally, in such systems the contribution of ionized gas to the total luminosity could be high [see e.g. @I97b; @P98; @P02], especially in the NIR range [see e.g. @Vanzi00; @Smith2009], and should be taken into account.
Recently, studies of the $L-Z$ relation were extended to larger volumes by including moderate- and high-redshift galaxies [@KobylZarit1999; @Contini2002; @Maier2004]. Variations of the $L-Z$ relation with redshift can provide a means to study the galaxy evolution with look-back time [see, e.g., @Kobulniky2003]. It was established in this study that the slopes and zero points of the $L-Z$ relation evolve smoothly with redshift. Its large dispersion has been attributed to galaxy evolution effects. However, these results and their comparison with those for nearby galaxies should be considered with caution. The high-redshift samples are biased by different selection criteria and metallicity calibrations as compared to the local galaxies. They consist on average of more luminous and higher metallicity galaxies. Star-forming dwarf galaxies in the relatively high-redshift (up to $z$ $\sim$ 1) samples are rare because of their intrinsic faintness. Moreover, due to the weakness of the \[O [iii]{}\]$\lambda$4363 emission line in the spectra of these galaxies, their abundance determinations are more uncertain and could lead to a large scatter in the $L-Z$ diagrams. This fact could be the reason for a larger scatter of high-redshift dwarf galaxies if the direct $T_{\rm e}$-method is used instead of the empirical R$_{23}$ one [e.g., @Kakazu07].
In summary, it is difficult to obtain reliable metallicities over a large luminosity range in a homogeneous manner, i.e. employing a unique technique (e.g. the direct $T_{\rm e}$-method), even for nearby galaxies. Therefore, different methods for abundance determination are applied for galaxies of different types. The direct method is mainly used for nearby low-metallicity galaxies, while various empirical methods are used for nearby high-metallicity galaxies and for almost all high-redshift galaxies. The variety of methods results in significant differences in the $L-Z$ relations obtained with the direct $T_{\rm e}$-method and those based on strong emission line ratio calibrations, such as $R_{23}$, $P$-method, N2 and O3N2 methods. These differences were reported by many authors [e.g., @Pil2004; @Shi2005; @Hoyos2005; @Kakazu07].
Large surveys, such as the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and Sloan Digital Sky Survey (SDSS), provide rich data sets for statistically improved studies of the $L-Z$ relation. For example, @Lamareille2004 using more than 6000 spectra of SF galaxies at $z$ $<$ 0.15 from the 2dFGRS have obtained an $L-Z$ relation that is much steeper than that for nearby irregulars and spiral galaxies. @Tremonti2004 studied the mass-metallicity ($M-Z$) relation for 53000 SF galaxies within $z$ $\sim$ 0.2 extracted from SDSS, using their stellar continuum and line fitting method. This method is applicable because the bulk of their emission-line galaxies show weak emission lines and strong stellar absorption features, and therefore the contribution of gaseous emission to the galaxy luminosity is low. The @Tremonti2004 $M-Z$ relation is relatively steep but it flattens for massive galaxies at masses above 10$^{10}$ M$_{\odot}$. On the contrary, @MelbourneSalzer2002 using 519 emission-line galaxies from the KPNO International Spectroscopic Survey (KISS) found that the slope of the $L-Z$ relation for luminous galaxies is steeper than that for dwarf galaxies. Nevertheless, @Pil2004 have compared the $L-Z$ relation based on more than 1000 published spectra of H [ii]{} regions in spiral galaxies to that for irregular galaxies. They found that the slope of the relation for spirals is slightly shallower than the one for irregular galaxies. Furthermore, using 72 star-forming galaxies, @Shi2005 have also shown that the slope of the $L-Z$ relation for luminous galaxies is slightly shallower than that for dwarf galaxies.
Is the slope of the $L-Z$ relation invariant for galaxies of different type, such as local dwarf and spiral galaxies and high-redshift galaxies? If differences in the $L-Z$ relations for intermediate- and high-redshift and local ones are present, they may yield important constraints on the Star Formation History (SFH) of galaxies. For this, an as accurate as possible $L-Z$ relation, based on homogeneous high-quality photometric and spectroscopic data is required for galaxies that covers a large range in metallicity and luminosity. In particular, probing the slope of the $L-Z$ relation in its low-metallicity end, i.e. in the range expected for unevolved low-mass galaxies in the faraway universe, is much needed. For this purpose, in this paper we focus our study on the lowest-metallicity galaxy candidates selected from large spectroscopic surveys, using deep follow-up spectroscopic observations.
Specifically, the objective of the work is to study the $L-Z$ relation for a large uniform sample of emission-line galaxies in the local Universe for which the element abundances are obtained with high precision. The main feature of our sample is that it is one of the richest currently available at the low-metallicity end. For the galaxy selection we used different surveys such as 2dFGRS, SDSS and others. Most of our sample galaxies currently undergo strong episodes of star formation (SF).
We performed 3.6m ESO spectroscopic observations of a sample of 38 H [ii]{} regions in 28 emission-line galaxies and 8.2m VLT spectroscopic observations of a sample of 28 H [ii]{} regions in 16 emission-line galaxies. We supplement our new data with our previous data collected from the MMT observations [@IT2007], and from the 3.6m ESO observations [@BJlarge2007; @Pap2008] of the low-metallicity emission-line galaxies selected from the SDSS, with the sample used by @IT04a to study the helium abundance in low-metallicity BCDs (henceforth referred to as the HeBCD sample) and with the MMT sample used by @TI2005 to study high-ionization emission lines in low-metallicity BCDs. Our MMT, 3.6m ESO and HeBCD low-metallicity galaxies were selected from different surveys [a more complete description of the MMT, 3.6m ESO and HeBCD subsamples can be found in @IT2007; @BJlarge2007; @Pap2008; @IT04a]. During past years we selected from the SDSS and performed follow-up spectroscopic observations with large telescopes of (i) BCDs with strong ongoing SF, i.e. galaxies with high EW(H$\beta$), blue colours, high ionisation parameter, and (ii) low-metallicity galaxies in a relatively quiescent phase of SF, i.e. galaxies with low EW(H$\beta$), low ionisation parameter or older starburst age. For this, we selected galaxies with weak or not detected \[O [iii]{}\]$\lambda$4363 emission line and with \[O [iii]{}\]$\lambda$4959/H$\beta$ $\la$ 1 and \[N [ii]{}\]$\lambda$6583/H$\beta$ $\la$ 0.05 [@IPGFT2006; @IT2007].
SDSS is an excellent source of both photometric and spectroscopic data for more than one million galaxies in its Data Release 7 (DR7) [@A09]. Despite that, our stringent selection criteria resulted in a very small sample of extremely metal-deficient emission-line galaxies with reliable abundance determinations. This sample is supplemented for the purpose of comparison by a sample of $\sim$ 9000 SDSS emission-line galaxies (SDSS sample) over a larger range of metallicities. The oxygen abundances for the galaxies from the SDSS sample are obtained using the direct $T_{\rm e}$-method. In addition, only high-quality spectra of SDSS galaxies with the non-detected \[O [iii]{}\]$\lambda$4363 emission line are included, for which oxygen abundances are derived by a semiempirical method [@IT2007].
Thus, we construct a large homogeneous sample with uniform selection criteria, uniform data reduction methods, and uniform techniques for the element abundance determinations. The apparent $g$ magnitudes for our entire data set are taken from the SDSS. They were used to derive absolute $g$ magnitudes which were corrected for the Galactic extinction and Virgo cluster infall, except for the comparison SDSS sample galaxies. For the latter galaxies the absolute magnitude was derived from the observed redshift, adopting a Hubble constant of $H_0$ = 75 km s$^{-1}$ Mpc$^{-1}$.
The paper is organized as follows. Observations and data reduction are described in Sect. 2. Physical conditions and element abundances in the galaxies from the new observations are presented in Sect. 3. We discuss the properties of the $L-Z$ relation in Sect. 4 and summarise our conclusions in Sect. 5.
Observations and data reduction
===============================
The new spectra of the 3.6m ESO sample were obtained on 14 - 16 September, 2007 with the spectrograph EFOSC2. The grism Gr\#7 and a long slit with the width of 12 were used yielding a wavelength coverage of $\lambda$$\lambda$3400–5200Å. The long slit was centered on the brightest part of each galaxy and simultaneously on another H [ii]{} regions, whenever present. The name of each galaxy with its different H [ii]{} regions, the coordinates R.A., Dec. (J2000.0), date of observation, exposure time, number of exposures for each observation, average airmass and seeing are given in Table \[obs36\]. All spectra were obtained at low airmass or with the slit oriented along the parallactic angle, so no corrections for atmospheric refraction have been applied.
The new VLT spectra were obtained during several runs in October - December, 2006 and January, 2007 with the spectrograph FORS2 mounted at the ESO VLT UT2. The observing conditions were photometric during the nights with seeing $<$ 15. Several observations were performed under excellent seeing conditions ($<$ 08). The grisms 600B ($\lambda$$\lambda$$\sim$3400–6200) and 600RI and filter GG435 ($\lambda$$\lambda$$\sim$5400–8620) for the blue and red parts of the spectrum, respectively, were used. A 1$\times$360 long slit was centered on the brightest H [ii]{} regions of each galaxy. In Table \[obsVLT\], the same parameters as in Table \[obs36\] are given for the VLT observations. Note that for each galaxy the first and the second lines are related to the observations in the blue and red ranges, respectively. Again, as for the EFOSC2 spectra, the observations were obtained at low airmass, and no corrections for atmospheric refraction were applied.
The data were reduced with the IRAF[^3] software package. This included bias–subtraction, flat–field correction, cosmic-ray removal, wavelength calibration, night sky background subtraction, correction for atmospheric extinction and absolute flux calibration of the two–dimensional spectrum. The spectra were also corrected for interstellar extinction using the reddening curve of @W58. One-dimensional spectra of one or several H [ii]{} regions in each galaxy were extracted from two-dimensional observed spectra. The flux- and redshift-calibrated one–dimensional EFOSC2 3.6m spectra of the H [ii]{} regions are shown in Fig. \[fig1\] for all galaxies given in Table \[obs36\]. One-dimensional VLT spectra are shown in Fig. \[fig2\] for 28 objects listed in Table \[obsVLT\]. For VLT spectra of the four background galaxies and H [ii]{} region No.2 in the galaxy J2354-0004 without a detectable \[O [iii]{}\]$\lambda$4363$\AA$ emission line, no abundance determination has been done.
Emission-line fluxes were measured using Gaussian profile fitting. The errors of the line fluxes were calculated from the photon statistics in the non-flux-calibrated spectra. They have been propagated in the calculations of the elemental abundance errors. The quality of the VLT data reduction could be verified by a comparison of He [i]{} $\lambda$5876 emission line fluxes measured in the blue and red spectra of the same object. We found that the fluxes of the He [i]{} $\lambda$5876 emission line in spectra of bright objects differ by no more than 1-2% indicating an accuracy in the flux calibration at the same level. For faint objects the difference between the flux of the He [i]{} $\lambda$5876 emission line in the blue and red spectra is higher, $\sim$5 – 10%, and is comparable to the statistical errors listed in Table \[t4\_1\_VLT\].
The extinction coefficient $C$(H$\beta$) and equivalent widths of the hydrogen absorption lines EW(abs) are calculated simultaneously, minimizing the deviations of corrected fluxes $I(\lambda)$/$I$(H$\beta$) of all hydrogen Balmer lines from their theoretical recombination values as $$\begin{aligned}
\frac{I(\lambda)}{I({\rm H}\beta)} & = &\frac{F(\lambda)}{F({\rm H}\beta)}
\frac{EW(\lambda)+|EW(abs)|}{EW(\lambda)}\frac{EW({\rm H}\beta)}{EW({\rm H}\beta)+|EW(abs)|} \\
& \times & 10^{C({\rm H}\beta)f(\lambda)}.\end{aligned}$$ Here $f$($\lambda$) is the reddening function normalized to the value at the wavelength of the H$\beta$ line, $F$($\lambda$)/$F$(H$\beta$) are the observed hydrogen Balmer emission line fluxes relative to that of H$\beta$, EW($\lambda$) and EW(H$\beta$) the equivalent widths of emission lines, and EW(abs) the equivalent widths of hydrogen absorption lines which we assumed to be the same for all hydrogen lines. For $f$($\lambda$) we adopted the reddening law by @W58. The extinction-corrected fluxes of emission lines other than hydrogen ones are derived from equation $$\begin{aligned}
\frac{I(\lambda)}{I({\rm H}\beta)} & = &\frac{F(\lambda)}{F({\rm H}\beta)}
\times 10^{C({\rm H}\beta)f(\lambda)} .\end{aligned}$$
The extinction-corrected emission line fluxes $I$($\lambda$) relative to the H$\beta$ fluxes multiplied by 100, the extinction coefficients $C$(H$\beta$), the equivalent widths EW(H$\beta$), the observed H$\beta$ fluxes $F$(H$\beta$) (in units 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$), and the equivalent widths of the hydrogen absorption lines are listed in Table \[t3\_1\_36\] (3.6m ESO observations) and in Table \[t4\_1\_VLT\] (VLT observations). $C$(H$\beta$) and EW(abs) are set to zero in Tables \[t3\_1\_36\] and \[t4\_1\_VLT\] if we do not have enough observational data or their values are negative.
Physical conditions and element abundances
==========================================
The electron temperature $T_{\rm e}$, the ionic and total heavy element abundances were derived following @Iz06. In particular, for the ions O$^{2+}$, Ne$^{2+}$ and Ar$^{3+}$ we adopt the temperature $T_{\rm e}$(O [iii]{}) directly derived from the \[O [iii]{}\] $\lambda$4363/($\lambda$4959 + $\lambda$5007) emission-line ratio. For $T_{\rm e}$(O [ii]{}) and $T_{\rm e}$(S [iii]{}) we use the relation between the electron temperatures $T_{\rm e}$(O [iii]{}) and the temperatures characteristic for ions O$^{+}$ and S$^{2+}$ obtained by @Iz06 from the H [ii]{} photoionization models based on recent stellar atmosphere models and improved atomic data [@Stasin2003].
We use $T_{\rm e}$(O [ii]{}) for the calculation of O$^{+}$, N$^{+}$, S$^{+}$ and Fe$^{2+}$ abundances and $T_{\rm e}$(S [iii]{}) for the calculation of S$^{2+}$, Cl$^{2+}$ and Ar$^{2+}$ abundances. The electron number densities for some H [ii]{} regions were obtained from the \[S [ii]{}\] $\lambda$6717/$\lambda$6731 emission line ratio. These lines were not observed or not measured in the remaining H [ii]{} regions. For the abundance determination in those H [ii]{} regions we adopt $N_{\rm e}$ = 10 cm$^{-3}$. The precise value of the electron number density makes little difference in the derived abundances since in the low-density limit which holds for the H [ii]{} regions considered here, the element abundances do not depend sensitively on $N_{\rm e}$. The electron temperatures $T_{\rm e}$(O [iii]{}), $T_{\rm e}$(O [ii]{}), the ionization correction factors ($ICF$s), the ionic and total O and Ne abundances are given in Table \[t5\_1\_36\] for 3.6m observations. The electron temperatures $T_{\rm e}$(O [iii]{}), $T_{\rm e}$(O [ii]{}), $T_{\rm e}$(S [iii]{}), electron number density $N_{\rm e}$(\[S [ii]{}\]), the ionization correction factors ($ICF$s), the ionic and total O, N, Ne, S, Cl, Ar and Fe abundances are given in Table \[t6\_1\_VLT\] for VLT observations.
The oxygen abundances 12 + log O/H in 61 H [ii]{} regions out of 66 obtained from the new 3.6m ESO and VLT observations range from 7.05 to 8.22. Among them, 27 H [ii]{} regions with 12 + log O/H $<$ 7.6 are found, including 10 H [ii]{} regions with 12 + log O/H $<$ 7.3. The combined sample consisting of the new observations, 43 BCDs from the HeBCD sample, 30 galaxies from our previous 3.6m ESO observations and 20 galaxies from the MMT observations yields a data set of 154 H [ii]{} regions. For comparison, we also use $\sim$9000 SDSS emission-line galaxies with the \[O [iii]{}\] $\lambda$4363 emission line detected at least at the 1$\sigma$ level, allowing abundance determination by the direct $T_{\rm e}$-method. In addition, SDSS galaxies with high-quality spectra where the \[O [iii]{}\]$\lambda$4363 emission line was not detected are used. In the latter case, the oxygen abundances were derived by the semiempirical method. SDSS galaxies from the comparison sample mostly populate the high-metallicity, high-luminosity ranges, as compared to the galaxies from our combined sample of low-metallicity emission-line galaxies (Figs. \[fig5\] - \[fig7\]). The considered galaxies span two dex in gas-phase oxygen abundance, from 12 + log O/H $\sim$ 7.0 through $\sim$ 9.0.
We use SDSS $g$ magnitudes for the determination of the absolute magnitude $M_g$ of all galaxies from our samples, while usually $B$ magnitudes and $M_B$ are considered in the literature. However, @Pap2008 have shown that for regions with ongoing bursts of star formation, which is the case for our sample galaxies, the $B$–$g$ colour index is of the order of 0.1 mag only and $<$0.3 mag during the first few Gyrs of galactic evolution. Therefore, we do not transform $M_g$ to $M_B$ and directly compare $M_g$’s for the galaxies from our samples with $M_B$’s for the galaxies available from the literature. The use of the SDSS $g$-band photometry for all our samples allows us to investigate the $L-Z$ relation over the $M_g$ range from –21 mag to the faintest magnitude of $\sim$ –12 mag at the low-metallicity end.
Results
=======
Luminosity-metallicity relation
-------------------------------
In order to illustrate the main properties of our sample we plot (a) the reddening parameter $C$(H$\beta$) obtained from the Balmer decrement and (b) the logarithm of the H$\beta$ equivalent width (Fig. \[fig3\]) and the logarithm of the H$\beta$ line luminosity (in erg s$^{-1}$) (Fig. \[fig4\]) as a function of absolute magnitude $M_g$. The new 3.6m telescope and VLT data are shown by open circles and stars, respectively. The metal-poor galaxies collected from previous 3.6m ESO observations are shown by filled triangles [@BJlarge2007; @Pap2008]. Filled circles denote the data from the HeBCD sample collected by @ING2004a and @IT04a. The MMT data [@IT2007] are shown by large filled circles. The comparison SDSS sample is represented by asterisks. From the latter sample H [ii]{} regions in nearby spiral galaxies are excluded, as are faint SDSS galaxies with $m_g$ $>$ 18, the nearest SDSS galaxies with the redshift $z$ $<$ 0.004 and all SDSS galaxies with $\sigma$\[$I(4363)$\]/$I(4363)$ $>$ 0.25, totaling 443 SDSS galaxies from the comparison sample.
Our sample does not show any trend with absolute magnitude of either $C$(H$\beta$) or EW(H$\beta$), contrary to what was obtained by @Salzer05 for the KISS sample. The extinction in our sample galaxies is low. Only a few galaxies have $C$(H$\beta$) $>$ 0.4. The range of EW(H$\beta$) $\sim$ 0 – 300 $\AA$ for the galaxies from our sample is similar to that for the KISS sample [@Salzer05] but it is higher than that for the high-redshift galaxies of @Kobulniky2003 where EW(H$\beta$) $\leq$ 60 $\AA$.
The logarithm of the H$\beta$ luminosity log $L$(${\rm{H}\beta}$) of our galaxies ranges from 36 to 42 (Fig. \[fig4\]). For comparison, the galaxies from the KISS sample by @Salzer05 and intermediate-redshift galaxies by @Kobulniky2003 have log $L$(${\rm{H}\beta}$) $\sim$ 39 – 43 and 39 – 42, respectively, i.e. low-luminosity galaxies are lacking.
In Fig. \[fig5\] we show the oxygen abundance - absolute magnitude $M_g$ relation for the galaxies with oxygen abundances calculated mainly with the $T_{\rm e}$-method. In this Figure, the same samples and symbols as in Fig \[fig3\] are used. The region denoted as “branch” is populated mainly by galaxies with relatively high redshifts ($z$ $>$ 0.02) and oxygen abundances derived by the $T_{\rm e}$-method. Note that selection effects could be present for “branch” high-redshift galaxies which are predominantly distant spirals. In these galaxies we select mainly low metallicity 2 regions with a detectable \[O [iii]{}\]$\lambda$4363 line while the abundance gradient is present in spirals. The dotted line is a mean least-squares fit to all our data and the solid line is a mean least-squares fit to our data excluding “branch” galaxies with M$_g$ $<$ –18.4 and systems with an oxygen abundance in the range 8.0 – 8.3. The dashed line is a mean least-squares fit to the local dwarf irregular galaxies by @Skillman1989. Our sample (including the SDSS subsample) shows the familiar trend of increasing metallicity with increasing luminosity. A linear least square fit to all data yields the relation
$${\rm 12+log(O/H)} = (5.706\pm 0.199) - (0.134\pm 0.012) {\rm M}_{g}$$
(dotted line in Fig. \[fig5\]). Excluding ”branch” galaxies we obtain the relation $${\rm 12+log(O/H)} = (5.076\pm 0.320) - (0.174\pm 0.200) {\rm M}_{g}$$ (solid line in Fig. \[fig5\]). We note that the Skillman et al. and Richer & McCall fits do not extend over the metallicity range of the present data. Therefore, we extrapolate the former fit in Fig. \[fig5\] (dashed line) to higher metallicities. Skillman’s and our fits are obviosuly very similar. The slopes of our $L-Z$ relation of 0.134 (0.174) are very close to the slope of 0.153 by @Skillman1989 and to the slope of 0.147 by @RicherMcC1995.
Our sample is well populated in the low-luminosity range, while less than 10 galaxies from the KISS sample [@Salzer05] which were used for the study of the $L-Z$ relation are fainter than $M_B$ = –15, and none of them has an oxygen abundance less than 7.6. Our sample, excluding the SDSS subsample, has a lower dispersion around the dotted line compared to all our data and shows a shift to lower metallicities or/and higher luminosities. This likely can be attributed to our selection criteria which are optimized for the search for very metal-poor emission-line galaxies. Additionally, our sample galaxies display significant to strong ongoing SF giving rise to a large contribution from young stars and ionized gas to the total light of the galaxy. @P96 [see also @P02], using surface brightness profile decomposition to separate the star-forming component from the underlying host galaxy of BCDs, found that SF regions provide on average 50% of the total $B$-band emission within the 25 $B$ [mag/$\sq\arcsec$]{} isophote, with several examples of more intense starbursts whose flux contribution exceeds 70%. As a result, a shift of BCDs by a $\Delta\,M \sim$–0.75 mag with respect to the relatively quiescent dIrr population is to be expected in Fig. \[fig5\] (see also Fig. \[fig9\]). A similar offset to lower metallicities or/and higher luminosities has been found by @Kakazu07 for their intermediate-redshift low-metallicity emission-line galaxies with strong SF activity. The mass estimate of the galaxy is less sensitive to the presence of star-forming regions as compared to its luminosity. This was demonstrated by @Ellison2008 who found that galaxies in close pairs show enhanced SF activity as compared to a control sample of isolated galaxies. At the same time galaxies in close pairs show a smaller offset in the mass-metallicity relation as compared to the luminosity-metallicity relation. Thus, the offset in Fig. \[fig5\] indicates that both higher luminosities and lower metallicities may contribute to the shift in the luminosity-metallicity diagram of our sample galaxies relative to more quiescent dIrrs.
In Fig. \[fig6\] we demonstrate that the region of “branch” galaxies is populated mainly by relatively high-redshift systems. The sample is the same as in Fig. \[fig5\] but in the left panel only SDSS galaxies with oxygen abundances derived with the $T_{\rm e}$-method are shown and in the right panel only relatively high-redshift galaxies with $z$ $>$ 0.02 are selected.
The location of the galaxies on the luminosity-metallicity diagram is also sensitive to the method used for the abundance determination. In order to illustrate its effect on the observed $L-Z$ relation, we compare in Fig. \[fig7\] the oxygen abundance of SDSS sample galaxies (dots) obtained with the direct $T_{\rm e}$-method (left panel) and with the semiempirical strong-line method (right panel). The abundances for other galaxies in Fig. \[fig7\] are derived with the $T_{\rm e}$-method. In this Figure we show the larger control sample of the SDSS ($N$=7964) as compared to Fig. \[fig5\]. Only H [ii]{} regions in nearby spiral galaxies and from the nearest SDSS galaxies with redshifts $z$ $<$ 0.004 were excluded from the $\sim$ 9000 SDSS sources while faint galaxies with $m_g$ $>$ 18 are included. Symbols in Fig. \[fig7\] are the same as in Fig. \[fig3\] except for SDSS galaxies which are shown by dots. The dotted line is a mean least-squares fit to all our data from Fig. \[fig5\], while the solid line is a mean least-squares fit to the same data excluding “branch” galaxies.
It can be seen from Fig. \[fig7\] that the oxygen abundance of a given galaxy obtained by different methods could differ by $\sim$0.3–0.5 dex, especially for luminous galaxies. This figure illustrates clearly above 12 + log O/H $\sim$ 8.5 and $M_g$ $<$ –19 - –20 significant discrepancies between oxygen abundances obtained from the $T_{\rm e}$-method and empirical methods. @Stas2002 emphasized that, at high metallicity, the $T_{\rm e}$ derived from \[O [iii]{}\] $\lambda$4363 would largely overestimate the temperature of the O$^{++}$ zone (and largely underestimate the metallicity) because cooling is dominated by the \[O [iii]{}\] $\lambda$52 $\mu$m and \[O [iii]{}\] $\lambda$88 $\mu$m lines. At the same time @Pil2007 demonstrated that there is an observational limit of the highest possible metallicities near 12 + log O/H $\sim$ 8.95. This maximum value was determined in the centers of the most luminous (–22.3 $\la$ $M_B$ $\la$ –20.3) galaxies using the semiempirical ff-method [@Pil2006]. Thus, although the main mechanisms determining the electron temperature in H [ii]{} nebulae have been known for a long time, there are still important unsolved problems.
The contribution of star-forming regions to the light of the galaxy can be quantified by the equivalent width EW(H$\beta$) of the H$\beta$ emission line which in turn depends on the age of the burst of star formation. In Fig. \[fig8\] we show the same samples as in Fig. \[fig5\] except for the SDSS galaxies now being split into two subsamples. In the left panel only those with high equivalent widths EW(H$\beta$) $>$ 80$\AA$ are shown while in the right panel only SDSS galaxies with low equivalent widths EW(H$\beta$) $<$ 20$\AA$ are plotted. The dotted line in the left and right panels is a mean least-squares fit to all our data shown in Fig. \[fig5\], while the solid line is a mean least-squares fit to the same data excluding “branch” galaxies. There is a clear difference between the two subsamples of the SDSS galaxies by $\sim$ 0.4 dex in oxygen abundance or, equivalently, by $\sim$ 3 mag in absolute magnitude. SDSS galaxies with EW(H$\beta$) $>$ 80$\AA$ nicely follow the relation for our dwarf low-metallicity emission-line galaxies shown as reference objects by filled and open circles, stars, filled triangles and large filled circles. On the other hand, the SDSS galaxies with EW(H$\beta$) $<$ 20$\AA$ are located systematically above the low-metallicity galaxies. We propose two possible explanations for such a difference between the two subsamples of SDSS galaxies: 1) the emission of the SDSS galaxies with high EW(H$\beta$) is dominated by star-forming regions, therefore they have higher luminosities compared to galaxies in a relatively quiescent stage; 2) SDSS galaxies with low EW(H$\beta$) are the ones with higher astration level, therefore they are more chemically evolved systems with higher oxygen abundances. Perhaps both of these explanations are tenable, accounting for the observed differences between SDSS galaxies with high and low EW(H$\beta$). Thus, the lowest-metallicity SDSS galaxies are found predominantly among galaxies with high EW(H$\beta$). On the other hand, no extremely low-metallicity SDSS galaxies are found among systems with EW(H$\beta$) $<$ 20$\AA$. Thus, mixing of the SDSS galaxies with EW(H$\beta$) $<$ 20$\AA$ and $>$ 80$\AA$ results in a significant increase of the dispersion of the luminosity-metallicity diagram.
The redshift of the galaxy could also play a role. In Fig. \[fig5\] the bulk of the galaxies with 12+log(O/H)=8.0 – 8.3 and absolute magnitudes between –19 and –21 mag (denoted as “branch” galaxies) is represented by higher-redshift systems as compared to other galaxies from the SDSS and a correction for redshift is required. Since “branch” galaxies are blue, a correction for redshift for systems with weak emission lines would increase their brightness by $\sim$ 0.1 - 0.3 mag. This will not be enough to remove the offset between “branch” galaxies and lower-redshift galaxies in Fig. \[fig5\]. The situation is more complicated for “branch” galaxies with strong emission lines since their effect on the apparent magnitudes of a galaxy in standard passbands will significantly depend on redshift [see e.g. @Z08]. Because of these reasons, we decided not to take into account corrections for redshift.
Comparison of our sample with other data
----------------------------------------
In Fig. \[fig9\] we compare our $L-Z$ relation with other published data for galaxies of different types. In this Figure, all of our galaxies from Fig. \[fig5\], including those from the comparison SDSS sample, are shown by small filled circles. Some well known metal-poor galaxies are depicted by large filled circles and are labelled. Their absolute magnitudes $M_B$ are taken from @Kewley2007. For comparison, 23 KISS emission-line galaxies by @Lee2004 are displayed with large open double circles. The abundances for these galaxies are derived with the $T_{\rm e}$-method, the $B$-band magnitudes are from @Salzer1989 and @GildePaz2003. With open double squares we show 25 nearby dIrrs with the 4.5$\mu$m [*Spitzer*]{} luminosities and compiled O/H abundances derived with the $T_{\rm e}$-method [@Lee45mu2006]. With large open circles and large crosses we respectively show 20 irregular galaxies from @Skillman1989 and 21 dwarf irregular galaxies from @RicherMcC1995 for which oxygen abundances are obtained mainly with the $R_{23}$ empirical method, and for a few objects only with the $T_{\rm e}$-method. The thick solid line is a mean least-squares fit to all our data. The thin solid line is a least-squares fit to the data by @RicherMcC1995 while the dotted line is a mean least-squares fit to the data by @Skillman1989. The dashed line is the luminosity-metallicity relation for local metal-poor BCDs obtained by @KunthOstlin2000.
Data for intermediate- and high-redshift galaxies are also shown. The most distant ($z$ $<$ 1) extremely metal-poor galaxies (XMPGs) [@Kakazu07] with the oxygen abundances derived with the empirical method are shown with filled squares, while relatively metal-poor luminous galaxies at $z$ $\sim$ 0.7 [@Hoyos2005] (O/H derived with the $T_{\rm e}$-method) with filled triangles. The remaining samples in Fig. \[fig9\] are the following: a) the large open circles correspond to the $z$ = 3.36 lensed galaxy [@Villar2004] and to the average position of luminous Lyman-break galaxies at redshifts $z$ $\sim$ 2.5 [@KobKoo2000] (O/H derived with the $R_{23}$ method); b) small open circles stand for 66 Canada-France Redshift Survey (CFRS) galaxies by @Lilly2003 in the redshift range of $\sim$ 0.5 – 1.0 (O/H derived with the $R_{23}$ empirical method); c) asterisks are for 204 GOODS-N (Great Observatories Origins Deep Survey - North) emission-line galaxies in the range of redshifts 0.3 $<$ $z$ $<$ 1.0 [@KobKew04] (O/H is derived with the $R_{23}$ empirical method); d) small open rombs indicate 64 emission-line field galaxies from the Deep Extragalactic Evolutionary Probe Groth Strip Survey (DGSS) in the redshift range of $\sim$ 0.3 – 0.8 [@Kobulniky2003] (O/H derived with the $R_{23}$ empirical method); e) open squares are for the gamma-ray burst (GRB) hosts by @Kewley2007. Small open squares are for galaxies with O/H derived with the empirical method [@KewleyDopita2002] and large open squares for the galaxies with O/H derived with the $T_{\rm e}$-method [@Kewley2007]; f) filled stars denote the 14 star-forming emission-line galaxies at intermediate redshifts (0.11 $<$ $z$ $<$ 0.5) by @KobylZarit1999 (O/H derived with the empirical $R_{23}$ method); g) open filled triangles are for 29 distant 15$\mu$m-selected luminous infrared galaxies (LIRGs) at $z$ $\sim$ 0.3 – 0.8 taken from the sample of @Liang2004 (O/H derived with the empirical method); and, finally, h) the large dotted rectangle depicts the position of Lyman break galaxies [LBGs, @Pettini2001] on the $L-Z$ diagram.
The location of our galaxies on the luminosity-metallicity diagram is similar to that obtained previously for local emission-line galaxies but is shifted to higher luminosities and/or lower metallicities compared to that obtained for quiescent irregular dwarf galaxies. For comparison, @Lee2004 have also demonstrated that their 54 H [ii]{} KISS galaxies with O/H derived with the $T_{\rm e}$-method follow the $L-Z$ relation with a slope similar to that for a more quiescent dIrrs but are shifted to higher brightness by 0.8 magnitudes. Furthermore, they have shown that H [ii]{} galaxies with disturbed irregular outer isophotes (likely due to the interaction) are shifted to a more luminous and/or more metal-poor region in the $L-Z$ diagram as compared to morphologically more regular galaxies. Note that their samples of H [ii]{} galaxies and of dIrrs are in the same luminosity range as our sample. @Pap2008 also note that in contrast to the majority ($>$90%) of BCDs, the extremely metal-poor SF dwarfs reveal more irregular and bluer hosts.
Thus, the difference in the zero point between our $L-Z$ relation for low-metallicity galaxies and for other galaxies seems to be primarily due to the differences in the intrinsic properties of the galaxies selected for different samples with various selection criteria.
A key question is whether a unique $L-Z$ relation does exist for galaxies of different types. The assessment of this issue is complicated by offsets of high-redshift galaxies with different look-back-times. In this context, @Kobulniky2003 have shown that both the slopes and zero points of the $L-Z$ relation exhibit a smooth evolution with redshift. A possible universal $L-Z$ relation for galaxies is also blurred by the fact that metallicity determinations of various galaxy samples, differing in their EW(H$\beta$), absolute magnitude and redshift, do not employ a unique technique. More specifically, several authors emphasize the presence of a well-known shift between the O/H ratio obtained by the direct $T_{\rm e}$-method and empirical strong-line methods. Oxygen abundances obtained by empirical methods are by 0.1 –0.25 dex [@Shi2005] and even by up to 0.6 dex [@Hoyos2005] higher than those obtained with the $T_{\rm e}$-method. For our sample we obtained an offset of $\sim$0.3–0.5 dex.
It can be seen from Fig. \[fig9\] that the high-redshift galaxies with an oxygen abundance derived by the $T_{\rm e}$-method have a shallower slope compared to local galaxies. On the other hand, oxygen abundances of high-redshift galaxies obtained with the $R_{23}$ empirical strong-line method [data in Fig. \[fig9\] by @Lilly2003; @KobKew04; @Liang2004] are higher and follow the relation for high-metallicity SDSS galaxies in Fig. \[fig7\]b despite the fact that oxygen abundances for the latter galaxies were calculated with the different semi-empirical strong-line method. Because of this agreement we decided not to re-calculate oxygen abundances of high-redshift galaxies with the semi-empirical method and adopted O/H values from the literature. Keeping in mind the systematic differences between oxygen abundances derived with the empirical and the $T_{\rm e}$-methods, it might be worth considering a decrease in oxygen abundance by $\sim$ 0.2 – 0.6 dex for all high-redshift galaxies with O/H derived with the empirical method. In that case, the position of high-redshift galaxies on the $L-Z$ diagram would be consistent with that of the “branch” galaxies. Such considerations add further support to the results obtained by @Pil2004 and @Shi2005 that the more luminous galaxies have a slope of the $L-Z$ relation more shallow than that of the dwarf galaxies.
We presume that our $L-Z$ relation could be useful as a local reference for studies of this relation for other types of local galaxies and/or of high-redshift galaxies.
Summary
=======
We present VLT spectroscopic observations of a new sample of 28 H [ii]{} regions from 16 emission-line galaxies and ESO 3.6m telescope spectroscopic observations of a new sample of 38 H [ii]{} regions from 28 emission-line galaxies. These galaxies have mainly been selected from the Data Release 6 (DR6) of the Sloan Digital Sky Survey (SDSS) as low-metallicity galaxy candidates.
Physical conditions and element abundances are derived with the $T_{\rm e}$-method for 38 H [ii]{} regions observed with the 3.6m telescope and for 23 H [ii]{} regions observed with the VLT.
From our new observations we find that the oxygen abundance in 61 out of the 66 observed H [ii]{} in our sample ranges from 12 + log O/H = 7.05 to 8.22. The oxygen abundance in 27 H [ii]{} regions is 12 + log O/H $<$ 7.6 and among them 10 H [ii]{} regions have an oxygen abundance less than 7.3.
This new data in combination with objects from our previous studies constitute a large uniform sample of 154 H [ii]{} regions with high-quality spectroscopic data which are used to study the luminosity-metallicity ($L-Z$) relation for the local galaxies with emphasis on its low-metallicity end.
As a comparison sample we use $\sim$ 9000 SDSS emission-line galaxies with higher oxygen abundances which are also obtained mainly by the direct $T_{\rm e}$-method. For all of our sample galaxies the $g$ magnitudes are taken from the SDSS while the distances are from the NED. The entire sample spans nearly two orders of magnitude with respect to its gas-phase metallicity, from 12 + log O/H $\sim$ 7.0 to $\sim$ 8.8, and covers an absolute magnitude range from $M_g$ $\sim$ –12 to $\sim$ –20.
We find that the metallicity-luminosity relation for our galaxies is consistent with previous ones obtained for objects of similar type. The local $L-Z$ relation obtained with high-quality spectroscopic data is useful for predictions of galaxy evolution models.
N. G. G. and Y. I. I. thank the Max Planck Institute for Radioastronomy (MPIfR) for hospitality, and acknowledge support through DFG grant No. Fr 325/57-1. P. P. thanks the Department of Astronomy and Space Physics at Uppsala University for its warm hospitality. K. J. Fricke thanks the MPIfR for Visiting Contracts during 2008 and 2009. This research was partially funded by project grant AYA2007-67965-C03-02 of the Spanish Ministerio de Ciencia e Innovacion. We acknowledge the work of the Sloan Digital Sky Survey (SDSS) team. Funding for the SDSS has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the US Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http:// www.sdss.org/.
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Thuan, T. X., & Izotov, Y. I. 2005, , 161, 240
Tremonti, C. A., Heckman, T. M., Kauffmann, G., et al. 2004, , 613, 898
Vanzi, L., Hunt, L.K., Thuan, T.X., & Izotov, Y.I. 2000, , 363, 493
Vaduvescu, O., McCall, M.L., & Richer, M.G. 2007, , 134, 604
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Villar-Martín, M., Cerviño, M, & González Delgado, R. M. 2004, MNRAS, 355, 1132
Whitford, A. E. 1958, , 63, 201
Zackrisson, E., Bergvall, N.,& Leitet, E. 2008, , 676, L9
[lcclcccc]{}
Galaxy & R.A., Dec. & Date & Exp. time & Airmass & seeing & redshift\
& (J2000) &(2007) & (sec) & & &\
J0015+0104 & [ 00$^{\rm h}$15$^{\rm m}$20$^{\rm s}$.7 ]{}, [$01$$^{\circ}$04$^{\rm '}$37$^{\rm ''}$]{}& 14 Sep & 4$\,\times\,$900 & 1.207 & 174 & 0.00689 $\pm$ 0.00021\
J0016+0108 & 00 16 28. 3, 01 08 02 & 15 Sep & 4$\,\times\,$675 & 1.306 & 190 & 0.01035 $\pm$ 0.00006\
J0029-0108 & 00 29 04. 7, –01 08 26 & 16 Sep & 4$\,\times\,$450 & 1.614 & 188 & 0.01313 $\pm$ 0.00006\
J0029-0025 & 00 29 49. 5, –00 25 40 & 14 Sep & 4$\,\times\,$900 & 1.142 & 172 & 0.01440 $\pm$ 0.00013\
0057-0022 & 00 57 12. 6, –00 21 58 & 16 Sep & 4$\,\times\,$900 & 1.525 & 147 & 0.00956 $\pm$ 0.00023\
J0107+0001 & 01 07 50. 8, 00 01 28 & 16 Sep & 3$\,\times\,$267 & 1.188 & 141 & 0.01835 $\pm$ 0.00012\
J0109+0107 & 01 09 08. 0, 01 07 16 & 15 Sep & 4$\,\times\,$900 & 1.174 & 154 & 0.00396 $\pm$ 0.00013\
J0126-0038 No.1 & 01 26 46. 1, –00 38 39 & 16 Sep & 4$\,\times\,$900 & 1.332 & 136 & 0.00632 $\pm$ 0.00010\
J0126-0038 No.2 & 01 26 46. 1, –00 38 39 & 16 Sep & 4$\,\times\,$900 & 1.332 & 136 & 0.00642 $\pm$ 0.00016\
J0135-0023 & 01 35 44. 0, –00 23 17 & 14 Sep & 4$\,\times\,$900 & 1.142 & 233 & 0.01708 $\pm$ 0.00006\
J0213-0002 No.1 & 02 13 57. 7, –00 02 56 & 16 Sep & 4$\,\times\,$900 & 1.252 & 152 & 0.03640 $\pm$ 0.00008\
J0213-0002 No.2 & 02 13 57. 7, –00 02 56 & 16 Sep & 4$\,\times\,$900 & 1.252 & 152 & 0.03635 $\pm$ 0.00011\
J0216+0115 No.1 & 02 16 29. 3, –01 15 21 & 15 Sep & 4$\,\times\,$900 & 1.223 & 202 & 0.00939 $\pm$ 0.00004\
J0216+0115 No.2 & 02 16 29. 3, 01 15 21 & 15 Sep & 4$\,\times\,$900 & 1.223 & 202 & 0.00940 $\pm$ 0.00004\
096632 & 02 51 47. 5, –30 06 32 & 15 Sep & 4$\,\times\,$900 & 1.400 & 228 & 0.00354 $\pm$ 0.00009\
J0252+0017 & 02 52 16. 8, 00 17 41 & 16 Sep & 3$\,\times\,$800 & 1.165 & 138 & 0.00527 $\pm$ 0.00011\
J0256+0036 & 02 56 28. 3, 00 36 28 & 14 Sep & 4$\,\times\,$900 & 1.150 & 247 & 0.00919 $\pm$ 0.00013\
J0303-0109 No.1 & 03 03 31. 3, –01 09 47 & 14 Sep & 2$\,\times\,$800 & 1.170 & 243 & 0.03055 $\pm$ 0.00011\
J0303-0109 No.2 & 03 03 31. 3, –01 09 47 & 14 Sep & 2$\,\times\,$800 & 1.170 & 243 & 0.03039 $\pm$ 0.00017\
J0341-0026 No.1 & 03 41 18. 1, -00 26 28& 16 Sep & 3$\,\times\,$800 & 1.198 & 146 & 0.03080 $\pm$ 0.00020\
J0341-0026 No.2 & 03 41 18. 1, –00 26 28 & 16 Sep & 3$\,\times\,$800 & 1.198 & 146 & 0.03045 $\pm$ 0.00008\
J0341-0026 No.3 & 03 41 18. 1, –00 26 28 & 16 Sep & 3$\,\times\,$800 & 1.198 & 146 & 0.03047 $\pm$ 0.00008\
G1815456-670126 & 18 15 46. 5, –67 01 23 & 14 Sep & 4$\,\times\,$900 & 1.280 & 200 & 0.01131 $\pm$ 0.00008\
G2052078-691229 No.1 & 20 52 07. 1, –69 12 30 & 16 Sep & 3$\,\times\,$800 & 1.328 & 150 & 0.00212 $\pm$ 0.00008\
G2052078-691229 No.2 & 20 52 07. 1, –69 12 30 & 16 Sep & 3$\,\times\,$800 & 1.328 & 150 & 0.00203 $\pm$ 0.00011\
J2053+0039 & 20 53 12. 6, 00 39 15 & 15 Sep & 4$\,\times\,$525 & 1.219 & 155 & 0.01328 $\pm$ 0.00007\
J2105+0032 No.1 & 21 05 08. 6, 00 32 23 & 14 Sep & 3$\,\times\,$800 & 1.408 & 145 & 0.01431 $\pm$ 0.00003\
J2105+0032 No.2 & 21 05 08. 6, 00 32 23 & 14 Sep & 3$\,\times\,$800 & 1.408 & 145 & 0.01436 $\pm$ 0.00013\
J2112-0016 No.1 & 21 12 00. 8, –00 16 49 & 15 Sep & 3$\,\times\,$800 & 1.441 & 132 & 0.01195 $\pm$ 0.00015\
J2112-0016 No.2 & 21 12 00. 8, –00 16 49 & 15 Sep & 3$\,\times\,$800 & 1.441 & 132 & 0.01215 $\pm$ 0.00021\
J2119-0732 & 21 19 42. 4, –07 32 24 & 14 Sep & 3$\,\times\,$800 & 1.130 & 178 & 0.00966 $\pm$ 0.00006\
J2120-0058 & 21 20 25. 9, –00 58 27 & 15 Sep & 4$\,\times\,$900 & 1.172 & 154 & 0.01979 $\pm$ 0.00004\
J2150+0033 & 21 50 32. 0, 00 33 05 & 15 Sep & 3$\,\times\,$800 & 1.155 & 232 & 0.01508 $\pm$ 0.00003\
G2155572-394614 & 21 55 57. 9, –39 46 14 & 15 Sep & 3$\,\times\,$900 & 1.241 & 190 & 0.00740 $\pm$ 0.00008\
J2227-0939 & 22 27 30. 7, –09 39 54 & 14 Sep & 3$\,\times\,$800 & 1.060 & 214 & 0.00528 $\pm$ 0.00007\
PHL 293B & 22 30 36. 8, –00 06 37 & 14 Sep & 4$\,\times\,$900 & 1.230 & 135 & 0.00537 $\pm$ 0.00004\
J2310-0109 No.1 & 23 10 42. 0, –01 09 48 & 16 Sep & 3$\,\times\,$800 & 1.471 & 145 & 0.01254 $\pm$ 0.00008\
J2310-0109 No.2 & 23 10 42. 0, –01 09 48 & 16 Sep & 3$\,\times\,$800 & 1.471 & 145 & 0.01232 $\pm$ 0.00009\
[lcclcccc]{}
Galaxy & R.A., Dec. & Date & Exp. time & Airmass & seeing & redshift\
& (J2000) & & (sec) & & &\
J0004+0025 No.1 & [ 00$^{\rm h}$04$^{\rm m}$21$^{\rm s}$.6 ]{}, [$00$$^{\circ}$25$^{\rm '}$36$^{\rm ''}$]{} & 20.10.2006& 2$\,\times\,$750& 1.246&060 & 0.01269 $\pm$ 0.00004\
& & &2$\,\times\,$540 & 1.181 & 037 & 0.01269 $\pm$ 0.00004\
J0004+0025 No.2 & 00 04 21. 6, 00 25 36 & 20.10.2006 &2$\,\times\,$750 & 1.246 & 060 & 0.01266 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.181 & 037 & 0.01266 $\pm$ 0.00003\
J0014-0044 No.1 & 00 14 28. 8, –00 44 44 & 19.11.2006 &2$\,\times\,$750 & 1.158 & 122 & 0.01361 $\pm$ 0.00004\
& & &2$\,\times\,$540 & 1.217 & 093 & 0.01361 $\pm$ 0.00004\
J0014-0044 No.2 & 00 14 28. 8, –00 44 44 & 19.11.2006 &2$\,\times\,$750 & 1.158 & 122 & 0.01379 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.217 & 093 & 0.01379 $\pm$ 0.00002\
J0202-0047 & 02 02 38. 0, –00 47 44 & 13.01.2007 &2$\,\times\,$750 & 1.211 & 185 & 0.03371 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.294 & 163 & 0.03371 $\pm$ 0.00003\
J0301-0059 No.1 & 03 01 26. 3, –00 59 26 & 13.12.2006 &2$\,\times\,$750 & 1.140 & 146 & 0.03841 $\pm$ 0.00000\
& & &2$\,\times\,$540 & 1.191 & 138 & 0.03841 $\pm$ 0.00000\
J0301-0059 No.2 & 03 01 26. 3, –00 59 26 & 13.12.2006 &2$\,\times\,$750 & 1.140 & 146 & 0.03822 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.191 & 138 & 0.03822 $\pm$ 0.00002\
J0301-0059 No.3 & 03 01 26. 3, –00 59 26 & 13.12.2006 &2$\,\times\,$750 & 1.140 & 146 & 0.03807 $\pm$ 0.00008\
& & &2$\,\times\,$540 & 1.191 & 138 & 0.03807 $\pm$ 0.00008\
J0315-0024 No.1 & 03 15 59. 9, –00 24 26 & 26.11.2006 &2$\,\times\,$1500 & 1.145 & 132 & 0.02247 $\pm$ 0.00005\
& & &2$\,\times\,$1080 & 1.197 & 108 & 0.02247 $\pm$ 0.00005\
J0315-0024 No.2 & 03 15 59. 9, –00 24 26 & 26.11.2006 &2$\,\times\,$1500 & 1.145 & 132 & 0.02261 $\pm$ 0.00006\
& & &2$\,\times\,$1080 & 1.197 & 108 & 0.02261 $\pm$ 0.00006\
J0338+0013 (BG) & 03 38 12. 2, 00 13 13 & 25.11.2006 &2$\,\times\,$750 & 1.207 & 076 & 0.39695 $\pm$ 0.00000\
& & &2$\,\times\,$540 & 1.284 & 070 & 0.39695 $\pm$ 0.00000\
J0338+0013 & 03 38 12. 2, 00 13 13 & 25.11.2006 &2$\,\times\,$750 & 1.207 & 076 & 0.04266 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.284 & 070 & 0.04266 $\pm$ 0.00002\
G0405204-364859 No.1 & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.00281 $\pm$ 0.00004\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.00281 $\pm$ 0.00004\
G0405204-364859 No.2 & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.00275 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.00275 $\pm$ 0.00003\
G0405204-364859 No.3 & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.00276 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.00276 $\pm$ 0.00003\
G0405204-364859 (BG) & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.17236 $\pm$ 0.00000\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.17236 $\pm$ 0.00000\
J0519+0007 & 05 19 02. 7, 00 07 29 & 16.11.2006 &2$\,\times\,$750 & 1.220 & 103 & 0.04438 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.166 & 127 & 0.04438 $\pm$ 0.00002\
J2104-0035 No.1 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00469 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00469 $\pm$ 0.00002\
J2104-0035 No.2 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00469 $\pm$ 0.00001\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00469 $\pm$ 0.00001\
J2104-0035 No.3 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00471 $\pm$ 0.00005\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00471 $\pm$ 0.00005\
J2104-0035 No.4 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00472 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00472 $\pm$ 0.00003\
J2302+0049 No.1 & 23 02 10. 0, 00 49 39 & 12.10.2006 &2$\,\times\,$750 & 1.170 & 118 & 0.03312 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.132 & 123 & 0.03312 $\pm$ 0.00002\
J2302+0049 No.2 & 23 02 10. 0, 00 49 39 & 12.10.2006 &2$\,\times\,$750 & 1.170 & 118 & 0.03311 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.132 & 123 & 0.03311 $\pm$ 0.00002\
J2324-0006 & 23 24 21. 3, –00 06 29 & 12.10.2006 &2$\,\times\,$750 & 1.102 & 155 & 0.00896 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.119 & 153 & 0.00896 $\pm$ 0.00002\
J2354-0005 No.1 & 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.00771 $\pm$ 0.00003\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.00771 $\pm$ 0.00003\
J2354-0005 No.2 & 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.00798 $\pm$ 0.00004\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.00798 $\pm$ 0.00004\
J2354-0005 (BG1)& 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.16534 $\pm$ 0.00000\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.16534 $\pm$ 0.00000\
J2354-0005 (BG2)& 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.16520 $\pm$ 0.00000\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.16520 $\pm$ 0.00000\
$^a$first line for each galaxy is related to the observation in the blue range and second line to the one in the red range.
[lrrrrrrrr]{}
\
\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&\
& [J0015$+$0104 ]{}&[J0016$+$0108 ]{}& [J0029$-$0108 ]{}&[J0029$-$0025 ]{}& [J0057$-$0022 ]{}&[J0107$+$0001 ]{}& [J0109$+$0107 ]{}&[J0126$-$0038 ]{}\
& & & & & & & & No.1\
3727 \[O [ii]{}\] & 118.7 $\pm$ 5.7 & 221.2 $\pm$ 7.8 & 171.6 $\pm$ 30.8 & 130.3 $\pm$ 7.8 & 219.7 $\pm$ 5.4 & 162.3 $\pm$ 13.6 & 209.3 $\pm$ 4.3 & 238.1 $\pm$ 4.1\
3868 \[Ne [iii]{}\] & ... & ... & ... & ... & 31.0 $\pm$ 2.0 & ... & 30.1 $\pm$ 1.0 & 38.5 $\pm$ 0.9\
3889 He [i]{} + H8 & ... & ... & ... & ... & 17.3 $\pm$ 1.8 & ... & 20.6 $\pm$ 1.6 & 17.2 $\pm$ 0.9\
3968 \[Ne [iii]{}\] + H7 & ... & ... & ... & ... & 16.1 $\pm$ 1.9 & ... & 25.6 $\pm$ 1.5 & 22.2 $\pm$ 0.9\
4101 H$\delta$ & ... & ... & ... & ... & 24.9 $\pm$ 1.9 & ... & 24.9 $\pm$ 1.3 & 24.8 $\pm$ 0.9\
4340 H$\gamma$ & 49.3 $\pm$ 3.5 & 47.0 $\pm$ 3.4 & ... & 43.9 $\pm$ 4.8 & 47.2 $\pm$ 1.9 & ... & 46.9 $\pm$ 1.4 & 46.9 $\pm$ 1.0\
4363 \[O [iii]{}\] & ... & ... & ... & ... & 5.1 $\pm$ 0.1 & ... & 3.7 $\pm$ 0.4 & 6.9 $\pm$ 0.4\
4471 He [i]{} & ... & ... & ... & ... & ... & ... & 2.6 $\pm$ 0.4 & 2.4 $\pm$ 0.3\
4658 \[Fe [iii]{}\] & ... & ... & ... & ... & ... & ... & ... & 1.5 $\pm$ 0.3\
4686 He [ii]{} & ... & ... & ... & ... & ... & ... & ... & 1.7 $\pm$ 0.3\
4861 H$\beta$ & 100.0 $\pm$ 4.9 & 100.0 $\pm$ 4.2 & 100.0 $\pm$ 15.0 & 100.0 $\pm$ 6.2 & 100.0 $\pm$ 2.6 & 100.0 $\pm$ 8.0 & 100.0 $\pm$ 2.1 & 100.0 $\pm$ 1.7\
4959 \[O [iii]{}\] & 20.1 $\pm$ 2.0 & 61.1 $\pm$ 2.8 & 48.1 $\pm$ 8.8 & 38.8 $\pm$ 3.2 & 76.1 $\pm$ 2.0 & 31.0 $\pm$ 3.3 & 102.5 $\pm$ 2.1 & 139.7 $\pm$ 2.3\
5007 \[O [iii]{}\] & 57.8 $\pm$ 2.8 & 190.6 $\pm$ 6.1 & 144.3 $\pm$ 22.7 & 168.6 $\pm$ 7.9 & 219.4 $\pm$ 4.8 & 93.1 $\pm$ 7.9 & 305.1 $\pm$ 5.6 & 409.4 $\pm$ 6.4\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.245 ]{}& [ 0.000 ]{}& [ 0.120 ]{}& [ 0.410 ]{}\
$F$(H$\beta$)$^b$ & [ 2.77 ]{}& [ 4.66 ]{}& [ 1.19 ]{}& [ 2.16 ]{}& [ 12.99 ]{}& [ 2.88 ]{}& [ 18.18 ]{}& [ 75.47 ]{}\
EW(H$\beta$) ($\AA$) & [ 57.5 ]{}& [ 20.4 ]{}& [ 6.8 ]{}& [ 53.8 ]{}& [ 44.6 ]{}& [ 87.7 ]{}& [ 50.1 ]{}& [ 48.7 ]{}\
EW(abs) ($\AA$)$^a$ & [ 0.00 ]{}& [ 0.00 ]{}& [ 0.00 ]{}& [ 2.00 ]{}& [ 0.25 ]{}& [ 2.00 ]{}& [ 2.15 ]{}& [ 0.70 ]{}\
&\
& [J0126$-$0038 ]{}&[J0135$-$0023 ]{}& [J0213$-$0002 ]{}&[J0213$-$0002 ]{}& [J0216$+$0115 ]{}&[J0216$+$0115 ]{}& [096632 ]{}&[J0252$+$0017 ]{}\
&No.2 & &No.1 &No.2 &No.1 &No.2 & &\
3727 \[O [ii]{}\] & 224.9 $\pm$ 6.3 & 177.9 $\pm$ 10.0 & 182.2 $\pm$ 9.8 & 291.0 $\pm$ 16.6 & 225.2 $\pm$ 8.7 & 236.5 $\pm$ 12.4 & 306.5 $\pm$ 6.9 & 257.0 $\pm$ 26.1\
3868 \[Ne [iii]{}\] & ... & ... & ... & ... & ... & ... & 8.4 $\pm$ 0.9 & ... \
4101 H$\delta$ & 24.8 $\pm$ 2.2 & ... & ... & ... & ... & ... & 29.7 $\pm$ 1.7 & ... \
4340 H$\gamma$ & 44.8 $\pm$ 2.2 & 44.0 $\pm$ 5.8 & 47.1 $\pm$ 5.0 & 47.3 $\pm$ 7.3 & 49.3 $\pm$ 3.9 & ... & 48.1 $\pm$ 1.8 & ... \
4861 H$\beta$ & 100.0 $\pm$ 3.0 & 100.0 $\pm$ 6.7 & 100.0 $\pm$ 6.6 & 100.0 $\pm$ 7.6 & 100.0 $\pm$ 4.6 & 100.0 $\pm$ 10.0 & 100.0 $\pm$ 2.4 & 100.0 $\pm$ 10.0\
4959 \[O [iii]{}\] & 59.3 $\pm$ 1.9 & 43.5 $\pm$ 3.5 & 97.0 $\pm$ 5.8 & 115.6 $\pm$ 7.2 & 83.8 $\pm$ 3.8 & 42.1 $\pm$ 3.9 & 35.2 $\pm$ 1.2 & 71.7 $\pm$ 8.5\
5007 \[O [iii]{}\] & 170.7 $\pm$ 4.3 & 155.9 $\pm$ 7.9 & 279.2 $\pm$ 13.1 & 332.9 $\pm$ 16.7 & 245.6 $\pm$ 8.3 & 126.9 $\pm$ 6.7 & 100.3 $\pm$ 2.3 & 270.5 $\pm$ 24.1\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}\
$F$(H$\beta$)$^b$ & [ 7.47 ]{}& [ 2.87 ]{}& [ 2.54 ]{}& [ 1.73 ]{}& [ 3.11 ]{}& [ 2.26 ]{}& [ 10.20 ]{}& [ 1.69 ]{}\
EW(H$\beta$) ($\AA$) & [ 45.6 ]{}& [ 14.0 ]{}& [ 52.1 ]{}& [ 36.7 ]{}& [ 17.2 ]{}& [ 24.3 ]{}& [ 48.3 ]{}& [ 11.1 ]{}\
EW(abs) ($\AA$)$^a$ & [ 2.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 0.00 ]{}& [ 0.00 ]{}& [ 0.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}\
&\
& [J0256$+$0036 ]{}&[J0303$-$0109 ]{}& [J0303$-$0109 ]{}&[J0341$-$0026 ]{}& [J0341$-$0026 ]{}&[J0341$-$0026 ]{}& [G1815-6701 ]{}&[G2052-6912 ]{}\
& &[ No.1 ]{}& [ No.2]{}&[ No.1 ]{}& [ No.2]{}&[ No.3 ]{}& &[ No.1 ]{}\
3727 \[O [ii]{}\] & 190.3 $\pm$ 5.1 & 137.9 $\pm$ 4.5 & 170.4 $\pm$ 22.5 & 227.5 $\pm$ 13.6 & 136.4 $\pm$ 5.9 & 87.5 $\pm$ 5.0 & 263.8 $\pm$ 4.4 & 137.6 $\pm$ 2.1\
3750 H12 & ... & ... & ... & ... & ... & ... & ... & 3.3 $\pm$ 0.2\
3771 H11 & ... & ... & ... & ... & ... & ... & ... & 3.8 $\pm$ 0.2\
3798 H10 & ... & ... & ... & ... & ... & ... & ... & 5.1 $\pm$ 0.2\
3835 H9 & ... & ... & ... & ... & ... & ... & ... & 7.1 $\pm$ 0.2\
3868 \[Ne [iii]{}\] & 31.7 $\pm$ 1.7 & 42.9 $\pm$ 2.2 & ... & ... & ... & ... & 52.3 $\pm$ 1.1 & 37.2 $\pm$ 0.6\
3889 He [i]{} + H8 & ... & 30.6 $\pm$ 2.8 & ... & ... & ... & ... & 17.5 $\pm$ 0.8 & 19.3 $\pm$ 0.4\
3968 \[Ne [iii]{}\] + H7 & ... & 36.4 $\pm$ 3.0 & ... & ... & ... & ... & 25.7 $\pm$ 0.8 & 27.2 $\pm$ 0.5\
4026 He [i]{} & ... & ... & ... & ... & ... & ... & ... & 1.5 $\pm$ 0.1\
4068 \[S [ii]{}\] & ... & ... & ... & ... & ... & ... & ... & 1.3 $\pm$ 0.1\
4101 H$\delta$ & ... & 33.8 $\pm$ 2.7 & ... & ... & ... & ... & 23.7 $\pm$ 0.8 & 25.9 $\pm$ 0.4\
4340 H$\gamma$ & 47.8 $\pm$ 2.2 & 47.0 $\pm$ 2.5 & ... & ... & 45.4 $\pm$ 4.4 & 50.0 $\pm$ 4.7 & 46.9 $\pm$ 1.0 & 46.6 $\pm$ 0.7\
4363 \[O [iii]{}\] & ... & 8.2 $\pm$ 1.0 & ... & ... & ... & ... & 8.7 $\pm$ 0.4 & 4.1 $\pm$ 0.1\
4471 He [i]{} & ... & ... & ... & ... & ... & ... & 3.5 $\pm$ 0.4 & 3.9 $\pm$ 0.1\
4658 \[Fe [iii]{}\] & ... & ... & ... & ... & ... & ... & 1.7 $\pm$ 0.3 & 0.3 $\pm$ 0.1\
4686 He [ii]{} & ... & ... & ... & ... & ... & ... & 1.6 $\pm$ 0.4 & 0.3 $\pm$ 0.1\
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & ... & ... & ... & ... & ... & 0.7 $\pm$ 0.1\
4861 H$\beta$ & 100.0 $\pm$ 3.0 & 100.0 $\pm$ 3.0 & 100.0 $\pm$ 13.0 & 100.0 $\pm$ 8.0 & 100.0 $\pm$ 4.7 & 100.0 $\pm$ 5.2 & 100.0 $\pm$ 1.7 & 100.0 $\pm$ 1.5\
4921 He [i]{} & ... & ... & ... & 18.9 $\pm$ 2.8 & ... & ... & ... & 0.9 $\pm$ 0.1\
4959 \[O [iii]{}\] & 110.1 $\pm$ 3.0 & 130.6 $\pm$ 3.5 & 26.6 $\pm$ 5.1 & 68.0 $\pm$ 4.9 & 77.0 $\pm$ 3.5 & 65.3 $\pm$ 3.3 & 171.5 $\pm$ 2.7 & 160.5 $\pm$ 2.3\
5007 \[O [iii]{}\] & 316.2 $\pm$ 7.3 & 368.4 $\pm$ 8.9 & 79.9 $\pm$ 11.6 & 253.6 $\pm$ 13.9 & 243.1 $\pm$ 8.4 & 186.1 $\pm$ 7.6 & 508.6 $\pm$ 7.9 & 481.2 $\pm$ 7.0\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.515 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.575 ]{}& [ 0.390 ]{}\
$F$(H$\beta$)$^b$ & [ 9.25 ]{}& [ 11.86 ]{}& [ 1.71 ]{}& [ 4.49 ]{}& [ 4.49 ]{}& [ 3.54 ]{}& [ 66.43 ]{}& [448.40 ]{}\
EW(H$\beta$) ($\AA$) & [ 26.2 ]{}& [ 95.0 ]{}& [ 21.2 ]{}& [ 27.5 ]{}& [ 85.2 ]{}& [ 145.8 ]{}& [ 35.9 ]{}& [ 221.4 ]{}\
EW(abs) ($\AA$)$^a$ & [ 0.00 ]{}& [ 1.50 ]{}& [ 2.00 ]{}& [ 0.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 0.40 ]{}& [ 0.75 ]{}\
&\
& [G2052-6912]{}&[J2053+0039 ]{}& [J2105+0032 ]{}&[J2105+0032 ]{}& [J2112-0016 ]{}&[J2112-0016 ]{}& [J2119-0732 ]{}&[J2120-0058 ]{}\
& [No.2]{}&[ ]{}& [ No.1]{}&[ No.2 ]{}& [ No.1]{}&[ No.2 ]{}& &[ ]{}\
3727 \[O [ii]{}\] & 104.5 $\pm$ 1.6 & 85.8 $\pm$ 8.2 & 109.0 $\pm$ 8.3 & 187.0 $\pm$ 10.8 & 106.6 $\pm$ 2.1 & 107.6 $\pm$ 4.6 & 161.7 $\pm$ 4.1 & 252.5 $\pm$ 5.3\
3750 H12 & 3.5 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
3771 H11 & 4.1 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
3798 H10 & 5.2 $\pm$ 0.2 & ... & ... & ... & ... & ... & ... & ... \
3820 He [i]{} & 1.1 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
3835 H9 & 7.2 $\pm$ 0.2 & ... & ... & ... & ... & ... & ... & ... \
3868 \[Ne [iii]{}\] & 47.9 $\pm$ 0.7 & ... & ... & ... & 42.7 $\pm$ 1.0 & 25.7 $\pm$ 2.0 & 38.5 $\pm$ 1.7 & 39.9 $\pm$ 1.4\
3889 He [i]{} + H8 & 18.1 $\pm$ 0.3 & ... & ... & ... & 18.4 $\pm$ 0.8 & 18.0 $\pm$ 3.1 & 24.5 $\pm$ 2.0 & 22.5 $\pm$ 1.7\
3968 \[Ne [iii]{}\] + H7 & 30.9 $\pm$ 0.5 & ... & ... & ... & 27.9 $\pm$ 1.0 & 37.0 $\pm$ 3.3 & ... & 20.4 $\pm$ 1.7\
4026 He [i]{} & 1.7 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4068 \[S [ii]{}\] & 1.0 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4101 H$\delta$ & 25.9 $\pm$ 0.4 & ... & ... & ... & 25.8 $\pm$ 1.0 & 41.1 $\pm$ 2.4 & 26.8 $\pm$ 1.8 & 27.5 $\pm$ 1.5\
4340 H$\gamma$ & 46.7 $\pm$ 0.7 & ... & ... & ... & 48.7 $\pm$ 1.1 & 46.9 $\pm$ 3.0 & 46.7 $\pm$ 1.8 & 47.2 $\pm$ 1.6\
4363 \[O [iii]{}\] & 6.7 $\pm$ 0.1 & ... & ... & ... & 5.3 $\pm$ 0.4 & 5.5 $\pm$ 0.9 & ... & 7.8 $\pm$ 0.8\
4387 He [i]{} & 0.6 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4471 He [i]{} & 4.1 $\pm$ 0.1 & ... & ... & ... & 3.7 $\pm$ 0.3 & ... & ... & ... \
4658 \[Fe [iii]{}\] & 0.2 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4711 \[Ar [iv]{}\] + He [i]{} & 1.4 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4740 \[Ar [iv]{}\] & 0.6 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4861 H$\beta$ & 100.0 $\pm$ 1.5 & 100.0 $\pm$ 9.0 & 100.0 $\pm$ 8.0 & 100.0 $\pm$ 6.0 & 100.0 $\pm$ 1.8 & 100.0 $\pm$ 3.5 & 100.0 $\pm$ 2.6 & 100.0 $\pm$ 2.2\
4921 He [i]{} & 1.1 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4959 \[O [iii]{}\] & 207.9 $\pm$ 3.0 & 75.2 $\pm$ 7.1 & 71.2 $\pm$ 5.3 & 53.4 $\pm$ 4.0 & 189.9 $\pm$ 3.2 & 152.6 $\pm$ 4.7 & 137.3 $\pm$ 3.1 & 102.4 $\pm$ 2.1\
5007 \[O [iii]{}\] & 623.9 $\pm$ 9.0 & 216.5 $\pm$ 14.3 & 263.8 $\pm$ 14.4 & 165.6 $\pm$ 8.1 & 565.0 $\pm$ 9.1 & 454.1 $\pm$ 12.5 & 413.1 $\pm$ 8.6 & 297.6 $\pm$ 5.6\
\
$C$(H$\beta$)$^a$ & [ 0.495 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.245 ]{}& [ 0.095 ]{}& [ 0.415 ]{}\
$F$(H$\beta$)$^b$ & [766.20 ]{}& [ 1.58 ]{}& [ 2.57 ]{}& [ 2.36 ]{}& [ 49.75 ]{}& [ 7.37 ]{}& [ 16.82 ]{}& [ 15.15 ]{}\
EW(H$\beta$) ($\AA$) & [ 395.9 ]{}& [ 24.4 ]{}& [ 18.2 ]{}& [ 121.7 ]{}& [ 95.2 ]{}& [ 176.1 ]{}& [ 52.8 ]{}& [ 40.1 ]{}\
EW(abs) ($\AA$)$^a$ & [ 1.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 0.00 ]{}& [ 4.15 ]{}& [ 2.00 ]{}& [ 0.90 ]{}\
&\
& [J2150+0033 ]{}&[G2155-3946 ]{}& [J2227-0939 ]{}&[PHL 293B ]{}& [J2310-0109 ]{}&[J2310-0109 ]{}\
& &[ ]{}& [ ]{}&[ ]{}& [ No.1]{}&[ No.2 ]{}& [ ]{}&[ ]{}\
3727 \[O [ii]{}\] & 190.0 $\pm$ 7.3 & 323.4 $\pm$ 10.2 & 197.0 $\pm$ 7.0 & 65.7 $\pm$ 1.1 & 128.0 $\pm$ 4.0 & 177.4 $\pm$ 6.0\
3750 H12 & ... & ... & ... & 5.0 $\pm$ 0.4 & ... & ... \
3771 H11 & ... & ... & ... & 6.4 $\pm$ 0.4 & ... & ... \
3798 H10 & ... & ... & ... & 8.2 $\pm$ 0.4 & ... & ... \
3835 H9 & ... & ... & ... & 8.4 $\pm$ 0.4 & ... & ... \
3868 \[Ne [iii]{}\] & ... & 16.6 $\pm$ 1.9 & 35.0 $\pm$ 3.0 & 41.0 $\pm$ 0.7 & 20.9 $\pm$ 1.4 & 27.8 $\pm$ 1.9\
3889 He [i]{} + H8 & ... & ... & 23.4 $\pm$ 7.2 & 21.4 $\pm$ 0.5 & ... & ... \
3968 \[Ne [iii]{}\] + H7 & ... & ... & 26.9 $\pm$ 4.9 & 28.9 $\pm$ 0.6 & ... & ... \
4101 H$\delta$ & ... & 22.8 $\pm$ 4.6 & 28.5 $\pm$ 4.4 & 25.7 $\pm$ 0.5 & ... & ... \
4340 H$\gamma$ & 39.9 $\pm$ 3.4 & 47.1 $\pm$ 3.0 & 47.3 $\pm$ 3.4 & 47.4 $\pm$ 0.8 & 38.3 $\pm$ 2.7 & 41.5 $\pm$ 3.4\
4363 \[O [iii]{}\] & 7.3 $\pm$ 1.3 & ... & ... & 12.3 $\pm$ 0.3 & ... & 8.7 $\pm$ 1.3\
4471 He [i]{} & ... & ... & ... & 3.5 $\pm$ 0.2 & ... & ... \
4686 He [ii]{} & ... & ... & ... & 1.4 $\pm$ 0.1 & ... & ... \
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & ... & 1.8 $\pm$ 0.1 & ... & ... \
4740 \[Ar [iv]{}\] & ... & ... & ... & 1.1 $\pm$ 0.1 & ... & ... \
4861 H$\beta$ & 100.0 $\pm$ 4.6 & 100.0 $\pm$ 3.8 & 100.0 $\pm$ 3.9 & 100.0 $\pm$ 1.5 & 100.0 $\pm$ 3.3 & 100.0 $\pm$ 3.8\
4959 \[O [iii]{}\] & 86.9 $\pm$ 3.5 & 59.3 $\pm$ 2.4 & 128.2 $\pm$ 4.3 & 157.1 $\pm$ 2.4 & 122.1 $\pm$ 3.4 & 115.5 $\pm$ 3.7\
5007 \[O [iii]{}\] & 255.9 $\pm$ 8.4 & 170.9 $\pm$ 5.1 & 373.0 $\pm$ 10.9 & 466.7 $\pm$ 7.0 & 369.4 $\pm$ 9.0 & 332.8 $\pm$ 9.2\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.305 ]{}& [ 0.000 ]{}& [ 0.000 ]{}\
$F$(H$\beta$)$^b$ & [ 4.59 ]{}& [ 10.53 ]{}& [ 21.39 ]{}& [107.50 ]{}& [ 14.00 ]{}& [ 9.26 ]{}\
EW(H$\beta$) ($\AA$) & [ 30.0 ]{}& [ 13.8 ]{}& [ 20.3 ]{}& [ 117.2 ]{}& [ 26.1 ]{}& [ 30.5 ]{}\
EW(abs) ($\AA$)$^a$ & [ 2.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 1.50 ]{}& [ 2.00 ]{}& [ 2.00 ]{}\
$^a$ zero value is assumed if a negative value is derived.
$^b$ in units of 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$.
[lrrrrrrrr]{}
\
\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&\
& [J0004$+$0025 ]{}&[J0004$+$0025 ]{}& [J0014$-$0044 ]{}&[J0014$-$0044 ]{}& [J0202$-$0047 ]{}&[J0301$-$0059 ]{}& [J0301$-$0059 ]{}&[J0301$-$0059 ]{}\
& No.1 & No.2 & No.1 & No.2 & & No.1 & No.2 & No.3\
3727 \[O [ii]{}\] & 214.05 $\pm$ 11.27 &253.26 $\pm$ 13.99 & 99.16 $\pm$ 1.73 &298.77 $\pm$ 9.30 &191.81 $\pm$ 3.50 &331.24 $\pm$ 6.29 &304.43 $\pm$ 6.86 &330.51 $\pm$ 21.26\
3750 H12 & ... & ... & 4.57 $\pm$ 0.56 & ... & ... & ... & ... & ... \
3771 H11 & ... & ... & 5.06 $\pm$ 0.52 & ... & 4.22 $\pm$ 0.66 & ... & ... & ... \
3798 H10 & ... & ... & 7.06 $\pm$ 0.52 & ... & 4.83 $\pm$ 0.67 & ... & ... & ... \
3835 H9 & ... & ... & 8.66 $\pm$ 0.46 & ... & 5.85 $\pm$ 0.58 & ... & ... & ... \
3868 \[Ne [iii]{}\] & ... & 33.25 $\pm$ 4.96 & 50.11 $\pm$ 0.85 & ... & 41.34 $\pm$ 0.92 & 24.06 $\pm$ 0.98 & 23.23 $\pm$ 1.72 & ... \
3889 He [i]{} + H8 & ... & ... & 21.72 $\pm$ 0.55 & ... & 22.44 $\pm$ 0.72 & 22.59 $\pm$ 1.80 & ... & ... \
3968 \[Ne [iii]{}\] + H7 & ... & ... & 32.07 $\pm$ 0.64 & ... & 28.21 $\pm$ 0.78 & 25.37 $\pm$ 1.37 & ... & ... \
4026 He [i]{} & ... & ... & 1.10 $\pm$ 0.11 & ... & ... & ... & ... & ... \
4101 H$\delta$ & ... & ... & 25.84 $\pm$ 0.51 & 28.41 $\pm$ 2.21 & 25.71 $\pm$ 0.70 & 28.75 $\pm$ 1.16 & 28.14 $\pm$ 2.37 & ... \
4340 H$\gamma$ & 47.25 $\pm$ 3.35 & 47.74 $\pm$ 3.21 & 46.79 $\pm$ 0.77 & 51.88 $\pm$ 1.98 & 48.61 $\pm$ 0.93 & 46.88 $\pm$ 1.15 & 45.15 $\pm$ 1.68 & ... \
4363 \[O [iii]{}\] & ... & ... & 8.87 $\pm$ 0.18 & ... & 8.97 $\pm$ 0.37 & 3.46 $\pm$ 0.42 & 4.43 $\pm$ 0.92 & ... \
4387 He [i]{} & ... & ... & 0.57 $\pm$ 0.08 & ... & ... & ... & ... & ... \
4471 He [i]{} & ... & ... & 3.83 $\pm$ 0.12 & ... & 3.49 $\pm$ 0.26 & ... & ... & ... \
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & 1.08 $\pm$ 0.09 & ... & ... & ... & ... & ... \
4740 \[Ar [iv]{}\] & ... & ... & 0.84 $\pm$ 0.07 & ... & ... & ... & ... & ... \
4861 H$\beta$ & 100.00 $\pm$ 3.77 &100.00 $\pm$ 3.98 &100.00 $\pm$ 1.49 &100.00 $\pm$ 2.52 &100.00 $\pm$ 1.60 &100.00 $\pm$ 1.81 &100.00 $\pm$ 2.17 &100.00 $\pm$ 17.01\
4921 He [i]{} & ... & ... & 0.94 $\pm$ 0.06 & ... & ... & ... & ... & ... \
4959 \[O [iii]{}\] & 37.84 $\pm$ 1.98 & 62.06 $\pm$ 2.64 &207.93 $\pm$ 3.04 & 59.47 $\pm$ 1.64 &161.71 $\pm$ 2.50 & 91.37 $\pm$ 1.62 & 90.51 $\pm$ 1.93 & 53.80 $\pm$ 4.75\
4988 \[Fe [iii]{}\] & ... & ... & 0.44 $\pm$ 0.01 & ... & ... & ... & ... & ... \
5007 \[O [iii]{}\] & 108.99 $\pm$ 3.45 &183.91 $\pm$ 5.64 &621.54 $\pm$ 9.04 &175.13 $\pm$ 3.94 &495.48 $\pm$ 7.53 &275.10 $\pm$ 4.60 &267.82 $\pm$ 5.17 &154.51 $\pm$ 8.64\
5015 He [i]{} & ... & ... & 0.90 $\pm$ 0.03 & ... & 2.20 $\pm$ 0.15 & ... & ... & ... \
5518 \[Cl [iii]{}\] & ... & ... & 0.38 $\pm$ 0.06 & ... & ... & ... & ... & ... \
5538 \[Cl [iii]{}\] & ... & ... & 0.61 $\pm$ 0.10 & ... & ... & ... & ... & ... \
5876 He [i]{} & ... & ... & 10.92 $\pm$ 0.19 & 13.81 $\pm$ 0.29 & 10.96 $\pm$ 0.27 & 9.94 $\pm$ 0.61 & 11.70 $\pm$ 0.82 & ... \
6300 \[O [i]{}\] & ... & ... & 1.92 $\pm$ 0.06 & ... & 3.11 $\pm$ 0.13 & 9.14 $\pm$ 0.52 & 9.57 $\pm$ 0.60 & ... \
6312 \[S [iii]{}\] & ... & ... & 1.73 $\pm$ 0.06 & ... & 1.74 $\pm$ 0.11 & 0.97 $\pm$ 0.02 & ... & ... \
6363 \[O [i]{}\] & ... & ... & 0.70 $\pm$ 0.05 & ... & 1.42 $\pm$ 0.02 & 2.73 $\pm$ 0.40 & 2.72 $\pm$ 0.05 & ... \
6548 \[N [ii]{}\] & ... & ... & 2.16 $\pm$ 0.06 & 10.39 $\pm$ 0.88 & 3.86 $\pm$ 0.13 & 6.66 $\pm$ 0.44 & 7.77 $\pm$ 0.53 & ... \
6563 H$\alpha$ & 269.07 $\pm$ 8.14 &266.58 $\pm$ 8.43 &283.02 $\pm$ 4.47 &288.13 $\pm$ 6.59 &281.45 $\pm$ 4.63 &284.02 $\pm$ 5.14 &281.73 $\pm$ 5.83 &288.79 $\pm$ 19.92\
6583 \[N [ii]{}\] & 15.55 $\pm$ 1.51 & 12.75 $\pm$ 1.07 & 5.73 $\pm$ 0.11 & 22.66 $\pm$ 0.99 & 9.59 $\pm$ 0.21 & 20.47 $\pm$ 0.54 & 26.57 $\pm$ 0.81 & 33.85 $\pm$ 3.43\
6678 He [i]{} & ... & ... & 3.05 $\pm$ 0.07 & ... & 2.69 $\pm$ 0.11 & 1.83 $\pm$ 0.35 & ... & ... \
6717 \[S [ii]{}\] & 32.18 $\pm$ 1.51 & 27.90 $\pm$ 1.41 & 8.04 $\pm$ 0.14 & 31.47 $\pm$ 1.01 & 16.63 $\pm$ 0.32 & 42.25 $\pm$ 0.88 & 47.03 $\pm$ 1.11 & 63.87 $\pm$ 4.85\
6731 \[S [ii]{}\] & 25.79 $\pm$ 1.51 & 22.66 $\pm$ 1.32 & 5.78 $\pm$ 0.11 & 24.87 $\pm$ 0.91 & 12.38 $\pm$ 0.26 & 29.45 $\pm$ 0.68 & 32.66 $\pm$ 0.86 & 44.23 $\pm$ 4.15\
7065 He [i]{} & ... & ... & 2.45 $\pm$ 0.06 & ... & 1.66 $\pm$ 0.09 & 1.58 $\pm$ 0.26 & ... & ... \
7136 \[Ar [iii]{}\] & ... & ... & 6.61 $\pm$ 0.12 & ... & 8.26 $\pm$ 0.19 & 6.60 $\pm$ 0.41 & 7.65 $\pm$ 0.41 & ... \
7281 He [i]{} & ... & ... & 0.51 $\pm$ 0.03 & ... & 0.92 $\pm$ 0.08 & ... & ... & ... \
7320 \[O [ii]{}\] & ... & ... & 1.51 $\pm$ 0.05 & ... & 2.16 $\pm$ 0.09 & ... & ... & ... \
7330 \[O [ii]{}\] & ... & ... & 1.36 $\pm$ 0.05 & ... & 1.85 $\pm$ 0.07 & ... & ... & ... \
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}&[ 0.000 ]{}&[ 0.210 ]{}&[ 0.030 ]{}&[ 0.305 ]{}&[ 0.215 ]{}&[ 0.185 ]{}&[ 0.110 ]{}\
$F$(H$\beta$)$^b$ & [ 0.76 ]{}&[ 0.80 ]{}&[ 20.90 ]{}&[ 0.81 ]{}&[ 5.47 ]{}&[ 5.40 ]{}&[ 3.59 ]{}&[ 0.38 ]{}\
EW(H$\beta$) ($\AA$) & [ 22.5 ]{}&[ 21.4 ]{}&[ 275.7 ]{}&[ 74.2 ]{}&[ 132.6 ]{}&[ 28.3 ]{}&[ 13.6 ]{}&[ 8.0 ]{}\
EW(abs) ($\AA$) & [ 2.00 ]{}&[ 2.00 ]{}&[ 2.20 ]{}&[ 2.00 ]{}&[ 0.50 ]{}&[ 1.95 ]{}&[ 1.75 ]{}&[ 2.00 ]{}\
&\
& [J0315$-$0024 ]{}&[J0315$-$0024 ]{}& [J0338$+$0013 ]{}&[G0405-3648]{}& [G0405-3648]{}&[G0405-3648]{}& [J0519$+$0007 ]{}&[J2104$-$0035 ]{}\
& No.1 & No.2 & & No.1 &No.2 & No.3 & & No.1\
3727 \[O [ii]{}\] & 174.28 $\pm$ 5.19 & 190.40 $\pm$ 9.12 & 60.18 $\pm$ 1.06 & 129.50 $\pm$ 5.31 & 160.45 $\pm$ 3.75 & 154.37 $\pm$ 3.90 & 27.98 $\pm$ 0.47 & 27.37 $\pm$ 0.97\
3750 H12 & ... & ... & 3.59 $\pm$ 0.30 & ... & ... & ... & 3.13 $\pm$ 0.18 & 4.11 $\pm$ 0.75\
3771 H11 & ... & ... & 3.80 $\pm$ 0.29 & ... & ... & ... & 4.07 $\pm$ 0.18 & 5.22 $\pm$ 0.73\
3798 H10 & ... & ... & 6.86 $\pm$ 0.30 & ... & ... & ... & 5.75 $\pm$ 0.18 & 6.80 $\pm$ 0.66\
3820 He [i]{} & ... & ... & 2.00 $\pm$ 0.18 & ... & ... & ... & 0.74 $\pm$ 0.09 & ... \
3835 H9 & ... & ... & 8.51 $\pm$ 0.28 & ... & ... & ... & 7.26 $\pm$ 0.19 & 8.01 $\pm$ 0.61\
3868 \[Ne [iii]{}\] & 8.88 $\pm$ 1.88 & ... & 41.54 $\pm$ 0.71 & 9.66 $\pm$ 1.46 & 11.59 $\pm$ 1.16 & 14.90 $\pm$ 1.09 & 35.86 $\pm$ 0.56 & 24.32 $\pm$ 0.53\
3889 He [i]{} + H8 & 20.76 $\pm$ 2.11 & ... & 18.64 $\pm$ 0.41 & 18.38 $\pm$ 3.37 & 18.70 $\pm$ 1.09 & 20.05 $\pm$ 1.24 & 17.21 $\pm$ 0.31 & 22.61 $\pm$ 0.66\
3968 \[Ne [iii]{}\] + H7 & 16.88 $\pm$ 1.83 & ... & 28.87 $\pm$ 0.53 & 17.38 $\pm$ 2.88 & 15.71 $\pm$ 1.02 & 20.71 $\pm$ 1.21 & 26.96 $\pm$ 0.44 & 23.31 $\pm$ 0.65\
4026 He [i]{} & ... & ... & 2.82 $\pm$ 0.17 & ... & ... & ... & 1.55 $\pm$ 0.08 & 2.12 $\pm$ 0.18\
4068 \[S [ii]{}\] & ... & ... & 1.16 $\pm$ 0.02 & ... & ... & ... & 0.65 $\pm$ 0.06 & ... \
4101 H$\delta$ & 21.87 $\pm$ 1.58 & ... & 26.20 $\pm$ 0.47 & 25.79 $\pm$ 1.85 & 21.62 $\pm$ 0.82 & 24.52 $\pm$ 0.99 & 25.67 $\pm$ 0.41 & 25.83 $\pm$ 0.63\
4227 \[Fe [v]{}\] & ... & ... & ... & ... & ... & ... & 0.86 $\pm$ 0.14 & ... \
4340 H$\gamma$ & 49.15 $\pm$ 1.56 & 50.73 $\pm$ 4.71 & 48.06 $\pm$ 0.76 & 47.97 $\pm$ 1.55 & 45.48 $\pm$ 1.00 & 49.12 $\pm$ 1.13 & 47.62 $\pm$ 0.71 & 48.28 $\pm$ 0.82\
4363 \[O [iii]{}\] & 2.04 $\pm$ 0.74 & ... & 14.66 $\pm$ 0.27 & 4.02 $\pm$ 0.76 & 4.03 $\pm$ 0.36 & 5.12 $\pm$ 0.50 & 15.04 $\pm$ 0.24 & 9.87 $\pm$ 0.20\
4387 He [i]{} & ... & ... & ... & ... & ... & ... & 0.37 $\pm$ 0.05 & ... \
4471 He [i]{} & ... & ... & 3.70 $\pm$ 0.13 & 2.18 $\pm$ 0.72 & 3.54 $\pm$ 0.43 & 2.63 $\pm$ 0.40 & 3.60 $\pm$ 0.08 & 3.65 $\pm$ 0.13\
4658 \[Fe [iii]{}\] & ... & ... & 0.49 $\pm$ 0.06 & ... & ... & ... & 0.49 $\pm$ 0.04 & ... \
4686 He [ii]{} & ... & ... & 1.15 $\pm$ 0.08 & ... & ... & ... & 1.52 $\pm$ 0.05 & 0.65 $\pm$ 0.10\
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & 2.35 $\pm$ 0.09 & ... & ... & ... & 2.33 $\pm$ 0.06 & 1.52 $\pm$ 0.09\
4740 \[Ar [iv]{}\] & ... & ... & 1.40 $\pm$ 0.07 & ... & ... & ... & 1.51 $\pm$ 0.06 & 0.74 $\pm$ 0.10\
4861 H$\beta$ & 100.00 $\pm$ 1.95 & 100.00 $\pm$ 4.69 & 100.00 $\pm$ 1.48 & 100.00 $\pm$ 2.03 & 100.00 $\pm$ 1.70 & 100.00 $\pm$ 1.77 & 100.00 $\pm$ 1.44 & 100.00 $\pm$ 1.50\
4921 He [i]{} & ... & ... & 1.27 $\pm$ 0.08 & ... & ... & ... & 0.98 $\pm$ 0.05 & 1.11 $\pm$ 0.08\
4959 \[O [iii]{}\] & 53.68 $\pm$ 1.19 & 51.95 $\pm$ 2.12 & 176.71 $\pm$ 2.58 & 70.62 $\pm$ 1.51 & 48.62 $\pm$ 0.91 & 45.69 $\pm$ 0.93 & 148.33 $\pm$ 2.13 & 96.10 $\pm$ 1.41\
4988 \[Fe [iii]{}\] & ... & ... & 0.54 $\pm$ 0.05 & ... & 1.96 $\pm$ 0.31 & ... & 0.65 $\pm$ 0.04 & 0.75 $\pm$ 0.13\
5007 \[O [iii]{}\] & 162.82 $\pm$ 2.95 & 151.44 $\pm$ 4.43 & 530.20 $\pm$ 7.71 & 206.51 $\pm$ 3.80 & 140.34 $\pm$ 2.29 & 137.62 $\pm$ 2.33 & 439.62 $\pm$ 6.30 & 288.91 $\pm$ 4.20\
5015 He [i]{} & ... & ... & 2.59 $\pm$ 0.08 & ... & ... & ... & 0.97 $\pm$ 0.04 & 1.64 $\pm$ 0.18\
5755 \[N [ii]{}\] & ... & ... & ... & ... & ... & ... & 0.15 $\pm$ 0.02 & ... \
5876 He [i]{} & 9.03 $\pm$ 0.55 & ... & 11.14 $\pm$ 0.21 & 8.41 $\pm$ 0.58 & 9.77 $\pm$ 0.48 & 5.72 $\pm$ 0.29 & 11.60 $\pm$ 0.18 & 9.38 $\pm$ 0.19\
6300 \[O [i]{}\] & ... & ... & 1.72 $\pm$ 0.07 & ... & 4.36 $\pm$ 0.36 & 2.75 $\pm$ 0.25 & 1.07 $\pm$ 0.03 & 0.49 $\pm$ 0.06\
6312 \[S [iii]{}\] & ... & ... & 1.01 $\pm$ 0.07 & ... & ... & ... & 0.78 $\pm$ 0.02 & 0.66 $\pm$ 0.06\
6363 \[O [i]{}\] & ... & ... & 0.53 $\pm$ 0.05 & ... & ... & ... & 0.37 $\pm$ 0.02 & ... \
6548 \[N [ii]{}\] & ... & ... & ... & ... & ... & ... & 1.17 $\pm$ 0.03 & ... \
6563 H$\alpha$ & 283.73 $\pm$ 5.33 & 288.03 $\pm$ 8.76 & 277.02 $\pm$ 4.37 & 279.40 $\pm$ 5.45 & 275.30 $\pm$ 4.73 & 272.50 $\pm$ 4.83 & 274.85 $\pm$ 4.27 & 274.51 $\pm$ 4.32\
6583 \[N [ii]{}\] & 6.88 $\pm$ 0.30 & 11.83 $\pm$ 1.44 & 1.64 $\pm$ 0.06 & 5.41 $\pm$ 0.37 & 6.48 $\pm$ 0.31 & 5.14 $\pm$ 0.23 & 3.10 $\pm$ 0.05 & 0.93 $\pm$ 0.06\
6678 He [i]{} & 1.89 $\pm$ 0.04 & ... & 2.69 $\pm$ 0.08 & ... & 2.31 $\pm$ 0.24 & 2.64 $\pm$ 0.22 & 2.95 $\pm$ 0.05 & 2.50 $\pm$ 0.07\
6717 \[S [ii]{}\] & 8.51 $\pm$ 0.36 & 13.97 $\pm$ 1.68 & 4.65 $\pm$ 0.10 & 13.04 $\pm$ 0.55 & 16.04 $\pm$ 0.40 & 14.92 $\pm$ 0.39 & 2.36 $\pm$ 0.04 & 2.10 $\pm$ 0.06\
6731 \[S [ii]{}\] & 9.99 $\pm$ 0.33 & 9.02 $\pm$ 0.97 & 3.55 $\pm$ 0.09 & 8.37 $\pm$ 0.42 & 12.20 $\pm$ 0.35 & 9.95 $\pm$ 0.30 & 2.19 $\pm$ 0.04 & 1.57 $\pm$ 0.06\
7065 He [i]{} & ... & ... & 4.94 $\pm$ 0.10 & 2.53 $\pm$ 0.33 & 1.35 $\pm$ 0.19 & 1.90 $\pm$ 0.19 & 6.54 $\pm$ 0.11 & 2.30 $\pm$ 0.06\
7136 \[Ar [iii]{}\] & ... & ... & 3.14 $\pm$ 0.08 & 3.69 $\pm$ 0.30 & 2.82 $\pm$ 0.20 & 2.33 $\pm$ 0.16 & 2.36 $\pm$ 0.04 & 1.25 $\pm$ 0.05\
7281 He [i]{} & ... & ... & ... & ... & ... & ... & ... & 0.72 $\pm$ 0.03\
7320 \[O [ii]{}\] & ... & ... & ... & ... & ... & ... & ... & 0.49 $\pm$ 0.03\
7330 \[O [ii]{}\] & ... & ... & ... & ... & ... & ... & ... & 0.25 $\pm$ 0.03\
\
$C$(H$\beta$)$^a$ & [ 0.135 ]{}& [ 0.015 ]{}& [ 0.245 ]{}& [ 0.005 ]{}& [ 0.000 ]{}& [ 0.075 ]{}& [ 0.285 ]{}& [ 0.235 ]{}\
$F$(H$\beta$)$^b$ & [ 1.98 ]{}& [ 0.42 ]{}& [ 15.97 ]{}& [ 3.05 ]{}& [ 4.21 ]{}& [ 3.55 ]{}& [ 81.87 ]{}& [ 19.30 ]{}\
EW(H$\beta$) ($\AA$) & [ 36.1 ]{}& [ 23.1 ]{}& [ 220.6 ]{}& [ 18.3 ]{}& [ 36.5 ]{}& [ 38.7 ]{}& [ 241.8 ]{}& [ 213.6 ]{}\
EW(abs) ($\AA$) & [ 1.15 ]{}& [ 2.00 ]{}& [ 0.80 ]{}& [ 1.25 ]{}& [ 0.65 ]{}& [ 1.30 ]{}& [ 0.65 ]{}& [ 0.60 ]{}\
&\
& [J2104$-$0035 ]{}&[J2104$-$0035 ]{}& [J2104$-$0035 ]{}&[J2302$+$0049 ]{}& [J2302$+$0049 ]{}&[J2324$-$0006 ]{}& [J2354$-$0004 ]{}\
& No.2 & No.3 &No.4 & No.1 &No.2 & & No.1\
3727 \[O [ii]{}\] & 114.48 $\pm$ 11.06 & 108.68 $\pm$ 4.99 & 88.02 $\pm$ 3.86 & 59.42 $\pm$ 1.00 & 190.27 $\pm$ 4.15 & 152.34 $\pm$ 2.43 & 138.52 $\pm$ 11.46\
3750 H12 & ... & ... & ... & 3.18 $\pm$ 0.25 & ... & 3.14 $\pm$ 0.33 & ... \
3771 H11 & ... & ... & ... & 4.04 $\pm$ 0.23 & ... & 3.83 $\pm$ 0.33 & ... \
3798 H10 & ... & ... & ... & 6.34 $\pm$ 0.24 & ... & 6.29 $\pm$ 0.32 & ... \
3820 He [i]{} & ... & ... & ... & 1.21 $\pm$ 0.14 & ... & ... & ... \
3835 H9 & ... & ... & ... & 7.43 $\pm$ 0.24 & ... & 7.53 $\pm$ 0.31 & ... \
3868 \[Ne [iii]{}\] & ... & ... & ... & 45.23 $\pm$ 0.72 & 40.05 $\pm$ 1.31 & 47.39 $\pm$ 0.75 & ... \
3889 He [i]{} + H8 & ... & 16.40 $\pm$ 1.55 & 22.24 $\pm$ 1.54 & 20.52 $\pm$ 0.39 & 22.92 $\pm$ 1.37 & 21.00 $\pm$ 0.42 & ... \
3968 \[Ne [iii]{}\] + H7 & ... & 17.76 $\pm$ 1.60 & 14.37 $\pm$ 1.37 & 29.75 $\pm$ 0.50 & 25.90 $\pm$ 1.29 & 31.37 $\pm$ 0.54 & ... \
4026 He [i]{} & ... & ... & ... & 1.64 $\pm$ 0.10 & ... & 1.84 $\pm$ 0.11 & ... \
4068 \[S [ii]{}\] & ... & ... & ... & ... & ... & 1.19 $\pm$ 0.09 & ... \
4076 \[S [ii]{}\] & ... & ... & ... & ... & ... & 0.41 $\pm$ 0.08 & ... \
4101 H$\delta$ & ... & 24.53 $\pm$ 1.22 & 24.61 $\pm$ 1.15 & 26.13 $\pm$ 0.43 & 30.30 $\pm$ 1.24 & 26.61 $\pm$ 0.45 & 25.66 $\pm$ 2.60\
4340 H$\gamma$ & 51.10 $\pm$ 3.95 & 49.03 $\pm$ 1.34 & 48.00 $\pm$ 1.33 & 47.28 $\pm$ 0.72 & 46.12 $\pm$ 1.24 & 47.16 $\pm$ 0.71 & 48.20 $\pm$ 2.01\
4363 \[O [iii]{}\] & ... & 3.59 $\pm$ 0.65 & 1.61 $\pm$ 0.69 & 14.90 $\pm$ 0.24 & 6.28 $\pm$ 0.54 & 9.47 $\pm$ 0.16 & 4.81 $\pm$ 1.57\
4387 He [i]{} & ... & ... & ... & ... & ... & 0.31 $\pm$ 0.08 & ... \
4471 He [i]{} & ... & ... & ... & 3.55 $\pm$ 0.09 & ... & 3.70 $\pm$ 0.08 & ... \
4658 \[Fe [iii]{}\] & ... & ... & ... & 0.49 $\pm$ 0.08 & ... & 0.59 $\pm$ 0.05 & ... \
4686 He [ii]{} & ... & ... & ... & 2.40 $\pm$ 0.08 & ... & 0.76 $\pm$ 0.05 & ... \
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & ... & 2.41 $\pm$ 0.08 & ... & 1.10 $\pm$ 0.05 & ... \
4740 \[Ar [iv]{}\] & ... & ... & ... & 1.41 $\pm$ 0.07 & ... & 0.42 $\pm$ 0.04 & ... \
4861 H$\beta$ & 100.00 $\pm$ 3.67 & 100.00 $\pm$ 2.01 & 100.00 $\pm$ 1.95 & 100.00 $\pm$ 1.45 & 100.00 $\pm$ 1.93 & 100.00 $\pm$ 1.44 & 100.00 $\pm$ 2.60\
4921 He [i]{} & ... & ... & ... & 0.89 $\pm$ 0.05 & ... & 1.02 $\pm$ 0.05 & ... \
4959 \[O [iii]{}\] & 18.99 $\pm$ 1.81 & 32.09 $\pm$ 0.90 & 18.23 $\pm$ 0.62 & 198.50 $\pm$ 2.86 & 105.26 $\pm$ 1.96 & 182.82 $\pm$ 2.62 & 59.53 $\pm$ 1.82\
4988 \[Fe [iii]{}\] & ... & ... & ... & 0.75 $\pm$ 0.05 & 3.20 $\pm$ 0.39 & 0.87 $\pm$ 0.04 & ... \
5007 \[O [iii]{}\] & 59.43 $\pm$ 2.53 & 94.72 $\pm$ 1.89 & 60.24 $\pm$ 1.23 & 567.92 $\pm$ 7.98 & 315.09 $\pm$ 5.54 & 543.75 $\pm$ 7.78 & 166.78 $\pm$ 3.88\
5015 He [i]{} & ... & ... & ... & 1.78 $\pm$ 0.06 & ... & 0.96 $\pm$ 0.05 & ... \
5518 \[Cl [iii]{}\] & ... & ... & ... & ... & ... & 0.25 $\pm$ 0.04 & ... \
5538 \[Cl [iii]{}\] & ... & ... & ... & ... & ... & 0.22 $\pm$ 0.03 & ... \
5876 He [i]{} & 5.84 $\pm$ 0.74 & 9.56 $\pm$ 0.57 & 9.39 $\pm$ 0.52 & 10.00 $\pm$ 0.18 & 8.03 $\pm$ 0.45 & 10.10 $\pm$ 0.16 & 9.03 $\pm$ 0.67\
6300 \[O [i]{}\] & ... & ... & ... & 1.30 $\pm$ 0.05 & ... & 2.69 $\pm$ 0.05 & ... \
6312 \[S [iii]{}\] & ... & ... & ... & 1.21 $\pm$ 0.06 & ... & 1.66 $\pm$ 0.04 & ... \
6363 \[O [i]{}\] & ... & ... & ... & 0.48 $\pm$ 0.04 & ... & 0.97 $\pm$ 0.03 & ... \
6548 \[N [ii]{}\] & ... & ... & ... & ... & ... & 1.46 $\pm$ 0.04 & ... \
6563 H$\alpha$ & 288.62 $\pm$ 8.99 & 273.23 $\pm$ 5.26 & 276.97 $\pm$ 5.20 & 277.79 $\pm$ 4.33 & 280.05 $\pm$ 5.31 & 281.56 $\pm$ 4.37 & 275.84 $\pm$ 6.36\
6583 \[N [ii]{}\] & 4.92 $\pm$ 0.56 & 4.28 $\pm$ 0.38 & 3.93 $\pm$ 0.26 & 1.70 $\pm$ 0.05 & 6.61 $\pm$ 0.45 & 4.94 $\pm$ 0.09 & 4.85 $\pm$ 0.46\
6678 He [i]{} & ... & 3.66 $\pm$ 0.33 & 2.53 $\pm$ 0.22 & 2.18 $\pm$ 0.05 & 2.49 $\pm$ 0.20 & 2.91 $\pm$ 0.06 & ... \
6717 \[S [ii]{}\] & 12.66 $\pm$ 1.06 & 8.30 $\pm$ 0.40 & 8.87 $\pm$ 0.34 & 4.47 $\pm$ 0.08 & 16.91 $\pm$ 0.51 & 11.75 $\pm$ 0.19 & 12.16 $\pm$ 0.69\
6731 \[S [ii]{}\] & 10.50 $\pm$ 1.04 & 4.19 $\pm$ 0.29 & 5.96 $\pm$ 0.35 & 3.48 $\pm$ 0.07 & 11.47 $\pm$ 0.39 & 8.79 $\pm$ 0.14 & 6.55 $\pm$ 0.56\
7065 He [i]{} & ... & ... & ... & 2.18 $\pm$ 0.05 & ... & 2.43 $\pm$ 0.05 & ... \
7136 \[Ar [iii]{}\] & ... & ... & ... & 3.41 $\pm$ 0.07 & 4.56 $\pm$ 0.09 & 5.80 $\pm$ 0.10 & ... \
7281 He [i]{} & ... & ... & ... & 0.52 $\pm$ 0.03 & ... & 0.59 $\pm$ 0.02 & ... \
7320 \[O [ii]{}\] & ... & ... & ... & 0.65 $\pm$ 0.03 & ... & 2.06 $\pm$ 0.04 & ... \
7330 \[O [ii]{}\] & ... & ... & ... & 0.55 $\pm$ 0.03 & ... & 1.67 $\pm$ 0.03 & ... \
\
$C$(H$\beta$)$^a$ & [ 0.015 ]{}& [ 0.225 ]{}& [ 0.205 ]{}& [ 0.260 ]{}& [ 0.195 ]{}& [ 0.250 ]{}& [ 0.080 ]{}\
$F$(H$\beta$)$^b$ & [ 1.10 ]{}& [ 1.81 ]{}& [ 2.02 ]{}& [ 32.24 ]{}& [ 2.59 ]{}& [ 55.62 ]{}& [ 1.05 ]{}\
EW(H$\beta$) ($\AA$) & [ 9.6 ]{}& [ 36.3 ]{}& [ 84.2 ]{}& [ 199.4 ]{}& [ 41.9 ]{}& [ 219.5 ]{}& [ 25.0 ]{}\
EW(abs) ($\AA$) & [ 2.00 ]{}& [ 0.65 ]{}& [ 0.85 ]{}& [ 0.10 ]{}& [ 2.55 ]{}& [ 0.15 ]{}& [ 0.90 ]{}\
$^a$ zero value is assumed if a negative value is derived.
$^b$ in units of 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$.
[lrrrrrrrr]{}
\
\
&\
Property & [J0015+0104 ]{} & [J0016+0108 ]{} & [J0029-0108 ]{} & [J0029-0025 ]{} & [J0057-0022 ]{} & [J0107+0001 ]{} & [J0109+0107 ]{} & [J0126-0038 ]{}\
& & & & & & & & No.1\
$T_{\rm e}$(O [iii]{}) (K) & $20000 \pm1020 $ & $17023 \pm1008 $ & $18580 \pm1188 $ & $19025 \pm1020 $ & $16258 \pm 215 $ & $20000 \pm1054 $ & $12337 \pm 558 $ & $14070 \pm 351 $\
$T_{\rm e}$(O [ii]{}) (K) & $16284 \pm1293 $ & $15379 \pm1217 $ & $15961 \pm1470 $ & $16083 \pm1273 $ & $15006 \pm 180 $ & $16284 \pm1336 $ & $12196 \pm 517 $ & $13624 \pm 313 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.084 \pm 0.017$ & $ 0.186 \pm 0.038$ & $ 0.129 \pm 0.037$ & $ 0.096 \pm 0.019$ & $ 0.199 \pm 0.008$ & $ 0.115 \pm 0.025$ & $ 0.381 \pm 0.052$ & $ 0.294 \pm 0.020$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.033 \pm 0.004$ & $ 0.153 \pm 0.022$ & $ 0.096 \pm 0.019$ & $ 0.099 \pm 0.013$ & $ 0.200 \pm 0.007$ & $ 0.053 \pm 0.007$ & $ 0.571 \pm 0.076$ & $ 0.534 \pm 0.036$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & ... & ... & ... & ... & $ 2.421 \pm 0.457$\
O/H ($\times$10$^4$) & $ 0.117 \pm 0.017$ & $ 0.339 \pm 0.044$ & $ 0.225 \pm 0.042$ & $ 0.194 \pm 0.023$ & $ 0.399 \pm 0.011$ & $ 0.168 \pm 0.026$ & $ 0.952 \pm 0.092$ & $ 0.852 \pm 0.042$\
12 + log(O/H) & $ 7.070 \pm 0.063$ & $ 7.530 \pm 0.056$ & $ 7.353 \pm 0.080$ & $ 7.289 \pm 0.052$ & $ 7.601 \pm 0.012$ & $ 7.226 \pm 0.066$ & $ 7.979 \pm 0.042$ & $ 7.931 \pm 0.021$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & ... & ... & ... & $ 0.660 \pm 0.048$ & ... & $ 1.498 \pm 0.220$ & $ 1.247 \pm 0.092$\
ICF & ... & ... & ... & ... & 1.213 & ... & 1.155 & 1.141\
log(Ne/O) & ... & ... & ... & ... & $-0.697 \pm 0.053$ & ... & $-0.740 \pm 0.101$ & $-0.777 \pm 0.049$\
&\
Property& [J0126-0038 ]{} & [J0135-0023 ]{} & [J0213-0002 ]{} & [J0213-0002 ]{} & [J0216+0115 ]{} & [J0216+0115 ]{} & [096632 ]{} & [J0252+0017 ]{}\
& No.2 & & No.1 &No.2 &No.1 &No.2 & &\
$T_{\rm e}$(O [iii]{}) (K) & $17255 \pm1005 $ & $18365 \pm1022 $ & $16036 \pm1017 $ & $14366 \pm1019 $ & $16076 \pm1009 $ & $17938 \pm1023 $ & $17425 \pm1005 $ & $15617 \pm1064 $\
$T_{\rm e}$(O [ii]{}) (K) & $15481 \pm1218 $ & $15895 \pm1260 $ & $14888 \pm1209 $ & $13839 \pm1182 $ & $14910 \pm1200 $ & $15750 \pm1253 $ & $15552 \pm1221 $ & $14650 \pm1256 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.185 \pm 0.037$ & $ 0.135 \pm 0.028$ & $ 0.169 \pm 0.037$ & $ 0.341 \pm 0.084$ & $ 0.208 \pm 0.045$ & $ 0.185 \pm 0.038$ & $ 0.249 \pm 0.049$ & $ 0.251 \pm 0.063$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.136 \pm 0.019$ & $ 0.102 \pm 0.014$ & $ 0.263 \pm 0.043$ & $ 0.413 \pm 0.080$ & $ 0.229 \pm 0.036$ & $ 0.091 \pm 0.013$ & $ 0.078 \pm 0.011$ & $ 0.255 \pm 0.048$\
O/H ($\times$10$^4$) & $ 0.321 \pm 0.042$ & $ 0.238 \pm 0.031$ & $ 0.432 \pm 0.057$ & $ 0.754 \pm 0.116$ & $ 0.437 \pm 0.058$ & $ 0.276 \pm 0.040$ & $ 0.327 \pm 0.051$ & $ 0.506 \pm 0.079$\
12 + log(O/H) & $ 7.506 \pm 0.057$ & $ 7.376 \pm 0.056$ & $ 7.636 \pm 0.057$ & $ 7.877 \pm 0.067$ & $ 7.641 \pm 0.057$ & $ 7.441 \pm 0.063$ & $ 7.515 \pm 0.067$ & $ 7.704 \pm 0.068$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & ... & ... & ... & ... & ... & $ 0.148 \pm 0.026$ & ... \
ICF & ... & ... & ... & ... & ... & ... & 1.408 & ... \
log(Ne/O) & ... & ... & ... & ... & ... & ... & $-1.195 \pm 0.233$ & ... \
&\
Property& [J0256+0036 ]{} & [J0303-0109 ]{} & [J0303-0109 ]{} & [J0341-0026 ]{} & [J0341-0026 ]{} & [J0341-0026 ]{} & [G1815-6701 ]{} & [G2052-6912]{}\
& &No.1 &No.2 &No.1 &No.2 &No.3 & & No.1\
$T_{\rm e}$(O [iii]{}) (K) & $15446 \pm1004 $ & $15850 \pm 960 $ & $20000 \pm1143 $ & $16136 \pm1023 $ & $17233 \pm1010 $ & $19005 \pm1014 $ & $14202 \pm 308 $ & $10991 \pm 110 $\
$T_{\rm e}$(O [ii]{}) (K) & $14548 \pm1183 $ & $14784 \pm 816 $ & $16284 \pm1449 $ & $14942 \pm1218 $ & $15472 \pm1223 $ & $16078 \pm1264 $ & $13721 \pm 273 $ & $11007 \pm 105 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.190 \pm 0.042$ & $ 0.131 \pm 0.020$ & $ 0.121 \pm 0.030$ & $ 0.209 \pm 0.046$ & $ 0.113 \pm 0.023$ & $ 0.064 \pm 0.013$ & $ 0.318 \pm 0.019$ & $ 0.370 \pm 0.014$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.327 \pm 0.055$ & $ 0.359 \pm 0.055$ & $ 0.046 \pm 0.008$ & $ 0.222 \pm 0.036$ & $ 0.189 \pm 0.027$ & $ 0.120 \pm 0.015$ & $ 0.645 \pm 0.038$ & $ 1.281 \pm 0.043$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & ... & ... & ... & $ 1.764 \pm 0.426$ & $ 0.611 \pm 0.106$\
O/H ($\times$10$^4$) & $ 0.517 \pm 0.069$ & $ 0.490 \pm 0.059$ & $ 0.166 \pm 0.031$ & $ 0.430 \pm 0.059$ & $ 0.302 \pm 0.036$ & $ 0.184 \pm 0.020$ & $ 0.981 \pm 0.043$ & $ 1.657 \pm 0.045$\
12 + log(O/H) & $ 7.713 \pm 0.058$ & $ 7.690 \pm 0.052$ & $ 7.221 \pm 0.082$ & $ 7.634 \pm 0.059$ & $ 7.480 \pm 0.051$ & $ 7.265 \pm 0.047$ & $ 7.992 \pm 0.019$ & $ 8.219 \pm 0.012$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & $ 0.780 \pm 0.138$ & $ 0.981 \pm 0.159$ & ... & ... & ... & ... & $ 1.645 \pm 0.105$ & $ 2.810 \pm 0.109$\
ICF & 1.138 & 1.092 & ... & ... & ... & ... & 1.126 & 1.125\
log(Ne/O) & $-0.765 \pm 0.121$ & $-0.660 \pm 0.102$ & ... & ... & ... & ... & $-0.724 \pm 0.042$ & $-0.720 \pm 0.023$\
&\
Property& [G2052-6912 ]{} & [J2053+0039 ]{} & [J2105+0032 ]{} & [J2105+0032 ]{} & [J2112-0016 ]{} & [J2112-0016 ]{} & [J2119-0732 ]{} & [J2120-0058 ]{}\
&No.2 & &No.1 &No.2 &No.1 &No.2 & &\
$T_{\rm e}$(O [iii]{}) (K) & $11865 \pm 89 $ & $18364 \pm1039 $ & $17397 \pm1027 $ & $17930 \pm1021 $ & $11339 \pm 320 $ & $12417 \pm 771 $ & $14591 \pm1003 $ & $17281 \pm 908 $\
$T_{\rm e}$(O [ii]{}) (K) & $11756 \pm 83 $ & $15894 \pm1282 $ & $15541 \pm1247 $ & $15747 \pm1251 $ & $11239 \pm 302 $ & $12268 \pm 713 $ & $13996 \pm1167 $ & $15492 \pm 736 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.218 \pm 0.006$ & $ 0.065 \pm 0.014$ & $ 0.089 \pm 0.019$ & $ 0.146 \pm 0.030$ & $ 0.264 \pm 0.025$ & $ 0.192 \pm 0.036$ & $ 0.183 \pm 0.043$ & $ 0.207 \pm 0.025$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 1.309 \pm 0.033$ & $ 0.150 \pm 0.021$ & $ 0.194 \pm 0.029$ & $ 0.119 \pm 0.017$ & $ 1.366 \pm 0.120$ & $ 0.834 \pm 0.151$ & $ 0.487 \pm 0.088$ & $ 0.235 \pm 0.030$\
O/H ($\times$10$^4$) & $ 1.527 \pm 0.034$ & $ 0.215 \pm 0.026$ & $ 0.283 \pm 0.035$ & $ 0.265 \pm 0.034$ & $ 1.630 \pm 0.123$ & $ 1.026 \pm 0.156$ & $ 0.669 \pm 0.098$ & $ 0.442 \pm 0.039$\
12 + log(O/H) & $ 8.184 \pm 0.009$ & $ 7.332 \pm 0.052$ & $ 7.451 \pm 0.053$ & $ 7.423 \pm 0.056$ & $ 8.212 \pm 0.033$ & $ 8.011 \pm 0.066$ & $ 7.826 \pm 0.064$ & $ 7.646 \pm 0.038$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & $ 2.728 \pm 0.079$ & ... & ... & ... & $ 2.871 \pm 0.284$ & $ 1.249 \pm 0.261$ & $ 1.116 \pm 0.214$ & $ 0.723 \pm 0.093$\
ICF & 1.064 & ... & ... & ... & 1.079 & 1.062 & 1.094 & 1.194\
log(Ne/O) & $-0.721 \pm 0.017$ & ... & ... & ... & $-0.721 \pm 0.058$ & $-0.888 \pm 0.124$ & $-0.739 \pm 0.123$ & $-0.710 \pm 0.096$\
&\
Property& [J2150+0033 ]{} & [G2155-3946 ]{} & [J2227-0939 ]{} & [PHL 293B ]{} & [J2310-0109 No.1 ]{} & [J2310-0109 No.2 ]{}\
& & & & & No.1 &No.2\
$T_{\rm e}$(O [iii]{}) (K) & $16314 \pm1008 $ & $16087 \pm1008 $ & $14707 \pm1007 $ & $17410 \pm 228 $ & $15417 \pm1005 $ & $15359 \pm1006 $\
$T_{\rm e}$(O [ii]{}) (K) & $15036 \pm 845 $ & $14915 \pm1199 $ & $14075 \pm1173 $ & $15546 \pm 184 $ & $14531 \pm1183 $ & $14495 \pm 867 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.171 \pm 0.026$ & $ 0.298 \pm 0.064$ & $ 0.219 \pm 0.051$ & $ 0.053 \pm 0.002$ & $ 0.128 \pm 0.028$ & $ 0.179 \pm 0.029$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.230 \pm 0.036$ & $ 0.160 \pm 0.025$ & $ 0.434 \pm 0.079$ & $ 0.360 \pm 0.012$ & $ 0.379 \pm 0.063$ & $ 0.348 \pm 0.059$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & $ 0.655 \pm 0.065$ & ... & ... \
O/H ($\times$10$^4$) & $ 0.401 \pm 0.044$ & $ 0.458 \pm 0.068$ & $ 0.653 \pm 0.094$ & $ 0.420 \pm 0.012$ & $ 0.507 \pm 0.069$ & $ 0.527 \pm 0.066$\
12 + log(O/H) & $ 7.603 \pm 0.048$ & $ 7.661 \pm 0.065$ & $ 7.815 \pm 0.062$ & $ 7.624 \pm 0.013$ & $ 7.705 \pm 0.060$ & $ 7.722 \pm 0.054$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & $ 0.363 \pm 0.070$ & $ 0.993 \pm 0.202$ & $ 0.729 \pm 0.026$ & $ 0.517 \pm 0.094$ & $ 0.695 \pm 0.127$\
ICF & ... & 1.319 & 1.122 & 1.046 & 1.086 & 1.124\
log(Ne/O) & ... & $-0.980 \pm 0.193$ & $-0.768 \pm 0.134$ & $-0.742 \pm 0.021$ & $-0.956 \pm 0.114$ & $-0.829 \pm 0.120$\
[lrrrrrrrr]{}
\
\
&\
Property& [J0004$+$0025 ]{} & [J0004$+$0025 ]{} & [J0014$-$0044 ]{} & [J0014$-$0044 ]{} & [J0202$-$0047 ]{} & [J0301$-$0059 ]{} & [J0301$-$0059 ]{} & [J0301$-$0059 ]{}\
&No.1 &No.2 &No.1 &No.2 & &No.1 &No.2 &No.3\
$T_{\rm e}$(O [iii]{}) (K) & $18630 \pm1019 $ & $16701 \pm1016 $ & $13195 \pm 123 $ & $16308 \pm1006 $ & $14607 \pm 277 $ & $12582 \pm 585 $ & $13999 \pm1236 $ & $16248 \pm1033 $\
$T_{\rm e}$(O [ii]{}) (K) & $15975 \pm1263 $ & $15229 \pm1221 $ & $12939 \pm 112 $ & $15033 \pm1201 $ & $14006 \pm 243 $ & $12416 \pm 539 $ & $13571 \pm1104 $ & $15001 \pm1232 $\
$T_{\rm e}$(S [iii]{}) (K) & $17163 \pm 846 $ & $15562 \pm 844 $ & $13067 \pm 102 $ & $15236 \pm 835 $ & $14307 \pm 230 $ & $12499 \pm 485 $ & $13785 \pm1026 $ & $15186 \pm 857 $\
$N_{\rm e}$(S [ii]{}) (cm$^{-3}$) & $ 193 \pm 128$ & $ 213 \pm 132$ & $ 33 \pm 28$ & $ 167 \pm 78$ & $ 78 \pm 38$ & $ 10 \pm 10$ & $ 10 \pm 10$ & $ 10 \pm 10$\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.164 \pm 0.033$ & $ 0.224 \pm 0.048$ & $ 0.146 \pm 0.005$ & $ 0.274 \pm 0.057$ & $ 0.218 \pm 0.011$ & $ 0.565 \pm 0.078$ & $ 0.381 \pm 0.089$ & $ 0.300 \pm 0.067$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.073 \pm 0.010$ & $ 0.156 \pm 0.024$ & $ 0.960 \pm 0.027$ & $ 0.158 \pm 0.024$ & $ 0.580 \pm 0.030$ & $ 0.485 \pm 0.065$ & $ 0.353 \pm 0.084$ & $ 0.141 \pm 0.023$\
O/H ($\times$10$^4$) & $ 0.237 \pm 0.034$ & $ 0.380 \pm 0.053$ & $ 1.106 \pm 0.028$ & $ 0.431 \pm 0.062$ & $ 0.798 \pm 0.032$ & $ 1.050 \pm 0.101$ & $ 0.734 \pm 0.123$ & $ 0.441 \pm 0.071$\
12 + log(O/H) & $ 7.374 \pm 0.063$ & $ 7.580 \pm 0.061$ & $ 8.044 \pm 0.011$ & $ 7.635 \pm 0.063$ & $ 7.902 \pm 0.017$ & $ 8.021 \pm 0.042$ & $ 7.866 \pm 0.073$ & $ 7.644 \pm 0.070$\
\
N$^+$/H$^+$ ($\times$10$^6$) & $ 1.015 \pm 0.155$ & $ 0.913 \pm 0.140$ & $ 0.573 \pm 0.013$ & $ 1.663 \pm 0.240$ & $ 0.811 \pm 0.029$ & $ 2.245 \pm 0.202$ & $ 2.399 \pm 0.373$ & $ 2.491 \pm 0.409$\
ICF & 1.377 & 1.675 & 7.041 & 1.534 & 3.647 & 1.892 & 1.930 & 1.408\
log(N/O) & $-1.229 \pm 0.094$ & $-1.396 \pm 0.091$ & $-1.438 \pm 0.015$ & $-1.228 \pm 0.090$ & $-1.431 \pm 0.023$ & $-1.393 \pm 0.057$ & $-1.200 \pm 0.099$ & $-1.099 \pm 0.102$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & $ 0.659 \pm 0.138$ & $ 1.989 \pm 0.064$ & ... & $ 1.196 \pm 0.067$ & $ 1.118 \pm 0.167$ & $ 0.764 \pm 0.198$ & ... \
ICF & ... & 1.273 & 1.047 & ... & 1.094 & 1.268 & 1.226 & ... \
log(Ne/O) & ... & $-0.656 \pm 0.184$ & $-0.725 \pm 0.019$ & ... & $-0.785 \pm 0.035$ & $-0.870 \pm 0.119$ & $-0.894 \pm 0.205$ & ... \
\
S$^{+}$/H$^+$ ($\times$10$^6$) & $ 0.510 \pm 0.059$ & $ 0.484 \pm 0.059$ & $ 0.176 \pm 0.003$ & $ 0.549 \pm 0.065$ & $ 0.318 \pm 0.010$ & $ 0.988 \pm 0.076$ & $ 0.920 \pm 0.121$ & $ 1.038 \pm 0.138$\
S$^{++}$/H$^+$ ($\times$10$^6$) & ... & ... & $ 1.393 \pm 0.056$ & ... & $ 1.036 \pm 0.079$ & $ 0.919 \pm 0.113$ & ... & ... \
ICF & ... & ... & 1.724 & ... & 1.142 & 0.897 & ... & ... \
log(S/O) & ... & ... & $-1.612 \pm 0.019$ & ... & $-1.712 \pm 0.031$ & $-1.788 \pm 0.052$ & ... & ... \
\
Cl$^{++}$/H$^+$ ($\times$10$^8$) & ... & ... & $ 3.474 \pm 0.410$ & ... & ... & ... & ... & ... \
ICF & ... & ... & 1.243 & ... & ... & ... & ... & ... \
log(Cl/O) & ... & ... & $-3.408 \pm 0.052$ & ... & ... & ... & ... & ... \
\
Ar$^{++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 3.416 \pm 0.075$ & ... & $ 3.606 \pm 0.116$ & $ 3.719 \pm 0.336$ & $ 3.569 \pm 0.440$ & ... \
Ar$^{+++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 1.213 \pm 0.103$ & ... & ... & ... & ... & ... \
ICF & ... & ... & 1.122 & ... & 1.141 & 1.097 & 1.043 & ... \
log(Ar/O) & ... & ... & $-2.460 \pm 0.020$ & ... & $-2.287 \pm 0.022$ & $-2.410 \pm 0.057$ & $-2.295 \pm 0.090$ & ... \
\
Fe$^{++}$/H$^+$($\times$10$^6$)(4658) & ... & ... & ... & ... & ... & ... & ... & ... \
Fe$^{++}$/H$^+$($\times$10$^6$)(4988) & ... & ... & $ 0.125 \pm 0.003$ & ... & ... & ... & ... & ... \
ICF & ... & ... & 10.398 & ... & ... & ... & ... & ... \
log(Fe/O) ($\lambda$4658) & ... & ... & ... & ... & ... & ... & ... & ... \
log(Fe/O) (4988) & ... & ... & $-1.932 \pm 0.015$ & ... & ... & ... & ... & ... \
&\
Property& [J0315$-$0024 ]{} & [J0315$-$0024 ]{} & [J0338$+$0013 ]{} & [G0405-3648]{} & [G0405-3648]{} & [G0405-3648]{} & [J0519$+$0007 ]{} & [J2104$-$0035 ]{}\
&No.1 &No.2 & &No.1 &No.2 &No.3 & &No.1\
$T_{\rm e}$(O [iii]{}) (K) & $12550 \pm1733 $ & $18111 \pm1012 $ & $17882 \pm 208 $ & $15013 \pm1293 $ & $18169 \pm 908 $ & $21295 \pm1349 $ & $20143 \pm 238 $ & $20194 \pm 294 $\
$T_{\rm e}$(O [ii]{}) (K) & $12387 \pm1597 $ & $15811 \pm1243 $ & $15729 \pm 165 $ & $14278 \pm1125 $ & $15830 \pm 712 $ & $16407 \pm 902 $ & $16306 \pm 171 $ & $16313 \pm 210 $\
$T_{\rm e}$(S [iii]{}) (K) & $12468 \pm1438 $ & $16732 \pm 840 $ & $16542 \pm 173 $ & $14645 \pm1073 $ & $16780 \pm 753 $ & $19375 \pm1119 $ & $18419 \pm 198 $ & $18461 \pm 244 $\
$N_{\rm e}$(S [ii]{}) (cm$^{-3}$) & $ 1099 \pm 206$ & $ 10 \pm 10$ & $ 116 \pm 50$ & $ 10 \pm 10$ & $ 108 \pm 57$ & $ 10 \pm 10$ & $ 483 \pm 60$ & $ 84 \pm 74$\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.339 \pm 0.138$ & $ 0.147 \pm 0.030$ & $ 0.048 \pm 0.002$ & $ 0.137 \pm 0.030$ & $ 0.125 \pm 0.014$ & $ 0.107 \pm 0.014$ & $ 0.021 \pm 0.001$ & $ 0.019 \pm 0.001$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.289 \pm 0.115$ & $ 0.108 \pm 0.014$ & $ 0.385 \pm 0.012$ & $ 0.228 \pm 0.051$ & $ 0.099 \pm 0.012$ & $ 0.069 \pm 0.010$ & $ 0.248 \pm 0.007$ & $ 0.162 \pm 0.006$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & $ 0.531 \pm 0.042$ & ... & ... & ... & $ 0.444 \pm 0.021$ & $ 0.126 \pm 0.019$\
O/H ($\times$10$^4$) & $ 0.628 \pm 0.179$ & $ 0.255 \pm 0.033$ & $ 0.438 \pm 0.012$ & $ 0.365 \pm 0.059$ & $ 0.224 \pm 0.018$ & $ 0.176 \pm 0.017$ & $ 0.273 \pm 0.007$ & $ 0.183 \pm 0.006$\
12 + log(O/H) & $ 7.798 \pm 0.124$ & $ 7.406 \pm 0.056$ & $ 7.641 \pm 0.012$ & $ 7.562 \pm 0.070$ & $ 7.351 \pm 0.036$ & $ 7.246 \pm 0.043$ & $ 7.437 \pm 0.012$ & $ 7.261 \pm 0.014$\
\
N$^+$/H$^+$ ($\times$10$^6$) & $ 0.768 \pm 0.202$ & $ 0.786 \pm 0.128$ & $ 0.110 \pm 0.004$ & $ 0.440 \pm 0.067$ & $ 0.430 \pm 0.036$ & $ 0.319 \pm 0.031$ & $ 0.195 \pm 0.004$ & $ 0.058 \pm 0.003$\
ICF & 1.847 & 1.713 & 8.642 & 2.686 & 1.785 & 1.616 & 12.223 & 8.833\
log(N/O) & $-1.646 \pm 0.169$ & $-1.277 \pm 0.091$ & $-1.662 \pm 0.019$ & $-1.490 \pm 0.096$ & $-1.465 \pm 0.051$ & $-1.535 \pm 0.061$ & $-1.059 \pm 0.015$ & $-1.551 \pm 0.029$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & $ 0.416 \pm 0.199$ & ... & $ 0.689 \pm 0.022$ & $ 0.258 \pm 0.071$ & $ 0.185 \pm 0.028$ & $ 0.164 \pm 0.024$ & $ 0.447 \pm 0.013$ & $ 0.301 \pm 0.011$\
ICF & 1.240 & ... & 1.040 & 1.143 & 1.251 & 1.286 & 1.031 & 1.037\
log(Ne/O) & $-1.085 \pm 0.385$ & ... & $-0.786 \pm 0.019$ & $-1.094 \pm 0.181$ & $-0.987 \pm 0.123$ & $-0.923 \pm 0.133$ & $-0.774 \pm 0.018$ & $-0.766 \pm 0.022$\
\
S$^{+}$/H$^+$ ($\times$10$^6$) & $ 0.291 \pm 0.066$ & $ 0.202 \pm 0.028$ & $ 0.073 \pm 0.002$ & $ 0.225 \pm 0.028$ & $ 0.250 \pm 0.016$ & $ 0.205 \pm 0.016$ & $ 0.040 \pm 0.001$ & $ 0.031 \pm 0.001$\
S$^{++}$/H$^+$ ($\times$10$^6$) & ... & ... & $ 0.387 \pm 0.027$ & ... & ... & ... & $ 0.223 \pm 0.009$ & $ 0.189 \pm 0.019$\
ICF & ... & ... & 1.947 & ... & ... & ... & 2.539 & 1.978\
log(S/O) & ... & ... & $-1.689 \pm 0.028$ & ... & ... & ... & $-1.612 \pm 0.018$ & $-1.623 \pm 0.040$\
\
Ar$^{++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 1.068 \pm 0.028$ & $ 1.544 \pm 0.201$ & $ 0.937 \pm 0.080$ & $ 0.618 \pm 0.052$ & $ 0.682 \pm 0.014$ & $ 0.357 \pm 0.015$\
Ar$^{+++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 0.919 \pm 0.051$ & ... & ... & ... & $ 0.762 \pm 0.032$ & $ 0.371 \pm 0.049$\
ICF & ... & ... & 1.867 & 1.058 & 1.048 & 1.060 & 2.455 & 1.898\
log(Ar/O) & ... & ... & $-2.342 \pm 0.027$ & $-2.349 \pm 0.090$ & $-2.358 \pm 0.051$ & $-2.430 \pm 0.056$ & $-2.213 \pm 0.025$ & $-2.431 \pm 0.064$\
\
Fe$^{++}$/H$^+$($\times$10$^6$)(4658) & ... & ... & $ 0.083 \pm 0.010$ & ... & ... & ... & $ 0.076 \pm 0.007$ & ... \
Fe$^{++}$/H$^+$($\times$10$^6$)(4988) & ... & ... & $ 0.091 \pm 0.009$ & ... & $ 0.327 \pm 0.059$ & ... & $ 0.101 \pm 0.007$ & $ 0.117 \pm 0.021$\
ICF & ... & ... & 12.634 & ... & 2.360 & ... & 18.224 & 12.932\
log(Fe/O) ($\lambda$4658) & ... & ... & $-1.623 \pm 0.054$ & ... & ... & ... & $-1.298 \pm 0.041$ & ... \
log(Fe/O) (4988) & ... & ... & $-1.579 \pm 0.043$ & ... & $-1.464 \pm 0.086$ & ... & $-1.170 \pm 0.031$ & $-1.081 \pm 0.079$\
&\
Property& [J2104$-$0035 ]{} & [J2104$-$0035 ]{} & [J2104$-$0035 ]{} & [J2302$+$0049 ]{} & [J2302$+$0049 ]{} & [J2324$-$0006 ]{} & [J2354$-$0004 ]{}\
&No.2 &No.3 & No.4 &No.1 &No.2 & &No.1\
$T_{\rm e}$(O [iii]{}) (K) & $20000 \pm1061 $ & $21471 \pm2559 $ & $17794 \pm4091 $ & $17281 \pm 179 $ & $15203 \pm 616 $ & $14313 \pm 128 $ & $18120 \pm3255 $\
$T_{\rm e}$(O [ii]{}) (K) & $16284 \pm1345 $ & $16411 \pm1693 $ & $15697 \pm3257 $ & $15492 \pm 145 $ & $14399 \pm 533 $ & $13801 \pm 113 $ & $15814 \pm2559 $\
$T_{\rm e}$(S [iii]{}) (K) & $18300 \pm 880 $ & $19521 \pm2124 $ & $16469 \pm3396 $ & $16043 \pm 148 $ & $14801 \pm 511 $ & $14057 \pm 106 $ & $16740 \pm2702 $\
$N_{\rm e}$(S [ii]{}) (cm$^{-3}$) & $ 255 \pm 248$ & $ 10 \pm 10$ & $ 10 \pm 10$ & $ 143 \pm 44$ & $ 10 \pm 10$ & $ 84 \pm 31$ & $ 10 \pm 10$\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.083 \pm 0.018$ & $ 0.075 \pm 0.019$ & $ 0.070 \pm 0.036$ & $ 0.050 \pm 0.001$ & $ 0.196 \pm 0.020$ & $ 0.182 \pm 0.005$ & $ 0.107 \pm 0.044$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.034 \pm 0.004$ & $ 0.047 \pm 0.012$ & $ 0.043 \pm 0.024$ & $ 0.450 \pm 0.012$ & $ 0.335 \pm 0.035$ & $ 0.675 \pm 0.018$ & $ 0.120 \pm 0.051$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & $ 1.342 \pm 0.062$ & ... & $ 0.715 \pm 0.051$ & ... \
O/H ($\times$10$^4$) & $ 0.117 \pm 0.019$ & $ 0.122 \pm 0.023$ & $ 0.113 \pm 0.043$ & $ 0.513 \pm 0.012$ & $ 0.531 \pm 0.041$ & $ 0.864 \pm 0.019$ & $ 0.227 \pm 0.067$\
12 + log(O/H) & $ 7.068 \pm 0.070$ & $ 7.088 \pm 0.080$ & $ 7.052 \pm 0.166$ & $ 7.710 \pm 0.010$ & $ 7.725 \pm 0.033$ & $ 7.937 \pm 0.009$ & $ 7.355 \pm 0.128$\
\
N$^+$/H$^+$ ($\times$10$^6$) & $ 0.310 \pm 0.051$ & $ 0.266 \pm 0.049$ & $ 0.265 \pm 0.095$ & $ 0.117 \pm 0.003$ & $ 0.528 \pm 0.045$ & $ 0.431 \pm 0.009$ & $ 0.322 \pm 0.092$\
ICF & 1.323 & 1.592 & 1.587 & 9.717 & 2.732 & 4.662 & 2.133\
log(N/O) & $-1.455 \pm 0.103$ & $-1.462 \pm 0.114$ & $-1.428 \pm 0.230$ & $-1.653 \pm 0.017$ & $-1.566 \pm 0.049$ & $-1.633 \pm 0.013$ & $-1.518 \pm 0.177$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & ... & ... & $ 0.819 \pm 0.024$ & $ 1.030 \pm 0.115$ & $ 1.457 \pm 0.043$ & ... \
ICF & ... & ... & ... & 1.040 & 1.139 & 1.072 & ... \
log(Ne/O) & ... & ... & ... & $-0.780 \pm 0.017$ & $-0.656 \pm 0.075$ & $-0.743 \pm 0.018$ & ... \
\
S$^{+}$/H$^+$ ($\times$10$^6$) & $ 0.198 \pm 0.026$ & $ 0.103 \pm 0.015$ & $ 0.132 \pm 0.038$ & $ 0.073 \pm 0.001$ & $ 0.293 \pm 0.018$ & $ 0.232 \pm 0.004$ & $ 0.164 \pm 0.037$\
S$^{++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & $ 0.508 \pm 0.027$ & ... & $ 1.045 \pm 0.032$ & ... \
ICF & ... & ... & ... & 2.124 & ... & 1.300 & ... \
log(S/O) & ... & ... & ... & $-1.619 \pm 0.023$ & ... & $-1.716 \pm 0.014$ & ... \
\
Cl$^{++}$/H$^+$ ($\times$10$^8$) & ... & ... & ... & ... & ... & $ 1.385 \pm 0.150$ & ... \
ICF & ... & ... & ... & ... & ... & 1.193 & ... \
log(Cl/O) & ... & ... & ... & ... & ... & $-3.719 \pm 0.048$ & ... \
\
Ar$^{++}$/H$^+$ ($\times$10$^7$) & ... & ... & ... & $ 1.222 \pm 0.027$ & $ 1.869 \pm 0.095$ & $ 2.613 \pm 0.053$ & ... \
Ar$^{+++}$/H$^+$ ($\times$10$^7$) & ... & ... & ... & $ 1.004 \pm 0.053$ & ... & $ 0.487 \pm 0.052$ & ... \
ICF & ... & ... & ... & 2.041 & 1.061 & 1.265 & ... \
log(Ar/O) & ... & ... & ... & $-2.313 \pm 0.024$ & $-2.428 \pm 0.040$ & $-2.417 \pm 0.015$ & ... \
\
Fe$^{++}$/H$^+$($\times$10$^6$)(4658) & ... & ... & ... & $ 0.086 \pm 0.014$ & ... & $ 0.139 \pm 0.012$ & ... \
Fe$^{++}$/H$^+$($\times$10$^6$)(4988) & ... & ... & ... & $ 0.131 \pm 0.010$ & $ 0.673 \pm 0.096$ & $ 0.204 \pm 0.010$ & ... \
ICF & ... & ... & ... & 14.309 & 3.639 & 6.495 & ... \
log(Fe/O) ($\lambda$4658) & ... & ... & ... & $-1.618 \pm 0.069$ & ... & $-1.982 \pm 0.039$ & ... \
log(Fe/O) (4988) & ... & ... & ... & $-1.437 \pm 0.034$ & $-1.336 \pm 0.071$ & $-1.815 \pm 0.024$ & ... \
[^1]: Based on observations collected at the European Southern Observatory, Chile, VLT and 3.6m telescopes.
[^2]: Tables 1 - 6 and Figures 1 - 2 are only available in electronic form in the online edition.
[^3]: IRAF is the Image Reduction and Analysis Facility distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation (NSF).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Nian-Sheng Tang[^1] Xiao-Dong Yan and Pu-Ying Zhao\
\
title: Exponentially tilted likelihood inference on growing dimensional unconditional moment models
---
[**Abstract**]{}: Growing-dimensional data with likelihood unavailable are often encountered in various fields. This paper presents a penalized exponentially tilted likelihood (PETL) for variable selection and parameter estimation for growing dimensional unconditional moment models in the presence of correlation among variables and model misspecification. Under some regularity conditions, we investigate the consistent and oracle properties of the PETL estimators of parameters, and show that the constrainedly PETL ratio statistic for testing contrast hypothesis asymptotically follows the central chi-squared distribution. Theoretical results reveal that the PETL approach is robust to model misspecification. We also study high-order asymptotic properties of the proposed PETL estimators. Simulation studies are conducted to investigate the finite performance of the proposed methodologies. An example from the Boston Housing Study is illustrated.\
[***Keywords***]{}: Growing-dimensional data analysis; Model misspecification; Moment uncondition models; Penalized exponentially tilted likelihood; Variable selection.
[GBK]{}[song]{}
Introduction
============
Exponentially tilted (ET) likelihood (Imbens, Spady and Johnson, 1998) is a useful nonparametric approach to evaluate estimates and confidence regions of unknown parameters in unconditional moment models of the form $E\{g(x;\theta)\}=0$, which provides a unified approach for parameter estimation in a class of statistical models with likelihood function unavailable, where $g(x;\theta)$ is a vector-valued nonlinear function of a random vector $x$ and a parameter vector $\theta$. The merits of the ET likelihood include (i) it behaves better than empirical likelihood under model misspecification (Schennach, 2007), that is, the ET likelihood is robust to model misspecification, (ii) it allows a computationally convenient treatment of misspecified models (Kitamura, 2000), and (iii) it is flexible in incorporating auxiliary information. Hence, several authors, for example, Schennach (2005, 2007), Zhu et al. (2009) and Caner (2010), discussed its properties and applications when the number of parameters is fixed and less than or equal to sample size.
Growing-dimensional parametric or semiparametric models are widely used to make statistical inference on complicated data sets such as longitudinal and panel data in econometrics (Fan and Peng, 2004). It is commonly assumed that only a small number of covariates actually contribute to the considered models, which leads to the well-known sparse models for helping interpretation and improving prediction accuracy (Bradic, Fan and Wang, 2011). To this end, many penalized methods have been developed to simultaneously select the important covariates and estimate parameters in various statistical models when the number of parameters diverges. For example, Fan and Peng (2004) investigated the nonconcave penalized likelihood with a growing number of nuisance parameters in a linear regression model; Lam and Fan (2008) presented a profile-kernel likelihood inference in a linear regression model; Wang, Li and Leng (2009) studied shrinkage tuning parameter selection; Zou and Zhang (2009) proposed an adaptive elastic-net procedure for a linear regression model; Li, Peng and Zhu (2011) investigated asymptotic properties of a nonconcave penalized M-estimator in a sparse, diverging-dimensional, linear regression model; Caner and Zhang (2014) extended the least squares based adaptive elastic net estimator of Zou and Zhang (2009) to generalized method of moments (GMMs); Caner, Han and Lee (2016) presented an adaptive elastic net GMM estimation for many invalid moment conditions. Recently, Leng and Tang (2012) presented a penalized empirical likelihood method in estimating equations, which can be used to improve the efficiency of parameter estimation by incorporating some auxiliary information when likelihood function is unavailable, with a diverging number of parameters, but their empirical likelihood method is sensitive to model misspecification. Also, to the best of our knowledge, there is little work done on extending the above mentioned approaches to unconditional moment models with a diverging number of parameters in the presence of model misspecification. More importantly, this extension is challenging in the presence of model misspecification and high correlation among variables because (i) the number of Lagrange multipliers used to obtain the solution to minimizing the ET likelihood function increases with sample size, (ii) the nonconvex optimization is involved (Leng and Tang, 2012), and (iii) there is a well-known ill-posed problem, i.e., the resulting estimator has very slow rate of convergence (see, e.g., Ai and Chen, 2003; Hall and Horowitz, 2005; Darolles, Fan, Florens and Renault, 2011; Chen and Pouzo, 2012).
In this paper, we develop a penalized ET (PET) likelihood procedure for parameter estimation, variable selection and statistical inference for unconditional moment models with a diverging number of parameters in the presence of model misspecification and high correlation among variables via the sieve method (Ai and Chen, 2003). With a proper penalty function and diverging rate of dimensionality, we demonstrate that (i) the resulting estimator possesses the advantages of the penalized likelihood approach, i.e., the PET method has the oracle properties (Fan and Li, 2001) that it identifies the true sparse structure of the considered model with probability tending to one and with the optimal efficiency; (ii) the resulting estimator has the advantages of the ET likelihood method, i.e., the PET method behaves better than the penalized empirical likelihood approach in the presence of model misspecification; (iii) the constrainedly profiled PET likelihood ratio statistic is asymptotically distributed as the chi-squared distribution indicating that the Wilks’ theorem holds, which can be used to test hypotheses and construct confidence regions of parameters of interest. In addition, we extend the high-order asymptotic properties of the ET estimator given in Schennach (2007) for a fixed number of parameters to the case that the number of parameters diverges; and we also establish selection consistency for NP dimension case.
The rest of this paper is organized as follows. Section 2 first introduces the PET likelihood, and then investigates the oracle properties of the proposed PET estimators, asymptotic chi-squared distribution, high-order asymptotic properties and selection consistency. Simulation studies are given in Section 3. An example from the Boston Housing Study is analyzed in Section 4. Some concluding remarks are given in Section 5. Proofs of Theorems are presented in Appendix.
Methods
=======
Exponentially tilted likelihood
-------------------------------
Suppose that $X_1,\ldots,X_n$ are independent and identically distributed (i.i.d.) random vectors from an unknown distribution $F(x)$ with $x\in \mathcal{X}\subset \mathcal{R}^{\iota}$. Without assuming a specific form of $F(x)$, we are interested in making inference on a $p\times 1$ vector of unknown parameters of interest, denoted by $\theta$, based on $r$ ($r\geq p$) functionally independent estimating functions $g(X_i;\theta)=(g_1(X_i;\theta),\ldots,g_r(X_i;\theta))^{{\!\top\!}}$ that satisfy the unconditional moment condition of the form: $E_{F_x}\{g(X_i;\theta_0)\}=0$ for $\theta_0\in\Theta\subset\mathcal{R}^p$ and $i=1,\ldots,n$, which is usually referred to as general estimating equations or unconditional moment models (Owen, 2001), where $\theta_0$ is the unique true value of $\theta$ and $E_{F_x}$ denotes the expectation taken with respect to $F(x)$. The selection of $g(X;\theta)$ is flexible and the details can refer to Leng and Tang (2012).
When $r=p$, one can obtain estimation of $\theta$ by solving the following unconditional moment conditions: $n^{-1}\sum_{i=1}^ng(X_i;\theta)=0$ (Leng and Tang, 2012). When $r>p$ and $p$ is fixed, one can employ empirical likelihood approach to obtain more efficient estimation of $\theta$ by combining available information (Qin and Lawless, 1994). However, when $r>p$ and $p$ is large, it is commonly assumed that only a small number of variables actually contribute to unconditional moment conditions, which leads to the sparsity pattern in unknown parameter vector $\theta$ and thus makes variable selection crucial (Bradic, Fan and Wang, 2011). To this end, Leng and Tang (2012) studied growing dimensional unconditional moment models via empirical likelihood approach, and presented a penalized empirical likelihood procedure for parameter estimation and variable selection. In what follows, we present an ET approach to investigate parameter estimation and variable selection for unconditional moment models with a growing number of parameters because the ET likelihood is a robust nonparametric tool to make statistical inference on unconditional moment models (Imbens et al., 1998; Owen, 2001) when unconditional moment models are misspecified.
For $i=1,\ldots,n$, let $w_i=dF(X_i)={\rm Pr}(\mathbb{X}_i=X_i)$, where $X_i$ is the observation of random vector $\mathbb{X}_i$. The ET likelihood can be defined as the Kullback-Leibler divergence between the empirical frequencies $1/n$ and $w_i$ subject to some restrictions. Following Imbens et al. (1998), the ET estimator $\hat\theta_{ET}$ of $\theta$ is the solution to the following Kullback-Leibler information criterion: ${\rm inf}_{w_1,\ldots,w_n,\theta}\sum_{i=1}^nw_i\log w_i$ subject to $\sum_{i=1}^nw_i=1$, $w_i\geq 0$ and $\sum_{i=1}^nw_ig(X_i;\theta)=0$. To wit, the ET likelihood for $\theta$ can be defined as $$\label{PET1}
L(\theta)={\rm inf}\left\{\prod_{i=1}^nw_i\log(w_i): w_i\geq 0, \sum_{i=1}^nw_i=1, \sum_{i=1}^nw_ig(X_i;\theta)=0\right\},$$ which is minimized at $w_i=\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}/\sum_{j=1}^n\exp\{\nu^{{\!\top\!}}g(X_j;\theta)\}$, where $\nu$ is the lagrange multiplier. Under some regularity conditions, it is easily shown that the profiled log-ET likelihood ratio function can be expressed as $$\label{PETL22}
\ell(\nu,\theta)=-\{\log L(\theta)+\log(n)\}=\log\left[\frac{1}{n}\sum\limits_{i=1}^{n}\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}\right],$$ where $\nu$ satisfies $Q_{n1}(\nu,\theta)=n^{-1}\sum_{i=1}^n\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}g(X_i;\theta)=0$. Thus, $\hat\theta_{ET}$ is the solution to the following nonlinear optimization problem: $\hat\theta_{ET}=\arg\max_{\theta\in\Theta}\inf_{\nu\in\widehat{\mathcal{V}}_n(\theta)} \ell(\nu,\theta)$, where $\widehat{\mathcal{V}}_n(\theta)=\{\nu: \nu^{{\!\top\!}}g(X_i;\theta)\in\mathcal{E},i=1,\cdots,n\}$ in which $\mathcal{E}$ is an open interval containing zero. Generally, under some regularity conditions, $\hat\theta_{ET}$ can be obtained by simultaneously solving the following equations: $$\label{ITEQ}
Q_{n1}(\nu,\theta)=0~~{\rm and}~~Q_{n2}(\nu,\theta)=n^{-1}\sum\limits_{i=1}^n\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}\nu^{{\!\top\!}}\partial_\theta g(X_i;\theta)=0,$$ where $\partial_\theta$ represents the partial derivative with respect to $\theta$.
Penalized exponentially tilted likelihood
-----------------------------------------
To identify the important covariates in growing-dimensional data analysis, following Fan and Li (2001), we consider the following profiled PET likelihood ratio function by combining Equation (\[PETL22\]) and some proper penalty function: $$\label{PETL13}
\ell_p(\theta)=\log\left[\frac{1}{n}\sum_{i=1}^{n}\exp\{\nu^{{\!\top\!}}(\theta)g(X_i;\theta)\}\right]-\sum_{j=1}^{p}p_{\gamma}(|\theta_j|),$$ where $\nu(\theta)=\inf_{\nu\in\widehat{\mathcal{V}}_n(\theta)} \ell(\nu,\theta)$, $p_{\gamma}(t)$ is some proper penalty function with a tuning parameter $\gamma$ controlling the trade-off between bias and model complexity.
When the number of parameters diverges, there is a well-known ill-posed problem (Chen and Pouzo, 2012) for our considered unconditional moment models, i.e., for any $\mathbb{C}> 0$, there are sequences $\{\theta^{(k)}\}_{k=1}^{\infty}$ of $\theta$ in $\Theta$ such that $\liminf_{k\rightarrow\infty}||\theta^{(k)}-\theta_0||\ge \mathbb{C}$ but $\liminf_{k\rightarrow\infty}E\{||g(X;\theta^{(k)})||^2\}=0$. To solve the ill-posed problem, we incorporate two types of regularization methods (e.g., the regularization by sieves and the regularization by penalization) into the PET procedure (\[PETL13\]). The commonly used sieves with sparsity constraints can be expressed as $$\label{sieve}
\Theta_{s(n)}=\Big\{\theta\in\Theta: |\mathbb{J}_n|\leq s(n)\Big\},$$ where $\mathbb{J}_n=\{j:\theta_j\neq 0\}$, $|\mathbb{J}_n|$ denotes the cardinality of $\mathbb{J}_n$, and $s(n)$ is some given positive integer associated with sample size $n$. The constraint $|\mathbb{J}_n|\leq s(n)$ reflects the prior sparsity information on $\theta_0\in\Theta$. Following Chen and Pouzo (2012), the penalty function $p_\gamma(t)$ in (\[PETL13\]) is typically some convex and/or lower semicompact. Here $p_\gamma(t)$ is taken to be the smoothly clipped absolute deviation (SCAD) penalty because the SCAD penalty function satisfies three desirable conditions for variable selection, i.e. asymptotic unbiasedness, sparsity and continuity of the estimated parameters (Fan and Li, 2001). The SCAD penalty is a function whose first derivative has the following form $$p_{\gamma}'(t)=\gamma\left\{I(t\leq \gamma)+\frac{(a\gamma-t)_+}{(a-1)\gamma}I(t>\gamma)\right\}$$ for some $a>2$, where $(s)_+=s$ for $s>0$ and $0$ otherwise. The corresponding PET likelihood ratio function is referred to as the SCAD-PET likelihood ratio function. Similar to Fan and Li (2001), we take $a=3.7$ in our simulation studies. Thus, under Equations (\[PETL13\]) and (\[sieve\]), the SCAD-PET estimator (denoted by $\hat\theta$) of $\theta$ can be obtained by minimizing $\ell_p(\theta)$, i.e., $$\label{sievepet}
\hat{\theta}=\min_{\theta\in\Theta_{s(n)}}\tilde{\ell}_p(\theta_{\mathbb{J}_n}),$$ where $\tilde{\ell}_p(\theta_{\mathbb{J}_n})=\log\left[\frac{1}{n}\sum_{i=1}^{n}\exp\{\tilde{\nu}^{{\!\top\!}}(\theta_{\mathbb{J}_n})
\tilde{g}(\tilde{X}_i;\theta_{\mathbb{J}_n})\}\right]-\sum_{j\in\mathbb{J}_n}p_{\gamma}(|\theta_j|)$, and $\tilde{\nu}$, $\tilde{g}(\cdot)$, $\tilde{X}_i$ are their corresponding reduced forms of Lagrange, estimating functions and covariates under the sieve space $\Theta_{s(n)}$ after ignoring the auxiliary variables excluding $\theta_1$, respectively. Thus, under the sparsity assumption, we can write $\theta=(\theta_1^{{\!\top\!}},\theta_2^{{\!\top\!}})^{{\!\top\!}}$, where $\theta_1\in \mathcal {R}^q$ and $\theta_2\in \mathcal {R}^{p-q}$ correspond to the nonzero and zero components of $\theta$, respectively. Once we have the prior sparsity structure of $\theta$, we can make statistical inference on $\theta_1$ based on the reduced estimating equations $\psi(Z_i;\theta_1)=(\psi_1(Z_i;\theta_1)$ $,\ldots,\psi_k(Z_i;\theta_1))^{{\!\top\!}}$ satisfying the following unconditional moment restrictions: $E_{F_z}\{\psi(Z_i;\theta_{10})\}=0$ for $\theta_{10}\in \mathcal{R}^q$, where $\psi(Z_i;\theta_{1})$ is some reduced version of $g(X_i;\theta)$ under $\theta_2=0$, $\theta_{10}$ is the true value of $\theta_1$, and $Z_i$ is the selected important covariates (i.e., $Z_i\subset X_i$) for $i=1,\ldots,n$. It is assumed that $Z_1,\ldots,Z_n$ are independent and identically distributed as an unknown distribution $F(z)$ with $z\in \mathcal{Z}\subset \mathcal{R}^q$. In this case, the corresponding constrainedly profiled PET likelihood ratio function can be defined as $$\label{cPET}
\bar{\ell}_p(\theta_1)=\log\left[\frac{1}{n}\sum_{i=1}^{n}\exp\{\lambda^{{\!\top\!}}(\theta_1)\psi(Z_i;\theta_1)\}\right]-\sum_{j=1}^{q}p_{\gamma}(|\theta_{1j}|),$$ where $\lambda(\theta_1)\in\widehat{\Lambda}_n(\theta_1)=\{\lambda:\lambda^{{\!\top\!}}\psi(Z_i;\theta_1)\in\mathcal{E},i=1,\ldots,n\}$ in which $\mathcal{E}$ is an open interval containing zero.
[**Example 1**]{}. As an illustration, we consider a mean regression model: $E(X_i)=\theta$ for $i=1,\ldots,n$. In this case, estimating equations: $g(X_i,\theta)=X_i-\theta$ can be used to make statistical inference on $\theta$. Clearly, estimating equations $g(\cdot)$ satisfy unconditional moment model: $E\{g(X_i,\theta)\}=0$. Under the assumption: $\theta_2^T=(\theta_{21},\ldots,\theta_{2,p-q})=0$, we can obtain the following reduced estimating equations: $\psi(Z_i,\theta_1)=(X_{i1}-\theta_{11},\ldots, X_{iq}-\theta_{1q})^{{\!\top\!}}$, where $Z_i=(X_{i1},\ldots,X_{iq})^T$ and $\theta_1=(\theta_{11},\ldots,\theta_{1q})^T$.
[**Example 2**]{}. We consider a linear regression model: $Y_i=\tilde{X}_i^T\theta+\varepsilon_i$ with $E(\varepsilon_i)=0$, where $\tilde{X}_i=(X_{i1},\ldots,X_{ip})^T$ and $\theta=(\theta_{11},\ldots,\theta_{1q},\theta_{21},\ldots,\theta_{2,p-q})^T$. Let $X_i=\{Y_i,\tilde{X}_i\}$. Thus, estimating equations: $g(X_i,\theta)=(X_{i1}(Y_i-\tilde{X}_i^T\theta),\ldots,X_{ip}(Y_i-\tilde{X}_i^T\theta))^T$ can be employed to make statistical inference on $\theta$. When $\theta_{21}=\cdots=\theta_{2,p-q}=0$, the reduced estimating equations are given by $\psi(Z_i,\theta_1)=(X_{i1}(Y_i-\tilde{Z}_i^T\theta_1),\ldots, X_{iq}(Y_i-\tilde{Z}_i^T\theta_1))^{{\!\top\!}}$, which satisfies $E\{\psi(Z_i,\theta_1)\}=0$, where $\tilde{Z}_i=(X_{i1},\ldots,X_{iq})^T$, $Z_i=\{Y_i,\tilde{Z}_i\}$ and $\theta_1=(\theta_{11},\ldots,\theta_{1q})^T$.
Selection consistency
---------------------
In this subsection, we investigate the consistency of the above presented model selection. Let $\mathbb{J}=\{j:\theta_{0j}\neq 0\} $ be the index set of nonzero components of the true parameter vector $\theta_0$, where $\theta_{0j}$ is the $j$th component of $\theta_0$ for $j=1,\ldots,p$. Denote the cardinality of $\mathbb{J}$ as $q=|\mathbb{J}|$, which is unknown. The true parameter vector $\theta_0$ has the following form $\theta_0=(\theta_{10}^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}$, where $\theta_{10}$ is the true value of $\theta_1$. The corresponding decomposition of $\hat\theta$ can be written as $\hat\theta=(\hat\theta_1^{{\!\top\!}},\hat\theta_2^{{\!\top\!}})^{{\!\top\!}}$. Let $\mathcal{D}_n=\{\theta:||\theta-\theta_{0}||\leq \mathbb{C}\sqrt{r/n}\}$ be neighborhoods of $\theta_{0}$ for some constant $\mathbb{C}>0$. $\Sigma(\theta)=E\{(g(X_i;\theta)-Eg(X_i;\theta))(g(X_i;\theta)-Eg(X_i;\theta))^{{\!\top\!}}\}$, $\Gamma(\theta)=E\{\partial_{\theta}g(X_{i};\theta)\}$, and $\Sigma=
\Sigma(\theta_0)$, $\Gamma=\Gamma(\theta_0)$.
Lv and Fan (2009) pointed out that the increase of $p^{'}_{\gamma}(s)$ with respect to $\gamma$ allows for $\gamma$ effectively controlling the overall strength of penalty. Therefore, we can take the tuning parameter $\gamma$ to be our required threshold. Specifically, the selection criterion is $\hat{\mathbb{J}}=\{j: |\hat{\theta}_{j}|>\gamma\}$, where $\hat{\theta}_{j}$ is the $j$th component of the PET estimator $\hat\theta$. Here, our main purpose is to show the selection consistency, i.e., ${\rm Pr}(\hat{\mathbb{J}}= \mathbb{J})\rightarrow 1$ as $n\rightarrow \infty$ even if $p$ diverges with $n$. Note that the event $\{\hat{\mathbb{J}}\neq \mathbb{J}\}$ is equivalent to the event $\{|\hat{\theta}_{j}|\leq \gamma$ for some $j\in \mathbb{J}\}\cup\{|\hat{\theta}_{j}|> \gamma$ for some $j\in \mathbb{J}^c\}$. Also, we have $$\begin{array}{llll}
{\rm Pr}(\{|\hat{\theta}_{j}|\leq \gamma\; {\rm for\; some\; j} \in \mathbb{J}\})&\leq& \sum_{j\in \mathbb{J}}{\rm Pr}(|\hat{\theta}_{j}|\leq \gamma)\\
&=&\sum_{j\in \mathbb{J}}{\rm Pr}(|\theta_{0j}|-|\hat{\theta}_{j}|\ge |\theta_{0j}|-\gamma)\\
&\leq&\sum_{j\in \mathbb{J}}{\rm Pr}(|\theta_{0j}-\hat{\theta}_{j}|\ge |\theta_{0j}|-\gamma)\\
&\leq&\sum_{j\in \mathbb{J}}{\rm Pr}(|\theta_{0j}-\hat{\theta}_{j}|\ge \min_{j\in \mathbb{J}}|\theta_{0j}|-\gamma)\\
&\leq&\sum_{j\in\mathbb{J}} {\rm Pr}(|\hat{\theta}_{j}-\theta_{0j}|\ge \gamma).
\end{array}$$ The last inequality holds because Assumption \[ass4\](i) leads to $\min_{j\in \mathbb{J}}|\theta_{0j}|>2\gamma$. Similarly, we can obtain ${\rm Pr}(\{|\hat{\theta}_{j}|> \gamma$ for some $j\in \mathbb{J}^c\})\leq\sum_{j\in \mathbb{J}^c}{\rm Pr}(|\theta_{0j}-\hat{\theta}_{j}|\ge \gamma)
\leq\sum_{j\in\mathbb{J}^c}{\rm Pr}(|\hat{\theta}_{j}-\theta_{0j}|\ge \gamma)$. Combining the above equations leads to ${\rm Pr}(\hat{\mathbb{J}}\neq \mathbb{J})\leq \sum_{j=1}^pP(|\hat{\theta}_{j}-\theta_{0j}|\ge \gamma)$. Thus, the error probability of selection consistency is affected by the inconsistency of parameter estimation. Let $\mathbb{E}(A)$ be the eigenvalue of a positive definite matrix $A$. The following assumptions are required to make statistical inference on $\hat{\theta}$.
\[ass1\] (Identification, Sieves) (i) The support $\Theta$ of $\theta$ is a compact set in $\mathcal{R}^p$, and $\theta_{0}=(\theta_{10}^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}\in \Theta$ is the unique solution to $E\{g(X_i;\theta)\}=0$ for $i=1,\ldots,n$; (ii) $\{\Theta_{s}: s\ge 1\}$ is a sequence of nonempty closed subsets satisfying $\Theta_{s}\subseteq\Theta_{s+1}\subseteq\Theta$, and there is $\Pi_n\theta_0\in\Theta_{s(n)}$ such that $||\Pi_n\theta_0-\theta_0||=O(\sqrt{r/n})$; (iii) $E\{(||g(X_i;\Pi_n\theta_0)||D(g)^{-1/2})^{\delta}\}<\infty$ for some $\delta>2$ and $D(g)^2n^{2/\delta-1}=o(1)$, where $D(g)$ is the number of moment conditions $g(\cdot)$.
\[ass3\] (Sample Moment Criterion) Let $a_0$ and $b_0$ be constants. (i) $a_0\leq \mathbb{E}\{\frac{1}{n}\sum_{i=1}^ng(X_i;\Pi_n\theta_0)g^{{\!\top\!}}(X_i;\Pi_n\theta_0)\}\leq b_0<\infty$ w.p.a.1; (ii) $a_0\leq \sup_{j}E\{g_j(X_i;\Pi_n\theta_0)\}^2\leq b_0<\infty$, $a_0\leq \sup_{j,l,}E\{g_j(X_i;\Pi_n\theta_0)
g_l(X_i;\Pi_n\theta_0)\}^2\leq b_0<\infty$ for $j,l=1,\ldots,r$; (iii) there are $\mathcal{K}_1<\infty$ and $\mathbb{K}_{1}(X_i)$ such that $\sup_{j,l,\theta\in\Theta_{s(n)}}|\partial g_j(X_i;\theta)/\partial\theta_{l}|\leq \mathbb{K}_{1}(X_i)$ and $E\{\mathbb{K}^2_{1}(X_i)\}\leq \mathcal{K}_1$ for $j=1,\ldots,r$ and $l=1,\ldots,p$; (iv) there are $\mathcal{K}_2<\infty$ and $\mathbb{K}_{2}(X_i)$ such that $\sup_{j,l_1,l_2,\theta\in\Theta_{s(n)}}|\partial^2g_j(X_i;\theta)/\partial\theta_{l_1}\partial\theta_{l_2}|\leq \mathbb{K}_{2}(X_i)$ and $E\{\mathbb{K}^2_{2}(X_i)\}\leq \mathcal{K}_2$ for $j=1,\ldots,r$ and $l_1=q+1,\ldots,p,l_2=1,\ldots,q$.
Assumption \[ass1\](i) ensures the existence and consistency of the maximizer of Equation (\[PETL13\]). It also implies that $\theta_{10}\in \Theta_1$ is the unique solution to $E\{\psi(Z_i;\theta_1)\}=0$ with the sparsity assumption $\theta_{20}=0$ for $i=1,\ldots,n$, where $\theta_{20}$ is the true value of $\theta_2$. Although, following Chen and Pouzo (2012), we give the similar definition for the sieve method in Assumption \[ass1\](ii), the PET procedure is quite a good fit for semiparametric estimation under slowly growing dimension. Assumption \[ass1\](iii) is proposed to control the tail probability behavior of unconditional moment models by considering the diverging rate of data. Similar to Chang, Chen and Chen (2015), we can use some function $h(r)>0$ to replace the factor $r^{1/2}$. Assumption \[ass1\](iii) also implies that $E\{\sup_{\theta_1\in\Theta_1}(||\psi(Z_i;\theta_1)||k^{-1/2})^{\delta}\}<\infty$ for some $\delta>2$. Assumption \[ass3\](i) allows for bounding eigenvalues of the corresponding sample matrices in probability. Assumption \[ass3\](ii) can be applied to unconditional moment constraints $\psi(Z_i;\theta_1)$. Assumption \[ass3\](iii) implies that for any $\theta_1\in\Theta_1$, there exist $\tilde{\mathcal{K}}_1<\infty$ and $\tilde{K}_{1}(Z_i)$ such that $|\partial \psi_j(Z_i;\theta_1)/\partial\theta_{1l}|\leq \tilde{K}_{1}(Z_i)$ and $E\{\tilde{K}^2_{1}(Z_i)\}\leq \tilde{\mathcal{K}}_1$ for $j=1,\ldots,k$ and $l=1,\ldots,q$. Assumption \[ass3\](iv) indicates that only $rq(p-q)$ twice derivatives of moments are bounded, which is less than the number assumed in Leng and Tang (2012). Under Assumption \[ass3\], we can estimate the true model well due to its well-posedness.
[**Example 3**]{} (Linear instrumental variable regression model). Following Staiger and Stock (1997), a linear instrumental variable (IV) regression model has the following structural equation: $y_i=Y_i^{{\!\top\!}}\theta+\epsilon_i$ for $i=1,\ldots,n$, where $Y_i$ is a $p\times1$ vector of endogenous regressors and $\theta$ is a $p\times 1$ vector of parameters of interest, together with the following reduced equation for $Y_i$: $Y_i=\mathbb{B} D_i+w_i$, where $D_i$ is a $K\times 1$ ($K\ge p$) vector of instrument variables, and $\mathbb{B}$ is a $p\times K$ matrix of nuisance parameters. It is assumed that $v_i =(\epsilon_i, w_i^{{\!\top\!}})^{{\!\top\!}}$ satisfies moment conditions $E(v_i|D_i)=0$ for $i=1,\ldots,n$, and $v_1|D_1,\ldots,v_n|D_n$ are i.i.d.. Here we consider the following estimating equations: $g(X_i;\theta)=\tilde{\mathcal{X}}_i^{{\!\top\!}}(y_i-\tilde{\mathcal{X}}_i\theta)$, where $\tilde{\mathcal{X}}_i=D_i^{{\!\top\!}}(D^{{\!\top\!}}D)^{-1}D^{{\!\top\!}}Y$, $D=(D_1,\ldots,D_n)^{{\!\top\!}}$, $Y=(Y_1,\ldots,Y_n)^{{\!\top\!}}$, and $X_i=\{y_i,\tilde{\mathcal{X}}_i\}$. Under the above assumption together with $E(D_i)=0$, we obtain $E\{g(X_i;\theta)\}=0$. To provide more primitive and transparent regular conditions, we assume ${\rm var}(D_iD_i^{{\!\top\!}})=I_K$. Thus, the identification condition and moment criterion corresponding to Assumptions $\ref{ass1}$ and $\ref{ass3}$ reduce to $E||K^{-1/2}D_i||^{4\kappa}<\infty$, $E|w_i^{{\!\top\!}}\theta_0+\epsilon_i|^{4\kappa}<\infty$, and $\mathbb{E}_{\max}(\mathbb{B}\mathbb{B}^{{\!\top\!}})<\infty$, where $\kappa$ is some positive integer.
\[th6\] Under Assumptions \[ass1\]-\[ass3\], $r^2=o(n)$ and $rp=o(n)$, for $\theta\in \mathcal{D}_n$, we have $$\label{selec2}
\ell_p(\theta)=-\frac{1}{2}(\theta-\hat{\theta})^{{\!\top\!}}\mathfrak{J}(\theta-\hat{\theta})+R_n,$$ where $\mathfrak{J}=J_{0}+J$, $\hat{\theta}=\mathfrak{J}^{-1}(J_{0}\theta_{0}+J\hat{\theta}_{ET})$, $R_n=o_p(r/n)$, $J=\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}$, and $\hat{\theta}_{ET}$ is the maximum ET likelihood estimator of parameter vector $\theta$, and $J_0$ is a diagonal matrix with the $j$th diagonal element being $J_{0}^{jj}$ for $j=1,\ldots,p$.
The $j$th nonzero diagonal element $J_0^{jj}$ can be obtained from the following quadratic approximation of $p_\gamma(|\theta_j|)$ at $\theta_{0j}$: $p_{\gamma}(|\theta_j|)=p_{\gamma}(|\theta_{0j}|)+\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)(\Pi_n\theta_{0})_j}{|(\Pi_n\theta_{0})_j|}(\theta_j-\theta_{0j})
\approx p_{\gamma}(|\theta_{0j}|)+\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)}{2|(\Pi_n\theta_{0})_j|}(\theta_j^2-\theta_{0j}^2)$ for $\theta_j\in\{\theta_j: |\theta_j-\theta_{0j}|\leq \mathbb{C}\sqrt{r/n}\}$, which leads to $J_0^{jj}=\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)}{|(\Pi_n\theta_{0})_j|}$. By Theorem \[th6\] and the selection criterion $\hat{\mathbb{J}}=\{j: |\hat{\theta}_{j}|>\gamma\}$, if the eigenvalues of matrix $J$ are limited to some finite interval, the condition $\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)}{|(\Pi_n\theta_{0})_j|}<\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)}{\min_{j\in\mathbb{J}}|(\Pi_n\theta_{0})_j|}\ll\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)}{\gamma}=o(\mathbb{E}_{\min}(J))=o(1)$ implies that $\hat{\theta}_{j}$ can be derived from $\hat{\theta}_{ET}^j$ for $j\in\mathbb{J}$ under Assumption \[ass4\](i), where $\hat\theta_{ET}^j$ is the $j$th element of $\hat\theta_{ET}$. This fact shows that we can obtain selection consistency of $\hat{\theta}_{\mathbb{J}}$ from Theorem \[th6\] and the selection criterion when the ET likelihood dominates the penalty function. On the other hand, when the penalty function dominates the ET likelihood and $\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)}{|(\Pi_n\theta_{0})_j|}>\frac{p_{\gamma} '(|(\Pi_n\theta_{0})_j|)}{\gamma}\gg\mathbb{E}_{\max}(J)$ for $j\in\mathbb{J}^c$, which leads to $\hat{\theta}_{j}=0$ because of the sparsity assumption of $\theta_{\mathbb{J}^c}=0$, we can obtain selection consistency of $\hat{\theta}_{\mathbb{J}^c}$. In what follows, we will discuss the error probability of the event $\hat{\mathbb{J}}\neq\mathbb{J}$, and bound it by using some special moment conditions. To this end, similar to Bondell and Reich (2012), we denote $Q^*$ as a $p\times p$ matrix whose column vectors are eigenvectors of matrix $\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}$ corresponding to its $p$ eigenvalues $d_1^*\ge\ldots\ge d_t^*>0=d_{t+1}^*=\ldots=d_p^*$. Denote $Q^*=(Q_1^*,Q_2^*)$, where $Q_1^*$ denotes the first $t$ columns of $Q^*$, those corresponding to the nonzero eigenvalues and $Q_2^*$ are the remaining $p-t$ columns of $Q^*$. Since $Q^*$ is an orthonormal basis, we have $\theta_0=Q^*\eta=Q_1^*\eta_1+Q_2^*\eta_2$, where $\eta_1$ and $\eta_2$ are the corresponding partition of $\eta$, i.e., $\eta=(\eta_1^{{\!\top\!}},\eta_2^{{\!\top\!}})^{{\!\top\!}}$. To obtain selection consistency, we need the following assumption.
\[ass4\] (Penalty Criterion) The penalty function $p_\gamma(t)$ is lower semicompact, and (i) $\max_j|\theta_{0j}|<\infty$, $\gamma$ satisfies $\gamma/\min_{j\in\mathbb{J}}|(\Pi_n\theta_{0})_j|\rightarrow 0$; (ii) $\gamma\rightarrow 0$ together with $\frac{\log p}{\gamma\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)}\rightarrow 0$ as $n\rightarrow \infty$; (iii) $\gamma\rightarrow 0$ together with $\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)\frac{\sqrt{q}}{\gamma^2}\rightarrow 0$ as $n\rightarrow \infty$; (iv) $||Q_2^*\eta_2||_{\infty}=$ $O(\frac{\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)\sqrt{q}}{\gamma})$ as $n\rightarrow \infty$.
Assumption \[ass4\](i) assumes that the true values of parameters are finite, and the rate of the threshold $\gamma$ decreasing to zero is faster than the rate on the magnitude of the true nonzero coefficients, which is guaranteed to remain identifiable from zero. Assumption \[ass4\](ii) gives the rate at which the threshold may decrease to zero, while still allowing for the exclusion of the unimportant predictors with enough large $\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)\gg \log(p)/\min_{j\in\mathbb{J}}|(\Pi_n\theta_{0})_j|$ by combining this Assumption and Assumption \[ass4\](i). If the threshold diverges too quickly, the bias of estimator will not enough quickly vanish. Because Assumption \[ass4\](iii) implies $\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)\frac{\sqrt{q}}{\gamma}\rightarrow 0$, we have $\min_{j\in\mathbb{J}}|(\Pi_n\theta_{0})_j|\gg(\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)\sqrt{q})^{1/2}$ under Assumptions \[ass4\](i) and \[ass4\](iii). Assumptions \[ass4\](iii) and \[ass4\](iv) show that the true parameter lies in a linear space spanned by $\Gamma\Sigma\Gamma^{{\!\top\!}}$, which is a basic identifiability condition for identifying the true nonzero parameters. Note that true parameters actually lie in a $q$-dimensional subspace. Thus, only if one function of true parameters is estimable within the linear space, we allow that $q/n$ may even diverge. We can allow for the case that $\theta$ may actually be sparse in some linear transformed space, but it is not sparse in the original space. Note that Assumption \[ass4\](iv) is only a possible formulation for which the assumption holds, but it is not the only way to satisfy the assumption.
\[th7\] Under Theorem \[th6\] and Assumption \[ass4\], as $n\rightarrow \infty$, we have $$P(\hat{\mathbb{J}}\neq \mathbb{J})\leq2\frac{p}{\sqrt{\gamma\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)}}\exp\left\{-\frac{\gamma\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)}{8}\right\}\rightarrow 0.$$
Theorem \[th7\] shows that the above proposed parameter selection procedure is consistent, i.e., $P(\hat{\mathbb{J}}= \mathbb{J})\rightarrow 1$. It also implies that $\hat{\theta}_2$=0 with probability tending to 1. In the following corollary, we obtain the rate of $\max_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)$ (hence the rate of $\gamma$) when Assumptions \[ass4\](i)-(iv) hold and $p$ diverges at its fastest possible rate.
\[cor2\] Suppose that $\frac{\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)}{\gamma}=O(\frac{\log(p)}{n})$, $\max_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)=O(\frac{n}{\sqrt{q}})$ and Assumptions \[ass4\](i)-(iv) hold. The PET thresholding parameter selection procedure is also consistent when $p$ satisfies $\log(p)=O((n/\sqrt{q})^c)$ for some $0<c<1$.
When $q$ does not grow with $n$, i.e., the true number of nonzero parameters is fixed, we can allow for an exponential growing case, for example, $\log(p)=O(n^c)$ for $0<c<1$. If $\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)/\gamma=O(\log(p)/n)$ and $\max_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)=O(n/\sqrt{q})$, Assumptions \[ass4\](ii) and \[ass4\](iii) are equivalent to $\log(p)\sqrt{q}/(n\gamma)$ $\rightarrow 0$, which corresponds to the rate of threshold $\gamma$.
Asymptotic properties of the SCAD-PET estimator
-----------------------------------------------
In this subsection, we discuss consistent and oracle properties of the SCAD-PET estimator $\hat{\theta}_1$ , and show that the constrainedly profiled PET likelihood ratio function like ET likelihood ratio function is asymptotically distributed as the chi-squared distribution.
Denote $\bar\psi(\theta_1)=n^{-1}\sum_{i=1}^n\psi(Z_i;\theta_1)$, $\Gamma_1(\theta_1)=E\{\partial_{\theta_1}\psi(Z_{i};\theta_1)\}$, $\Sigma_1(\theta_1)=E\{(\psi(Z_i;\theta_1)$ $-E\psi(Z_i;\theta_1))(\psi(Z_i;\theta_1)-E\psi(Z_i;\theta_1))^{{\!\top\!}}\}$, $\Sigma(\theta)=E\{(g(X_i;\theta)-Eg(X_i;\theta))(g(X_i;\theta)-Eg(X_i;\theta))^{{\!\top\!}}\}$, $K_1(\theta_1)=\{\Gamma_1^{{\!\top\!}}(\theta_1)\Sigma^{-1}_1(\theta_1)\Gamma_1(\theta_1)\}^{-1}$. Let $\mathcal{D}_{1n}=\{\theta_1:||\theta_1-\theta_{10}||\leq \mathbb{C}\sqrt{k/n}\}$ be neighborhoods of $\theta_{10}$ for some constant $\mathbb{C}>0$.
\[ass5\] (i) There are two positive constants $a_0$ and $b_0$ such that (i) the eigenvalue of $\Gamma_1^{{\!\top\!}}(\theta_1)\Gamma_1(\theta_1)$ satisfies $a_0\leq \mathbb{E}(\Gamma_1^{{\!\top\!}}(\theta_1)\Gamma_1(\theta_1))\leq b_0<\infty$ for all $\theta_1\in \mathcal{D}_{1n}$; (ii) for any $\theta_1\in \mathcal{D}_{1n}$, there are $\mathcal{K}_3<\infty$ and $\mathbb{K}_{3}(Z_i)$ such that $|\partial^3\psi_j(Z_i;\theta_1)/\partial\theta_{1l_1}\partial\theta_{1l_2}\partial\theta_{1l_3}|\leq \mathbb{K}_{3}(Z_i)$ and $E\{\mathbb{K}^2_{3}(Z_i)\}\leq \mathcal{K}_3$ for $j=1,\ldots,k$ and $l_1,l_2,l_3=1,\ldots,q$.
\[ass6\] (i) There are two positive functions $\zeta_1(k,q)$ and $\zeta_2(\varepsilon)$ such that for any $\varepsilon$, $\inf_{\{\theta_1\in\Theta_1:||\theta_1-\theta_{10}||\ge\varepsilon\}}||E\psi(Z_i;\theta_1)||\ge\zeta_1(k,q)
\zeta_2(\varepsilon)>0$, where $\lim\inf_{k,q\rightarrow\infty}\zeta_1(k,q)>0$; (ii) $\sup_{\theta_1\in\Theta_1}||\bar{\psi}(\theta_1)-E\psi(Z_i;\theta_1)||=o_p\{\zeta_1(k,q)\}$.
\[ass7\] Suppose the penalty function $p_\gamma(t)$ is lower semicompact and satisfies $\max_{j\in\mathbb{J}}p_{\gamma}(|\theta_{0j}|)\leq \mathbb{C}\gamma$, where $\mathbb{C}$ is some constant and $\gamma=O(k/\{nq\})$.
Assumption \[ass5\](i) shows that the constaints on eigenvalues of matrix $\Gamma_1(\theta_1)\Gamma_1^{{\!\top\!}}(\theta_1)$ is a relaxed condition of Chang et al. (2015). Assumption \[ass5\](ii) is used to control the order of the remainder term when taking the third-order expansion of the objective function. Assumption \[ass6\](i) is the population identification condition for the diverging parameter space. Assumption \[ass6\](ii) is an extension of the uniform convergence, whose detailed interpretation can refer to Chang et al. (2015). The lower semicompact penalty in Assumption \[ass7\] has been used by Chen and Pouzo (2012), and implies that the effective parameter space converts an ill-posed problem into a well-posed one. Assumptions \[ass4\] and \[ass7\] hold for many penalty functions such as the SCAD penalty function and the hard-threshold penalty. However, for $L_1$ penalty, $\gamma=p'_\gamma(|\theta_{0j}|)=O(\frac{k}{nq})$ in Assumption \[ass7\] is in conflict with $\gamma$ supposed in Assumption \[ass4\], which implies that the PET likelihood estimator with the $L_1$ penalty generally cannot achieve the consistency rate of $O_p(\sqrt{k/n})$ established in Theorem \[th1\], and has not the oracle property established in Theorem \[th2\] when the number of parameters diverges with sample size $n$. In fact, the above mentioned issue has been pointed out by Fan and Li (2001) and Zou (2006) even for the finite $p$.
\[th1\][(]{}Consistency of PET Estimator of $\hat{\theta}_1$[)]{} Under Assumptions \[ass1\]-\[ass7\], there is a strict local maximizer $\hat{\theta}=(\hat{\theta}_1^{{\!\top\!}},\hat{\theta}_2^{{\!\top\!}})$ of the PET likelihood $\ell_p(\theta)$ such that $\hat{\theta}_2=0$ with probability tending to 1 as $n\rightarrow \infty$ and $||\hat{\theta}_1-\theta_{10}||=O_p(\sqrt{k/n})$.
Theorem \[th1\] establishes the consistency of the proposed PET estimator $\hat{\theta}_1$, that is, there is a root-($n/k$)-consistent PET estimator of $\theta_1$. It also shows that the sparsity property of the proposed PET estimator $\hat\theta$ is still valid, that is, zero components in $\theta_0$ are estimated as zero with probability tending to one under Theorem \[th1\]. Generally, it is right to assume $q/k<1$ in Theorem \[th1\] because our main goal is to select nonzero parameters.
\[ass8\] The tuning parameter $\gamma$ and second derivatives of the penalty function $p_\gamma(t)$ satisfy $\gamma=o(n^{1/4})$ and $\max_{j\in\mathbb{J}}p_{\gamma}^{''}(|\theta_{10j}|)=o(1/\sqrt{kq})$, respectively.
Under Assumption \[ass4\](iii) and $\gamma=o(n^{1/4})$, we have $\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|\theta_{10j}|)=o(1/\sqrt{nq})$, which is a useful conclusion in the proof of Theorem \[th2\]. Clearly, Assumption \[ass8\] holds for the SCAD penalty function.
\[th2\][(]{}Oracle Property[)]{} Under Assumptions \[ass1\]-\[ass8\], we have
[(i) (]{}Sparsity[)]{} $\hat\theta_2=0$ with probability tending to one. [(ii) (]{}Asymptotic normality[)]{} $\sqrt{n}G_n\mathcal{K}^{-1/2}(\hat{\theta}_1-\theta_{10})\stackrel{{\cal L}}{\rightarrow}\mathcal {N}(0,V)$ when $k^2(k+q)^3=o(n)$, where $G_n$ is a $d\times q$ matrix such that $G_nG_n^{{\!\top\!}}\rightarrow V$, $V$ is a $d\times d$ nonnegative symmetric matrix with the fixed $d$, $\mathcal{K}=K_1(\theta_{10})$, and $\stackrel{\cal L}{\rightarrow}$ denotes convergence in distribution.
Theorem \[th2\] shows that the sparsity and asymptotic normality of the proposed PET estimator still hold when the number of parameters diverges. Under different assumptions, we can obtain different diverging rates in Theorems \[th2\](i) and \[th2\](ii) by controlling the remainder term of the Taylor’s expansion. Note that when $k$ grows with $q$ and $q/k\rightarrow \kappa\in(0,1]$, Theorem \[th2\](ii) is similar to that given in Leng and Tang (2012) for the penalized empirical likelihood estimator with $q=o(n^{1/5})$.
Let $\mathcal{P}=\Theta\times\widehat{\Lambda}_n$, $S(\Delta)=\bar{\ell}_p(\theta_1)$ in which $\Delta=\{\lambda,\theta_1\}$, $S_1(\Delta)=\partial S(\Delta)/\partial\lambda$, $S_2(\Delta)=\partial S(\Delta)/\partial\theta_1$, $S_{l,j}(\Delta)$ be the $j$th component of $S_l(\Delta)$ for $l=1,2$ and $j=1,\ldots,k+q$, and $\mathbb{E}_{\max}(A)$ be the maximum eigenvalue of matrix $A$. If Assumptions \[ass1\]-\[ass4\], \[ass6\]-\[ass8\], $\sup_{\Delta\in\mathcal{P}}\mathbb{E}_{\max}$ $\{\partial_\Delta^{2}S_{l,j}(\Delta)\}=O_p(1)$ in probability for $l=1,2$ and $j=1,\ldots,k+q$ hold and $q/k\rightarrow \kappa\in(0,1]$, the asymptotic normality of Theorem \[th2\](ii) is still valid when $k^2(k+q)=o(n)$ or $q=o(n^{1/3})$, which is assumed in Fan and Lv (2011).
The above results are established on the basis of the corrected specification of unconditional moment models. However, in some applications, unconditional moment models may be misspecified. Hence, it is necessary to study the asymptotic properties of the PET estimators of parameters in the presence of misspecified unconditional moment models. To this end, we need the following regularity conditions.
\[ass9\] The function $Q(\theta)=\log E\exp\{\nu^{*{\!\top\!}}(\theta)g(X_i;\theta)\}-\sum_{j=1}^{p}p_{\gamma}(|\theta_j|)
$ is maximized at a unique “pseudo-true" value $\theta^{*}=(\theta_1^{*{\!\top\!}},\theta_2^{*{\!\top\!}})^{{\!\top\!}}\in {\rm int}(\Theta)$ of $\theta$, where $\theta_2^*=0$ and ${\rm int}(\Theta)$ is the inner set of the compact set $\Theta$.
\[ass10\] Functions $g(X_i;\theta)$ and $p_\gamma(|\theta|)$ are continuous with respect to parameter vector $\theta\in\Theta$ (or components of $\theta$).
\[ass11\] There are a function $H(X_i)$ and a finite constant $H^*<\infty$ such that $\sup_{\theta\in\Theta}\sup_{\nu\in\widehat{\mathcal{V}}_n(\theta)}\exp\{\nu^{{\!\top\!}}g(X_i;$ $\theta)\}<H(X_i)$ and $E(H(X_i))^2\leq H^*$, where $\widehat{\mathcal{V}}_n(\theta)$ is a compact set such that $\nu^{*}(\theta)\in {\rm int}(\widehat{\mathcal{V}}_n(\theta))$.
\[ass12\] Functions $\Omega_{jl}(X_i;\theta)=\partial^2g(X_i;\theta)/\partial\theta_j\partial\theta_l$ are continuous with respect to $\theta$ in the neighborhood $\mathcal{Q}^*$ of $\theta^*$ for $j,l=1,\ldots,p$.
\[ass13\] As $n\rightarrow \infty$, $\lim \inf_{\gamma\rightarrow 0}\lim \inf_{\theta\rightarrow 0^+}p_{\gamma}^{'}(\theta)/\gamma>0$ and $\gamma$ satisfies $||\nu(\theta)||=o(\gamma)$.
\[ass14\] There is a function $f(X_i)$ satisfying $$E\left\{\sup_{\theta\in\cal Q^*}\sup_{\nu\in\widehat{\mathcal{V}}_n(\theta)}\exp\{k_1\nu^{{\!\top\!}}g(X_i;\theta)\}(f(X_i))^{k_2}\right\}<\infty~~{\rm for}~k_1, k_2=0,1,2$$ such that $||g(X_i;\theta)||\leq f(X_i)$, $||\Gamma(X_i;\theta)||\leq f(X_i)$, $||\Omega_{jl}(X_i;\theta)||\leq f(X_i)$ for $\forall X_i\in\mathcal{R}^{\iota}$, $\forall \theta\in\cal Q^*$ and $j,l=1,\ldots,p$, where $\Gamma(X_i;\theta)=\partial_{\theta}g(X_i;\theta)$. The first and second derivatives of the penalty function $p_\gamma(t)$ satisfy $\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|\theta_{0j}|)\leq \mathbb{C}\gamma$ and $\max_{j\in\mathbb{J}}p_{\gamma}^{''}(|\theta_{0j}|)\leq \mathbb{C}\gamma$.
Assumption \[ass9\] is employed to ensure the existence of the PET estimator maximizing the objective function $\ell_p(\theta)$ with the sparsity condition. The similar sparsity condition has been adopted in Lu, Goldberg and Fine (2012), which showed that there are zero components in the “pseudo-true" value, and the notion of an oracle estimator is defined in terms of such sparseness. Assumptions \[ass10\]-\[ass12\] for the continuity and boundness satisfying the conditions of Slutsky Theorem are adopted to ensure the consistency of the proposed PET estimator under misspecification. Assumption \[ass13\] shows that the order of the lagrange multiplier of the PET likelihood is controlled by that of the tuning parameter in penalty function, which plays an important role on the local concave optimization problem of the PET likelihood. This ensures the sparsity property of the PET estimator in the presence of misspecified unconditional moment restrictions. The boundness in Assumption \[ass14\] is designed to satisfy the conditions given in Theorem \[th3\].4 of Newey and McFadden (1994) for asymptotic normality under the just-identified case.
\[th3\] (large sample properties under misspecification). Under Assumptions \[ass9\]-\[ass14\], as $n\rightarrow \infty$, we have
[(i) (]{}Consistency[)]{} $\hat{\theta}\stackrel{{\cal P}}{\rightarrow}\theta^{*}$, where $\stackrel{\cal P}{\rightarrow}$ denotes convergence in probability;
[(ii) (]{}Sparsity[)]{} $\hat\theta_2=0$ with probability tending to one;
[(iii) (]{}Asymptotic normality[)]{} Let $\mathcal{G}=E\{\partial_\phi\Psi(X_i;\phi)\}_{\phi=\phi^{*}}$, $\Xi=E\{\Psi(X_i;\phi^{*})\Psi^{{\!\top\!}}(X_i;\phi^{*})\}$. If $\mathcal{G}$ is a nonsingular matrix, we have $n^{1/2}(\hat{\phi}-\phi^{*})\stackrel{{\cal L}}{\rightarrow}\mathcal {N}(0,\mathcal{G}^{-1}\Xi\mathcal{G}^{-{\!\top\!}})$, where $\Psi$ and $\phi$ are defined in Lemma \[lem5\] of the Appendix.
Theorem \[th3\](i) indicates that the nonexistence of convergence rate is due to misspecified unconditional moment restrictions. The consistency holds only when the objective PET likelihood function converges to its population form satisfying some continuity and boundness assumptions. The sparsity of Theorem \[th3\](ii) is derived from the existence of zero components in “pseudo-true" value of $\theta$. Combining Theorems \[th3\](i) and \[th3\](ii) yields $\hat{\theta}_1\stackrel{{\cal P}}{\rightarrow} \theta_1^*$. Theorem \[th3\](iii) establishes the asymptotic normality of nonzero components of $\theta$ or $\phi$ based on the objective function defined in Lemma \[lem5\].
Constrainedly PET likelihood ratio test
---------------------------------------
In this subsection, similar to Fan and Peng (2004) and Leng and Tang (2012), we consider testing linear hypotheses of the form $$H_0: B_n\theta_{10}=0 ~~{\rm versus}~~ H_1: B_n\theta_{10}\neq 0,$$ where $B_n$ is a user-specified $d\times q$ matrix such that $B_nB_n^{{\!\top\!}}=I_d$ for a fixed $d$, and $I_d$ is a $d\times d$ identity matrix. The above hypotheses include testing individual and multiple components of $\theta_{10}$ as special cases (Fan and Peng, 2004; Leng and Tang, 2012). For example, the null hypothesis $H_{0j}: \theta_{10j}=0$ for some $j\in\{1,\ldots,q\}$ can be written as the null hypothesis $H_{0j}: B_n\theta_{10}=0$ in which $B_n$ is a $1\times q$ vector with the $j$th element being 1 and 0 elsewhere, where $\theta_{10j}$ is the $j$th component of $\theta_{10}$. Inspired by Fan and Peng (2004) and Leng and Tang (2012), we consider the following constrainedly PET likelihood ratio statistic $$\label{PETL23}
\hat{\ell}(B_n)=2n\{\bar{\ell}_p(\hat{\theta}_1)-\max_{\theta_{1},B_n\theta_{1}=0}\bar{\ell}_p(\theta_{1})\}$$ for testing $H_0$: $B_n\theta_{10}=0$.
\[th4\] Under the null hypothesis and Assumptions \[ass1\]-\[ass8\], we have $\hat{\ell}(B_n)\stackrel{{\cal L}}{\rightarrow} \chi_d^2$ as $n\rightarrow \infty$, where $\chi_d^2$ denotes the chi-squared distribution with $d$ degrees of freedom.
Theorem \[th4\] establishes the asymptotic distribution of the above presented test statistic $\hat\ell(B_n)$ under the null hypothesis $H_0$: $B_n\theta_{10}=0$, which indicates that the well-known Wilk’s phenomenon for the likelihood, empirical likelihood (Owen, 2001) and adjusted ET likelihood (Zhu et al., 2009) functions is still valid for the PET likelihood with a growing number of parameters. Thus, we extend the result given in Zhu et al. (2009) to unconditional moment models with a growing number of parameters. The above asymptotic result can be used to simultaneously test statistically significance of several covariates by taking some specific matrix $B_n$ (Fan and Peng, 2004). To wit, we can use the above asymptotic result to identify zero and nonzero components of $\theta_1$. Also, it can be adopted to construct asymptotic confidence region of $B_n\theta_1$. The PET-likelihood-ratio-test-based $100(1-\alpha)\%$ approximate confidence region for $B_n\theta_{1}$ is given by $$\label{PETCI}
R_{\alpha}=\left\{\xi:2n\{\bar{\ell}_p(\hat{\theta}_1)-\max_{\theta_1,B_n\theta_1=\xi}\bar{\ell}_p(\theta_1)\}\leq\mathcal {\chi}_d^2(1-\alpha)\right\},$$ where $\mathcal {\chi}_d^2(1-\alpha)$ is the $(1-\alpha)$-quantile of the chi-squared distribution with $d$ degrees of freedom.
High-order asymptotic properties
--------------------------------
In this subsection, we present the high-order asymptotic property of the proposed PET estimator under some proper regularity conditions, which is an extension of the high-order results for the ET estimator in Schennach (2007).
Let $\eta=(\theta_1^{{\!\top\!}},\lambda^{{\!\top\!}})^{{\!\top\!}}$, $\eta_0=(\theta_{10}^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}$, $\hat{\eta}$ be the constrained PET estimator of $\eta$, $\Gamma_{1i}(\theta_1)=\partial \psi(Z_i;\theta_1)/\partial\theta_1^{{\!\top\!}}$ and $\psi_i(\theta_1)=\psi(Z_i;\theta_1)$. Theorem \[th2\] has established the following first-order conditions: $$\label{high0}
0=\frac{1}{n}\sum\limits_{i=1}^nm(Z_i;\hat{\eta})=\frac{1}{n}\sum\limits_{i=1}^n\rho_i(\hat{\eta})
\begin{pmatrix}
\Gamma_{1i}(\hat{\theta}_1)\hat{\lambda} \\
\psi_i(\hat{\theta}_1)
\end{pmatrix}
-\begin{pmatrix}
w(\hat{\theta}_1)\\
0
\end{pmatrix},$$ where $\rho_i(\eta)=n\pi_i=n\exp\{\lambda^{{\!\top\!}}\psi(Z_{i};\theta_1)\}/\sum_{j=1}^{n}\exp\{\lambda^{{\!\top\!}}\psi(Z_j;\theta_1)\}$, and the $j$th component of vector $w(\theta_1)$ has the form of $p_{\gamma}^{'}(|\theta_{1j}|){\rm sign}(\theta_{1j})$ for $j=1,\ldots,q$. By (\[high0\]), we derive the stochastic expansion of the constrained PET estimator based on the following additional smoothness and moment conditions.
\[ass15\] (i) Let $\mathcal{S}=k+q$, and $m(Z_i;\eta)$ be four-order continuously differentiable with respect to $\eta\in\{\eta:||\eta-\eta_0||\leq \sqrt{k/n}\}$; (ii) there are $\mathcal{M}_t<\infty$ and $M_{t}(Z_i)$ such that $\partial^tm_j(Z_i;\eta)/\partial\eta_{l_1}\cdots\partial\eta_{l_t}\leq M_{t}(Z_i)$ and $E\{M^2_{t}(Z_i)\}\leq \mathcal{M}_t$ for $j=1,\ldots,\mathcal{S}$, $l_1,\ldots,l_t=1,\ldots,\mathcal{S}$, $t=1,\ldots,4$; (iii) $\mathbb{M}=E\partial m(Z_i;\eta_0)/\partial\eta^{{\!\top\!}}$ exists and has finite eigenvalue.
Although Assumption \[ass15\] is a general condition in restricting the boundness of derivatives and eigenvalues, it has an overlap with Assumptions \[ass3\] and \[ass5\]. Then, we obtain
\[th5\] Under Theorems \[th1\] and \[th2\], Assumption \[ass15\] and $q/k\rightarrow \kappa$, where $\kappa$ is some constant, as $n\rightarrow \infty$, we have $$\label{high01}
\hat{\eta}-\eta_0=\frac{\tilde{\upsilon}}{\sqrt{n}}+\frac{Q_1(\tilde{\upsilon})+Q_2(\tilde{\upsilon},\tilde{A})}{n}
+\frac{Q_3(\tilde{\upsilon},\tilde{A})+
Q_4(\tilde{\upsilon},\tilde{A})+Q_5(\tilde{\upsilon})+Q_6(\tilde{\upsilon})}{n\sqrt{n}}+R_n,$$ where $\tilde{\upsilon}=-\frac{1}{\sqrt{n}}\sum_{i=1}^n\mathbb{M}^{-1}m(Z_i;\eta_0)$, $\tilde{A}=\frac{1}{\sqrt{n}}\sum_{i=1}^n\partial m(Z_i;\eta_0)/\partial\eta^{{\!\top\!}}-\sqrt{n}\mathbb{M}$, $Q_1(\tilde{\upsilon})=-(2\mathbb{M})^{-1}\sum_{j=1}^{\mathcal{S}}\tilde{\upsilon}_jM_j^*\tilde{\upsilon}$, $\tilde{\upsilon}_j$ is the $j$th element of $\tilde{\upsilon}$ and $M_j^*=E\{\partial^2m(Z_i;\eta_0)/\partial\eta_j\partial\eta^{{\!\top\!}}\}$, $Q_2(\tilde{\upsilon},\tilde{A})=-\mathbb{M}^{-1}\tilde{A}\tilde{\upsilon}$, $Q_3(\tilde{\upsilon},\tilde{A})=-(4\mathbb{M})^{-1}$ $\{-4\tilde{A} \mathbb{M}^{-1}\tilde{A}\tilde{\upsilon}-\sum_{j=1}^{\mathcal{S}}(\tilde{a}_j+2\tilde{b}_j)M_j^*\tilde{\upsilon}+2\sum_{j=1}^{\mathcal{S}}\tilde{\upsilon}_j\tilde{B}_j\tilde{\upsilon}\}$, $\tilde{B}_j=\frac{1}{\sqrt{n}}\sum_{i=1}^n\partial^2m(Z_i;\eta_0)/\partial\eta_j\partial\eta^{{\!\top\!}}-\sqrt{n}M_j^*$, $\tilde{a}_j$ and $\tilde{b}_j$ are the $j$th elements of $\tilde{a}=\mathbb{M}^{-1}\sum_{j=1}^{\mathcal{S}}\tilde{\upsilon}_jM_j^*\tilde{\upsilon}$ and $\tilde{b}=\mathbb{M}^{-1}\tilde{A}\tilde{\upsilon}$, respectively, $Q_4(\tilde{\upsilon},\tilde{A})=(2\mathbb{M})^{-1}\{\tilde{A}\mathbb{M}^{-1}$ $\sum_{j=1}^{\mathcal{S}} \tilde{\upsilon}_jM_j^*\tilde{\upsilon}+\sum_{j=1}^{\mathcal{S}}\tilde{\upsilon}_jM_j^*\tilde{b}\}$, $Q_5(\tilde{\upsilon})=-(6\mathbb{M})^{-1}\sum_{j,l=1}^{\mathcal{S}}\tilde{\upsilon}_j\tilde{\upsilon}_lM_{jt}^*\tilde{\upsilon}$ with $M_{jt}^*=E\{\partial^3m(Z_i;$ $\eta_0)/\partial\eta_j\partial\eta_t\partial\eta^{{\!\top\!}}\}$, $Q_6(\tilde{\upsilon})=-(2\mathbb{M})^{-1}\sum_{j=1}^{\mathcal{S}}\tilde{\upsilon}_jM_j^*$ $Q_1(\tilde{\upsilon})$ and $R_n=O_p(q^{9/2}/n^2)$.
By Theorem \[th5\], we have $||Q_1(\tilde{\upsilon})||=||Q_3(\tilde{\upsilon},\tilde{A})||=O_p(q^{5/2})$, $||Q_2(\tilde{\upsilon},\tilde{A})||=O_p(q^{3/2})$, $||Q_4(\tilde{\upsilon},\tilde{A})||=O_p(q^{7/2})$, $||Q_5(\tilde{\upsilon},\tilde{A})||=O_p(q^{4})$, and $||Q_6(\tilde{\upsilon},\tilde{A})||=O_p(q^{9/2})$. Because $||Q_1(\tilde{\upsilon})/n||$ has the largest order $O_p(q^{5/2}/n)$ among all terms except for $\tilde{\upsilon}/\sqrt{n}$, the conditions $q=o(n^{1/5})$ and $q/k\rightarrow \kappa<1$ given in remark of Theorem \[th2\](ii) become the sufficient condition of $O_p(q^{5/2}/n)=o_p(n^{-1/2})$. Furthermore, it follows from Theorem \[th5\] that asymptotic (higher-order) bias of the proposed PET estimator for nonzero parameter vector is given by $${\rm Bias}(\hat{\theta}_1)=E\{Q_1(\tilde{\upsilon})+Q_2(\tilde{\upsilon},\tilde{A})\}/n.$$ To investigate the precision of the bias, we introduce the following notations. Let $\mathbb{H}=\mathcal{K}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}$, $A=\mathcal{K}\mathcal{W}^2\mathcal{K}^{{\!\top\!}}$, $B^{{\!\top\!}}=\mathcal{K}\mathcal{W}^2\mathbb{H}$, $\mathcal{C}=\mathbb{H}^{{\!\top\!}}\mathcal{W}^2\mathbb{H}$, $\mathcal{W}=\mathcal{W}(\theta_{10})={\rm diag}(\omega_{11},\ldots,\omega_{qq})$ be an $q\times q$ diagonal matrix, where $\mathcal{K}$ is defined in Theorem \[th2\], and $\omega_{jj}=p'_\gamma(|\theta_{10j}|)$ for $j=1,\ldots,q$. Denote $\mathcal{A}_1=b_1+a_2+b_2$, $\mathcal{A}_2=c_1+2c_2+d_2$, where the $j$th component of $b_1$ is $b_{1j}={\rm tr}(B^{{\!\top\!}}E\{\partial^2\psi_i(\theta_{10})/\partial\theta_{1j}\partial\theta_1^{{\!\top\!}}\})/2$ for $j=1,\ldots,q$, $a_2=\sum_{j=1}^kE\{B^{{\!\top\!}}e_j\partial^2 \psi_{ij}/\partial\theta_1\partial\theta_1^{{\!\top\!}}\}/2$ in which $\psi_{ij}$ is the $j$th element of $\psi_i$ and $e_j$ is a $k\times 1$ vector whose $j$th element is 1 and 0 elsewhere, $b_2=E({\Gamma_{1i}^j}^{{\!\top\!}}\mathcal{C}\psi_i)$ in which $\Gamma_{1i}^j=\partial^2\psi_i(\theta_{10})/\partial\theta_{1j}\partial\theta_1^{{\!\top\!}}$, the $j$th element of $c_1$ is $c_{1j}={\rm tr}(AE\{\partial^2\psi_{ij}(\theta_0)/\partial\theta_1\partial\theta_1^{{\!\top\!}}\})/2$ for $j=1,\ldots,k$, $c_2=E(\Gamma_{1i}^jB^{{\!\top\!}}\psi_i)$, and $d_2=-E(\psi_i\psi_i^{{\!\top\!}}\mathcal{C}\psi_i)/2$.
\[ass16\] For $1\leq j\leq q$, the second and third derivatives of the penalty function satisfy $\max_{j}p''(|\theta_{10j}|)=O_p(1/n)$ and $p'''(|\theta_{10j}|)=0$, respectively.
A lot of penalty functions, for example, Lasso, SCAD (Fan and Li, 2001) and MCP (Zhang, 2010), satisfy Assumption \[ass16\].
\[cor1\] Under Theorem \[th5\] and Assumption \[ass16\], we have $${\rm Bias}(\hat{\theta}_1)=\{\mathcal{K}\mathcal{A}_1+\mathbb{H}\mathcal{A}_2\}/n+{\rm Bias}(\hat{\theta}_{1ET}),$$ where ${\rm Bias}(\hat{\theta}_{1ET})$ is derived from Theorem \[th4\].2 of Newey and Smith (2004).
When there is no penalty function in deriving ET estimator of $\theta_1$ or $\mathcal{W}=0$, we have ${\rm Bias}(\hat{\theta}_1)={\rm Bias}(\hat{\theta}_{1ET})$ because of $\mathcal{A}_1=\mathcal{A}_2=0$ for the considered case. Thus, ${\rm Bias}(\hat{\theta}_1)={\rm Bias}(\hat{\theta}_{1ET})$ for enough large nonzero parameters for the SCAD penalty function because of $p'_\gamma(|\theta_{10j}|)=\gamma\neq 0$.
Implementation
--------------
Similar to Leng and Tang (2012), a nonlinear optimization procedure can be employed to maximize $\ell_p(\theta)$ given in Equation (\[PETL13\]). It is quite difficult to implement the nonlinear optimization procedure because of the nonconcave penalty function $p_\gamma(|\theta_j|)$ involved. To address the issue, we consider the following local quadratic approximation to the penalty function (Fan and Li, 2001) at a fixed value $\theta_j^{(m)}$ of $\theta_j$: $p_\gamma(|\theta_j|)\approx p_\gamma(|\theta_j^{(m)}|)+\frac{1}{2}\{p_\gamma '(|\theta_j^{(m)}|)/|\theta_j^{(m)}|\}\{\theta_j^2-(\theta_j^{(m)})^2\}$, where $\theta_j^{(m)}$ is the estimated value of $\theta_j$ at the $m$th step and $\theta_j$ is the $j$th component of $\theta$. Thus, the nonlinear optimization algorithm given in Owen (2001) can be adopted to maximize Equation (\[PETL13\]) based on the above local quadratic approximation of $p_\gamma(|\theta_j|)$. Repeating the nonlinear optimization procedure until convergence yields the PET estimate $\hat\theta$ of $\theta$.
To obtain the PET estimate of $\theta$, it is also necessary to find a data-driven approach to select the penalty parameter $\gamma$. To select an appropriate penalty parameter $\gamma$, we consider the following adjusted aBIC criterion: ${\rm aBIC}(\gamma)=-2\ell(\hat\theta_\gamma)+C_n\frac{\log(n)}{n}{\rm df}_\gamma$, where $\ell(\theta)=\ell(\nu,\theta)$ is given in Equation (\[PETL22\]), $\hat\theta_\gamma$ is the PET estimator of $\theta$ depending on the tuning parameter $\gamma$, df$_\gamma$ is the number of nonzero components in $\theta$ representing the “degrees of freedom" of the estimated unconditional moment models, $C_n$ is a scaling factor diverging to infinity at a slow rate as $p\rightarrow \infty$. When $p$ is fixed, we set $C_n=1$, otherwise we take $C_n={\rm max}\{\log\log p,1\}$ (Tang and Leng, 2010). The rigorous proof of the consistency of the aBIC for the PET likelihood function is worth of further investigating. Also, the following GCV criterion (Fan and Li, 2001) can be used to select $\gamma$ in a linear regression model: ${\rm GCV}(\gamma)=n^{-1}||Y-X\theta_{\gamma}||^2/\{1-e(\gamma)/n\}^2$, where $\theta_{\gamma}=\arg\max_{\theta}\{{\rm GCV}(\gamma)\}$, $e(\gamma)={\rm tr}\{P_X(\theta_{\gamma})\}$ and $P_X(\theta_{\gamma})=X\{X^{{\!\top\!}}X+n\mathcal{B}(\theta_{\gamma})\}^{-1}X^{{\!\top\!}}$ and $\mathcal{B}(\theta_{\gamma})={\rm diag}\{p_{\gamma}^{'}(|\theta_{1\gamma}|)/|\theta_{1\gamma}|,\ldots,p_{\gamma}^{'}(|\theta_{p\gamma}|)/|\theta_{p\gamma}|\}$.
Simulation studies
==================
In this section, three simulation studies are conducted to investigate the finite sample performance of our proposed methodologies.
[**Experiment 1 (The Population Mean Vector)**]{}. In this experiment, we first generate independent and identically distributed random vector $Z_i\in \mathcal{R}^p$ whose components independently follow the $\chi_1^2$ distribution (called as a correctly specified model ‘CM’) or $\chi_{1.2}^2$ distribution (called as a misspecified model ‘MS’, which is used to investigate the robustness of our proposed PET procedure), and then set $X_i=\theta+\mathbb{R}^{1/2}(Z_i-{\bf 1}_p)$, where the true value $\theta_0$ of parameter vector $\theta\in \mathcal{R}^p$ is set to be $\theta_0=(1,0.6,0.3,0,\ldots,0)^{{\!\top\!}}$, and the true values of components in $\mathbb{R}=(\rho_{jl})$ are set to be $\rho_{jj}=1$ and $\rho_{jl}=0.3$ or $0.7$ for $j\not= l$, respectively, which are used to investigate the performance of our presented PET estimator under different correlated structure, and ${\bf 1}_p$ is a $p\times 1$ vector whose elements are one. Under model CM, we have $E(X_i)=\theta$ and ${\rm Var}(X_i)=\mathbb{R}$, which implies that components of $X_i$ are not independent of each other. To illustrate our proposed methods, we consider the following estimating equations: $g(X_i;\theta)=X_i-\theta$, which satisfy the unconditional moment restrictions: $E\{g(X_i;\theta_0)\}=0$ under model CM, but $E\{g(X_i;\theta_0)\}\not= 0$ under model MS. Clearly, in the experiment, we have $r=p$.
We consider the following four combinations of dimensionality $p$ and sample size $n$: $(n,p)=(50,7)$, $(100,10)$, $(200,14)$ and $(500,19)$, where $p$ is taken to be the integer of $8(3n)^{1/5.1}-14$, which is used to make comparison with Tang and Leng (2012). For each of four combinations, $2000$ repetitions are conducted to investigate the accuracy of our proposed estimators in terms of root mean square errors (RMS) and the performance of our proposed variable selection procedure. For each replication, $\hat\theta$ (representing the ‘PET’ estimator of $\theta$) is evaluated by maximizing $\ell_p(\theta)$ given in Equation (\[PETL13\]) via the optimization procedure introduced in Section 2.7 with the initial value of $\theta$ taken to be $\theta^{(0)}=n^{-1}\sum_{i=1}^nZ_i$ for model CM and $\theta^{(0)}=(1,0.6,0.3,0.01,\ldots,0.01)^{{\!\top\!}}$ for model MS. Similar to Fan and Li (2001), we set a component of $\hat\theta$ to be zero whenever it is less than some threshold value, such as 0.001, which is close to zero.
For comparison, we compute the sample mean estimator $\bar{X}=n^{-1}\sum_{i=1}^nX_i$ (denoted as ‘Mean’ method), the hard-threshold estimator $\hat{\theta}^{HT}_j=\bar{X}_jI(\bar{X}_j<\gamma_1)$ (denoted as ‘HT’ method), the soft-threshold estimator $\hat{\theta}^{ST}_j={\rm sign}(\bar{Z}_j)\{|\bar{Z}_j|-\gamma_2\}_{+}$ (denoted as ‘ST’ method), and a quadratic-loss-based estimator $\hat{\theta}_{QL}=\arg\min\limits_\theta\{(\bar{X}-\theta)^{{\!\top\!}}\mathcal{W}_n^{-1}(\bar{X}-\theta)+\gamma_3\sum_{i=1}^{p}|\theta_j|\}$ (denoted as ‘QL’ method), where $\bar{X}_j$ is the $j$th component of $\bar{X}$, $\gamma_1,\gamma_2$ and $\gamma_3$ are the tuning parameters, which can be obtained by using a five-fold cross-validation method to minimize the squared predictive error for the mean vector, $\{t\}_+=t$ for $t>0$ and $0$ otherwise, and $\mathbb{W}_n=n^{-1}\sum_{i=1}^{n}(X_i-\bar{X})(X_i-\bar{X})^{{\!\top\!}}$. The aBIC criterion introduced in Section 2.7 is adopted to select the tuning parameter $\gamma$ in the penalized function (\[PETL13\]). We evaluate RMS values of nonzero components in $\theta_0$ for the above presented five estimators, and their corresponding average numbers of zero coefficients that are correctly and incorrectly identified.
Results are presented in Table 1. Examination of Table 1 shows that (i) all the above mentioned four approaches (i.e., Mean, HT, ST and QL methods) to select zero coefficients yield a relatively small average number of false zero coefficients regardless of values of $p$, $n$ and $\rho_{jl}$; (ii) the PET variable selection method and the hard-threshold variable selection method behave satisfactory in the sense that their corresponding average numbers of correctly estimated zero components are quite close to the true number $p-3$ of zero components, whilst their corresponding average numbers of incorrectly estimated zero coefficients approach 0; (iii) the RMS value of our proposed PET estimator is smaller than those of other estimators for our considered highly correlated data; (iv) increasing sample size or correlation among components can improve efficiency in terms of the RMS values and the average numbers of false or true zero coefficients. These results demonstrate that our proposed PET method behaves better than others in terms of variable selection and parameter estimation, especially for highly correlated data, which indicates that our empirical results are consistent with those given in Theorem 1.
Also, to compare the performance of our-used adjusted BIC (aBIC) criterion with the traditional BIC (i.e., ${\rm BIC}(\gamma)=-2\ell(\hat\theta_\gamma)+\frac{\log(n)}{n}{\rm df}_\gamma$) and AIC (i.e., ${\rm AIC}(\gamma)=-2\ell(\hat\theta_\gamma)+\frac{2}{n}{\rm df}_\gamma$) criteria in growing dimensionality, we computed the average model size (i.e., the average value of the number of non-zero coefficients, ‘AMS’) and the percentage of the correctly identified true model (‘PCIM’) for the above generated 2000 datasets. Intuitively, a good model selection procedure should be a procedure whose AMS value is quite close to the true model size $q$ and PCIM value is close to 1. Results are given in Table 2. Examination of Table 2 shows that (i) the AIC method fails to identify the true model because of its over-fitting effect in large samples when the true model is of finite dimension; (ii) the PCIM value of the aBIC method approaches $100\%$ and its AMS value is close to $q=3$ when $p$ is moderate or large; (iii) the PCIM and AMS values of the BIC method increase as $p$ increases, and are close to $100\%$ and $q=3$ when $p$ is moderate or large, respectively, whilst the aBIC method slightly outperforms the BIC method. In a word, the aBIC consistently outperforms the BIC and AIC criteria.
To investigate the performance of our proposed PET-likelihood-ratio-based confidence interval of parameter of interest, we only evaluate the $95\%$ confidence interval of parameter $\theta_2$ for each of the above generated $2000$ data sets. Table 3 presents the empirical frequencies of $\theta_2\notin R_{\alpha}$ for various true values of $\theta_2$. Examination of Table 3 shows that (i) the frequency of $\theta_2\notin R_{\alpha}$ at the true value of $\theta_2=0.6$ is quite close to the pre-specified significant level $\alpha=0.05$ as $n$ (or $\rho_{jl}$) is large, for example, $n=500$ (or $\rho_{jl}=0.7$), which is consistent with the conclusion given in Theorem \[th4\]; (ii) power increases as $n$ or correlation coefficient among components increases or $\theta_2$ deviates more from the true value $0.6$ of $\theta_2$. These observations show that our presented PET-likelihood-ratio-based test procedure performs well.
To investigate the robustness of the proposed PET procedure to misspecified unconditional moment models (i.e., Model MS), we first generate 2000 data sets from Model MS with the same parameter settings as given in the above simulation study, and then calculate our proposed PET estimates and the penalized empirical likelihood estimates (Leng and Tang, 2012) of $\theta$ and the corresponding average numbers of zero coefficients that are correctly and incorrectly identified for our proposed variable selection procedure and Leng and Tang’s (2012) procedure (denoted as ‘PEL’ method) for each of 2000 data sets based on estimating equations: $g(X_i,\theta)=X_i-\theta$, which do not satisfy unconditional moment restrictions: $E\{g(X_i,\theta)\}=0$ for $i=1,\ldots,n$ under Model CM. To wit, the fitted unconditional moment restrictions are misspecified. The values of Bias, RMS and SD for $\theta$ with $(n,p)=(50,7)$ are presented in Table 4, where ‘Bias’ is the difference between the true value and the mean of the estimates based on 2000 replications, and ‘SD’ is the standard deviation of 2000 estimates. From Table 4, we observe that (i) the PET estimates of parameters are robust to misspecified unconditional moment models in terms of Bias, whilst the penalized empirical likelihood estimates of parameters are sensitive to misspecified unconditional moment models in the sense that their corresponding Biases deviate from zero; (ii) the values of ‘RMS’ and ‘SD’ are almost identical under our considered cases, which indicates that the estimated standard deviation is rather reliable regardless of the PET method or the PEL method; (iii) the PET method behaves better than the PEL method in terms of RMS values when unconditional moment models are misspecified; (iv) the accuracy of the PET estimator can be improved as the correlation among components of $X_i$ increases; (v) the PET variable selection procedure behaves better than the PEL variable selection method in the sense that the average number of correctly identifying nonzero components for the PET method is quite close to the true number (i.e., $3$) of nonzero components even when unconditional moment models are misspecified.
[**Experiment 2 (Linear regression model)**]{}. In the experiment, we consider the following linear regression model: $Y_i=Z_i^{{\!\top\!}}\theta+\epsilon_i$ for $i=1,\ldots,n$, where $Z_i=(z_{i1},\ldots,z_{ip})^{{\!\top\!}}$ is assumed to follow a multivariate normal distribution with zero mean and covariance matrix $\mathbb{R}=(\rho_{jl})$ with $\rho_{jl}=0.5^{|j-l|}$, and $\epsilon_i$ follows the standard normal distribution $\mathcal{N}(0,1)$. The true value of $\theta\in\mathcal {R}^p$ is taken to be $\theta_0=(3,1.5,0,0,2,0,\ldots,0)$ including three nonzero components and $p-3$ zero components. To illustrate our proposed approach to over-identified moment condition models, we introduce an instrumental variable $U_i=(u_{i1},\ldots,u_{ip})^{{\!\top\!}}$, which are independently generated from $u_{ij}\stackrel{i.i.d.}{\sim}z_{ij}+\mathcal {N}(0,1)$ for $j=1,\ldots,p$, and consider the following unconditional moment restrictions: $$g(X_i;\theta)=(z_{i1}(Y_i-Z_i^{{\!\top\!}}\theta),\ldots,z_{ip}(Y_i-Z_i^{{\!\top\!}}\theta),u_{i1}(Y_i-Z_i^{{\!\top\!}}\theta),\ldots,u_{ip}(Y_i-Z_i^{{\!\top\!}}\theta))^{{\!\top\!}},$$ which satisfy $E\{g(X_i;\theta)\}=0$, where $X_i=(Z_i^{{\!\top\!}},Y_i)^{{\!\top\!}}$ for $i=1,\ldots,n$. In this case, $r=2p$.
Similar to Experiment 1, $2000$ data sets $\{X_i: i=1,\ldots,n\}$ are independently generated from the above specified linear model to evaluate RMS values of nonzero parameters in $\theta$ and the corresponding average numbers of nonzero coefficients correctly and incorrectly identified. The tuning parameter $\gamma$ in the PET likelihood (\[PETL13\]) is selected via the GCV criterion introduced in Section 2.7. For comparison, we evaluate the RMS values of the least squares estimators of parameters in $\theta$ under the assumption that the true sparsity of the model is known. Results corresponding to Table 1 are given in Table 5.
Examination of Table 5 shows that (i) the PET method performs well for variable selection in the sense that its corresponding average number of zero coefficients correctly identified is quite close to $p-3$; (ii) the RMS values of the PET estimators are slightly larger than those of the generalized least squares estimators when $n$ is small, whilst their corresponding RMS values are almost identical when $n$ is large, for example, $n=500$; (iii) the RMS values of two estimators decrease as $n$ increases.
[**Experiment 3 (Nonparametric structural equation model)**]{}. In the experiment, we consider the following structural equation models: $$\label{ExEE1}
Y_i=\mathbb{Z}\omega_i+\epsilon_i,~~~\omega_i=\mathbb{U}\omega_i+\zeta_i,~~i=1,\ldots,n,$$ where $Y_i$ is a $p_y\times 1$ vector of manifest variables, $\omega_i$ is a $q_\omega\times 1$ vector of latent variables, $\mathbb{Z}$ is a $p_y\times q_\omega$ factor loading matrix, $\mathbb{U}$ is a $q_\omega\times q_\omega$ coefficient matrix used to identify the correlation structure among latent variables, and it is assumed that measurement error $\epsilon_i$ is distributed as the multivariate normal distribution with zero mean and covariance $\Phi_\epsilon$, i.e., $\epsilon_i\sim \mathcal{N}(0,\Phi_\epsilon)$ in which $\Phi_\epsilon={\rm diag}(\phi_1,\ldots,\phi_{p_y})$, measurement error $\zeta_i$ follows the multivariate normal distribution $\mathcal{N}(0,\Psi_{\zeta})$ with $\Psi_{\zeta}$=diag$\{\tau_1,\cdots,\tau_{q_\omega}\}$, and $p_y=2q_\omega$. The data set $\{Y_i: i=1,\ldots,n\}$ is generated from model (\[ExEE1\]) with the following specifications of $\mathbb{Z}$, $\mathbb{U}$, $\Phi_\epsilon$ and $\Psi_\zeta$: $$\mathbb{Z}=\left(
\begin{array}{ccccccc}
1& b_{21} & 0 & 0 &\cdots & 0& 0 \\
0& 0 & 1&b_{42}& \cdots&0& 0\\
\vdots&\vdots&\vdots&\vdots& \ddots &\vdots &\vdots \\
0&0 &0 & 0& \cdots &1&b_{p_yq_\omega}\\
\end{array}
\right),~~
\mathbb{U}=\left(\begin{array}{cccccc}
0& \varphi_{12} &\cdots &\varphi_{1,q_\omega-1}& \varphi_{1,q_\omega} \\
\varphi_{21}& 0 &\cdots & \varphi_{2,q_\omega-1}& \varphi_{2,q_\omega}\\
\vdots&\vdots & \ddots &\vdots &\vdots \\
\varphi_{q_\omega,1}&\varphi_{q_\omega,2} & \cdots &\varphi_{q_\omega,q_\omega-1} &0\\
\end{array}
\right)$$ in which $1$ and $0$ in $\mathbb{Z}$ and $\mathbb{U}$ are known parameters for model identification. The true values of $b_{2l,l}$, $\phi_j$ and $\tau_l$ are set to be $0.8$, $0.8$ and $0.8$ for $l=1,\ldots,q_\omega$ and $j=1,\ldots,p_y$, respectively; and the true value of $\varphi_{j_1,j_2}$ is taken to $0.8$ for $|j_1-j_2|=1$ and 0 otherwise. Thus, there are $q_y^2+3q_\omega$ unknown parameters in $\theta$={$\mathbb{Z}$, $\mathbb{U}$, $\Phi_{\epsilon}$, $\Psi_{\zeta}$}. To illustrate our proposed method, we consider the following unconditional moment restrictions: $g(X_i;\theta)={\rm vech}\{Y_iY_i^{{\!\top\!}}-\mathbb{O}(\theta)\}$, which satisfy $E\{g(Y_i;\theta_0)\}=0$, where $\theta_0$ is the true value of $\theta$, $X_i=Y_i$, $\mathbb{O}(\theta)=\mathbb{Z}(I-\mathbb{U})^{-1}\Psi_{\zeta}(I-\mathbb{U})^{-1}\mathbb{Z}^{{\!\top\!}}+\Phi_{\epsilon}$ and ${\rm vech}(A)$ represents the half-vectorization of matrix $A$. Thus, the number of unconditional moment restrictions is $r=p_y(p_y+1)/2=q_\omega(2q_\omega+1)$. Clearly, the above considered unconditional moment model is an over-identification case when $q_\omega>2$. To solve the nonlinear optimization problem related to $\ell_p(\theta)$ given in Equation (\[PETL13\]), an essential pre-requisite is that zero vector is the interior point of the convex hull of $\{g(X_i;\theta): i=1,\ldots,n\}$. Following Zhu et al. (2009), an adjusted PET likelihood $\ell_{ap}(\theta)$ can be used to evaluate the PET estimates of parameters in $\theta$. To this end, we define $g_{n+1}(\theta)=g(X_{n+1};\theta)=-\frac{a}{n}\sum_{i=1}^ng(X_{i};\theta)$, where $a=\max\{1, \log(n)/2\}$, and $X_{n+1}$ is introduced purely for notational simplicity. Thus, the corresponding adjusted PET likelihood ratio function is given by $$\label{EXPLAP}
\ell_{ap}(\theta)=\log\left[\frac{1}{n+1}\sum\limits_{i=1}^{n+1}\exp\{\nu^{{\!\top\!}}(\theta)\mathrm{vech}(Y_iY_i^{{\!\top\!}}-\mathbb{O}(\theta))\}\right]-\sum\limits_{j=1}^{q_\omega}\sum\limits_{l=1,l\neq j}^{q_\omega}p_{\gamma}(|\varphi_{jl}|),$$ which indicates that there are $2q_\omega^2+q_\omega=q_\omega(2q_\omega+1)$ moment restrictions but only $\varrho=q_\omega^2-q_\omega=q_\omega(q_\omega-1)$ parameters penalized.
Based on the above presented settings, we consider the following three combinations of the number of the penalized parameters $\varrho$ (i.e., $q_\omega$) and sample size $n$: ($n,q_\omega$) = (185, 3), (392, 4) and (919, 5), where $n$ is taken to be the integer of $\frac{1}{3}((\varrho+32)/11)^{5.1}$ for $q_\omega$=3, 4 and 5. For comparison, we calculate results corresponding to the penalized empirical likelihood method. To evaluate estimates of parameters in $\theta$, we choose the sieve space via the following procedure: (i) in the inner loop, given the current estimates $\hat{\mathbb{U}}$, $\hat{\Phi}_\epsilon$ and $\hat{\Psi}_\zeta$ of $\mathbb{U}$, $\Phi_\epsilon$ and $\Psi_\zeta$, we evaluate estimates $\hat{\mathbb{Z}}$ and $\hat\lambda(\hat{\mathbb{Z}})$ of $\mathbb{Z}$ and $\lambda$ based on the selected sieve space: $\Theta_{s(n)}^{\mathbb{U}}\times\Theta_{s(n)}^{\Phi_{\epsilon}}\times\Theta_{s(n)}^{\Psi_{\zeta}}$ by applying the Newton-Raphison optimization algorithm to the adjusted PET likelihood $\hat{\ell}_{ap}(\mathbb{Z},\lambda(\mathbb{Z})|\hat{\mathbb{U}}, \hat{\Phi}_{\epsilon}, \hat{\Psi}_{\zeta})$, which is defined in Equation (\[EXPLAP\]) with $\mathbb{U}$, $\Phi_\epsilon$ and $\Psi_\zeta$ replaced by $\hat{\mathbb{U}}$, $\hat{\Phi}_\epsilon$ and $\hat{\Psi}_\zeta$, where $\{\Theta_{s(n)}^{\mathbb{F}}\}_{s(n)=1}^{\infty}$ is a sieve space and a sequence of subsets of $\Theta$ for $\mathbb{F}=\mathbb{U}$, $\Phi_\epsilon$ and $\Psi_\zeta$; we can similarly evaluate estimates $\hat{\mathbb{U}}$ and $\hat\lambda(\hat{\mathbb{U}})$ of $\mathbb{U}$ and $\lambda$, estimates $\hat{\Phi}_\epsilon$ and $\hat\lambda(\hat{\Psi}_\epsilon)$ of $\Phi_\epsilon$ and $\lambda$, and estimates $\hat{\Psi}_\zeta$ and $\hat\lambda(\hat{\Psi}_\zeta)$ of $\Psi_\zeta$ and $\lambda$; (ii) in outer loop, we evaluate $(\hat{\mathbb{U}}, \hat{\lambda}(\hat{\mathbb{U}}))\to(\hat{\Phi}_{\epsilon}, \hat{\lambda}(\hat{\Phi}_{\epsilon}))\to (\hat{\Psi}_{\zeta}, \hat{\lambda}(\hat{\Psi}_{\zeta}))\to(\hat{\mathbb{Z}}, \hat{\lambda}(\hat{\mathbb{Z}}))\to\ldots$. Repeating the above procedure until the algorithm convergence yields the PET estimate $\hat\theta$ and penalized empirical likelihood estimate $\hat\theta_{ET}$ of $\theta$. The SD and RMS values of nonzero parameter estimators, the average numbers of correctly identified zero coefficients and incorrectly identified zero coefficients for 2000 replications are presented in Table 6. Inspection of Table 6 shows that (i) the SD and RMS values of two estimators decrease as $n$ increases; (ii) the average number of correctly identified zero coefficients approaches $q_\omega(q_\omega-1)$ for each of our considered two methods, and the average number of incorrectly identified zero coefficients decreases as $n$ increases.
An example
==========
In this section, an example taken from the Boston Housing Study is used to illustrate our proposed PET method in R package mlbench. The data set, which has even been analyzed by Harrison and Rubinfeld (1978), consists of $506$ observations on $14$ variables. The main purpose of this study is to identify the effect of clean air on house prices. Here, we take the logarithm of the median value (LMV) of owner occupied homes to be response variable ($y$), and other 13 variables to be covariates. These covariates include per capita crime rate by town (CRIM, $x_1$), proportion of residential land zoned for lots over 25,000 sq.ft (ZN, $x_2$), proportion of non-retail business acres per town (INDUS, $x_3$), Charles river dummy variable which is $1$ if it is tract bounds river and 0 otherwise (CHAS, $x_4$), nitric oxides concentration (parts per 10 million, NOX, $x_5$), average number of rooms per dwelling (RM, $x_6$), proportion of owner-occupied units built prior to 1940 (AGE, $x_7$), weighted distances to five Boston employment centers (DIS, $x_8$), index of accessibility to radial highways (RAD, $x_9$), full-value property-tax rate per 10,000 (TAX, $x_{10}$), pupil-teacher ratio by town (OTRATIO, $x_{11}$), $1000(bk-0.63)^2$ in which bk is the proportion of blacks by town (B, $x_{12}$), and proportion of population that has a lower status (LSTAT, $x_{13}$).
Following Harrison and Rubinfeld (1978), we consider the following linear model for the above introduced Boston Housing data set: $y_i=x_i^{{\!\top\!}}\theta+\varepsilon_i$, where $x_i=(1,x_{i,1},\ldots,x_{i,91})^{{\!\top\!}}$ in which $x_{i,1},\ldots,x_{i,13}$ are the above mentioned 13 covariates and $x_{i,14},\ldots,x_{i,91}$ are the interaction effects of any two covariates among 13 covariates, $\theta=(\theta_0,\theta_1,\ldots,\theta_{91})^{{\!\top\!}}$, and $\varepsilon_i$ is the random error whose distribution is assumed to be unknown but $E(\varepsilon_i)=0$. Generally, the least square method can be employed to estimate $\theta$. To illustrate our proposed method, we consider the following unconditional moment restrictions: $g(X_i;\theta)=x_i(y_i-x_i^{{\!\top\!}}\theta)$, where $X_i=(y_i,x_i^{{\!\top\!}})^{{\!\top\!}}$ for $i=1,\ldots,n$. Under the above given model assumption, we have $E\{g(X_i;\theta)\}=0$ for $i=1,\ldots,n$ with $n=506$. In this case, the number of moment restrictions is $r=92$.
The above presented PET method is used to evaluate estimate of $\theta=(\theta_0,\theta_1,\ldots,\theta_{91})^{{\!\top\!}}$ and select significant covariates with the initial value of $\theta$ taken to be its least square estimate. For comparison, we calculate the penalized empirical likelihood estimate of $\theta$. The GCV method introduced in Section 2.7 is adopted to select the tuning parameter $\gamma$ in the PET likelihood function (\[PETL13\]). Similar to Fan and Li (2001), we set a component of $\hat\theta$ to be zero whenever its estimate is less than the threshold value 0.001. Estimates of nonzero regression coefficients in $\theta$ identified by our proposed PET method are presented in Table 7. Inspection of Table 7 shows that (i) covariates $x_1$, $x_3$, $x_4$, $x_6$, $x_7$, $x_9$, $x_{11}$, $x_{12}$ and $x_{13}$ are the most significant covaraites in which variables CHAS, RM, AGE, RAD, TAX, OTRATIO and LSTAT have a positive effect on LMV of owner occupied homes, whilst variables CRIM, INDUS have a negative effect on LMV of owner occupied homes; (ii) interaction effects $x_1x_4$, $x_1x_6$, $x_1x_{10}$, $x_2x_4$, $x_3x_5$, $x_3x_{6}$, $x_7x_9$ and $x_{10}x_{11}$ have a positive effect on LMV, whilst interaction effects $x_1x_3$, $x_1x_5$, $x_1x_9$, $x_3x_{11}$, $x_4x_5$, $x_4x_6$, $x_5x_7$, $x_6x_7$, $x_6x_{10}$, $x_6x_{11}$, $x_6x_{13}$, $x_7x_{11}$, $x_7x_{12}$, $x_7x_{13}$, $x_9x_{11}$, and $x_{10}x_{13}$ have a negative effect on LMV according to our presented PET method; (iii) although variable $x_5$ is not detected to be the significant covariate, interaction effects $x_1x_5$, $x_3x_5$ and $x_4x_5$ related to $x_5$ are identified to be the significant covariates; (iv) the estimated standard errors (SEs) of the proposed PET estimators are smaller than those of the penalized empirical likelihood estimators; (v) the PET estimators have shorter confidence intervals than the penalized empirical likelihood estimators.
Discussion
==========
This paper presents a PET likelihood procedure for variable selection and parameter estimation in unconditional moment models with a diverging number of parameters in the presence of correlation among variables and model misspecification. We show that the PET likelihood possesses some properties analogous to the penalized likelihood such as the oracle properties, and the PET approach is robust to model misspecification like the exponentially tilted method. Under some regularity conditions, we show that the constrained PET likelihood ratio statistic for testing contrast hypothesis is asymptotically distributed as the central chi-squared distribution. Simulation studies are conducted to investigate the finite sample performance of the proposed methodologies, and an example from the Boston Housing Study is used to illustrate our proposed methodologies. Empirical results evidence that the penalized empirical likelihood method leads to an inappropriate conclusion when unconditional moment models are misspecified, but our proposed PET method leads to a desirable conclusion even if unconditional moment models are misspecified.
The proposed PET method in this paper is developed for the completely observed data. In many applications such as economics, data are often subject to missingness due to nonresponse or dropout of participants. In this case, it is interesting to consider the PET likelihood inference on growing dimensional unconditional moment models with missing data. It is well-known that influence analysis is an important step in data analysis. When there are influential observations in a data set, an important issue is how to identify these influential observations for our considered growing dimensional unconditional moment models with missing data. We are working on the two topics.
The developed PET theories in this paper are derived on the basis of the assumption: $p/r<1$. There are theories on generalized EL estimators that allow $p/r>1$ (e.g., see Shi, 2014). Hence, it is interesting to extend the proposed PET theories to the case: $p>r$. To wit, it is interesting to develop some theories and methods to simultaneously select parameter and moment restrictions as done in Cancer, Han and Lee (2016). Also, as a referee pointed out that it is difficult to imagine that the “pseudo-true” value $\theta^*$ of parameter vector $\theta$ is also fixed as sample size $n$ varies because the number of moments $r$ increases with sample size $n$. The issue can be incorporated to moment selection problems as done in Cancer, Han and Lee (2016).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to the Editor, an Associate Editor and three referees for their valuable suggestions and comments that greatly improved the manuscript. The research was supported by grants from the National Natural Science Foundation of China (Grant No.: 11165016).
Appendix: Proofs of Theorems {#appendix-proofs-of-theorems .unnumbered}
============================
\[lem1\] Under Assumption \[ass1\](iii), if $\nu\in\mathcal{V}_n=\{\nu\in\mathcal{R}^{r}:||\nu||\leq \pi_n\}$ and $\lambda\in\Lambda_n=\{\lambda\in\mathcal{R}^{k}:||\lambda||\leq \rho_{n}\}$, where $\pi_n=o_p(r^{-1/2}n^{-1/\delta})$ and $\rho_{n}=o_p(k^{-1/2}n^{-1/\delta})$, we have $\max_{1\leq i\leq n}\sup_{\theta\in\Theta_{s(n)}}|\nu^{{\!\top\!}}g(X_i;\theta)|=o_p(1)$ and $\max_{1\leq i\leq n}\sup_{\theta_1\in\Theta_1}|\lambda^{{\!\top\!}}\psi(Z_i;\theta_1)|=o_p(1)$. Also, [w.p.a.1]{}, $\mathcal{V}_n\subseteq\widehat{\mathcal{V}}_n(\theta)$ for all $\theta\in\Theta_{s(n)}$, and $\Lambda_n\subseteq\widehat{\Lambda}_n(\theta_1)$ for all $\theta_1\in\Theta_1$.
***Proof***. Assumption \[ass1\](iii) implies that $\max_{1\leq i\leq n}\sup_{\theta\in\Theta_{s(n)}}||g(X_i;\theta)||=O_p(n^{1/\delta}r^{1/2})$, also w.p.a.1 $\max_{1\leq i\leq n}\sup_{\theta_1\in\Theta_1}||\psi(Z_i;\theta_1)||=O_p(n^{1/\delta}k^{1/2})$. Then, we have $$\begin{array}{llll}
\max_{1\leq i\leq n}\sup_{\theta\in\Theta_{s(n)}}|\nu^{{\!\top\!}}g(X_i;\theta)|
\leq \pi_n\max_{1\leq i\leq n}\sup_{\theta\in\Theta_{s(n)}}||g(X_i;\theta)||=o_p(1),\\
\max_{1\leq i\leq n}\sup_{\theta_1\in\Theta_1}|\lambda^{{\!\top\!}}\psi(Z_i;\theta_1)|
\leq \rho_n\max_{1\leq i\leq n}\sup_{\theta_1\in\Theta_1}||\psi(Z_i;\theta_1)||=o_p(1).
\end{array}$$ So, w.p.a.1 $\nu^{{\!\top\!}}g(X_i;\theta)\in \mathcal{E}$ for all $\theta\in\Theta_{s(n)} $ and $||\nu||\leq \pi_n$, and $\lambda^{{\!\top\!}}\psi(Z_i;\theta_1)\in \mathcal{E}$ for all $\theta_1\in\Theta_1 $ and $||\lambda||\leq \rho_{n}$.
\[lem6\] Suppose that Assumptions \[ass1\] and \[ass3\] hold. Then, for unconditional moment restrictions $g(X;\theta)$ uniformly in $\theta\in$ $\mathcal{D}_n$, we have
[(i)]{} $||\frac{1}{n}\sum\limits_{i=1}^ng(X_i;\Pi_n\theta_0)-E(g(X_i;\Pi_n\theta_0))||=O_p(\sqrt{r/n})$;
[(ii)]{} $ ||\frac{1}{n}\sum\limits_{i=1}^n\frac{\partial g(X_i;\Pi_n\theta_0)}{\partial\theta}-\Gamma(\Pi_n\theta_0)||=O_p(\sqrt{rp/n})$;
[(iii)]{} $ ||\frac{1}{n}\sum\limits_{i=1}^ng(X_i;\Pi_n\theta_0)g(X_i;\Pi_n\theta_0)^{{\!\top\!}}-\Sigma(\Pi_n\theta_0)||=O_p(r/\sqrt{n})$;
[(iv)]{} $ ||\frac{1}{n}\sum\limits_{i=1}^ng(X_i;\Pi_n\theta_0)-E(g(X_i;\Pi_n\theta_0))-\frac{1}{n}\sum\limits_{i=1}^ng(X_i;\theta_0)+E(g(X_i;\theta_0))||=o_p(\sqrt{r/n})$;
[(v)]{} $||\frac{1}{n}\sum\limits_{i=1}^ng(X_i;\Pi_n\theta_0)g(X_i;\Pi_n\theta_0)^{{\!\top\!}}-\Sigma(\Pi_n\theta_0)-\frac{1}{n}\sum\limits_{i=1}^ng(X_i;\theta_0)g(X_i;\theta_0)^{{\!\top\!}}+
\Sigma(\theta_0)||=o_p(r/\sqrt{n})$.
[**Proof**]{}. Since the collection of components of unconditional moment restrictions $g(X_i;\theta)$ is P-Donsker class (Kosorok, 2008), thus (i) and (ii) hold. Again, the collection of any products of components of unconditional moment restrictions $g(X_i;\theta)$ is also P-Glivenko-Cantelli (P-G-C) class (Kosorok, 2008), thus (iii) holds. In fact, (i)-(iii) are the standard uniform consistency results, which are obtained by the law of large numbers. While (iv) and (v) are Bahadur type modulus of continuity results.
\[lem7\] Suppose that Assumptions \[ass1\](iii) and \[ass3\](i) hold and $r^2=o_p(n)$. Then, for any $\theta\in D_n$, $\nu(\theta)=\arg\min_{\nu \in\widehat{\mathcal{V}}_n(\theta)}\ell(\nu,\theta)$ exists, and $\nu(\theta)=-\{\Sigma(\theta)\}^{-1}\bar{g}(\theta)+o_p(\sqrt{r/n})$, where $\bar{g}(\theta)=n^{-1}\sum_{i=1}^ng(X_i;\theta)$.
[**Proof**]{}. Taking $\pi_{n}=o_p(r^{-1/2}n^{-1/\delta})$ yields $\sqrt{r/n}=o_p(\pi_{n})$ because of $r^{2}n^{2/\delta-1}=o_p(1)$. Let $\bar{\nu}=\arg\inf_{\nu\in\mathcal{V}_n}\ell(\nu, \theta)$, where $\mathcal{V}_n$ is defined in Lemma \[lem1\]. Then, we have $\max_{1\leq i\leq n}\sup_{\theta\in\Theta_{s(n)}}|\nu^{{\!\top\!}}g(X_i;\theta)|=o_p(1)$. Set $v_i=\nu^{{\!\top\!}}g(X_i;\theta)$, $\partial\ell(v_i)/\partial v_i=\frac{1}{n}\sum_{i=1}^n\rho_1(v_i)$, $\partial^2\ell(v_i)/\partial v_i^2=\frac{1}{n}\sum_{i=1}^n\rho_2(v_i)$. For any $\dot{\nu}$ on the line joining $\bar{\nu}$ and 0, it follows from Lemma \[lem1\] and $\rho_2(0)=1-1/n$ that w.p.a.1 $\rho_2(\dot{\nu}^{{\!\top\!}}g(X_i;\theta))\ge C$. Thus, by Assumption \[ass3\](i) and the Taylor expansion at $\nu=0$, we obtain $$\begin{array}{llll}
0=\ell(0, \theta)\ge\ell(\bar{\nu}, \theta)&=&\rho_1(0)\bar{\nu}^{{\!\top\!}}\bar{g}(\theta)+\frac{1}{2}\bar{\nu}^{{\!\top\!}}\left\{\frac{1}{n}\sum\limits_{i=1}^n\rho_2(\dot{\nu}^{{\!\top\!}}g(X_i;\theta))g(X_i;\theta)g^{{\!\top\!}}(Z_i;\theta)\right\}
\bar{\nu}\\
&\ge&-||\bar{\nu}||||\bar{g}(\theta)||+C||\bar{\nu}||^2,
\end{array}$$ which leads to $||\bar{\nu}||\leq ||\bar{g}(\theta)||$ w.p.a.1. By Lemma \[lem6\](i) and (iv), we have $||\bar{g}(\theta)||=O_p(\sqrt{r/n})$. Combining the above equations yields $||\bar{\nu}||=O_p(\sqrt{r/n})=o_p(\pi_{n})$. Therefore, w.p.a.1 $\bar{\nu}\in{\rm int}(\mathcal{V}_n)$, which indicates $\partial\ell(\bar{\nu},\theta)/\partial\nu=0$. By Lemma \[lem1\] and the convexity of $\ell(\nu,\theta)$ and $\widehat{\mathcal{V}}_n(\theta)$, it follows that $\bar{\nu}=\nu(\theta)$ and $\arg\inf_{\nu\in\widehat{\mathcal{V}}_n(\theta)}\ell(\nu, \theta)$ exists.
Taking the first-order partial derivative of $\ell(\nu,\theta)$ with respect to $\nu$ yields $$\partial\ell(\nu,\theta)/\partial\nu=\frac{1}{n}\sum\limits_{i=1}^n\frac{\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}}{1/n\sum_{j=1}^n\exp\{\nu^{{\!\top\!}}g(X_j;\theta)\}}g(X_i;\theta)=0.$$ Since $\max_{1\leq i\leq n}\sup_{\theta\in\Theta_{s(n)}}|\nu^{{\!\top\!}}g(X_i;\theta)|=o_p(1)$, the above equation is equivalent to $$\sum_{i=1}^n\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}g(X_i;\theta)=\sum\limits_{i=1}^n\{1+\nu^{{\!\top\!}}g(X_i;\theta)(1+o_p(1)\}g(X_i;\theta)=0.$$ Therefore, it follows from Lemma \[lem6\](iii) and $r^2=o(n)$ that $\nu(\theta)=-\Sigma^{-1}(\theta)\bar{g}(\theta)+o_p(\sqrt{r/n})$.
\[lem8\] If Assumptions \[ass1\](iii) and \[ass3\](i) hold and $r^2=o(n)$, $rp=o(n)$, for any sequence of sets $\{\theta:||\theta-\theta_0||=O(\sqrt{r/n})\}$, we have $\ell(\theta)=-(\theta-\theta_0)^{{\!\top\!}}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\theta-\theta_0)+2(\theta-\theta_0)^{{\!\top\!}}\Gamma\Sigma^{-1}\bar{g}(\theta_0)
-\bar{g}^{{\!\top\!}}(\theta_0)\Sigma^{-1}\bar{g}(\theta_0)+o_p(r/n)$, and $\hat{\theta}_{ET}-\theta_0=(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}})^{-1}\Gamma\Sigma^{-1}\bar{g}(\theta_0)$, where $\hat{\theta}_{ET}$ is the ET estimator of $\theta$.
***Proof***. It follows from Lemma \[lem7\] that the ET likelihood $\ell(\theta)$ can be written as $\ell(\theta)=-\bar{g}^{{\!\top\!}}(\theta)\Sigma^{-1}(\theta)\bar{g}(\theta)+o_p(r/n)$. For any $\theta\in \mathcal{D}_n$, considering the expansion of $\bar{g}(\theta)$ at $\theta_0$ and using Lemmas \[lem6\](i), \[lem6\](ii) and \[lem6\](v), we obtain $\bar{g}(\theta)=\bar{g}(\theta_0)+\Gamma^{{\!\top\!}}(\theta-\theta_0)+o_p(\sqrt{r/n})$. Then, it follows from Lemmas \[lem6\](iii) and \[lem6\](v) that Lemma \[lem8\] holds.
***Proof of Theorem \[th6\]***. From Lemma \[lem8\], we obtain $\ell(\theta)=-(\theta-\theta_0)^{{\!\top\!}}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\theta-\theta_0)+2(\theta-\theta_0)^{{\!\top\!}}\Gamma\Sigma^{-1}\bar{g}(\theta_0)
-g^{{\!\top\!}}(\theta_0)\Sigma^{-1}\bar{g}(\theta_0)+o_p(r/n)$, and $(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}})(\hat{\theta}_{ET}-\theta_0)=\Gamma\Sigma^{-1}\bar{g}(\theta_0)$. Combining the above equations yields $$\begin{array}{llll}
\ell(\theta)&=&-(\theta-\theta_0)^{{\!\top\!}}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\theta-\theta_0)+2(\theta-\theta_0)^{{\!\top\!}}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\theta-\theta_0)\\[1mm]
&&-\bar{g}^{{\!\top\!}}(\theta_0)\Sigma^{-1}\bar{g}(\theta_0)+o_p(r/n)\\[1mm]
&=&-(\theta-\theta_0)^{{\!\top\!}}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\theta-2\hat{\theta}_{ET}+\theta_0)-\bar{g}^{{\!\top\!}}(\theta_0)\Sigma^{-1}\bar{g}(\theta_0)+o_p(r/n)\\[1mm]
&=&-(\theta-\hat{\theta}_{ET})^{{\!\top\!}}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\theta-\hat{\theta}_{ET})+o_p(r/n).
\end{array}$$ It follows from the local quadratic approximation to the penalty function given in Section 2.7 that $p_\gamma(\theta)\propto(\theta-\theta_{0})^{{\!\top\!}}J_{0}(\theta-\theta_{0})+o_p(r/n)$ for any $\theta\in \mathcal{D}_n$. Then, we have $$\begin{array}{llll}
\ell_p(\theta)&=&\ell(\theta)-p_\gamma(\theta)\\
&=&-(\theta-\hat{\theta}_{ET})^{{\!\top\!}}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\theta-\hat{\theta}_{ET})-(\theta-\theta_{0})^{{\!\top\!}}J_{0}(\theta-\theta_{0})+R_n\\
&\propto& -(\theta-\hat{\theta})^{{\!\top\!}}\mathfrak{J}(\theta-\hat{\theta})+C_n+R_n,
\end{array}$$ where $\mathfrak{J}=J_{0}+\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}$, $\hat{\theta}=\mathfrak{J}^{-1}(J_{0}\theta_{0}+\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}\hat{\theta}_{ET})$, $R_n=o_p(r/n)$, and $C_n=-\theta_{0}^{{\!\top\!}}J_{0}\theta_{0}-\hat{\theta}_{ET}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}\hat{\theta}_{ET}+\hat{\theta}^{{\!\top\!}}\mathfrak{J}\hat{\theta}$ is some constant that dose not depend on $\theta$.
It follows from Equation (\[selec2\]) and Lemma \[lem8\] that $\hat{\theta}=(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}\hat{\theta}_{ET})$ and $\hat{\theta}_{ET}=(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}})^{-1}\Gamma\Sigma^{-1}\bar{g}(\theta_0)+\theta_0$. Then, we obtain $\hat{\theta}-\theta_0=-(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}J_{0}\theta_0+
(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}\Gamma\Sigma^{-1}\bar{g}(\theta_0)$, which yields $E(\hat{\theta}-\theta_0)=-(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}J_{0}\theta_0=m$ and var$(\hat{\theta}-\theta_0)=(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}=V$. Let $m_j$ be the $j$th component of $m$, $V_{jj}$ be the $j$th diagonal element of $V$, and $\mathcal{L}$ be standard normal random variable. Denote $m^*=\max_j\{|m_j|\}$ and $V^*=\mathbb{E}_{\rm max}\{V\}$. Then, we have $${\label{tail}}
\begin{aligned}
\sum_{j=1}^p{\rm Pr}(|\hat{\theta}_{j}-\theta_{0j}|\ge \gamma)&=\sum_{j=1}^p{\rm Pr}\left((|\hat{\theta}_{j}-\theta_{0j}|-|m_j|)/\sqrt{V^*}\ge (\gamma-|m_j|)/\sqrt{V^*}\right)\\
&\leq\sum_{j=1}^p{\rm Pr}\left((|\hat{\theta}_{j}-\theta_{0j}-m_j|)/\sqrt{V_{jj}}\ge (\gamma-m^*)/\sqrt{V^*}\right)\\
&\leq p{\rm Pr}\left(|\mathcal{L}|\ge (\gamma-m^*)/\sqrt{V^*}\right)\\
&\leq \frac{2 p\sqrt{V^*}}{\gamma-m^*}\exp\left\{-\frac{(\gamma-m^*)^2}{2V^*}\right\}.
\end{aligned}$$ Note that it is necessary to assume $\gamma>m^*$ because of ${\rm Pr}(|\hat{\theta}_{j}-\theta_{0j}-m_j|/\sqrt{V_{jj}}\ge (\gamma-m_j^*)/\sqrt{V^*})=1$ when $\gamma\leq m^*$, which indicates that the bound of $\sum_{j=1}^p{\rm Pr}(|\hat{\theta}_{j}-\theta_{0j}|\ge \gamma)$ is $p$ and we can not obtain the selection consistency. Although $m^*$ should not be larger than $\gamma$, we want to gain the smaller tail probability by increasing $m^*$. The same situation appears for $V^*$. Hence, it is necessary to get the bounds of the maximum bias $m^*$ and the maximum variance $V^*$.
Note that $V=\{(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}-J_{0}(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-2}\}$. Thus, we have $V^*=\mathbb{E}_{\max}\{(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}-J_{0}(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-2}\}
\leq\mathbb{E}_{\max}\{(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}\}\leq \mathcal{E}_n^{-1}$, where $\mathcal{E}_n$ is the smallest eigenvalue of matrix $\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+\phi_nI$ in which $\phi_n$ is some diagonal element of $J_{0}$. If $\mathcal{E}_n\ge \phi_n$, thus we have $V^*\leq \phi_n^{-1}$, which indicates that we should select the larger $\phi_n=\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)/\gamma$, otherwise, $V^*$ will be magnified to $1$ that would not have selection consistency. Hence, we choose $V^*\leq \gamma/\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)$.
To obtain the bound of the bias $m^*$, we note that $m^*=||(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+J_{0})^{-1}J_{0}\theta_0||_{\infty}=
||(\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}+\phi_nI)^{-1}\phi_n\theta_0||_{\infty}=||Q^*\mathbb{D}_n^*Q^{*{\!\top\!}}\theta_0||_{\infty}$, where $Q^*$ is orthogonal matrix derived from the eigenvalue decomposition of $\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}$, $\mathbb{D}_n^*$=Diag$(\frac{\phi_n}{d_1^*+\phi_n},\ldots,\frac{\phi_n}{d_p^*+\phi_n})$, and $d_1^*\geq d_2^*\geq \cdots\geq d_p^*$ are $p$ eigenvalues of matrix $\Gamma\Sigma^{-1}\Gamma^{{\!\top\!}}$. Further, we obtain $$\begin{array}{llll}
||Q^*\mathbb{D}_n^*Q^{*{\!\top\!}}\theta_0||_{\infty}&=&||Q^*\mathbb{D}_n^*Q^{*{\!\top\!}}Q_{1}^*\eta_1+Q^*\mathbb{D}_nQ^{*{\!\top\!}}Q_{2}^*\eta_2||_{\infty}\\
&\leq&||Q^*\mathbb{D}_n^*Q^{*{\!\top\!}}Q_{1}^*\eta_1||_{\infty}+||Q^*\mathbb{D}_n^*Q^{*{\!\top\!}}Q_{2}^*\eta_2||_{\infty}\\
&=&||Q^*\mathbb{D}_n^*(\eta_1^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}||_{\infty}+||Q^*\mathbb{D}_n^*(0^{{\!\top\!}},\eta_2^{{\!\top\!}})^{{\!\top\!}}||_{\infty}\\
&=&||Q^*\mathbb{D}_n^*(\eta_1^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}||_{\infty}+||Q_{2}^*\eta_2||_{\infty}
\end{array}$$ By Assumption \[ass4\](iv), we have $||Q_{2}^*\eta_2||_{\infty}=O_p(\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)\sqrt{q}/\gamma)$. For the first term, we have $$\begin{array}{llll}
||Q^*\mathbb{D}_n^*(\eta_1^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}||_{\infty}&\leq& ||Q^*\mathbb{D}_n^*(\eta_1^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}||=||\mathbb{D}_n^*(\eta_1^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}||\\
&\leq& \frac{\phi_n}{d_{t}^*+\phi_n}||\eta_1||=\frac{\phi_n}{d_{t}^*+\phi_n}||Q_{1}^*\eta_1||\leq\frac{\phi_n}{d_{t}^*+\phi_n}||\theta_0||.
\end{array}$$ Because all components of $\theta_0$ are finite and there are $q$ nonzero components in $\theta_0$, we should choose $\phi_n=\max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)/\gamma$, otherwise, $\phi_n/(d_{q}^*+\phi_n)$ will be magnified to $1$ that would not have selection consistency. Thus, we should choose $m^*\leq \max_{j\in\mathbb{J}}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)\sqrt{q}/\gamma$. It follows from Assumption \[ass4\](ii) and \[ass4\](iii) that $\gamma>2m^*$ should be selected for sufficiently large $n$, and Equation (\[tail\]) can be rewritten as $$\begin{aligned}
{\rm Pr}(\hat{\mathbb{J}}\neq \mathbb{J})&\leq2\frac{p\sqrt{V^*}}{\gamma/2}\exp\left\{-\frac{(\gamma/2)^2}{2V^*}\right\}\\
&\leq2\frac{p}{\sqrt{\gamma\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)}}
\exp\left\{-\frac{\gamma\min_{j\in\mathbb{J}^c}p_{\gamma}^{'}(|(\Pi_n\theta_{0})_j|)}{8}\right\}\rightarrow 0,
\end{aligned}$$ which implies that Theorem \[th7\] holds.
\[lem2\] Suppose that Assumptions \[ass1\] and \[ass3\](i) hold. Then, we have $\lambda(\theta_{10})=\arg\inf_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}(\lambda, \theta_{10})$ [w.p.a.1.]{}, $\inf_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}(\lambda,\theta_{10})= O_p(k/n)$, $\lambda(\theta_{10})=O_p(\sqrt{k/n})$ and $\hat{\nu}(\theta_{0})=O_p(\sqrt{r/n})$.
***Proof***. Taking $\rho_{n}=o_p(k^{-1/2}n^{-1/\delta})$ yields $\sqrt{k/n}=o_p(\rho_{n})$ because of $k^{2}n^{2/\delta-1}=o_p(1)$, which is derived from Assumption \[ass1\](iii): $r^{2}n^{2/\delta-1}=o_p(1)$. Let $\bar{\lambda}=\arg\inf_{\lambda\in\Lambda_n}\bar{\ell}(\lambda, \theta_{10})$, where $\Lambda_n$ is defined in Lemma \[lem1\]. Thus, we have $\max_{1\leq i\leq n}\sup_{\theta_1\in\Theta_1}|\lambda^{{\!\top\!}}\psi(Z_i;\theta_1)|=o_p(1)$. Denote $\tilde{v}_i=\lambda^{{\!\top\!}}\psi(Z_i;\theta_1)$, $\partial\bar{\ell}(\tilde{v}_i)/\partial \tilde{v}_i=\frac{1}{n}\sum_{i=1}^n\tilde\rho_1(\tilde{v}_i)$ and $\partial^2\bar{\ell}(\tilde{v}_i)/\partial \tilde{v}_i^2=\frac{1}{n}\sum_{i=1}^n\tilde\rho_2(\tilde{v}_i)$. For any $\lambda_d$ lying in the joining line between $\bar{\lambda}$ and the original point $0$, it follows from Lemma \[lem1\] and $\tilde\rho_2(0)=1-1/n$ that w.p.a.1 $\tilde\rho_2(\lambda_d^{{\!\top\!}}\psi(Z_i;\theta_{10}))\ge C$. Then, by Assumption \[ass3\](i), we obtain $a_0\leq \sup_{\theta_1\in \Theta_1}\mathbb{E}\{\frac{1}{n}\sum_{i=1}^n\psi(Z_i;\theta_1)\psi^{{\!\top\!}}(Z_i;\theta_1)\}\leq b_0<\infty$ w.p.a.1. The Taylor expansion at $\lambda=0$ with Lagrange remainder leads to $$\label{EQUA1}
\begin{array}{llll}
0=\bar{\ell}(0, \theta_{10})&\ge&\bar{\ell}(\bar{\lambda}, \theta_{10})=\tilde\rho_1(0)\bar{\lambda}^{{\!\top\!}}\bar{\psi}(\theta_{10})\\
&&+\frac{1}{2}\bar{\lambda}^{{\!\top\!}}
\left\{\frac{1}{n}\sum\limits_{i=1}^n\tilde\rho_2(\lambda_d^{{\!\top\!}}\psi(Z_i;\theta_{10}))\psi(Z_i;\theta_{10})\psi^{{\!\top\!}}(Z_i;\theta_{10})\right\}\bar{\lambda}\\
&\ge&-||\bar{\lambda}||||\bar{\psi}(\theta_{10})||+C||\bar{\lambda}||^2,
\end{array}$$ which yields $||\bar{\lambda}||\leq ||\bar{\psi}(\theta_{10})||$ w.p.a.1. Therefore, $||\bar{\psi}(\theta_{10})||=O_p(\sqrt{k/n})$, which yields $||\bar{\lambda}||=O_p(\sqrt{k/n})=o_p(\rho_n)$. Thus, w.p.a.1 $\bar{\lambda}\in{\rm int}(\Lambda_n)$, which yields $\partial\bar{\ell}(\bar{\lambda},\theta_{10})/\partial\lambda=0$. By Lemma \[lem1\] and the convexity of $\bar{\ell}(\lambda,\theta_{10})$ and $\widehat{\Lambda}_n(\theta_{10})$, it follows that $\bar{\lambda}=\lambda(\theta_{10})$, and $\arg\inf_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}(\lambda, \theta_{10})$ exists. By the last inequality given in Equation (\[EQUA1\]), we obtain $\inf_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}(\lambda, \theta_{10})= O_p(k/n)$.
By the same argument as done above, it follows from Lemma \[lem1\] and Assumption \[ass3\](i) that $\hat{\nu}(\theta_{0})=O_p(\sqrt{r/n})$.
\[lem3\] If Assumptions \[ass1\], \[ass3\](i) and \[ass7\] hold, we have $||\bar{\psi}(\hat{\theta}_1)||=O_p(\sqrt{k/n})$ and $||\lambda(\hat{\theta}_1)||=O_p(\sqrt{k/n})$.
***Proof***. For $\rho_{n}$ defined in Lemma \[lem1\], let $\tilde{\lambda}=-\rho_{n}\bar{\psi}(\hat{\theta}_1)/||\bar{\psi}(\hat{\theta}_1)||$, which yields $\tilde{\lambda}\in\Lambda_n$. Then, we have $\max_{1\leq i\leq n}|\tilde{\lambda}^{{\!\top\!}}\psi(Z_i;\hat{\theta}_1)|=o_p(1)$ and $\tilde{\lambda}\in\widehat{\Lambda}_n(\hat{\theta}_1)$ w.p.a.1. For any $\lambda_d$ lying in the joining line between $\tilde{\lambda}$ and 0, it follows from Lemma \[lem1\] that w.p.a.l. $\max_{1\leq i\leq n}\tilde\rho_2(\lambda_d^{{\!\top\!}}\psi(Z_i;\hat{\theta}))\leq C^*$. Taking Taylor’s expansion of $\bar{\ell}(\tilde{\lambda},\hat{\theta}_1)$ yields $$\begin{array}{llll}
\bar{\ell}(\tilde{\lambda}, \hat{\theta}_{1})&=&\tilde{\lambda}^{{\!\top\!}}\bar{\psi}(\hat{\theta}_{1})+\frac{1}{2}\tilde{\lambda}^{{\!\top\!}}
\left\{\frac{1}{n}\sum\limits_{i=1}^n\tilde\rho_2(\lambda_d^{{\!\top\!}}\psi(Z_i;\hat{\theta}_{1}))\psi(Z_i;\hat{\theta}_{1})\psi^{{\!\top\!}}(Z_i;\hat{\theta}_{1})\right\}\tilde{\lambda}\\
&\leq&-\rho_n||\bar{\psi}(\hat{\theta}_{1})||+C^*\rho_{n}^2.
\end{array}$$ On the other hand, we have $\bar{\ell}_p(\bar{\lambda}, \hat{\theta}_{1})\ge\inf_{\lambda\in\widehat{\Lambda}_n(\hat{\theta}_1)}\bar{\ell}_p(\lambda, \hat{\theta}_{1})\ge \inf_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}_p(\lambda, \theta_{10})$. By $\gamma=O_p(k/(nq))$ given in Assumption \[ass7\] and Lemma \[lem2\], we obtain $$\inf\limits_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}_p(\lambda, \theta_{10})=\inf\limits_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}(\lambda, \theta_{10})-\sum\limits_{j=1}^qp_\gamma(|\theta_{10j}|)=O_p(k/n).$$ It follows from $\bar{\ell}_p(\lambda, \theta_{1})\leq \bar{\ell}(\lambda, \theta_{1})$ for any $\theta_1\in\Theta_1$ and $\lambda\in\widehat{\Lambda}_n(\theta_{1})$ that $$\begin{array}{llll}
-\rho_n||\bar{\psi}(\hat{\theta}_{1})||+C^*\rho_n^2\ge \bar{\ell}(\bar{\lambda}, \hat{\theta}_{1})\ge \inf\limits_{\lambda\in\widehat{\Lambda}_n(\theta_{10})}\bar{\ell}_p(\lambda, \theta_{10})= O_p(k/n),
\end{array}$$ which indicates $||\bar{\psi}(\hat{\theta}_{1})||= O_p(\rho_n)$. Now we consider any $\varepsilon_n\rightarrow 0$. Let $\ddot{\lambda}=-\varepsilon_n\bar{\psi}(\hat{\theta}_{1})$. It follows from $\ddot{\lambda}=o_p(\rho_n)$ that $\ddot{\lambda}\in\Lambda_n$ w.p.a.1. Using the same argument given above yields $O_p(k/n)\leq\ddot{\lambda}^{{\!\top\!}}\bar{\psi}(\hat{\theta}_{1})+C||\ddot{\lambda}||^2
=-\varepsilon_{n}||\bar{\psi}(\hat{\theta}_{1})||^2+C\varepsilon_{n}^2||\bar{\psi}(\hat{\theta}_{1})||^2$. For enough large $n$, $1-\varepsilon_nC$ is bounded away from zero. Thus, it follows that $\varepsilon_{n}||\bar{\psi}(\hat{\theta}_{1})||^2=O_p(k/n)$, which leads to $||\bar{\psi}(\hat{\theta}_{1})||=O_p(\sqrt{k/n})$.
Using the same augment given in Lemma \[lem2\], it follows from $||\bar{\psi}(\hat{\theta}_1)||=O_p(\sqrt{k/n})$ that $||\lambda(\hat{\theta}_1)||=O_p(\sqrt{k/n})$.
\[lem4\] If Assumptions \[ass1\], \[ass3\](i)and \[ass7\] hold, we have $||\nu(\hat{\theta})||=O_p(\sqrt{r/n})$, where $\hat{\theta}$ is defined in Theorem \[th1\].
***Proof***. By $\max_{1\leq i\leq n}\sup_{\theta\in\Theta_{s(n)}}|\nu^{{\!\top\!}}g(X_i;\theta)|=o_p(1)$ given in Lemma \[lem1\] and the similar proof of Lemma \[lem2\], it follows from Assumptions \[ass1\] and \[ass3\](i) that $\inf_{\nu\in\widehat{\mathcal{V}}_n(\theta_{0})}\ell(\nu, \theta_{0})= O_p(r/n)$. Following the same argument as given in Lemma \[lem3\] and by Assumption \[ass7\]: $\gamma=O_p(k/(nq))=o_p(r/(nq))$, we can obtain $||\bar{g}(\hat{\theta})||=O_p(\sqrt{r/n})$. Again, following the same arguments as given in Lemma \[lem2\] and Lemma \[lem3\], we have $||\nu(\hat{\theta})||=O_p(\sqrt{r/n})$.
***Proof of Theorem \[th1\]***. Following Fan and Lv (2011), we divide the proof procedure of Theorem \[th1\] into two steps. First, we prove the consistency of the proposed PET estimator in the $q$-dimensional subspace. To this end, we consider restricting $\ell_p(\theta)$ into the $q$-dimensional subspace $\{\theta\in\mathcal{R}^p:\theta_{\mathbb{J}^c}=0\}$ of $\mathcal{R}^p$. The corresponding constrained PET likelihood function is $\bar{\ell}_p(\theta_1)$ given in Equation (\[cPET\]). For the constrained subspace, it follows from $p_\gamma(0)=0$ and $L(\theta)=L(\theta_1)$ that $\ell_p(\theta)=\bar{\ell}_p(\theta_1)$, where $L(\theta)$ is defined in Equation (2.1). Hence, $(\hat\theta_1,0)^{{\!\top\!}}$ is the maximizer of $\ell_p(\theta)$ on the constrained subspace, where $\hat{\theta}_{\mathbb{J}}=\hat{\theta}_1=\arg\max_{\theta_1\in\Theta_1}\bar{\ell}_p(\theta_1)$. From Lemma \[lem3\], we have $||\bar{\psi}(\hat{\theta}_1)||=O_p(\sqrt{k/n})$. Following the argument of Chang, Chen and Chen (2013), if $||\hat{\theta}_1-\theta_{10}||$ does not converge to zero in probability, there exists a subsequence $\{n^{*},k^{*},q^{*}\}$ such that $||\hat{\theta}_{1n^{*}}-\theta_{10}||\ge\varepsilon$ a.s. for some positive constant $\varepsilon$. By Assumption \[ass6\], we have $||E\{\psi(Z_i;\hat{\theta}_{1n^{*}})\}||=o_p\{\zeta_1(k^{*},q^{*})\}+O_p(\sqrt{k^{*}/n^{*}})$, which is in conflict with $||E\{\psi(Z_i;\hat{\theta}_{1n^{*}})\}||\ge\zeta_1(k^{*},q^{*})\zeta_2(\varepsilon)$ because of $\lim\inf_{k,q\rightarrow\infty}\zeta_1(k,q)>0$. Therefore, we obtain $||\hat{\theta}_1-\theta_{10}||\rightarrow 0$ as $n\rightarrow \infty$. Assumption \[ass5\](i) implies $||\bar{\psi}(\hat{\theta}_1)-\bar{\psi}(\theta_{10})||\ge C||\hat{\theta}_1-\theta_{10}||$ w.p.a.1. Then, we have $||\hat{\theta}_1-\theta_{10}||=O_p(\sqrt{k/n})$.
On the other hand, the sparsity property of the proposed PET estimator can be concluded from Theorem \[th7\], Hence, we have proved Theorem \[th1\].
***Proof of Theorem \[th2\]***. By Theorem \[th1\], we only need to prove asymptotic normality of $\hat{\theta}_1$. Let $S(\lambda,\theta_1)=\log n^{-1}\sum_{i=1}^{n}\exp\{\lambda^{{\!\top\!}}\psi(Z_i;\theta_1)\}-\sum_{j=1}^{q}p_{\gamma}(|\theta_{1j}|)$, where $\theta_{1j}$ is the $j$th component of $\theta_1$. Then, the constrained PET Likelihood $\bar{\ell}_{p}(\theta_1)$ given in Equation (\[cPET\]) can be written as $\bar{\ell}_{p}(\theta_1)=S(\lambda,\theta_1)$. Let $S_{1}(\lambda,\theta_1)=\partial S(\lambda,\theta_1)/\partial \lambda=\sum_{i=1}^{n}\pi_i\psi(Z_{i};\theta_1)$, $S_{2}(\lambda,\theta_1)=\partial S(\lambda,\theta_1)/\partial \theta_1=\sum_{i=1}^{n}\pi_i\{\partial_{\theta_1}
\psi(Z_{i};\theta_1)\}^{{\!\top\!}}\lambda-W(\theta_1)$, where $\partial_{\theta_1}\psi(Z_i;\theta_1)=\partial\psi(Z_i;\theta_1)/\partial\theta_1^{{\!\top\!}}$, $\pi_i=\exp\{\lambda^{{\!\top\!}}\psi(Z_{i};\theta_1)\}/$ $\sum_{j=1}^{n}\exp\{\lambda^{{\!\top\!}}\psi(Z_j;\theta_1)\}$, and the $j$th component of vector $W(\theta_1)$ is $p_{\gamma}^{'}(|\theta_{1j}|){\rm sign}(\theta_{1j})$ for $j=1,\ldots,q$. Thus, it follows from the definitions of $\hat{\lambda}$ and $\hat{\theta}_1$ that $\hat{\lambda}$ and $\hat{\theta}_1$ satisfy $S_{k}(\hat{\lambda},\hat{\theta}_1)=0$ for $k=1,2$.
Let $\bar{\Sigma}_1(\theta_{10})=\frac{1}{n}\sum_{i=1}^{n}\psi(Z_{i};\theta_{10})\psi^{{\!\top\!}}(Z_{i};\theta_{10})$ and $\bar{\Gamma}_1(\theta_{10})
=\frac{1}{n}\sum_{i=1}^{n}\partial_{\theta_1}^{{\!\top\!}}\psi(Z_{i};\theta_{10})$. Then, we have $$\begin{array}{llll}
S_{11}(0,\theta_{10})=\partial S(0,\theta_{10})/\partial \lambda\lambda^{{\!\top\!}}=\bar{\Sigma}_1(\theta_{10})-n^{-2}\left\{\sum\limits_{i=1}^{n}\psi(Z_i;\theta_{10})\right\}\left\{\sum\limits_{i=1}^{n}\psi(Z_i;\theta_{10})\right\}^{{\!\top\!}},\\
S_{12}(0,\theta_{10})=\partial S(0,\theta_{10})/\partial\lambda\theta_1^{{\!\top\!}}=\bar{\Gamma}_1^{{\!\top\!}}(\theta_{10}),~~~
S_{21}(0,\theta_{10})=\partial S(0,\theta_{10})/\partial\theta_1\lambda^{{\!\top\!}}=\bar{\Gamma}_1(\theta_{10}),\\
S_{22}(0,\theta_{10})=\partial S(0,\theta_{10})/\partial\theta_1\theta_1^{{\!\top\!}}=0.
\end{array}$$ Let $\Sigma_1=E\{\bar{\Sigma}_1(\theta_{10})\}$ and $\Gamma_1=E\{\bar{\Gamma}_1(\theta_{10})\}$. Taking Taylor’s expansion of $S_{k}(\hat{\lambda},\hat{\theta}_1)=0$ ($k=1,2$) at $(0,\theta_{10})$ yields $$\label{PETLA1}
\left(
\begin{array}{*{8}c}
-S_1(0,\theta_{10})\\
0
\end{array}
\right)
=
\left(
\begin{array}{*{8}c}
\Sigma_1&\Gamma_1^{{\!\top\!}}\\
\Gamma_1&0
\end{array}
\right)
\left(
\begin{array}{*{8}c}
\hat{\lambda}-0\\
\hat{\theta}_1-\theta_{10}
\end{array}
\right)
+R_n,$$ where $R_{n}=\sum_{j=1}^{5}R_{jn}$, $R_{1n}=(R_{1n}^{{\!\top\!}(1)}$, $R_{1n}^{{\!\top\!}(2)})^{{\!\top\!}}$ in which $R_{1n}^{(1)}\in\mathcal{R}^{k}$ and $R_{1n}^{(2)}\in\mathcal{R}^{q} $ and the $j$th component of $R_{1n}^{(l)}$ is $R_{1n,j}^{(l)}=\frac{1}{2}(\hat{\Delta}-\Delta_{0})^{{\!\top\!}}\partial_\Delta^{2}S_{l,j}(\Delta^*)(\hat{\Delta}-\Delta_{0})$ for $l=1,2$ and $j=1,\ldots,k+q$, $\Delta=(\lambda^{{\!\top\!}}, \theta_1^{{\!\top\!}})^{{\!\top\!}}$, $\partial_\Delta^2S_l=\partial^2S_l/\partial\Delta\partial\Delta^{{\!\top\!}}$ and $\Delta^*=({\lambda^*}^{{\!\top\!}}, {\theta_1^*}^{{\!\top\!}})^{{\!\top\!}} $ satisfying $ ||\lambda^*||\leq||\hat{\lambda} ||$ and $||\theta_1^*-\theta_{10}||\leq||\hat{\theta}_1-\theta_{10}||$. Other terms $R_{2n},\ldots,R_{5n}$ are shown as follows.
Define $$Q_n=
\left(
\begin{array}{*{8}c}
S_{11}(0,\theta_{10})&S_{12}(0,\theta_{10})\\
S_{21}(0,\theta_{10})&S_{22}(0,\theta_{10})
\end{array}
\right),~~~
Q=
\left(
\begin{array}{*{8}c}
\Sigma_1&\Gamma_1^{{\!\top\!}}\\
\Gamma_1&0
\end{array}
\right).$$ Following the argument of Fan and Peng (2004), it is easily shown that $$P(||Q_n-Q||>\epsilon)\leq \frac{1}{\epsilon^2}\sum_{i,j=1}^{k+q}E\left\{ \frac{\partial^2S(0,\theta_{10})}{\partial\Delta_i\partial\Delta_j}-E\frac{\partial^2S(0,\theta_{10})}{\partial\Delta_i\partial\Delta_j}\right\}^2= O_p\left(\frac{(k+q)^2}{n}\right).$$ Let $\tilde{G}_n=(0_{d\times k},G_n\mathcal{K}^{-1/2})$, where $G_n$ is defined in Theorem \[th2\], and $\mathbb{S}^*=(-S_1^{{\!\top\!}}(0,\theta_{10}),0^{{\!\top\!}})^{{\!\top\!}}$. From $Q_n^{-1}-Q^{-1}=-Q_n^{-1}(Q_n-Q)Q^{-1}$, we have $$\begin{array}{llll}
||\sqrt{n}\tilde{G}_n(Q_n^{-1}-Q^{-1})\mathbb{S}^*||^2&\leq&
n\mathbb{E}_{\rm max}(G_nG_n^T)\mathbb{E}_{\rm max}(\mathcal{K}^{-1})\mathbb{E}_{\rm max}^{-2}(Q_n^{-1})||(Q_n-Q)Q^{-1}\mathbb{S}^*||^2\\
&\leq&n\mathbb{E}_{\rm max}(G_nG_n^T)\mathbb{E}_{\rm max}(\mathcal{K}^{-1})\mathbb{E}_{\rm max}^{-2}(Q_n^{-1})||(Q_n-Q)||^2||Q^{-1}\mathbb{S}^*||^2\\
&=&O_p(k(k+q)^4/n^2)=o_p(1).
\end{array}$$ From the implicit theorem and envelope theorem, we have $\hat{\lambda}=\lambda(\hat{\theta}_1)=O_p(\sqrt{k/n})$. It follows from the Cauchy-Schwarz inequality and $k^2(k+q)^3=o_p(n)$ that $$||R_{1n}^{(1)}||^2\le n^{-2}||\hat{\Delta}-\Delta_0||^4n^2\sum_{i,j,l=1}^{k+q}\partial^2_{\Delta}S_{1,l}(\Delta^*)/\partial\Delta_i\partial\Delta_j=O_p(k^2(k+q)^3/n^2)=o_p(1/n).$$ By the definitions of $R_{1n}^{(1)}$ and $R_{1n}^{(2)}$, we obtain $||R_{1n}^{(2)}||^2=||R_{1n}^{(1)}||^2$, which leads to $||R_{1n}||=o_p(\sqrt{1/n})$.
When $k\sqrt{k+q}=o_p(\sqrt{n})$, which is a relaxed condition of $k^2(k+q)^3=o_p(n)$, it follows from $\sup_{\Delta\in\mathcal{P}}\mathbb{E}_{\max}\{\partial_\Delta^{2}S_{l,j}(\Delta)\}=O_p(1)$ given in the remark of Theorem \[th2\](ii) that $||R_{1n}^{(l)}||_\infty\leq\sup_{\Delta\in\mathcal{P}}\max_{j\in \{1,\ldots,k+q\}}\mathbb{E}_{\max}\{\partial_\Delta^{2}S_{l,j}(\Delta)\}
||\hat{\Delta}-\Delta_0||^2=O_p(k\sqrt{k+q}/n)=o_p(1/\sqrt{n})$ for $l=1,2$.
By Assumption \[ass8\], we obtain $||R_{2n}||=||(0,W^{{\!\top\!}}(\theta_{10}))^{{\!\top\!}}||=o_p(\sqrt{1/n})$ and $||R_{3n}||=||(0,W^{'}(\theta_1^*)(\hat{\theta}_1-\theta_{10}))^{{\!\top\!}}||=o_p(\sqrt{1/n})$. Following Assumption 4, Lemma \[lem3\], Theorem \[th1\] and $k^2(k+q)^3=o_p(n)$, we have $||R_{4n}||=||(\{(\bar{\Sigma}_1(\theta_{10})-\Sigma_1)\hat{\lambda}\}^{{\!\top\!}}+
\{(\bar{\Gamma}_1(\theta_{10})-\Gamma_1)(\hat{\theta}_1-\theta_{10})\}^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}||=o_p(\sqrt{1/n})$ and $||R_{5n}||=||(0^{{\!\top\!}},\hat{\lambda}^{{\!\top\!}}(\bar{\Gamma}_1(\theta_{10})-\Gamma_1))^{{\!\top\!}}||=o_p(\sqrt{1/n})$. Combining the above equations yields $||R_{n}||=o_p(\sqrt{1/n})$.
It follows from Equation (\[PETLA1\]) that $$\label{PETLA2}
\left(
\begin{array}{*{8}c}
\hat{\lambda}-0\\
\hat{\theta}_1-\theta_{10}
\end{array}
\right)
=Q^{-1}
\left\{\left(
\begin{array}{*{8}c}
-S_1(0,\theta_{10})\\
0\\
\end{array}
\right)+R_n
\right\},$$ which leads to $$\label{PETLA4}
\hat\theta_1-\theta_{10}=-\mathcal{K}\Gamma_{1}\Sigma_{1}^{-1}\{S_1(0,\theta_{10})+R_{2n}\},$$ where $||R_{2n}||=o_p(\sqrt{1/n})$, $\mathcal{K}=(\Gamma_{1}\Sigma_{1}^{-1}\Gamma_{1}^{{\!\top\!}})^{-1}$ and $S_1(0,\theta_{10})=\bar{\psi}=\frac{1}{n}\sum_{i=1}^{n}\psi(Z_{i};\theta_{10})$.
Let $B=-\mathcal{K}\Gamma_{1}\Sigma_{1}^{-1}$, $X_{ni}=n^{-1/2}G_n\mathcal{K}^{-1/2}B\psi(Z_i;\theta_{10})=n^{-1/2}Y_{ni}$. Then, for any $\epsilon>0$, we have $$\begin{array}{llll}
\sum_{i=1}^{n}E||X_{ni}||^2I(||X_{ni}||>\epsilon)
&=&nE||X_{n1}||^2I(||X_{n1}||>\epsilon)\\
&\leq&n\left\{E||X_{n1}||^4\right\}^{1/2}\left\{P(||X_{n1}||>\epsilon\right\}^{1/2}.
\end{array}$$ It follows from $G_nG_n^{{\!\top\!}}\rightarrow V$ that $P(||X_{n1}||>\epsilon)\le E||Y_{n1}||^2/(n\epsilon^2)=O_p(n^{-1})$ and $$\begin{array}{llll}
E||X_{n1}||^4&=&n^{-2}E\left\{\psi^{{\!\top\!}}(Z_i;\theta_1)B\mathcal{K}^{-1/2}G_n^{{\!\top\!}}G_n\mathcal{K}^{-1/2}B\psi(Z_i;\theta_1)\right\}^2\\
&\leq&n^{-2}\mathbb{E}^2_{\rm max}(G_nG_n^{{\!\top\!}})\mathbb{E}^2_{\rm max}(\mathcal{K}^{-1})E(\psi^{{\!\top\!}}\psi)^2\\
&=& O_p(\frac{q^2}{n^2}).
\end{array}$$ Combining the above equations yields $\sum_{i=1}^{n}E||X_{ni}||^2I(||X_{ni}||>\epsilon)=O_p(\frac{1}{\sqrt{n}})=o_p(1)$.
Since $\sum_{i=1}^{n}{\rm cov}(X_{ni})=n{\rm cov}(X_{n1})=
G_nG_n^{{\!\top\!}}\rightarrow V$ as $n\rightarrow \infty$ and $BS_1(0,\theta_{10})=(\hat{\theta}_1-\theta_{10})$, it follows from the central limit theorem that $\sqrt{n}G_n\mathcal{K}^{-1/2}(\hat{\theta}_1-\theta_{10})\stackrel{{\cal L}}{\rightarrow}\mathcal {N}(0,V)$. Thus, we have proved Theorem \[th2\].
***Proof of Theorem \[th3\]***. The method given in Theorem \[th1\]0 of Schennach (2007) can be used to prove Theorem \[th3\](i) based on $\ell_p(\theta)=\log\frac{1}{n}\sum_{i=1}^{n}\exp\{\nu^{{\!\top\!}}(\theta)g(X_i;\theta)\}-\sum_{j=1}^{p}p_{\gamma}(|\theta_j|)$ and the assumption on the continuity of the penalty function $p_\gamma(\cdot)$.
Now we prove the sparsity of the proposed PET estimator. The first-order partial derivative of the PET likelihood $\ell_p(\theta)$ with respect to $\theta_j$ for $j\notin\mathbb{J}$ is given by $$\begin{array}{llll}
\partial\ell_{p}(\theta)/\partial\theta_j
&=&\frac{\frac{1}{n}\sum\limits_{i=1}^{n}\exp\{\nu^{{\!\top\!}}g(X_{i};\theta)\}\partial_{\theta_j}g^{{\!\top\!}}(X_i;\theta)}
{\frac{1}{n}\sum\limits_{i=1}^{n}\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}}\nu-p_{\gamma}^{'}(|\theta_{j}|){\rm sign}(\theta_{j})\\
&=&\frac{\{h_{\nu j}+O_p(\sqrt{1/n})\}\nu}{E\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}+O_p(\sqrt{1/n})}-p_{\gamma}^{'}(|\theta_{j}|){\rm sign}(\theta_{j})\\[1mm]
&\stackrel{\Delta}{=}&\mathcal{J}_{1}+\mathcal{J}_{2},
\end{array}$$ where $\partial_{\theta_j}g^{{\!\top\!}}(X_i;\theta)=\partial g^{{\!\top\!}}(X_i;\theta)/\partial\theta_j$, and $h_{\nu j}=E[\exp\{\nu^{{\!\top\!}}g(X_{i};\theta)\}\partial_{\theta_j}g^{{\!\top\!}}(X_i;\theta)]$. By Assumption \[ass14\], combining the above equations yields $\mathcal{J}_{1}\leq O_p(1)||\nu||$. Thus, for $j\notin\mathbb{J}$, we have $$\partial\ell_{p}(\theta)/\partial\theta_j
=O_p(1)||\nu||-p_{\gamma}^{'}(|\theta_{j}|){\rm sign}(\theta_{j})\stackrel{\Delta}{=}\gamma\{-\frac{p_{\gamma}^{'}(|\theta_{j}|)}{\gamma}{\rm sign}(\theta_{j})\}+\frac{||\nu||}{\gamma}O_p(1)\},$$ which leads to $\partial\ell_p(\theta)/\partial\theta_j=\gamma\{-p_{\gamma}^{'}(|\theta_j|){\rm sign}(\theta_j)/\gamma+o_p(1)\}$ implying that the sign of $\partial\ell_{p}(\theta)/\partial\theta_j$ is dominated by the sign of $\theta_j$. Then, for any $j\notin\mathbb{J}$ and as $n\rightarrow\infty$, we have $\partial\ell_{p}(\theta)/\partial\theta_j<0$ when $\theta_j>0$, and $\partial\ell_{p}(\theta)/\partial\theta_j>0$ when $\theta_{j}<0$ with probability tending to one, which means $\hat{\theta}_{2}=0$ with probability tending to one. Thus, Theorem \[th3\](ii) holds.
To prove Theorem \[th3\](iii), we require the following Lemma. For convenience, we define $I_p =(H^{{\!\top\!}}_1, H^{{\!\top\!}}_2)^{{\!\top\!}}$, where $H_1\in\mathcal{R}
^{q\times p}$ and $H_2\in\mathcal{R}^{(p-q)\times p}$.
\[lem5\] Let $\hat{\phi}=(\hat{\rho},\hat{\tau},\hat{\nu},\hat{\theta})$ be the PET estimator of $\phi=(\rho,\tau,\nu,\theta)$, where $\rho=n^{-1}\sum_{i=1}^n\rho_i$ with $\rho_i=\exp(\nu^{{\!\top\!}}g(X_i;\theta))$. Then, $\hat\phi$ is the solution to $n^{-1}\sum_{i=1}^n\Psi(X_i;\phi)=0$, where $\Psi^{{\!\top\!}}(X_i;\phi)=(\rho_i-\rho,\rho_iH_2\theta,\rho_ig(X_i;\theta),\rho_i\Gamma_i^{*{\!\top\!}}\nu-\rho_iW(\theta)+\rho_iH_2^{{\!\top\!}}\tau)$ with $\Gamma_i^*=\partial_{\theta}g(X_i;\theta)$.
***Proof***. Define $S(\nu,\theta,\tau)=\log n^{-1}\sum_{i=1}^{n}\exp\{\nu^{{\!\top\!}}g(X_i;\theta)\}-\sum_{j=1}^{p}p_{\gamma}(|\theta_{j}|)+\tau^{{\!\top\!}}H_{2}\theta$. Following Tang and Leng (2012) and using the sparsity of $\hat{\theta}$, we obtain that $\hat{\theta}$ satisfies $\sum_{i=1}^{n}\hat{\pi}_i\hat{\Gamma}_i^{*{\!\top\!}}\hat{\nu}-W(\hat{\theta})+H_2^{{\!\top\!}}\hat{\tau}=0$, which leads to $\sum_{i=1}^n(\hat\rho_i\hat\Gamma_i^{*{\!\top\!}}\hat\nu-\hat\rho_iW(\hat\theta)+\hat\rho_iH_2^{{\!\top\!}}\hat\tau)=0$, where $\pi_i=\rho_i/\sum_{j=1}^{n}\rho_j$; and $\hat\nu$ and $\hat\tau$ satisfy $\sum_{i=1}^{n}\hat\pi_ig(X_i;\hat\theta)=0$ and $H_2\hat\theta=0$, which lead to $\sum_{i=1}^n\hat\rho_ig(X_i;\hat\theta)=0$ and $\sum_{i=1}^n\hat\rho_iH_2\hat\theta=0$, respectively. Also, it follows from the definition of $\rho$ that $\sum_{i=1}^n(\rho_i-\rho)=0$. The above equations show that $\hat\phi$ is the solution to $n^{-1}\sum_{i=1}^n\Psi(X_i;\phi)=0$.
***Proof of Theorem \[th3\](iii)***. Lemma \[lem5\] presents a just-identified GMM estimator, thus we can apply Theorem \[th3\].4 of Newey and McFadden (1994) to the just-identified case if we can show that (i) $E\{\sup_{\phi\in\mathcal{N}_\Phi}||\partial\Psi^{{\!\top\!}}(X_i;\phi)/\partial\phi||\}<\infty$ holds for some neighborhood $\mathcal{N}_\Phi$ of $\phi$, and (ii) $E\{\Psi(X_i;\phi)\Psi^{{\!\top\!}}(X_i;\phi)\}$ exists.
Consider components of matrix $\partial\Psi^{{\!\top\!}}(X_i;\phi)/\partial\phi$, which is given by $\omega\exp\{k_{1}\nu^{{\!\top\!}}g(X_i;\theta)\}$ $g^{k_{g}}\Gamma^{k_{\Gamma}}\Omega^{k_{\Omega}}$ for $0\leq k_{g}+
k_{\Gamma}+k_{\Omega}\leq 2$ and $k_{1}=0,1$, where $g, \Gamma$ and $\Omega$ denote elements of $g(X_i;\theta),\Gamma(X_i;\theta)$ and $\Omega_{jl}(X_i;\theta)$, respectively, and $\omega$ denotes the product of elements of $\phi$ that is necessarily bounded for $\phi\in\mathcal{N}_\Phi$ and also includes the first and second partial derivatives of penalty function such as $p_\gamma^{'}(\theta)$ and $p_\gamma^{''}(\theta)$. By Assumption \[ass14\], we can obtain (i). It follows from $\exp\{k_{1}\nu^{{\!\top\!}}g(X_i;\theta)\}|g|^{k_{g}}|\Gamma|^{k_{\Gamma}}|\Omega|^{k_{\Omega}}\leq\exp\{k_{1}\nu^{{\!\top\!}}g(X_i;\theta)\}
|f(X_i)|^{k_{g}+k_{\Gamma}+k_{\Omega}}$ that $$\begin{array}{lll}
E[\sup_{\phi\in\mathcal{N}_\Phi}\exp\{k_{1}\nu^{{\!\top\!}}g(X_i;\theta)\}|g|^{k_{g}}|\Gamma|^{k_{\Gamma}}|\Omega|^{k_{\Omega}}]\leq E[\sup_{\phi\in\mathcal{N}_\Phi}\exp\{k_{1}\nu^{{\!\top\!}}g(X_i;\theta)\}|f(X_i)|^{k_2}]\\
=E[\sup_{\theta\in\cal N}\sup_{\nu\in\Lambda(\theta)}\exp\{k_{1}\nu^{{\!\top\!}}g(X_i;\theta)\}(f(X_i))^{k_2}]<\infty.
\end{array}$$ Matrix $\Psi(X_i;\phi)\Psi^{{\!\top\!}}(X_i;\phi)$ has elements of the form $\omega\exp\{k_{1}\nu^{{\!\top\!}}g(X_i;\theta)\}g^{k_{g}}\Gamma^{k_{\Gamma}}$ with $k_{1}=0,1,2$, and $0\leq
k_{g}+k_{\Gamma}\leq 2$. Similar argument implies (ii).
***Proof of Theorem \[th4\]***. It follows from the argument of Theorem \[th2\] that $\hat{\lambda}$ and $\hat{\theta}_1$ can be obtained by maximizing $\bar{\ell}_p(\theta_1)$ under $H_0\cup H_1$, which indicates that $$\label{PETL7}
\hat\theta_1-\theta_{10}=-\mathcal{K}\Gamma_{1}^{{\!\top\!}}\Sigma_{1}^{-1}\bar{\psi}+R_{2n},~~~
\hat{\lambda}=\{\Sigma_1^{-1}\Gamma_1\mathcal{K}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}-\Sigma_1^{-1}\}\bar{\psi}+R_{1n}.$$ Let $\tilde{w}_i=\hat{\lambda}^{{\!\top\!}}\psi(z_i;\hat{\theta}_1)$. It follows from Lemma \[lem1\] that $\max_{1\leq i\leq n}|\tilde{w}_i|=o_p(1)$. Taking Taylor expansion of $\bar{\ell}(\hat{\theta}_1)$ at $\tilde{w}_i$ leads to $$\bar{\ell}(\hat{\theta}_1)=\log\left\{1+\frac{1}{n}\sum\limits_{i=1}^{n}\tilde{w}_i(1+o_p(1))\right\}=\frac{1}{n}\sum\limits_{i=1}^{n}\tilde{w}_i(1+o_p(1)).$$ Substituting expressions of $\hat{\lambda}$ into $\bar{\ell}_p(\hat{\theta}_1)$ yields $2n\bar{\ell}_p(\hat{\theta}_1,\hat{\lambda})=-n\bar{\psi}^{{\!\top\!}}\{\Sigma_1^{-1}\Gamma_1\mathcal{K}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}-\Sigma_1^{-1}\}\bar{\psi}+o_p(1)$, where $o_p(1)$ includes penalty function. It follows from $B_{n}\theta_{1}=0$ and $B_nB_n^{{\!\top\!}}=I_d$ and Theorem 3 of Tang and Leng (2012) that the constrained PET estimators $\tilde{\lambda}$ and $\tilde{\theta}_1$ of $\lambda$ and $\theta_1$ under $H_0$ can be obtained by maximizing $\tilde{\ell}_p(\lambda,\theta_1,\tau) =\log n^{-1}\sum_{i=1}^{n}\exp\{\lambda^{{\!\top\!}}\psi(Z_i$; $\theta_1)\}-\sum_{j=1}^{q}p_{\gamma}(|\theta_{1j}|)+\tau^{{\!\top\!}}B_{n}\theta_1$. It is easily shown that $\tilde{\lambda}=(\Sigma_1^{-1}\Gamma_1\mathbb{P}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}-\Sigma_1^{-1})\bar{\psi}+\tilde{R}_{1n}$, where $\mathbb{P}=\mathcal{K}B_{n}^{{\!\top\!}}(B_{n}\mathcal{K}B_{n}^{{\!\top\!}})^{{\!\top\!}}B_{n}\mathcal{K}-\mathcal{K}$. Thus, the constrained maximum PET likelihood is given by $2n\bar{\ell}_p(\tilde{\theta}_1, \tilde{\lambda})=-n\bar{\psi}^{{\!\top\!}}(\Sigma_1^{-1}\Gamma_1\mathbb{P}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}-\Sigma_1^{-1})\bar{\psi}+o_p(1)$. Then, the constrained PET likelihood ratio statistic for testing $H_0:B_n\theta_{10}=0$ is given by $$\hat{\ell}(B_n)=2n\bar{\ell}_p(\hat{\theta}_1, \hat{\lambda})-2n\bar{\ell}_p(\tilde{\theta}_1, \tilde{\lambda})=n\bar{\psi}^{{\!\top\!}}\Sigma_1^{-1/2}(\mathbb{T}_1-\mathbb{T}_2)\Sigma_1^{-1/2}\bar{\psi}+o_p(1),$$ where $\mathbb{T}_1=\Sigma_1^{-1/2}\Gamma_1\mathbb{P}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1/2}$ and $\mathbb{T}_2=\Sigma_1^{-1/2}\Gamma_1\mathcal{K}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1/2}$. It is easily shown that $\mathbb{T}_1$ and $\mathbb{T}_2$ are symmetric idempotent matrices, and the rank of matrix $\mathbb{T}_1-\mathbb{T}_2$ is $d$, which indicates that there is a matrix $\mathcal{T}$ such that $\mathbb{T}_1-\mathbb{T}_2=\mathcal{T}^{{\!\top\!}}\mathcal{T}$ and $\mathcal{T}\mathcal{T}^{{\!\top\!}}=I_d$ (Fan and Peng, 2004). Also, it follows from the center limit theorem that $\sqrt{n}\mathcal{T}\Sigma^{-1/2}\bar{\psi}\stackrel{{\cal L}}{\rightarrow}\mathcal {N}(0,I_d)$, which leads to $n\bar{\psi}^{{\!\top\!}}\Sigma^{-1/2}(\mathbb{T}_1-\mathbb{T}_2)\Sigma^{-1/2}\bar{\psi} \stackrel{{\cal L}}{\rightarrow}\mathcal {\chi}_d^2$.
***Proof of Theorem \[th5\]***. For simplicity, we denote $M_j^*=E\{\partial^2m(Z_i;\eta_0)/\partial\eta_j\partial\eta^{{\!\top\!}}\}$ in which $\eta_j$ is the $j$th element of $\eta$, $M_{jt}^*=E\{\partial^3m(Z_i;\eta_0)/\partial\eta_j\partial\eta_t\partial\eta^{{\!\top\!}}\}$, $\tilde{A}=\frac{1}{\sqrt{n}}\sum_{i=1}^n\partial m(Z_i;\eta_0)/\partial\eta^{{\!\top\!}}$ $-\sqrt{n}\mathbb{M}$, $\tilde{B}_j=\frac{1}{\sqrt{n}}\sum_{i=1}^n\partial^2m(Z_i;\eta_0)/\partial\eta_j\partial\eta^{{\!\top\!}}-\sqrt{n}M_j^*$, $\tilde{\upsilon}=-\frac{1}{\sqrt{n}}\sum_{i=1}^n\mathbb{M}^{-1}m(Z_i;\eta_0)$, $\tilde{a}=\mathbb{M}^{-1}\sum_{j=1}^\mathcal{S}\tilde{\upsilon}_jM_j^*\tilde{\upsilon}$ in which $\tilde{\upsilon}_j$ is the $j$th component of $\tilde{\upsilon}$, $\tilde{b}=\mathbb{M}^{-1}\tilde{A}\tilde{\upsilon}$, $\widehat{M}(\eta)=\frac{1}{n}\sum_{i=1}^n\partial m(Z_i;\eta)/\partial\eta^{{\!\top\!}}$.
Taking the Taylor expansion of $n^{-1}\sum_{i=1}^nm(Z_i;\hat\eta)$ at $\eta_0$ yields $$\label{high1}
\begin{array}{llll}
0=&\widehat{m}(\eta_0)+\widehat{M}(\eta_0)(\hat{\eta}-\eta_0)+\frac{1}{2}\sum\limits_{j=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})\{\partial \widehat{M}(\eta_0)/\partial\eta_j\}(\hat{\eta}-\eta_0)\\
&+\frac{1}{6}\sum\limits_{j,t=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})(\hat{\eta}_t-\eta_{0t})\{\partial^2 \widehat{M}(\bar{\eta})/\partial\eta_j\eta_t\}(\hat{\eta}-\eta_0),
\end{array}$$ where $\widehat{m}(\eta_0)=n^{-1}\sum_{i=1}^nm(Z_i;\eta_0)$ and $\bar{\eta}$ lies in the jointing line between $\hat\eta$ and $\eta_0$. Let $\widehat{M}=\widehat{M}(\eta_0)$. Then, it is easily shown from Equation (\[high1\]) that $$\label{high2}
\begin{array}{llll}
\hat{\eta}-\eta_0&=&\tilde{\upsilon}/\sqrt{n}-\mathbb{M}^{-1}\Big\{\tilde{A}(\hat{\eta}-\eta_0)/\sqrt{n}+\frac{1}{2}\sum\limits_{j=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})
M_j^*(\hat{\eta}-\eta_0)\\
&&+\frac{1}{2}\sum\limits_{j=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})\frac{\tilde{B}_j}{\sqrt{n}}(\hat{\eta}-\eta_0)\\
&&+\frac{1}{6}\sum\limits_{j,t=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})(\hat{\eta}_t-\eta_{0t})M_{jt}^*(\hat{\eta}-\eta_0)\\
&&+\frac{1}{6}\sum\limits_{j,t=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})(\hat{\eta}_t-\eta_{0t})(\partial^2 \widehat{M}(\bar{\eta})/\partial\eta_j\eta_t-M_{jt}^*)(\hat{\eta}-\eta_0)\Big\}.\\
\end{array}$$ Denote $\hat{\eta}-\eta_0=\tilde{\upsilon}/\sqrt{n}-\mathbb{M}^{-1}\Big\{\mathcal{S}_1^*+\mathcal{S}_2^*+\mathcal{S}_3^*+\mathcal{S}_4^*+\mathcal{S}_5^*\Big\}$. By the definitions of $\mathcal{S}_1^*,\ldots,\mathcal{S}_6^*$, we have $||\mathcal{S}_1^*||\leq ||\tilde{A}/\sqrt{n}||\cdot ||\hat{\eta}-\eta_0||=O_p(\sqrt{q^2/n})O_p(\sqrt{q/n})=O_p(\frac{q^{3/2}}{n})$, $||2\mathcal{S}_2^*||\leq ||\hat{\eta}-\eta_0||\surd\{\sum_{l_1,l_2=1}^{\mathcal{S}}(\sum_{j=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})M_{jl_1l_2}^*)^2\}
\leq ||\hat{\eta}-\eta_0||\surd\{\sum_{l_1,l_2=1}^{\mathcal{S}}||\hat{\eta}-\eta_0||^2(\sum_{j=1}^{\mathcal{S}}M_{jl_1l_2}^{*2})\}=O_p(\frac{q^{5/2}}{n})$, $||2\mathcal{S}_3^*||\leq ||\hat{\eta}-\eta_0||\surd\{\sum_{l_1,l_2=1}^{\mathcal{S}}||\hat{\eta}-\eta_0||^2(\sum_{j=1}^{\mathcal{S}}(\widehat{M}_{jl_1l_2}^*-M_{jl_1l_2}^*)^2)\}
=O_p(\frac{q^{5/2}}{n\sqrt{n}})$, $||6\mathcal{S}_4^*||\leq ||\hat{\eta}-\eta_0||\surd\{\sum_{l_1,l_2=1}^{\mathcal{S}}(\sum_{j,t=1}^{\mathcal{S}}(\hat{\eta}_j-\eta_{0j})(\hat{\eta}_t-\eta_{0t})M_{jtl_1l_2}^*)^2\}
\leq ||\hat{\eta}-\eta_0||\surd\{\sum_{l_1,l_2=1}^{\mathcal{S}}||\hat{\eta}-\eta_0||^4(\sum_{j=1}^{\mathcal{S}}(\sum_{t=1}^{\mathcal{S}}M^{*2}_{jtl_1l_2})^2)\}
=O_p(\frac{q^{4}}{n\sqrt{n}})$, and $||6\mathcal{S}_5||\leq||\hat{\eta}-\eta_0||\surd\{\sum_{l_1,l_2=1}^{\mathcal{S}}||\hat{\eta}-\eta_0||^4(\sum_{j=1}^{\mathcal{S}}$ $(\sum_{t=1}^{\mathcal{S}}
(\widehat{M}_{jtl_1l_2}^*-M_{jtl_1l_2}^*)^2)^2)\}=O_p(\frac{q^{5}}{n^2})$, where $M_{jl_1l_2}^*$ is the $(l_1,l_2)$th element of matrix $M_{j}^*$.
Based on the first and second-order conditions for $q=o_p(n^{1/5})$ and $q/k\rightarrow \kappa$, we can obtain the consistency and oracle properties of the proposed PET estimator. Thus, the order of $\mathcal{S}_2^*$ is the largest among $\mathcal{S}_1^*,\ldots,\mathcal{S}_5^*$. Combining the above results yields $\hat{\eta}-\eta_0=\tilde{\upsilon}/\sqrt{n}+O_p(\frac{q^{5/2}}{n})$. Using $\tilde{\upsilon}/\sqrt{n}$ to replace $\hat{\eta}-\eta_0$ in $\mathcal{S}_1^*$ and $\mathcal{S}_2^*$ yields $\hat{\eta}-\eta_0=\tilde{\upsilon}/\sqrt{n}-\mathbb{M}^{-1}\sum_{j=1}^{\mathcal{S}}\tilde{\upsilon}_jM_j^*\tilde{\upsilon}/2n-\mathbb{M}^{-1}
\tilde{A}\tilde{\upsilon}/n+O_p(\frac{q^4}{n\sqrt{n}})$. Replacing $\hat{\eta}-\eta_0$ in $\mathcal{S}_3^*$ and $\mathcal{S}_4^*$ by $\tilde{\upsilon}/\sqrt{n}-\mathbb{M}^{-1}\sum_{j=1}^{\mathcal{S}}\tilde{\upsilon}_jM_j^*\tilde{\upsilon}/2n-\mathbb{M}^{-1}\tilde{A}\tilde{\upsilon}/n$ and in $\mathcal{S}_1^*$ and $\mathcal{S}_2^*$ by $\tilde{\upsilon}/\sqrt{n}$ leads to Equation (\[high01\]).
[**Proof of Corollary \[cor1\]**]{}. Let $\hat{\theta}_1$ be the PET estimator of nonzero parameter vector $\theta_1$. Denote $\eta=(\theta_1^{{\!\top\!}},\lambda^{{\!\top\!}})^{{\!\top\!}}$, $\eta_0=(\theta_{10}^{{\!\top\!}},0^{{\!\top\!}})^{{\!\top\!}}$, $\Gamma_{1i}(\theta_1)=\partial \psi_i(\theta_1)/\theta_1$, and $${\label{coro1}}
\begin{aligned}
m(Z_i;\eta)=\rho_1(\lambda^{{\!\top\!}}\psi_i(\theta_1))
\begin{pmatrix}
\Gamma_{1i}(\theta_1)\lambda \\
\psi_i(\theta_1)
\end{pmatrix}
-\begin{pmatrix}
W(\theta_1)\\
0
\end{pmatrix},
\end{aligned}$$ where $\rho_1(\lambda^{{\!\top\!}}\psi_i(\theta_1))=n\pi_i=n\exp\{\lambda^{{\!\top\!}}\psi(Z_{i};\theta_1)\}/\sum_{j=1}^{n}\exp\{\lambda^{{\!\top\!}}\psi(Z_j;\theta_1)\}$, and the components of vector $W(\theta_1)$ is $p_{\gamma}^{'}(|\theta_{1j}|){\rm sign}(\theta_{1j})$ for $j=1,\ldots,q$. Let $\rho_1=1$, $\rho_2=1-1/n$, $\rho_3=1-3/n+2/n^2$, and $${\label{coro2}}
\begin{aligned}
\frac{\partial m(Z_i;\eta_0)}{\partial\eta}=
\begin{pmatrix}
-\dot{W}(\theta_{10})&\Gamma_{1i}^{{\!\top\!}} \\
\Gamma_{1i}&\psi_i\psi_i^{{\!\top\!}}
\end{pmatrix}, \mathbb{M}=\begin{pmatrix}
-\dot{W}(\theta_{10})&\Gamma_1^{{\!\top\!}} \\
\Gamma_1&\Sigma_1
\end{pmatrix}, \mathbb{M}^{-1}=\begin{pmatrix}
\mathcal{K}&\mathbb{H}^{{\!\top\!}} \\
\mathbb{H}&P
\end{pmatrix}+O_p(1/n),
\end{aligned}$$ where $\dot{W}(\theta_{10})_{jj}=\partial^2 p_\gamma(\theta_{10j})/\partial\theta_{1j}^2.$ It follows from Assumption \[ass16\] that $\max_{1\leq j\leq q}\dot{W}(\theta_{10})_{jj}$ $=O_p(1/n)$. Denote $\mathcal{K}=(\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}\Gamma_1)^{-1}$, $\mathbb{H}=\mathcal{K}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}$, and $P=\Sigma_1^{-1}-\Sigma_1^{-1}\Gamma_1 \mathcal{K}\Gamma_1^{{\!\top\!}}\Sigma_1^{-1}$. Let $\Gamma^j_{1i}=\partial^2\psi_i(\theta_{10})/\partial\theta_{1j}\partial\theta_1^{{\!\top\!}}$, $\psi_i^j=\partial \psi_i(\theta_{10})/\partial\theta_{1j}$, $w'^{j}=p'''(|\theta_{10j}|)=0$, $\tau=j-q$ for $j>q$, $e_\tau$ be a $k\times 1$ vector whose $\tau$th component is 1 and 0 elsewhere. Here, $\psi_{i\tau}$ is the $\tau$th component of $\psi_i$. Then, we have $${\label{coro3}}
\begin{aligned}
\frac{\partial^2 m(Z_i;\eta_0)}{\partial\eta_j\partial\eta}=\left\{
\begin{array}{llll}
\begin{pmatrix}
0&{\Gamma_{1i}^j}^{{\!\top\!}} \\
\Gamma_{1i}^j&\psi_i^j\psi_i^{{\!\top\!}}+\psi_i{\psi_i^j}^{{\!\top\!}}
\end{pmatrix} & {\rm if}~ j\leq q,\\ [5mm]
\begin{pmatrix}
\partial^2\{e_\tau^{{\!\top\!}}\psi_i(\theta_{10})\}/\partial\theta_1\partial\theta_1^{{\!\top\!}}&{\Gamma_{1i}^j}^{{\!\top\!}}e_\tau\psi_i^{{\!\top\!}}+\psi_{i\tau}{\Gamma_{1i}^j}^{{\!\top\!}} \\
\psi_ie_\tau^{{\!\top\!}}\Gamma_{1i}^j+\psi_{i\tau}\Gamma_{1i}^j&-\rho_3\psi_{i\tau}\psi_i\psi_i^{{\!\top\!}}
\end{pmatrix} & {\rm if}~ j> q.
\end{array}
\right.
\end{aligned}$$ Let $\upsilon_i=-\mathbb{M}^{-1}m(Z_i;\eta_0)$, $\mathbb{V}_i=\partial m(Z_i;\eta_0)/\partial\eta-E\{\partial m(Z_i;\eta_0)/\partial\eta\}$ and $\mathcal{W}=\mathcal{W}(\theta_{10})$. It follows from Equation (\[coro2\]) that $$E(\upsilon_i\upsilon_i^{{\!\top\!}})=
\begin{pmatrix}
\mathcal{K}\mathcal{W}^2\mathcal{K}^{{\!\top\!}}+\mathcal{K} &\mathcal{K}\mathcal{W}^2\mathbb{H}\\
\mathbb{H}^{{\!\top\!}}\mathcal{W}^2\mathcal{K}^{{\!\top\!}} &\mathbb{H}^{{\!\top\!}}\mathcal{W}^2\mathbb{H}+P
\end{pmatrix}
=\begin{pmatrix}
A&B^{{\!\top\!}} \\
B&\mathcal{C}
\end{pmatrix}
+\begin{pmatrix}
\mathcal{K}&0 \\
0&P
\end{pmatrix},$$ $$E(\mathbb{V}_i\upsilon_i)=
\begin{pmatrix}
-E(\Gamma_{1i}^jP\psi_i)\\
-E(\Gamma_{1i}^j\mathbb{H}\psi_i+\psi_i\psi_i^{{\!\top\!}}P\psi_i)
\end{pmatrix}=
\begin{pmatrix}
\varphi\\
f
\end{pmatrix},$$ where $A=\mathcal{K}\mathcal{W}^2\mathcal{K}^{{\!\top\!}}$, $B=\mathbb{H}^{{\!\top\!}}\mathcal{W}^2\mathcal{K}^{{\!\top\!}}$, $\mathcal{C}=\mathbb{H}^{{\!\top\!}}\mathcal{W}^2\mathbb{H}$, $\varphi=-E(\Gamma_{1i}P\psi_i)$ and $f=-E(\Gamma_{1i}^j\mathbb{H}\psi_i+\psi_i\psi_i^{{\!\top\!}}P\psi_i)$.
Combining the above equations yields $$\begin{array}{llll}
\sum\limits_{j=1}^{q+k}M_j^*E(\upsilon_i\upsilon_i^{{\!\top\!}})e_j/2&=&\frac{1}{2}\sum\limits_{j=1}^{q}M_j^*(A^{{\!\top\!}},B)^{{\!\top\!}}e_j+
\frac{1}{2}\sum\limits_{j=1}^{k}M_{j+q}^*(B^{{\!\top\!}},\mathcal{C}^{{\!\top\!}})^{{\!\top\!}}e_j\\
&&+\frac{1}{2}\sum\limits_{j=1}^{q}M_j^*(\mathcal{K},0)^{{\!\top\!}}e_j+
\frac{1}{2}\sum\limits_{j=1}^{k}M_{j+q}^*(0,P)^{{\!\top\!}}e_j\\
&=&
\begin{pmatrix}
a_1+b_1\\
c_1+d_1
\end{pmatrix}+
\begin{pmatrix}
a_2+b_2\\
c_2+d_2
\end{pmatrix}+
\begin{pmatrix}
E(\Gamma_{1i}^jP\psi_i)\\
\tilde{d}+\rho_3E(\psi_i\psi_i^{{\!\top\!}}P\psi_i)/2
\end{pmatrix},
\end{array}$$ where $a_{1}=0$, the $j$th component of $b_1$ is $b_{1j}={\rm tr}(B^{{\!\top\!}}E\{\partial^2\psi_i(\theta_{10})/\partial\theta_{1j}\partial\theta_1^{{\!\top\!}}\})/2$ for $j=1,\ldots,q$, $b_2=E\{{\Gamma_{1i}^j}^{{\!\top\!}}\mathcal{C}\psi_i\}$, $a_2=\sum_{j=1}^kE\{\partial^2 \psi_{ij}/\partial\theta_1\partial\theta_1^{{\!\top\!}}B^{{\!\top\!}}e_j\}/2$, the $j$th component of $c_1$ is $c_{1j}={\rm tr}(AE\{\partial^2\psi_{ij}(\theta_{10})/\partial\theta_1\partial\theta_1^{{\!\top\!}}\})/2$ for $j=1,\ldots,k$, $d_1=c_2=E\{\Gamma_{1i}^jB^{{\!\top\!}}\psi_i\}$, $d_2=-E(\psi_i\psi_i^{{\!\top\!}}\mathcal{C}\psi_i)/2$, the $j$th component of $\tilde{d}$ is $\tilde{d}_{j}={\rm tr}(\mathcal{K}E\{\partial^2\psi_{ij}(\theta_{10})/\partial\theta_1\partial\theta_1^{{\!\top\!}}\})/2$ for $j=1,\ldots,k$. Then, Bias$(\hat{\theta}_1)$ is the first $p$ elements of $\mathbb{L}_B=E\{Q_1(\tilde{\upsilon})+Q_2(\tilde{\upsilon},\tilde{A})\}/n$, which is given by $$\begin{aligned}
\mathbb{L}_B&=-(n\mathbb{M})^{-1}\left(E(\mathbb{V}_i\upsilon_i)+\sum\limits_{j=1}^{q+k}M_j^*E(\upsilon_i\upsilon_i^{{\!\top\!}})e_j/2\right)\\
&=-(n\mathbb{M})^{-1}\left\{
\begin{pmatrix}
\mathcal{A}_1\\
\mathcal{A}_2
\end{pmatrix}- \begin{pmatrix}
0\\
\tilde{d}-E(\Gamma_{1i}^j\mathbb{H}\psi_i)+(\rho_3/2-1)E(\psi_i\psi_i^{{\!\top\!}}P\psi_i)
\end{pmatrix}\right\},
\end{aligned}$$ where $\mathcal{A}_1=a_1+b_1+a_2+b_2$ and $\mathcal{A}_2=c_1+d_1+c_2+d_2$. Therefore, Bias$(\hat{\theta}_1)=\{\mathcal{K}\mathcal{A}_1+\mathbb{H}\mathcal{A}_2\}/n$+Bias$(\hat{\theta}_{1ET}).$
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0.04 in
[Table 2. Performance of the SCAD-PET under different Criteria of tuning parameter selection.]{}\
-- ---------- -- ----- --------- -- ----- -------- -- ------ -------- -- -- -- -- --
(n,p) MS CM MC CM MS CM
(50,7) 2.4 $75\%$ 2.2 $70\%$ 5.2 $82\%$
(100,10) 2.7 $86\%$ 2.5 $81\%$ 6.8 $78\%$
(200,14) 2.9 $94\%$ 2.6 $92\%$ 10.3 $70\%$
(500,19) 3 $100\%$ 2.9 $95\%$ 12.0 $51\%$
-- ---------- -- ----- --------- -- ----- -------- -- ------ -------- -- -- -- -- --
\
[^1]: Correspondence to: Dr. Nian-Sheng Tang, Key Lab of Statistical Modeling and Data Analysis of Yunnan Province, Yunnan University, Kunming 650091, P. R. of China. Tel: 86-871-5032416 Fax: 86-871-5033700 E-mail: nstang@ynu.edu.cn
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
*Parameterized quantum circuits (PQC, aka, variational quantum circuits) are among the proposals for a computational advantage over classical computation of near-term (not error corrected) digital quantum computers. PQCs have to be “trained” — i.e., the expectation value function has to be maximized over the space of parameters.*
This paper deals with the number of samples (or “runs” of the quantum computer) which are required to train the PQC, and approaches it from an information theoretic viewpoint. The main take-away is a disparity in the large amount of information contained in a single exact evaluation of the expectation value, vs the exponentially small amount contained in the random sample obtained from a single run of the quantum circuit.
**Keywords:** Near-term quantum computing; parameterized quantum circuits.
author:
- 'Evgenii Dolzhkov$^{a}$'
- 'Bahman Ghandchi$^{a,b}$'
- |
Dirk Oliver Theis$^{a,b}$\
$^a$ Theoretical Computer Science, University of Tartu, Estonia\
$^b$ Ketita Labs [OÜ]{}, Tartu, Estonia\
`ghandchi@`{`ketita.com`, `ut.ee`}, `dotheis@`{`ketita.com`, `ut.ee`}
date: |
Version: Fri Mar 29 17:49:52 CET 2019\
Compiled:
title: Information content of queries in training Parameterized Quantum Circuits
---
Introduction {#sec:intro}
============
Hybrid quantum-classical computing with parameterized (or variational) quantum circuits (PQCs) works by alternately running the parameterized quantum circuit on a digital, gate-based quantum computer, and updating parameters in classical hard- and software. The hybrid process aims to find parameter settings which optimize some objective function derived from the measurement results of the PQC, for example with the goal to find the ground state of the measurement Hamiltonian. This process has become known as “training” the PQC.
For the purpose of this paper, a Parameterized Quantum Circuit consists of a sequence of quantum operations, applied to a known initial state which we denote by $\ket0\bra0$, and followed by a measurement. Some of the quantum operations are unitaries of the form $$\label{eq:ham-unitary}
\rho \mapsto e^{-\pi i x_j H_j}\rho e^{\pi i x_j H_j}, \text{ $j=1,\dots,n$,}$$ where the $H_j$ are hermitian operators and $x\in\RR^n$ is the vector of parameters. For simplicity(!), we assume that the $H_j$ have $\pm1$ eigenvalues. (We allow more general dependence on the parameters in Section \[sec:eval-queries\].) We also assume that the observable in the final measurement has eigenvalues $\pm1$. Hence, a single run of the quantum circuit (with measurement) with parameters set to $x$ yields a random number in $\pm1$, whose expectation we denote by $f(x)$, and refer to as the *expectation value function* of the PQC. In this simplified setting, the training problem is this: $$\text{maximize } f(x) \text{ over $x\in\RR^n$.}$$ (Note that $n$ is the number of parameters, not the number of qubits.)
Even though, in applications, a good local maximum is often sufficient, training PQCs is known to be difficult for a variety of reasons. The least of it is that, as a non-concave maximization[^1] problem, the training problem is likely to be NP-hard: But classical neural network training has the same property, and it is not a huge problem there. More specific to the quantum case is the existence of “plateaus”: large regions of the parameter space where the gradient is close to 0 [@McClean-Boixo-Smelyanskiy-Babbush-Neven:barren:2018]. While training seems to work fine in practice with a small number of qubits, the exponential dependence on the number of qubits of indicators of “trouble” are worrysome. In this paper, we add one new worrying perspective to the discussion: The information content of the random output of a run of the PQC. For that we consider a setting which is very generous to the designer of a training algorithm: The algorithm is only ever used on a fixed $n$, and a fixed, finite number (depending on $n$) of functions $f_c$, $c\in\mathcal C$, all of which are known to the algorithm. The algorithm itself can be randomized. The algorithm has infinite computational resources; e.g., it can represent real numbers exactly, and make instantaneous computations on them (for parameters and expectation values).
Formally, we define the following. A *sample query* consists of the training algorithm setting the parameters to an $x\in\RR^n$ *ad libitum*, and then running once the quantum circuit with this setting, retrieving the resulting random number $F \in \{\pm1\}$ with $\Exp F=f(x)$. In contrast, in an *evaluation query* after setting the parameters *ad libitum,* the algorithm is given the real number $f(x)$ exactly.
The success of the algorithm is only measured in some definition of average-case — not worst-case — over $c\in \mathcal C$ and over its internal randomness. The following algorithm and theorem underlines just how ridiculously generous the compuational model is.
Pick a random parameter setting $x\in\RR^n$ \[step:random-x\]\
Query $f(x)$.\
Iterating over all $c\in \mathcal C$, find one with $f_c(x) = f(x)$\
Look up the parameter setting $x^*$ maximizing $f_c$ in a table\
output $x^*$.
\[thm:super\] Let $\mu$ be an arbitrary absolutely continuous probability measure on $\RR^n$. If in Step \[step:random-x\], $x$ is drawn according to $\mu$, then the Omnipotent Algorithm succeeds with probability 1.
The proof of this theorem, in Section \[sec:eval-queries\], will show that with probability 1 over the choice of $x$, the mapping $f_c \mapsto (x,f_c(x))$ is one-to-one.
In infomation theoretic terms, if $C$ is randomly chosen in $\mathcal C$, then a single evaluation query of $f_C$ at a random point contains all information about $f_C$: $$\label{eq:eval-q:cond-ent-0}
{\mathbb H}\bigl(f_A \mid (X,f_A(X)) \bigr) =0.$$
Our *Superman* algorithms with infinite memory and tables with worked-out solutions hit Kryptonite, when we replace evaluation-query access by sample-query access — even when we allow the output to be off by a significant amount from the true maximum. We propose the the following definition.
Let be a Las-Vegas (randomized) algorithm which is given sample-query access to one (unknown) element in a family $\mathcal C$ of PQCs with $n$ parameters. For $\alpha\in\RR_+$, we say that *$\alpha$-succeeds* in the training problem, if it outputs a parameter setting $x^*$ with $f(x^*) \ge \max_x f(x) -\alpha$, after performing a number of sample quries whose number depends on: the internal randomness of ; the random choice of $c$ uniformly in $\mathcal C$; and the randomness in the sample query results.
\[thm:sample\] There are constants $c>1$ and $\alpha \approx 1$ and a family of simple PQCs ($3^n$ for each number of qubits $n$) such that every $\alpha$-successful training algorithm, with probability $1-c^{-n}$, requires at least $c^n$ sample queries.
In terms of information content of queries: For any $m \ll c^n$, for the queries at the (random) parameter settings $X_1,\dots,X_m$ that a randomized training algorithm performs, and the corresponding sample query results $Q_1,\dots,Q_m$, if $C \in \mathcal C$ is chosen uniformly at random, we have $$\label{eq:sample-q:mutinf-exp-small}
{\mathbb I}\bigl(C : (X_1,Q_1,\dots,X_m,Q_m) \bigr) \le m 2^{-\Omega(n)}{\mathbb H}(C).$$
Clearly, the difference in the perfect performance of the algorithms for evaluation and sample queries lies in the information content of the queries, vs . The contribution of this paper lies in bringing the consideration of information content of queries to the table with regards to algorithms and lower bounds for PQC training.
#### This paper is organized as follows
In Section \[sec:eval-queries\], we prove Theorem \[thm:super\], Section \[sec:sample-queries\] is dedicted to the proof of Theorem \[thm:sample\]. We conclude with an outlook on related questions. We shoved some technicalities out of the weary eye of the reader into the appendix.
Evaluation queries are powerful {#sec:eval-queries}
===============================
We consider a family of parameterized quantum circuits which is more general than in the rest of the paper: The only restriction is that the parameters $x\in\RR^m$ occur only in unitary quantum operations of the form $$\ket{\psi} \mapsto e^{-i \sum_j \theta_j(x) H_j} \ket{\psi},$$ for arbitrary Hermitian operators $H_j$ and $H$ on no matter how many qubits, and arbitrary analytic functions $\theta_j\colon\RR^m \to \RR$.
In this section, we prove the following theorem
Let $\mathscr C$ be a countable family of parameterized quantum circuits, each taking exactly $m$ parameters, and such that the expectation value functions of the circuits are all distinct.
If $x\in\RR^m$ is chosen randomly according to an absolutely continuous measure on $\RR^m$, then, with probability 1, the circuit $C\in\mathscr C$ is uniquely determined by a single query of its expectation value (evaluation query) with parameters $x$.
The expectation value function is real analytic. Real analytic function have the property that zero-sets are lower-dimensional sub-varieties. Hence, their measure, wrt any absolutely continuous probability measure, is $0$. Consequently, the set of points $x$ for which two such functions coincide also has measure zero. We refer to [@Dolzhkov:MSc:2020] for the details.
Sample-queries are poor {#sec:sample-queries}
=======================
#### We will need some notation
It can be seen [@GilVidal-Theis:CalcPQC:2018] that $f$ is periodic in each coordinate, so we set ${\TT}:= \RR/\ZZ$; note that ${\TT^{n}}= (\RR/\ZZ)^n = \RR^n/\ZZ^n$. For a function $f$ with range in $[-1,+1]$, we write ${{\mathcal{F}}}_f(x)$ for the probability distribution with support contained in $\{\pm1\}$ and mean $f(x)$. Modeling the behavior of parameterized quantum circuits, we assume that, conditioned on $f$ and $x$, samples from ${{\mathcal{F}}}$ are independent from each other and from all other random variables.
For any function $f_0\colon{\TT^{n}}\to\RR$ and $a \in {\{0,\nfrac13,\nfrac23\}^{n}}\subset {\TT^{n}}$, we define the function $f_a := f_0({\boxempty}-a)$; similarly, for a set $P_0 \subset {\TT^{n}}$, we define $P_a := \{ x-a \mid x \in P_0\}$.
We are now well equipped to describe the technical approach.
The “plateau game” {#ssec:sample-queries:game}
------------------
For $n\in \NN$, consider the following *plateau game*[^2], played between Alice and Bob: $$\label{TheGame}\tag{$\mathscr G_n$}
\parbox{.85\linewidth}{\sl There is a set $P_0 \subset {\TT^{n}}$ with $0\notin P_0$, which is known to both Alice and Bob.
At the beginning of the game, Alice chooses an\footnotemark{} $a\in {\{0,\nfrac13,\nfrac23\}^{n}}$, hidden from Bob.
Then, the players proceed in rounds. In each round, Bob chooses an element $x\in{\TT^{n}}$, and asks Alice, ``Is $x \in P_{a}$?'', to which Alice answers truthfully.
The game ends when Bob queries a point which is not in $P_{a}$, in which case Bob wins; otherwise the game proceeds forever (and Alice wins).
}$$ Bob can make randomized queries, i.e., Alice cannot “change her mind” about her choice of $a$ after the first round begins.
Suppose that Bob makes randomized queries and wins after $M_a$ rounds, in the case that Alice picks $a$ as her hidden point ($M_a$ is a random variable). Suppose further that Alice chooses her hidden point uniformly at random in ${\{0,\nfrac13,\nfrac23\}^{n}}$; since it is a random variable, we denote it by a capital letter, $A$. Finally, let $$\label{eq:def-p}
p := \sup_{x\in{\TT^{n}}} \Prb\bigl( P_A \not\ni x \bigr)$$ denote the highest probability of Bob winning in the first round.
Here is the lower bound on the number of rounds it takes Bob to succeed.
\[lem:stupid-game\] $\displaystyle \Prb\bigl( M_A \le m ) \le pm
$
Before we give the proof, a remark. While the setting of the game allows Bob to adaptively choose his next move based on Alice’s answers to his previous questions, that adaptivity is not really present, as Bob — unless he has already won — (knows that he) always gets the same answer: “Yes”. Indeed, the transition from the training problem to the game (performed in Lemma \[lem:tech\] below) takes the adaptivity out of the equation.
Hence, Bob’s strategy is a single probability measure, which he may just as well choose before Alice picks her point. We refer to Appendix \[apx:non-adapt\] for the formal justification.
Let $m \in\NN$. To decide $M_A \le m$, we need to observe the game only for the first (at most) $m$ rounds. For $x\in ({\TT^{n}})^m$, $a\in{\{0,\nfrac13,\nfrac23\}^{n}}$, we let $\beta(a,x) := 1$, if Bob wins in the first $m$ rounds with the questions $x_1,x_2,\dots,x_m$, i.e., if there is a $j$ with $x_j\not\in P_a$; and $\beta(a,x) := 0$ otherwise.
By the above remark, the set of Bob’s strategies is the set of probability distributions $\mu$ on $({\TT^{n}})^m$, and $\mu$ does not depend on $a$. Now we just compute (the sums are over all $\in{\{0,\nfrac13,\nfrac23\}^{n}}$): $$\begin{aligned}
\Prb\bigl( M_A \le m )
&=
\tfrac{1}{3^n} \sum_a \int_{({\TT^{n}})^m} \beta(a,x) \,d\mu(x)
\\
&=
\int_{({\TT^{n}})^m} \tfrac{1}{3^n}\sum_a \beta(a,x) \,d\mu(x)
\\
&=
\int_{({\TT^{n}})^m} \Prb( \exists j\colon x_j\notin P_a ) \,d\mu(x)
\\
&\le
m\cdot \sup_{x\in ({\TT^{n}})^m} \Prb(x_j\notin P_a )
\\
&=
m p.
\end{aligned}$$ This completes the proof of Lemma \[lem:stupid-game\].
From the training problem to the plateau game {#ssec:sample-queries:core-lemma}
---------------------------------------------
Now we come to the technical lemma which connects the PQC-training problem to the plateau game .
\[lem:tech\] Let $n \in \NN$, $f_0\colon {\TT^{n}}\to \lt[ -1 , +1 \rt]$ a continuous function, and $P_0\subset{\TT^{n}}$. Further, let $\eta \in [-1,+1]$, $0\le \delta < \alpha$ (we will need $\delta \ll 1$ and $\alpha \gg \delta$ for the conclusion to make sense). We assume
1. \[lem:tech:cond:near-const\] $\displaystyle |f_0(x) - \eta| < \delta$ for all $x\in P_0$
2. \[lem:tech:cond:max\] $\displaystyle f_0(0) = \max_x f_0(x)$;
3. \[lem:tech:cond:diff\] $\max_{x\in P_0} f_0(x) + \alpha < f_0(0)$ for an $\alpha \gg \delta$.
Consider an $\alpha$-successful algorithm for maximizing a function $f\in \{f_a \mid a\in {\{0,\nfrac13,\nfrac23\}^{n}}\}$, which has sample-query access to $f$. With $A$ chosen uniformly at random from ${\{0,\nfrac13,\nfrac23\}^{n}}$, denote the number of sample queries performed by the algorithm for maximizing $f=f_A$ before it terminates by the random variable $T_A$.
With $p$ as in , we have that $$\label{eq:ub-on-nqueries}
\Prb( T_A \le m ) \le (p+\delta/2) m +1$$
The proof fills the remainder of this subsection. The idea is that we want to compare what the algorithm does when it samples according to ${{\mathcal{F}}}_f$ to what it does when it samples according to ${{\mathcal{F}}}_{\bar f}$, where $$\bar f :=
\begin{cases}
f(x), &\text{ if $x \notin P_0$} \\
\eta, &\text{ if $x \in P_0$.}
\end{cases}$$
For that, we have to run two “copies” of the algorithm in parallel, but the random decisions must be coupled; cf Appendix \[apx:coupling\] for the technical details. We say that the two runs *diverge,* if the algorithms’ inner states differ, or one of the queries result in a different answer.
\[lem:divergence\] Under the conditions of Lemma \[lem:tech\], the probability that the coupled runs of the algorithm diverge at or before the $m$th query is at most $\delta m/2$.
As the random decisions of the two runs are coupled, for the runs of the algorithm to diverge, at least one sample-query has to give a different result for ${{\mathcal{F}}}_f$ from ${{\mathcal{F}}}_{\bar f}$. If the first $j$ query points and query results are the same, then, by coupling, the $(j+1)$th query point $x_{j+1}$ is the same, too. Hence, the probability that the $(j+1)$th query result differs is $\abs{f(x) - \bar f(x)}/2$, which by by condition \[lem:tech:cond:near-const\] in Lemma \[lem:tech\], is at most $\delta/2$.
By induction, the probability that no query point or query result is different in any of the first $m$ queries is at least $$(1-\delta/2)^m$$ which, by Bernoulli’s inequality, is at least $1 - \delta m/2$.
Now we can finish the proof of Lemma \[lem:tech\].
W.l.o.g., we impose on algorithm that it queries (at least once) the point it eventually ouputs. This might lead to an additional query, which we account for by the “$+1$” on the RHS of .
With this modification, the for to output a point $x^*$ with $f(x^*)\ge \max_x f(x) - \alpha$, by conditions \[lem:tech:cond:max\] and \[lem:tech:cond:diff\], it is necessary that the algorithm queries at least 1 point $x\notin P_A$. We will lower bound the random variable $T'_A$ which counts the number of queries of until (and including) the first query point outside of $P_0$ is requested.
As explained above, we now synchronize the run of with a coupled run where the samples are taking according to ${{\mathcal{F}}}_{\bar f}$ instead of ${{\mathcal{F}}}_f$.
Let $m\in\NN$. We distinguish tow cases.
1. \[sdlkfj:div\] The two runs diverge before or at the $m$th query;
2. \[sdlkfj:not-div\] The two runs do not diverge before or at the $m$th query.
In case \[sdlkfj:div\], nothing can be said about $T'_A$. The probability of this happening is at most $\delta m/2$ by Lemma \[lem:divergence\].
As for the case \[sdlkfj:not-div\], note that with samples according to ${{\mathcal{F}}}_{\bar f}$ is a randomized strategy for Bob playing the plateau game: After each “yes” answer from Alice, he throws a coin with probability of heads $(1-\eta)/2$, and proceeds dependant on the outcome. Hence, by Lemma \[lem:stupid-game\], the probability of querying a point not in $P_0$ is at most $pm$.
This concludes the proof of Lemma \[lem:tech\].
A simple family of PQCs
-----------------------
We will now give a family of PQCs $C^{(n)}$, $n\in\NN$, on $n$ qubits, which implement functions $f_0^{(n)}\colon {\TT^{n}}\to\RR$ to which will apply Lemma \[lem:tech\]. In this section, the “PQC” will stand synonymous with “PQC with parameters given in the form and measurement is of an observable with $\pm1$-eigenvalues.”
We start with a simple observation for easy reference.
\[lem:tensor-PQC\] Let $g\colon{\TT^{n}}\to\RR$, $h\colon{\TT^{m}}\to\RR$, and suppose that the expectation value functions of PQCs $C_f$ and $C_g$ are $f$ and $g$ resp. Then $f\colon{\TT^{n+m}}\to\RR\colon (x,y)\to g(x)h(y)$ can be obtained as the expectation value function of a PQC.
Let $C_g$ ($C_h$) use $q_g$ ($q_h$) qubits, perform the quantum operation ${\mathcal E}_g(x)$ on parameter setting $x$ (${\mathcal E}_h(y)$ on parameter setting $y$), and ultimately measure the observable $M_g$ ($M_h$). With $\rho_0 := \ket{0}\bra{0}$: $$\begin{aligned}
g(x) &= \tr(M_g \; {\mathcal E}_g(x) \; \rho_0) \\
h(y) &= \tr(M_h \; {\mathcal E}_h(y) \; \rho_0).
\end{aligned}$$ The function $f$ is realized as an expectation value function by taking $q_g+q_h$ qubits, staring in $\rho_0\otimes \rho_0$, applying the quantum operation ${\mathcal E}_g\otimes {\mathcal E}_h$, and utlimately measuring the observable $M_g\otimes M_h$.
In view of Lemma \[lem:tensor-PQC\], we first give a PQC for the function $$\label{eq:delta:1}
h
\colon {\TT}\to \RR
\colon x \mapsto \nfrac13 +\cos(2\pi x)/3.$$ As for the first part, it can easily be seen that there exists a $\phi\in \lt]0,\pi/4\rt[$ such that $$R := e^{-2\pi i(\cos\phi X + \sin\phi Z)}$$ has the property that, with $x:=\nfrac13$, the quantum operation $\rho \mapsto R^x\rho R^{-x}$ maps $Z$ into the $X$-$Y$-plane. The same is then the case for $x=\nfrac23$. Now take the following PQC $C_1$: $$\label{eq:atomic-circuit}
\Qcircuit{
\lstick{\ket0} & \gate{R^{x}} & \meter \\
}$$ where the measurement at the end is of the observable $Z$. The expectation value function of and $h$ coincide for $x=0,\nfrac13,\nfrac23$, and since both are trigonometric polynomials of degree 1 (cf. [@GilVidal-Theis:CalcPQC:2018]), they are the same for every $x\in\RR$. Thus we have realized the function $h$ in as the expectation value function of the parameterized quantum circuit .
So we can realize the function $h$ defined in . Proving that $f$ as defined in has the required properties requires a little work.
Second, we invoke Lemma \[lem:tensor-PQC\], to give us a PQC, $C_n$, whose expectation value function is equal to $$\label{eq:delta:n}
f^{(n)}_0
\colon {\TT^{n}}\to \RR
\colon x \mapsto \prod_{j=1}^n h(x_j).$$
Conclusion of the proof of Therorem \[thm:sample\]
--------------------------------------------------
We are now ready to conclude the proof of Theorem \[thm:sample\].
Just one more definition: For $x\in{\TT^{n}}$ and $a\in{\{0,\nfrac13,\nfrac23\}^{n}}$, we denote by $d(a,x)$ the number of $j$ with $|x_j-a_j|<\nfrac16$, where for $y\in\RR$, $|y|$ denotes the *Bohr “norm”:* the smallest element in $y+\ZZ$. The quantity $d(a,x)$ is the Hamming distance between $a$ and the point resulting from “rounding” $x$ to the closest element in ${\{0,\nfrac13,\nfrac23\}^{n}}$ (if ties are broken properly).
With $f_0$ defined as in , we let[^3] $$\label{eq:def:P_0}
P_0 := \lt\{ x\in{\TT^{n}}\mid d(0,x) > n/2 \rt\}.$$
We note the following simple fact-plus-definition (of $\delta$) for easy reference.
\[lem:def-+-ieq–delta\] For all $x\in P_0$, $$\abs{ f_0(x) } < (2/3)^{n/2} =: \delta$$
This follows directly from the definitions of $f_0$ in , the fact that $\abs{h(t)} \le \nfrac23$ holds for all $t\in {\TT}$ with $|t|\ge\nfrac16$, and the definition of $P_0$.
The other important quantity for the use of Lemma \[lem:tech\] is $p$, as defined in . Another short lemma-plus-definition.
\[lem:def-+-ieq–p\] For all $x\in{\TT^{n}}$, if $A$ is chosen unformly at random in ${\{0,\nfrac13,\nfrac23\}^{n}}$, then $$\Prb\bigl( P_A \not\ni x \bigr) \le e^{-n/36} =: p.$$
Fix an arbitrary $x\in{\TT^{n}}$, and consider the Bernoulli random variables $$D_j :=
\begin{cases}
1, & \text{if $|x_j - A_j| < \nfrac16$},\\
0, & \text{otherwise.}
\end{cases}$$ Note that the $x\notin P_A$ is equivalent to $$D := \sum_{j=1}^n D_j > n/2.$$
Since the $A_j$ are independent, the $D_j$ are, too. Moreover, we have $\Prb( D_j = 1 ) \in \{0,\nfrac13\}$. By an appropriate version of Hoeffding’s inequality, we find that $$\begin{aligned}
\Prb\bigl( P_A \not\ni x \bigr)
&= \Prb( D > n/2 )
\\
&\le \Prb( D > \Exp D + n/6 )
\\
&\le e^{-n/36}.
\end{aligned}$$
We are now ready to put it all together.
With $f_0,P_0,\delta,p$ as above and $\eta := 0$, $\alpha := 1-2\delta$, the conditions of Lemma \[lem:tech\] are satisfied, with $\max_x f_0(x) = 1$. Hence, for any $\alpha$-successful algorithm we have for the number of sample queries, $T_A$, satisfies \[eq:ub-on-nqueries\]. By the definitions of $\delta,p$, we can find an absolute constant $c > 1$ such that $$\label{eq:prb-bd-numq}
\Prb( T_A < c^n ) \le c^{-n}.$$ This concludes the proof of the theorem.
### Notes about Theorem \[thm:sample\] {#notes-about-theoremthmsample .unnumbered}
It should be pointed out that in our setup, the algorithm is given the magic ability to know whether it has queried a point with a close-to-maximal function value. Hence, our definition of “omnipotence” may be modified to include that ability.
Theorem \[thm:sample\] is set in Las Vegas; it can readily be moved to Monte Carlo by just rearranging the proof.
The bound on the mutual information, , can be obtained from using standard tools; we leave that to the reader.
Outlook
=======
It is argued in [@Harrow-Napp:low-depth-gradient:2019], that sampling from derivatives gives and advantage in terms of convergence of gradient descent algorithms, under some assumptions. It remains to be seen whether querying samples from derivatives of $f$ removes the exponentially poor information content of sample queries in setting.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors would like to thank Vitaly Skachek for a discussion enabling the proof of Lemma \[lem:stupid-game\], which simplified the whole §\[ssec:sample-queries:game\].
This research was supported by the Estonian Research Council, ETAG (*Eesti Teadusagentuur*), through PUT Exploratory Grant \#620. Ironically, BG was partly supported by United States Air Force Office of Scientific Research (AFOSR) via AOARD Grant “Verification of Quantum Cryptography” (FA2386-17-1-4022).
APPENDIX {#appendix .unnumbered}
========
Non-adaptivity of the Plateau Game {#apx:non-adapt}
==================================
Formally, the set of *adaptive* strategies of Bob would be a sequence $\mu_j$ of mapps from the set $$\Bigl( {\TT^{n}}\times\{\texttt{Yes},\texttt{No}\} \Bigr)^{j-1}$$ of his previous question and Alice’s answers to the set of probability distributions on ${\TT^{n}}$. Since the right-hand entries (Alice’s answers) are always the same (unless Bob has won), this collapses to: for every $j\in\NN$ and $x\in({\TT^{n}})^{j-1}$, a probability distribution on ${\TT^{n}}$ — in other words, a probability distribution on $({\TT^{n}})^j$.
Coupling of the two runs of the algorithm {#apx:coupling}
=========================================
We assume an infinite stack of independent random numbers $R_0,R_1,R_2,R_3,\dots$, each uniformly distributed in $[-1,+1]$. Based on them, the sampling from ${{\mathcal{F}}}$ is performed and the random decisions of the algorithm are made:
- When querying a sample from ${{\mathcal{F}}}_g(x)$, the top of the stack, $R_0$ is considered. If $R_0 < g(x)$, the query result is $+1$, otherwise it is $-1$. $R_0$ is then popped from the stack.
- Similarly, when the randomized algorithm tosses a coin, the top $R_0$ is consulted in the obvious manner, and popped.
We run the algorithm simultanously with two different sample-query distributions: In one of the answer the queries according to the distributions ${{\mathcal{F}}}_f$, in the other run we answer according to ${{\mathcal{F}}}_{\bar f}$. In the two runs, we use two stacks the same infinite sequence of random numbers.
[^1]: Sorry. Next time, we’ll do “minimize” and “convex”.
[^2]: This isn’t really a game, it’s an active learning problem — but games are so much more fun…
[^3]: No attempts have been made to optimize the constants — not even the base of the exponentially many queries.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators ${{ \slashed{D} }}$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2({{\mathbb{C}}})$, and also applies to the standard $q$-sphere and the $q$-disk with the right classical limit and all properties holding except for ${{\mathcal{J}}}$ now being a twisted isometry. We also describe a noncommutative Chern construction from holomorphic bundles which in the $q$-sphere case provides the relevant bimodule connection.'
address: |
Dept of Mathematics, Swansea University\
Singleton Parc, Swansea SA2 8PP\
[ ]{}+\
Queen Mary, University of London\
School of Mathematics, Mile End Rd, London E1 4NS, UK
author:
- 'Edwin Beggs & Shahn Majid'
title: Spectral triples from bimodule connections and Chern connections
---
Introduction
============
A main difference between the well-known Connes approach to noncommutative geometry coming out of cyclic cohomology and the more constructive ‘quantum group’ approach to noncommutative geometry lies in the attitude towards the Dirac operator. In Connes’ approach this is defined axiomatically as an operator ${{ \slashed{D} }}$ on a Hilbert space which plays the role of Dirac operator on a spinor bundle and which is the starting point for Riemannian geometry, while in the quantum groups approach one builds up the geometry layer by layer starting with the differential algebra structure and often (but not necessarily) guided by quantum group symmetry, and arrives at ${{ \slashed{D} }}$ as an endpoint, normally after the Riemannian structure. This approach also should contain $q$-deformed and quantum group-related examples but it is known that these may take us beyond Connes axioms if we want to have the correct classical limit. For example, for the standard $q$-sphere where the construction in [@DS-sphere] meets Connes axioms at some algebraic level but has spectral dimension 0 (the eigenvalues of the Dirac operator distributes in a typical manner for a zero dimensional manifold) and hence do not have the correct classical limit.
The present paper joins up these two approaches, namely we show how within the constructive approach we can naturally obtain spectral triples, at least up to issues of functional analysis, from a bimodule connection on a chosen vector bundle (thought of as a ‘spinor bundle’), having fixed a first order differential calculus for our space and a ‘Clifford action’ ${{\triangleright}}$ of its 1-forms on the bundle. The latter plays the role of the Clifford structure. Our construction is still quite general and we don’t assume that the bundle is associated to a quantum frame bundle and connection induced by a quantum ‘spin’ connection on it as per the classical case, although that will be the case in the $q$-sphere example.
An outline of the paper is as follows. In Section 2.1, we recall Connes’ axioms [@Con; @ConMar] for a real spectral triple. Then in Section 2.2 we provide our main result, Theorem \[sptripres\], which constructs examples of these from bimodule connections at an algebraic level, i.e. before worrying about adjoints. Section 2.3 establishes further constraints on the bimodule connection and inner product data to have ${{ \slashed{D} }}$ hermitian and ${{\mathcal{J}}}$ an (antilinear) isometry. Section 2.4 completes the general theory with an explanation of how varying the bimodule connection amounts to an inner fluctuation of the spectral triple in the sense of Connes[@ConMar].
One of the first ingredients in Section 2.2 is that the ‘commutativity condition’ in Connes’ axioms (see (4) in our recap below) can be seen as making the Hilbert space ${{\mathcal{S}}}$ a bimodule [@LordDirac], see also [@Barrmatrix]. However, our notion of bimodule connection means a single (say, left) connection $\nabla$ which admits a modified right-connection rule via a generalised braiding [@Mou; @DV1; @DV2; @MMM; @Sitarz; @BegMa3; @BegMa4; @BegMa5; @MaTao]. This allows for connections on tensor products of bimodules which will be critical for what follows and is very different from what is meant by ‘bimodule connection’ in [@LordDirac], which comes from [@CuntzQuillen] and uses two unrelated connections, one left and one right, on a bimodule. Classically, the latter reduces to defining two unrelated connections on the same bundle and is not what we need. Specifically, the lack of relation between the left and right structures means that the antilinear ${{\mathcal{J}}}$ operator for the reality condition for Connes’ definition of Dirac operator could not be studied. In the context of what we mean by bimodule connections, another main tool in Section 2.2 is a conjugate bimodule whereby the antilinear map ${{\mathcal{J}}}:{{\mathcal{S}}}\to {{\mathcal{S}}}$ is formulated in terms of a linear map $j:{{\mathcal{S}}}\to \overline{{{\mathcal{S}}}}$. We use our previous work [@BegMa3] for the conjugate bimodule connection and related matters. Although one could view the use of bar categories here as a bookkeeping device to keep explicit track of anti/linearity, it is essential for tensor product operations like ${{\rm id}}{\otimes}j$ to make sense. In the context of general monoidal categories, the idea of bar category can be less trivial [@BegMa2], but it is very useful even in the present case of complex vector spaces and someantilinear maps.
Section 3 shows how the theory works on three examples. Section 3.1 covers the finite geometry of $2\times 2$ matrices $M_2({{\mathbb{C}}})$ as ‘coordinate algebra’. This is of course very well studied and we refer to [@Barrmatrix] for a recent treatment of spectral triples here. In our approach we start with a natural $*$-differential calculus $\Omega^1$ which is 2-dimensional over the algebra. As it happens we take the same bimodule for ${{\mathcal{S}}}$, i.e. 2-spinors. We take a natural choice of ${{\triangleright}}$ in this context and fixing this data we find a unique bimodule connection that meets our requirements of Section 2. This results in a single spectral triple which we compute as ${{ \slashed{D} }}={1\over 2}\gamma^2{\otimes}[\gamma^2,\ ]-{1\over 2}\gamma^1{\otimes}[\gamma^1,\ ]$ where $\gamma^i={\mathrm{i}}\sigma^i$ in terms of Pauli matrices. The commutators are inner derivations or ‘vector fields’ on $M_2({{\mathbb{C}}})$ and uniqueness means that fluctuations of this would entail a change of either the differential structure or the Clifford structure.
Section 3.2 covers the $q$-sphere ${{\mathbb{C}}}_q[S^2]$ with the geometrically correct spin bundle ${{\mathcal{S}}}={{\mathcal{S}}}_+\oplus{{\mathcal{S}}}_-$ given by $q$-monopole sections of charges $\pm 1$ as used in [@Ma:rieq]. This uses the standard 2D differential calculus coming from the 3D one[@Wor] on ${{\mathbb{C}}}_q[SU_2]$, a Clifford action ${{\triangleright}}$ given by the holomorphic structure introduced in [@Ma:rieq] and a $q$-monopole principal connection [@BrzMaj:gau], all of which led to a $q$-deformed ${{ \slashed{D} }}$ in a quantum frame bundle approach. Our new result is that the relevant covariant derivative on ${{\mathcal{S}}}$ is in fact a bimodule connection and we find a ${{\mathcal{J}}}$ and inner product (given by the Haar integral) so that all the axioms (1)-(6) of a real spectral triple of dimension 2 are satisfied at the pre-functional analysis level except for one: we find that ${{\mathcal{J}}}$ is necessarily not an isometry but some kind of twisted $q$-isometry in the sense $${{\langle}}\!{{\langle}}{{\mathcal{J}}}(\phi),{{\mathcal{J}}}(\psi){{\rangle}}\!{{\rangle}}=q^{\pm 1} {{\langle}}\!{{\langle}}\varsigma^{-1}(\psi),\phi{{\rangle}}\!{{\rangle}},\quad\forall \phi,\psi\in {{\mathcal{S}}}_\pm$$ where the brackets are the Hilbert space inner product and $\varsigma$ is the automorphism that makes the Haar integral a twisted trace in the sense of [@Murphy]. We identified ${{\mathcal{S}}}_\pm$ with degree $\mp1$ subspaces of ${{\mathbb{C}}}_q[SU_2]$ under the $U(1)$ action of the quantum principal bundle. More precisely, we obtain a 1-parameter family of ${{ \slashed{D} }}$ where a parameter $\beta$ extends the Clifford action from the canonical choice $\beta=1$ in [@Ma:rieq]. Our construction is different from another attempt at the $q$-sphere Dirac operator with 2D spinor space [@DLPS-sphere], where the ‘first order condition’ (see (6) in our recap below) had to be weakened to hold up to compact operators, which is not our case. Section 3.3 is our final example, the quantum disk ${{\mathbb{C}}}_q[D]$ as in [@klilesdisk], where we find again that everything works up to completions to give a dimension 2 spectral triple, for our choice of bimodule connection, aside from ${{\mathcal{J}}}$ being required to be a twisted isometry. The Clifford structure is similar the the $q$-sphere case and we again obtain a moduli space of examples as we vary a real parameter.
Section 4 returns to the general theory with a noncommutative framework for holomorphic bundles and their associated Chern connections along the lines of [@Cherncomplex]. At least in the nice case of the $q$-sphere this provides a more direct geometric route to the bimodule connection that we used for the ${{ \slashed{D} }}$ operator as well as the Levi-Civita connection in [@Ma:rieq] (i.e. without going through the frame bundle theory). This is computed along with the other examples in Section 5.
We also note [@RieffelResist] which has a similar starting point of a differential algebra equipped with a metric and which contains some steps towards a more analytic treatment.
Connes spectral triple
======================
This section starts with a short recap of Connes’ axioms of a spectral triple and then proceeds to our main results about the construction of these from bimodule connections.
Real spectral triples at an algebraic level
-------------------------------------------
At the pre-functional analysis level, a real spectral triple in dimension $n$ mod 8 consists of certain data [@ConMar] which we list as follows:
(1a) A Hilbert space ${{\mathcal{H}}}$, with inner product ${{\langle}}\!{{\langle}}\ ,\ {{\rangle}}\!{{\rangle}}$ antilinear in the 1st argument. A faithful representation of the a $*$-algebra $A$ on ${{\mathcal{H}}}$ such that for all $a\in A$ and $\phi,\psi\in {{\mathcal{H}}}$, $${{\langle}}\!{{\langle}}a^*.\psi,\phi{{\rangle}}\!{{\rangle}}={{\langle}}\!{{\langle}}\psi, a.\phi{{\rangle}}\!{{\rangle}}$$
(1b) Operators $\gamma, {{ \slashed{D} }}$ (both linear) and ${{\mathcal{J}}}$ (antilinear) on ${{\mathcal{H}}}$ obeying ${{\langle}}\!{{\langle}}{{\mathcal{J}}}\psi, {{\mathcal{J}}}\phi{{\rangle}}\!{{\rangle}}={{\langle}}\!{{\langle}}\phi,\psi{{\rangle}}\!{{\rangle}}$, $\gamma^*=\gamma$ and ${{ \slashed{D} }}^*={{ \slashed{D} }}$. (For odd dimension we may take $\gamma$ to be the identity.)
\(2) ${{\mathcal{J}}}^2={\epsilon}$, ${{\mathcal{J}}}\gamma={\epsilon}''\gamma {{\mathcal{J}}}$, $\gamma^2=1$, $[\gamma, a]=0$
\(3) ${{ \slashed{D} }}\gamma=(-1)^{n-1}\gamma {{ \slashed{D} }}$.
\(4) $[a, {{\mathcal{J}}}b{{\mathcal{J}}}^{-1}]=0$ for all $a,b\in A$
\(5) ${{\mathcal{J}}}{{ \slashed{D} }}={\epsilon}' {{ \slashed{D} }}{{\mathcal{J}}}$
\(6) $[[{{ \slashed{D} }},a],{{\mathcal{J}}}b{{\mathcal{J}}}^{-1}]=0$ for all $a,b\in A$
The signs ${\epsilon},{\epsilon}',{\epsilon}''$ in $\{+1,-1\}$ are taken from a table according to $n$ mod 8:
$n$ 0 1 2 3 4 5 6 7
-------------- --- ------ ------ ------ ------ ------ ------ ---
$\epsilon$ 1 1 $-1$ $-1$ $-1$ $-1$ 1 1
$\epsilon'$ 1 $-1$ 1 1 1 $-1$ 1 1
$\epsilon''$ 1 $-1$ 1 $-1$
We have grouped the axioms here into (1) that relate to the Hilbert space structure and ultimately to functional analysis, and the remainder which are more algebraic.
Construction of spectral triples from connections
-------------------------------------------------
Take a star algebra $A$ with a star differential calculus $(\Omega,{{\rm d}},\wedge)$, and a left $A$-module ${{\mathcal{S}}}$. The first proposition is also the starting point of [@LordDirac].
\[condd1\] Suppose we are given an antilinear map ${{\mathcal{J}}}:{{\mathcal{S}}}\to {{\mathcal{S}}}$ and a linear map $\gamma:{{\mathcal{S}}}\to {{\mathcal{S}}}$ satisfying properties (2) and (4). Then there is a bimodule structure on ${{\mathcal{S}}}$, with right action, for $\psi\in{{\mathcal{S}}}$ and $a\in A$ $$\psi.a= {{\mathcal{J}}}a^*{{\mathcal{J}}}^{-1}.\psi\ ,$$ and $\gamma$ is a bimodule map.
Property (4) states that the left and right actions commute. The condition $[\gamma,a]=0$ shows that $\gamma$ is a left module map. Then $$\gamma{{\mathcal{J}}}a^*{{\mathcal{J}}}^{-1}=\epsilon''\,{{\mathcal{J}}}\gamma a^*{{\mathcal{J}}}^{-1}=\epsilon''\,{{\mathcal{J}}}\,a^*\, \gamma {{\mathcal{J}}}^{-1}={{\mathcal{J}}}\,a^*{{\mathcal{J}}}^{-1} \gamma \ ,$$ so $[\gamma,{{\mathcal{J}}}a^*{{\mathcal{J}}}^{-1}]=0$, so $\gamma$ is a right module map.
Now we assume the conditions for Proposition \[condd1\], and examine some of the other conditions, given a particular construction for ${{ \slashed{D} }}$. However first, we need to define a bimodule connection.
A left connection $\nabla_{{\mathcal{S}}}:{{\mathcal{S}}}\to\Omega^1{\otimes}_A {{\mathcal{S}}}$ on ${{\mathcal{S}}}$ is a linear map obeying the left Leibniz rule $$\nabla_{{\mathcal{S}}}(a.\phi)={{\rm d}}a{\otimes}\phi+a.\nabla_{{\mathcal{S}}}(\phi)\ ,$$ for $a\in A$ and $\phi\in {{\mathcal{S}}}$. A left bimodule connection is a pair $(\nabla_{{\mathcal{S}}},\sigma_{{\mathcal{S}}})$ where $\nabla_{{\mathcal{S}}}$ is a left connection and $\sigma_{{\mathcal{S}}}:{{\mathcal{S}}}{\otimes}_A \Omega^1\to\Omega^1{\otimes}_A {{\mathcal{S}}}$ is a bimodule map obeying $$\sigma_{{\mathcal{S}}}(\phi{\otimes}{{\rm d}}a)=\nabla_{{\mathcal{S}}}(\phi.a)-\nabla_{{\mathcal{S}}}(\phi).a\ .$$
Note that we have a single connection, with a left Leibniz rule and a modified right Leibniz rule. This is the definition of bimodule connection used in [@Mou; @DV1; @DV2; @MMM; @Sitarz; @BegMa3; @BegMa4; @BegMa5; @MaTao] among others, and is defined in that manner so as to enable the tensor product of connections.
\[Diracnabla\] Suppose that $(\nabla_{{\mathcal{S}}},\sigma_{{\mathcal{S}}})$ is a left bimodule connection on ${{\mathcal{S}}}$, and that ${{\triangleright}}:\Omega^1{\otimes}_A {{\mathcal{S}}}\to
{{\mathcal{S}}}$ is a left module map. If we define ${{ \slashed{D} }}={{\triangleright}}\circ\nabla_{{\mathcal{S}}}$ then $[{{ \slashed{D} }},a]\phi={{\rm d}}a{{\triangleright}}\phi$. Then (6) is equivalent to ${{\triangleright}}$ being a bimodule map, and (5) is equivalent to $$\begin{aligned}
\epsilon'\, {{\mathcal{J}}}[{{ \slashed{D} }},a^*]{{\mathcal{J}}}^{-1}\phi={{\triangleright}}(\sigma_{{\mathcal{S}}}(\phi{\otimes}{{\rm d}}a))\ .\end{aligned}$$
**Proof:** The first statement is given by $${{\triangleright}}\circ\nabla_{{\mathcal{S}}}(a.\phi)={{\triangleright}}({{\rm d}}a{\otimes}\phi+a.\nabla_{{\mathcal{S}}}\phi)
={{\rm d}}a{{\triangleright}}\phi+a.({{\triangleright}})(\nabla_{{\mathcal{S}}}\phi)\ ,$$ as $\nabla_S$ is a connection, and the comment on bimodule maps is then immediate. If $\nabla_S$ is a bimodule connection we have ${{ \slashed{D} }}(\phi.a)={{\triangleright}}(\sigma_{{\mathcal{S}}}(\phi{\otimes}{{\rm d}}a)+({{ \slashed{D} }}\phi).a$. On the other hand, using (5) for the 2nd equality, $$\begin{aligned}
{{ \slashed{D} }}(\phi.a)=&\ {{ \slashed{D} }}({{\mathcal{J}}}a^*{{\mathcal{J}}}^{-1}\phi)=\epsilon'\, {{\mathcal{J}}}{{ \slashed{D} }}a^*{{\mathcal{J}}}^{-1} \phi = \epsilon'\, {{\mathcal{J}}}[{{ \slashed{D} }},a^*]{{\mathcal{J}}}^{-1}\phi+\epsilon'\, {{\mathcal{J}}}a^*{{ \slashed{D} }}{{\mathcal{J}}}^{-1}\phi \cr
=&\ \epsilon'\, {{\mathcal{J}}}[{{ \slashed{D} }},a^*]{{\mathcal{J}}}^{-1}\phi+ {{\mathcal{J}}}a^*{{\mathcal{J}}}^{-1}{{ \slashed{D} }}\phi
= \epsilon'\, {{\mathcal{J}}}[{{ \slashed{D} }},a^*]{{\mathcal{J}}}^{-1}\phi+ ({{ \slashed{D} }}\phi).a\ . \qquad\largesquare\end{aligned}$$ so the condition stated follows from (5). The argument is clearly reversible and (5) holds if the condition stated holds for all $a,\phi$.
Now we examine how to satisfy the conditions on $\gamma$ in terms of the connection and the ‘Clifford action’ ${{\triangleright}}$:
If there is a bimodule map $\gamma:{{\mathcal{S}}}\to {{\mathcal{S}}}$ with $\gamma^2={{\rm id}}$, which intertwines the connection $\nabla_{{\mathcal{S}}}$ (i.e. $\nabla_{{\mathcal{S}}}\gamma=({{\rm id}}{\otimes}\gamma)\nabla_{{\mathcal{S}}}$), and has $$\gamma\circ{{\triangleright}}=-{{\triangleright}}\circ({{\rm id}}{\otimes}\gamma):\Omega^1{\otimes}_A {{\mathcal{S}}}\to {{\mathcal{S}}}\ ,\
{{\mathcal{J}}}\circ\gamma=\epsilon''\,\overline{\gamma}\circ {{\mathcal{J}}}:{{\mathcal{S}}}\to{{\mathcal{S}}}\ ,$$ then $({{ \slashed{D} }},{{\mathcal{J}}},\gamma)$ satisfies all the conditions which include $\gamma$ in (2-6) for an even dimensional spectral triple.
**Proof:**There is only one nontrivial thing to check, $$\gamma\,{{ \slashed{D} }}(\phi)=\gamma\circ({{\triangleright}})\nabla_{{\mathcal{S}}}\phi=-({{\triangleright}})\circ({{\rm id}}{\otimes}\gamma)\nabla_{{\mathcal{S}}}\phi=
-({{\triangleright}})\circ\nabla_{{\mathcal{S}}}\gamma\phi=-{{ \slashed{D} }}(\phi)\ .\quad\largesquare$$
At first sight it might seem that satisfying the conditions for ${{\mathcal{J}}}$ would be very similar to the case for $\gamma$. However, this is not the case. The problem is that $({{\rm id}}{\otimes}{{\mathcal{J}}})\nabla_{{\mathcal{S}}}$ is not even defined. In a tensor product over the complex numbers, we have $\mathrm{i}\,\xi{\otimes}\phi=\xi{\otimes}\mathrm{i}\, \phi\in\Omega^1{\otimes}_A {{\mathcal{S}}}$. Now applying $({{\rm id}}{\otimes}{{\mathcal{J}}})$ to this gives $\mathrm{i}\,\xi{\otimes}{{\mathcal{J}}}\phi=-\xi{\otimes}\mathrm{i}\, {{\mathcal{J}}}\phi$, a contradiction unless both sides vanish. All this is before we actually look at the ${\otimes}_A$ part, and find more problems in that elements of $A$ are multiplied on the wrong side. Basically, tensor products and antilinear maps do not mix. Our problems are resolved if we are more careful and use the *conjugate* of a bimodule.
[@BegMa3] The conjugate of an $A$-bimodule $E$ is written $\overline{E}$, and is identical to $E$ as a set with addition. We denote an element of $\overline{E}$ by $\overline{e}$ where $e\in E$, so that they are not confused. The complex vector space structure, for $e,g\in E$ and $\lambda\in{{\mathbb{C}}}$ is $$\overline{e}+\overline{g}=\overline{e+g}\ ,\quad \lambda\,\overline{e}=\overline{\lambda^*\,e}\ .$$ The $A$-bimodule structure is given by a change of side, for $a\in A$, $$a.\overline{e}=\overline{e.a^*}\ ,\quad \overline{e}.a=\overline{a^*.e}\ .$$
For bimodules $E,F$, a bimodule map $\theta:E\to F$ gives another bimodule map $\overline{\theta}:\overline{E}\to \overline{F}$ by $\overline{\theta}(\overline{e})=\overline{\theta(e)}$. There is also a well defined bimodule map $\Upsilon:\overline{E{\otimes}_A F}\to \overline{F}{\otimes}_A \overline{E}$ flipping the order, defined by, for $e\in E$ and $f\in F$, given by $$\Upsilon(\overline{e{\otimes}f})=\overline{f}{\otimes}_A \overline{e}$$
Note that we *do not use bar as a complex conjugation operation*, on elements it is purely a bookkeeping notation for the antilinear identity map. In fact, if we want to take the complex conjugate of $\lambda\in{{\mathbb{C}}}$, as above, we write it as $\lambda^*\in{{\mathbb{C}}}$ to avoid confusion. The alternative, as stated above, is to be incapable of incorporating antilinear maps into tensor products. With this notation, an antilinear map can be regarded as a linear map into the conjugate. The flip map $\Upsilon$ simply implements a change of order implicit in taking conjugates.
To illustrate this, we again consider the antilinear map ${{\mathcal{J}}}$, but mapping into the conjugate $\overline{{{\mathcal{S}}}}$. Define a map $j:{{\mathcal{S}}}\to \overline{{{\mathcal{S}}}}$ by $j(\phi)=\overline{{{\mathcal{J}}}\phi}$, and this is a linear bimodule map, as we now show, for $a\in A$ and $\phi\in{{\mathcal{S}}}$, $$\begin{aligned}
&j(a.\phi)=\overline{{{\mathcal{J}}}(a.\phi)}=\overline{{{\mathcal{J}}}a{{\mathcal{J}}}^{-1}{{\mathcal{J}}}\phi}=\overline{{{\mathcal{J}}}(\phi).a^*}=a.\overline{{{\mathcal{J}}}(\phi)}=a.j(\phi)\ ,\cr
&j(\phi.a)=j({{\mathcal{J}}}a^* {{\mathcal{J}}}^{-1}\phi)=\overline{{{\mathcal{J}}}^2 a^*{{\mathcal{J}}}^{-1}\phi}=\epsilon\, \overline{ a^*{{\mathcal{J}}}^{-1}\phi}=
\epsilon\, \overline{ {{\mathcal{J}}}^{-1}\phi}.a= \overline{ {{\mathcal{J}}}\phi}.a=j(\phi).a\ .\end{aligned}$$ The other antilinear map we will need is the star operation on $\Omega$, extending the star operation on $A$. We define the bimodule map $\star:\Omega\to \overline{\Omega}$ by $\star\,\xi=\overline{\xi^*}$ for $\xi\in\Omega$.
Recall next that given a left bimodule connection $({{\mathcal{S}}},\nabla_{{\mathcal{S}}},\sigma_{{\mathcal{S}}})$ where $\sigma_{{\mathcal{S}}}$ is invertible, we have a canonical left bimodule connection $\nabla_{\overline{{{\mathcal{S}}}}}$ on $\overline{{{\mathcal{S}}}}$ given by [@BegMa3] $$\begin{aligned}
\label{riconbar}
\nabla_{\overline{{{\mathcal{S}}}}}(\overline{\phi})=(\star^{-1}{\otimes}{{\rm id}})\Upsilon\,\overline{\sigma_{{\mathcal{S}}}{}^{-1}\nabla_{{\mathcal{S}}}\phi}\ .\end{aligned}$$
The condition for $j$ to intertwine the left connections is $({{\rm id}}{\otimes}j)\nabla_{{\mathcal{S}}}=\nabla_{\overline{{{\mathcal{S}}}}}\,j$, or $$\begin{aligned}
\label{jpreserves}
({{\rm id}}{\otimes}j)\nabla_{{\mathcal{S}}}\phi=\nabla_{\overline{{{\mathcal{S}}}}}\,j(\phi)=\nabla_{\overline{{{\mathcal{S}}}}}(\overline{{{\mathcal{J}}}\phi})=
(\star^{-1}{\otimes}{{\rm id}})\Upsilon\,\overline{\sigma_{{\mathcal{S}}}{}^{-1}\nabla_{{\mathcal{S}}}({{\mathcal{J}}}\phi)}\ .\end{aligned}$$ The difference $({{\rm id}}{\otimes}j)\nabla_{{\mathcal{S}}}-\nabla_{\overline{{{\mathcal{S}}}}}\,j$ is a left module map, so to check the difference is zero, it is enough to do so on a set of left generators for ${{\mathcal{S}}}$. Using (\[jpreserves\]) we can calculate $$\begin{aligned}
\overline{{{ \slashed{D} }}{{\mathcal{J}}}\phi}=&\ \overline{{{\triangleright}}\,\nabla_{{\mathcal{S}}}({{\mathcal{J}}}\phi)}=\overline{{{\triangleright}}\,\sigma_{{\mathcal{S}}}}\Upsilon^{-1}(\star{\otimes}j)\nabla_{{\mathcal{S}}}\phi\ ,\cr
\overline{{{\mathcal{J}}}{{ \slashed{D} }}\phi}=&\ j\,{{ \slashed{D} }}\phi = j\,({{\triangleright}})\,\nabla_{{\mathcal{S}}}\phi\ .\end{aligned}$$ To satisfy property (5), we need $ j\,({{\triangleright}})=\epsilon'\, \overline{{{\triangleright}}\,\sigma_{{\mathcal{S}}}}\Upsilon^{-1}(\star{\otimes}j)$, which we can restate, using $\xi\in\Omega^1$ as $$\begin{aligned}
\label{lapreserves}
{{\mathcal{J}}}(\xi{{\triangleright}}\phi)=\epsilon'\, ({{\triangleright}})\sigma_{{\mathcal{S}}}({{\mathcal{J}}}\phi{\otimes}\xi^*)\ .\end{aligned}$$ The reader may complain that we have used antilinear maps in the tensor product in (\[lapreserves\]), but we have used them in *both* positions with a swap, which is legal. As long as we keep up the bookkeeping, all conjugates and antilinear maps stay legal.
We summarise the above results in the following theorem, stated in bimodule language. Note that we have not yet discussed the Hilbert space structure, we only refer to conditions (2)-(6). We denote by $\mathrm{bb}$ the canonical identification $s\mapsto \overline{\overline{s}}$ of a bimodule ${{\mathcal{S}}}$ with its double conjugate.
\[sptripres\] Suppose that ${{\mathcal{S}}}$ is an $A$-bimodule and $j:{{\mathcal{S}}}\to\overline{{{\mathcal{S}}}}$ a bimodule map obeying $\overline{j}\,j=\epsilon\,\mathrm{bb}:
{{\mathcal{S}}}\to\overline{\overline{{{\mathcal{S}}}}}$. Suppose that $({{\mathcal{S}}},\nabla_{{\mathcal{S}}},\sigma_{{\mathcal{S}}})$ is a left bimodule connection, where $\sigma_{{\mathcal{S}}}$ is invertible, and that $({{\rm id}}{\otimes}j)\nabla_{{\mathcal{S}}}=\nabla_{\overline{{{\mathcal{S}}}}}\,j$ for $\nabla_{\overline{{{\mathcal{S}}}}}$ the canonical left connection on ${\overline{{{\mathcal{S}}}}}$. Suppose that ${{\triangleright}}:\Omega^1{\otimes}_A {{\mathcal{S}}}\to
{{\mathcal{S}}}$ is a bimodule map obeying $ j\,({{\triangleright}})=\epsilon'\, \overline{{{\triangleright}}\,\sigma_{{\mathcal{S}}}}\Upsilon^{-1}(\star{\otimes}j)$. Then ${{ \slashed{D} }}={{\triangleright}}\circ\nabla:{{\mathcal{S}}}\to {{\mathcal{S}}}$ and ${{\mathcal{J}}}:{{\mathcal{S}}}\to{{\mathcal{S}}}$ defined by $j(\phi)=\overline{{{\mathcal{J}}}(\phi)}$ satisfy conditions (2)-(6) for an odd spectral triple.
If there is a bimodule map $\gamma:{{\mathcal{S}}}\to {{\mathcal{S}}}$ with $\gamma^2={{\rm id}}$, which intertwines the connection $\nabla_{{\mathcal{S}}}$, and has $$\gamma\circ{{\triangleright}}=-{{\triangleright}}\circ({{\rm id}}{\otimes}\gamma):\Omega^1{\otimes}_A {{\mathcal{S}}}\to {{\mathcal{S}}}\ ,\
j\circ\gamma=\epsilon''\,\overline{\gamma}\circ j:{{\mathcal{S}}}\to\overline{{{\mathcal{S}}}}\ ,$$ then $({{ \slashed{D} }},{{\mathcal{J}}},\gamma)$ satisfies the conditions (2)-(6) for an even spectral triple.
The complex valued inner product {#dirachermitian}
--------------------------------
A hermitian inner product is antilinear in one position (in this case the first) and linear in the other, so it may be guessed that it appears rather more natural when we use conjugates. If we take a linear map ${{\langle}}\!{{\langle}}\ ,\ {{\rangle}}\!{{\rangle}}:\overline{{{\mathcal{S}}}}{\otimes}_A {{\mathcal{S}}}\to {{\mathbb{C}}}$ then we have the right antilinearity properties, and *explicitly writing the antilinear identity* we have, for $a\in A$ and $\phi,\psi\in{{\mathcal{S}}}$, $${{\langle}}\!{{\langle}}\overline{\psi} , a.\phi {{\rangle}}\!{{\rangle}}= {{\langle}}\!{{\langle}}\overline{\psi}.a , \phi {{\rangle}}\!{{\rangle}}= {{\langle}}\!{{\langle}}\overline{a^*.\psi} , \phi {{\rangle}}\!{{\rangle}}\ ,$$ which is the equation in property (1a), with explicit conjugates added. We have used the standard comma for inner product, but with the conjugate modules notation we could equally consistently have written ${{\langle}}\!{{\langle}}\overline{\psi} {\otimes}\phi {{\rangle}}\!{{\rangle}}$ instead of ${{\langle}}\!{{\langle}}\overline{\psi} , \phi {{\rangle}}\!{{\rangle}}$.
As we have an antilinear map ${{\mathcal{J}}}$, we can define a bilinear inner product, rather than a hermitian inner product, by $(\!(,)\!)={{\langle}}\!{{\langle}},{{\rangle}}\!{{\rangle}}\circ(j{\otimes}{{\rm id}}):{{\mathcal{S}}}{\otimes}_A{{\mathcal{S}}}\to {{\mathbb{C}}}$. This is now complex linear on both sides while the property of ${{\langle}}\!{{\langle}}\ ,\ {{\rangle}}\!{{\rangle}}$ under complex conjugation of the output appears now as $(\!(\psi,\phi)\!)^*={\epsilon}(\!({{\mathcal{J}}}\phi,{{\mathcal{J}}}\psi)\!)$. Then the isometry condition ${{\langle}}\!{{\langle}}\overline{{{\mathcal{J}}}\psi},{{\mathcal{J}}}\phi{{\rangle}}\!{{\rangle}}={{\langle}}\!{{\langle}}\overline{\phi},\psi{{\rangle}}\!{{\rangle}}$ for ${{\mathcal{J}}}$ is now equivalent to $(\!(\psi,{{\mathcal{J}}}\phi)\!) = \epsilon\, (\!({{\mathcal{J}}}\phi,\psi)\!)$, and this reduces to, for all $\phi,\psi\in{{\mathcal{S}}}$, $$\begin{aligned}
(\!(\psi,\phi)\!) = \epsilon\, (\!(\phi,\psi)\!)\ .\end{aligned}$$
Meanwhile, ${{\langle}}\!{{\langle}}\overline{{{ \slashed{D} }}\psi},\phi{{\rangle}}\!{{\rangle}}={\epsilon}(\!({{\mathcal{J}}}{{ \slashed{D} }}\psi,\phi)\!{{)}}$ and ${{\langle}}\!{{\langle}}\overline{\psi},{{ \slashed{D} }}\phi{{\rangle}}\!{{\rangle}}={\epsilon}(\!({{\mathcal{J}}}\psi,{{ \slashed{D} }}\phi)\!{{)}}$, so assuming (5) and relabelling $\psi$, we see that ${{ \slashed{D} }}$ being hermitian is equivalent to showing that $$\label{roundDherm} {\epsilon}'(\!( {{ \slashed{D} }}\psi,\phi)\!{{)}}=(\!( \psi,{{ \slashed{D} }}\phi)\!{{)}}.$$ In the examples we shall deal directly with the definition of ${{ \slashed{D} }}$ being hermitian, but it is interesting to note some conditions on the bilinear inner product which would imply that ${{ \slashed{D} }}$ is hermitian. Note that bimodule connections extend canonically to tensor products.
For ${{ \slashed{D} }}$ constructed as in Theorem \[sptripres\], suppose $$0=(\!( ,)\!{{)}}\circ ({{\triangleright}}{\otimes}{{\rm id}})\nabla_{{{\mathcal{S}}}{\otimes}{{\mathcal{S}}}}:{{\mathcal{S}}}\otimes_A{{\mathcal{S}}}\to {{\mathbb{C}}}\ ,$$ and also that $$(\!( ,)\!{{)}}\circ (({{\triangleright}})\sigma_{{\mathcal{S}}}{\otimes}{{\rm id}})=- \, \epsilon'\, (\!( ,)\!{{)}}\circ ({{\rm id}}{\otimes}{{\triangleright}}): {{\mathcal{S}}}\otimes_A\Omega^1 \otimes_A {{\mathcal{S}}}\to {{\mathbb{C}}}\ .$$ Then ${{ \slashed{D} }}$ is hermitian.
**Proof:**By definition of the connection on tensor products, the first equation is explicitly $$0=(\!( ,)\!{{)}}\circ ({{ \slashed{D} }}{\otimes}{{\rm id}})+ (\!( ,)\!{{)}}\circ (({{\triangleright}})\sigma_{{\mathcal{S}}}{\otimes}{{\rm id}})({{\rm id}}{\otimes}\nabla_{{\mathcal{S}}})\ ,$$ and application of the second displayed equation gives (\[roundDherm\]). $\largesquare$
Note that we do not require that ${{\langle}}\!{{\langle}}\ ,\ {{\rangle}}\!{{\rangle}}$ is the composition of a positive linear functional with an $A$-valued hermitian inner product and $\nabla_S$ hermitian metric compatible (but both of these further features will apply in the $q$-sphere example).
Inner fluctuations
------------------
Given a bimodule $L$, there is a functor $\mathcal{G}_L$ from the category ${}_A\mathcal{M}_A$ of $A$-bimodules to itself given by $\mathcal{G}_L(E)=\overline{L}{\otimes}_A E{\otimes}_A L$, sending a bimodule map $\theta:E\to F$ to ${{\rm id}}{\otimes}\theta{\otimes}{{\rm id}}$. If we have a given isomorphism $L{\otimes}_A \overline{L}\, \cong\, A$ of $A$-bimodules, then $$\mathcal{G}_L(E){\otimes}_A \mathcal{G}_L(F)=
\overline{L}{\otimes}_A E{\otimes}_A L {\otimes}_A \overline{L}{\otimes}_A F{\otimes}_A L\, \cong\,\mathcal{G}_L(E{\otimes}_A F)\ ,$$ so the functor preserves the tensor product.
The description of Morita contexts can be found in [@BassK] (and a $C^*$-algebra description in [@RieffelMorita]), and involves a bimodule $L$ so that the tensor product of $L$ with its dual, both ways round, is isomorphic to $A$, and the two isomorphisms obey associativity conditions. The special case we have is where the dual of the bimodule is its conjugate, and we get a non-degenerate inner product. In [@NCline] this case is shown to give rise to an integer graded star algebra which is $L{\otimes}_A \dots{\otimes}_A L$ in positive degrees and $\overline{L}{\otimes}_A \dots{\otimes}_A \overline{L}$ in negative degrees. This star algebra can be thought of as the algebra of functions on a principal circle bundle on the noncommutative space, an idea defined formally in terms of a quantum principal bundle or Hopf-Galois extension [@BrzMaj:gau].
Take $\overline{c}\in\overline{L}$ and $x\in L$ which are inverses under the product, i.e.$\overline{c}{\otimes}x$ corresponds to $1\in A$. It will be convenient to write this identification as an inner product ${{\langle}},{{\rangle}}_L:\overline{L}{\otimes}_A L\to A$. Then there is a linear map $\overline{c}{\otimes}-{\otimes}x:E\to \mathcal{G}_L(E)$ given by $e\mapsto \overline{c}{\otimes}e{\otimes}x$, which is not necessarily a bimodule map. However it does have the tensorial property $$\xymatrix{
E\otimes_A F \ar[rr]^{ (\overline{c}{\otimes}-{\otimes}x){\otimes}(\overline{c}{\otimes}-{\otimes}x) \ \ \ \ \ } \ar[dr]^{\overline{c}{\otimes}-{\otimes}x } & & \mathcal{G}_L(E) \otimes_A \mathcal{G}_L(F) \\
& \mathcal{G}_L(E\otimes_A F) \ar[ur]^{\cong}
}$$
Now suppose we have a left bimodule connection $\nabla_L: L \to \Omega^1{\otimes}_A L$ and invertible $\sigma_L: L{\otimes}_A\Omega^1 \to \Omega^1{\otimes}_A L$. If the connection preserves the inner product we get $$\nabla_{ \overline{L} }{\otimes}{{\rm id}}+(\sigma_{ \overline{L} }{\otimes}{{\rm id}})({{\rm id}}{\otimes}\nabla_L)={{\rm d}}\circ {{\langle}},{{\rangle}}:\overline{L}{\otimes}_A L\to \Omega^1\ ,$$ and from this, remembering that the inner product is invertible, $$\sigma_{ \overline{L} }{}^{-1}\,\nabla_{ \overline{L} }(\overline{c}){\otimes}x=-\, \overline{c}{\otimes}\nabla_L (x)\ .$$ As $x$ is invertible, we can write $\nabla_L(x)=\kappa{\otimes}x$ for some $\kappa\in\Omega^1$, and then we deduce $\sigma_{ \overline{L} }{}^{-1}\,\nabla_{ \overline{L} }(\overline{c})=-\,\overline{c}{\otimes}\kappa$.
Now return to the assumptions and notations on the left bimodule connection $({{\mathcal{S}}},\nabla_{{\mathcal{S}}},\sigma_{{\mathcal{S}}})$ and the action ${{\triangleright}}:\Omega^1{\otimes}_A {{\mathcal{S}}}\to {{\mathcal{S}}}$ which earlier we related to the Dirac operator. Define an action of $\Omega^1$ on $\overline{L}{\otimes}_A {{\mathcal{S}}}{\otimes}_A L$ by $$\xi{{\triangleright}}(\overline{y}{\otimes}\phi{\otimes}x)=({{\rm id}}{\otimes}{{\triangleright}}{\otimes}{{\rm id}})(\sigma_{\overline{L}}{}^{-1}(\xi{\otimes}\overline{y}){\otimes}\phi{\otimes}x)$$ and the Dirac operator ${{ \slashed{D} }}_{\mathcal{G}_L({{\mathcal{S}}})}$ is the composition of this with the standard tensor product covariant derivative, $$\nabla_{\overline{L}}{\otimes}{{\rm id}}{\otimes}{{\rm id}}+(\sigma_{\overline{L}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})\big(
({{\rm id}}{\otimes}\nabla_{{\mathcal{S}}}{\otimes}{{\rm id}})+({{\rm id}}{\otimes}\sigma_{{\mathcal{S}}}{\otimes}{{\rm id}})({{\rm id}}{\otimes}{{\rm id}}{\otimes}\nabla_L)\big)\ .$$ On taking the composition we get some simplification, giving $$\begin{aligned}
{{ \slashed{D} }}_{\mathcal{G}_L({{\mathcal{S}}})} =&\ (\sigma_{\overline{L}}{}^{-1}\nabla_{\overline{L}}){{\triangleright}}{{\rm id}}{\otimes}{{\rm id}}+{{\rm id}}{\otimes}{{ \slashed{D} }}_{{\mathcal{S}}}{\otimes}{{\rm id}}+
({{\rm id}}{\otimes}({{\triangleright}})\sigma_{{\mathcal{S}}}{\otimes}{{\rm id}})({{\rm id}}{\otimes}{{\rm id}}{\otimes}\nabla_L)\ .\end{aligned}$$ Now consider the commutative diagram, $$\xymatrix{
{{\mathcal{S}}}\ar[r]^{\overline{c}{\otimes}- {\otimes}x\ \ \ } \ar[d]^{{{ \slashed{D} }}_{{\mathcal{S}}}} & \mathcal{G}_L({{\mathcal{S}}}) \ar[d]^{ {{ \slashed{D} }}_{\mathcal{G}_L({{\mathcal{S}}})} } \\
{{\mathcal{S}}}\ar[r] & \mathcal{G}_L({{\mathcal{S}}})
}$$ where the bottom line is $$\phi \longmapsto \overline{x}{\otimes}{{ \slashed{D} }}_{{\mathcal{S}}}(\phi){\otimes}x- \overline{c}{\otimes}\kappa{{\triangleright}}\phi {\otimes}x+ \overline{c}{\otimes}({{\triangleright}})\sigma_{{\mathcal{S}}}(\phi{\otimes}\kappa){\otimes}x\ .$$ This can be rewritten as $$\xymatrix{
{{\mathcal{S}}}\ar[r]^{\overline{c}{\otimes}- {\otimes}x\ \ \ } \ar[d]^{{{ \slashed{D} }}_{{\mathcal{S}}}+\hat\kappa } & \mathcal{G}_L({{\mathcal{S}}}) \ar[d]^{ {{ \slashed{D} }}_{\mathcal{G}_L({{\mathcal{S}}})} } \\
{{\mathcal{S}}}\ar[r]^{\overline{c}{\otimes}- {\otimes}x\ \ \ } & \mathcal{G}_L({{\mathcal{S}}})
}$$ where $\hat\kappa:{{\mathcal{S}}}\to {{\mathcal{S}}}$ for $\kappa\in\Omega^1$ is given by $$\hat\kappa(\phi)= ({{\triangleright}})\sigma_{{\mathcal{S}}}(\phi{\otimes}\kappa) - \kappa{{\triangleright}}\phi\ .$$ Rewriting (\[lapreserves\]) gives ${{\mathcal{J}}}(\kappa^* {{\triangleright}}{{\mathcal{J}}}^{-1} \phi)=\epsilon'\, ({{\triangleright}})\sigma_{{\mathcal{S}}}(\phi{\otimes}\kappa)$, so $$\begin{aligned}
\hat\kappa(\phi)= \epsilon'\, {{\mathcal{J}}}(\kappa^* {{\triangleright}}{{\mathcal{J}}}^{-1} \phi) -\kappa{{\triangleright}}\phi \ .\end{aligned}$$ If we follow [@ConMar] and specialise to the case where $L$ is $A$, then we can choose $x$ to be a unitary, in which case $\kappa^*=-\kappa$, and we have $$\begin{aligned}
\hat\kappa(\phi)= -\, \epsilon'\, {{\mathcal{J}}}(\kappa {{\triangleright}}{{\mathcal{J}}}^{-1} \phi) -\kappa{{\triangleright}}\phi \ ,\end{aligned}$$ in agreement with the usual formula for inner fluctuations. In [@ConMar] it is explained that the inner fluctuations of the standard model of particle physics correspond to the gauge bosons other than the graviton, and arise via the mechanism of Morita equivalences.
Examples of bimodule connections and Dirac operators
====================================================
Now we shall give three examples of our geometrical construction of Dirac operators from bimodule connections, on a matrix algebra, a quantum sphere, and a quantum disk.
A Dirac operator on $M_2({{\mathbb{C}}})$ {#SecMatrix}
-----------------------------------------
Take the algebra $A=M_2({{\mathbb{C}}})$, with calculus $$\Omega^1=\Omega^{1,0}\oplus \Omega^{0,1}=M_2 \oplus M_2\ ,$$ which we also write as $\Omega^{1,0}=M_2\,s$ and $\Omega^{1,0}=M_2\,t$, where $s=I_2\oplus 0$ and $t=0\oplus I_2$ are central elements (including $st=ts$), and to have a two dimensional calculus we impose $s^2=t^2=0$. The differential ${{\rm d}}$ is the graded commutator $[E_{12}s+E_{21}t,-\}$ (i.e. the commutator when applied to even forms, and the anticommutator on odd forms). This is a star calculus, where we use the usual star on matrices and $s^*=-t$.
Take an ansatz for a particular Dirac operator on the left module $\mathcal{S}=M_2({{\mathbb{C}}})\oplus M_2({{\mathbb{C}}})$, with the action of matrix product on each summand. The Hilbert space inner product is ${{\langle}}\!{{\langle}}\overline{x\oplus u},y\oplus v{{\rangle}}\!{{\rangle}}=\mathrm{Tr}(x^*y+u^*v)$. Define ${{ \slashed{D} }}$ by the following formula, $${{ \slashed{D} }}(x\oplus u)=(d_1u+uc_1) \oplus (d_2x+xc_2)\ ,$$ for matrices $c_i,d_i$. Now $$\begin{aligned}
{{\langle}}\!{{\langle}}\overline{{{ \slashed{D} }}(x\oplus u)},y\oplus v{{\rangle}}\!{{\rangle}}=&\
{{\langle}}\!{{\langle}}\overline{(d_1u+uc_1) \oplus (d_2x+xc_2)},y\oplus v{{\rangle}}\!{{\rangle}}\cr
=&\ \mathrm{Tr}(u^*d_1^*y+c_1^*u^*y+x^*d_2^*v+c_2^*x^*v)\ ,\cr
{{\langle}}\!{{\langle}}\overline{x\oplus u},{{ \slashed{D} }}(y\oplus v){{\rangle}}\!{{\rangle}}=&\
{{\langle}}\!{{\langle}}\overline{x\oplus u},(d_1v+vc_1) \oplus (d_2y+yc_2){{\rangle}}\!{{\rangle}}\cr
=&\ \mathrm{Tr}(u^*d_2y+u^*yc_2+x^*d_1v+x^*vc_1)\ .\end{aligned}$$ To have ${{ \slashed{D} }}$ hermitian we need $d_2=d_1^*$ and $c_2=c_1^*$. Define ${{\mathcal{J}}}:\mathcal{S} \to \mathcal{S}$ by ${{\mathcal{J}}}(x\oplus u)=(-u^*)\oplus x^*$, so $\epsilon=-1$. Now we have $
{{\mathcal{J}}}b{{\mathcal{J}}}^{-1}(x\oplus u)=x\,b^*\oplus u\,b^*$. Next $$\begin{aligned}
{{\mathcal{J}}}{{ \slashed{D} }}(x\oplus u)=&\ (-(d_1^*x+xc_1^*)^*) \oplus (d_1u+uc_1)^*\ ,\cr
{{ \slashed{D} }}{{\mathcal{J}}}(x\oplus u)=&\ (d_1x^*+x^*c_1) \oplus (-d_1^*u^*-u^*c_1^*)\end{aligned}$$ so $c_1=-d_1$ gives $\epsilon'=1$. Now the grading operator $\gamma(x\oplus u)=(-x)\oplus u$ completes the set of operators for dimension $n=2$ with ${\epsilon}''=-1$. We calculate $$[{{ \slashed{D} }},a](x\oplus u)=[d_1,a]u \oplus [d_1^*,a]x\ .$$ To fit with the differential structure we set $d_1=E_{12}$, and seek ${{\triangleright}}$ so that ${{\rm d}}a{{\triangleright}}(x\oplus u)=[{{ \slashed{D} }},a](x\oplus u)$ or $$([E_{12},a]\oplus [E_{21},a]){{\triangleright}}(x\oplus u)=[E_{12},a]u \oplus [E_{21},a]x$$ which we solve by defining the action of $\Omega^1$ as $(p\oplus q){{\triangleright}}(x\oplus u)=p\,u\oplus q\,x$. The required connection $\nabla_{{\mathcal{S}}}$ is then $$\nabla_{{\mathcal{S}}}(x\oplus u) = {{\rm d}}x{\otimes}(1\oplus 0) +
{{\rm d}}u{\otimes}(0\oplus 1)\ ,$$ and a little calculation gives, for $\xi\in\Omega^1=M_2\oplus M_2$, $$\sigma_{{\mathcal{S}}}((x\oplus u){\otimes}\xi) = x.\xi{\otimes}(1\oplus 0) +
u.\xi{\otimes}(0\oplus 1)\ .$$ With these choices we then verify the condition in Proposition \[Diracnabla\], $$\begin{aligned}
{{\mathcal{J}}}[{{ \slashed{D} }},b]{{\mathcal{J}}}^{-1}(x\oplus u) =&\ u[E_{12},b^*]\oplus x [E_{21},b^*]
={{\triangleright}}\sigma\big((x\oplus u) {\otimes}([E_{12},b^*]\oplus [E_{21},b^*])\big)\ .\end{aligned}$$ so this proposition applies. Similarly, we can check directly that $${{\langle}}\!{{\langle}}\overline{ {{\mathcal{J}}}(x\oplus u) },{{\mathcal{J}}}(y\oplus v){{\rangle}}\!{{\rangle}}=
{{\langle}}\!{{\langle}}\overline{ (-v^*)\oplus y^* },(-u^*)\oplus x^*{{\rangle}}\!{{\rangle}}=
\mathrm{Tr}(vu^*+yx^*) = {{\langle}}\!{{\langle}}\overline{x\oplus u},y\oplus v{{\rangle}}\!{{\rangle}}$$ so ${{\mathcal{J}}}$ is an isometry. We can also recover this and that ${{ \slashed{D} }}$ is hermitian from our deduced data and application of Section \[dirachermitian\].
To compare this with the known classification of spectral triples on matrix algebras, we refer to [@Barrmatrix]. We write ${{\mathcal{S}}}={{\mathbb{C}}}^2{\otimes}M_2({{\mathbb{C}}})$ by writing $x\oplus u$ as a vector, where $x,u\in M_2({{\mathbb{C}}})$. We take the signature $(0,2)$ Clifford algebra with $\{\gamma^i,\gamma^j\}=-2\delta_{ij}$ given by $\gamma^i={\mathrm{i}}\sigma^i$ in terms of Pauli matrices, $i=1,2$. We set $\gamma={\mathrm{i}}^{3}\gamma^1\gamma^2=-\sigma^3$ which agrees with the one above. We also need an antilinear $C$ such that $C^2={\epsilon}=-1$, $(Cv, Cw)=(w,v)$ for the standard left-antilinear inner product on ${{\mathbb{C}}}^2$, and $C\gamma^i={\epsilon}'\gamma^iC=\gamma^iC$. The operation $$C\begin{pmatrix} v_1 \cr v_2\end{pmatrix}=\begin{pmatrix} -\overline{v_2} \cr \overline{v_1}\end{pmatrix}$$ does the job and ${{\mathcal{J}}}=C{\otimes}(\ )^*$ then gives the same ${{\mathcal{J}}}$ as above. Finally, our Dirac operator can now be written as $${{ \slashed{D} }}=-{1\over 2}\left(\gamma^1{\otimes}[\gamma^1,\ ]-\gamma^2{\otimes}[\gamma^2,\ ]\right)$$ which is a specific member of the general class of spectral triple here (where commutators in general are by arbitrary antihermitian matrices). We have seen how this arises naturally from an action ${{\triangleright}}$ and a bimodule connection.
A Dirac operator on the noncommutative Hopf fibration {#ncHopfDirac}
-----------------------------------------------------
We follow the construction of the $q$-Dirac operator on the standard $q$-sphere as a framed quantum homogeneous space in [@Ma:rieq] but with a couple of constant parameters (which can be seen as normalisations) and now with consideration of $*$, ${{\mathcal{J}}}$ and an inner product which we not covered there. We recall that the algebra ${{\mathbb{C}}}_q[SU_2]$ has generators $a,b,c,d$, which are assigned grades $|a|=|c|=1$ and $|b|=|d|=-1$ and we use the conventions where $ba=qab$ etc. The standard $q$-sphere $A={{\mathbb{C}}}_q[S^2]$ is the subalgebra of grade zero elements in ${{\mathbb{C}}}_q[SU_2]$. The usual 3D calculus for ${{\mathbb{C}}}_q[SU_2]$ in [@Wor] has basis 1-forms $e^0,e^\pm$ of grades $|e^0|=0$ and $|e^\pm|=\pm 2$ and bimodule commutation relations $e^0x=q^{2|x|}x e^0$ central and $e^\pm x=q^{ |x|}x e^\pm$. For a calculus on the sphere, we take the horizontal forms (with basis $e^\pm$ of degree $|e^\pm|=\pm 2$), and then the grade zero submodule. This means that $\Omega^{1,0}$ and $\Omega^{0,1}$ for the cotangent bundle on the $q$-sphere can be identified with the degree $\mp 2$ subspaces of ${{\mathbb{C}}}_q[SU_2]$ respectively. Note also in this construction that both $\Omega^1$ and the horizontal forms $\Omega^1_{hor}$ on ${{\mathbb{C}}}_q[SU_2]$ are free modules (with basis $e^\pm,e^0$ and $e^\pm$ respectively) so we have a canonical projection $\pi:\Omega^1\to \Omega^1_{hor}$ of free left ${{\mathbb{C}}}_q[SU_2]$ which will be useful in computations, given by $e^0\to 0$. This is the set-up for the quantum Riemannian geometry of the standard $q$-sphere as a quantum homogeneous space from the quantum Hopf fibration [@Ma:rieq]; the $q$-monopole connection on the quantum principal bundle induces a canonical choice of ‘quantum Levi-Civita’ connection on $\Omega^1=\Omega^{1,0}\oplus \Omega^{0,1}$.
For the spin bundle we similarly set generators $f^\pm$ with grades $|f^\pm|=\pm1$, and ${{\mathcal{S}}}_\pm$ to be the grade zero elements in ${{\mathbb{C}}}_q[SU_2].f^\pm$, with ${{\mathcal{S}}}={{\mathcal{S}}}_+\oplus {{\mathcal{S}}}_-$. Suppose that the generators commute with all grade zero algebra elements. In other words, ${{\mathcal{S}}}_\pm$ can be identified with the degree $\mp1$ subspace of ${{\mathbb{C}}}_q[SU_2]$ and as a (bi)-module over ${{\mathbb{C}}}_q[S^2]$ (which means that $f^\pm$ commute with elements of $A$). This is again the set-up used in [@Ma:rieq] for the spin bundle as charge $\pm1$ $q$-monopole sections and again the $q$-monopole induces a covariant derivative $\nabla_S:{{\mathcal{S}}}\to \Omega^1{\otimes}_A{{\mathcal{S}}}$. This is well-known and given explicitly by $$\nabla_{{\mathcal{S}}}(x\,f^++y\,f^-)=\pi{{\rm d}}x.a {\otimes}d.f^+ -q^{-1}\, \pi{{\rm d}}x. c{\otimes}b .f^+ +\pi {{\rm d}}y. d {\otimes}a .f^- - q\,\pi{{\rm d}}y.b {\otimes}c .f^-$$ One can check that this is a bimodule connection with $$\sigma_S((x f^++ yf^-){\otimes}f e^{\pm})=xf e^{\pm }(a{\otimes}d - q^{-1}c{\otimes}b)f^+ + y f e^{\pm }(d{\otimes}a - q b{\otimes}c)f^-$$ for $f$ of degree $\mp 2$.
For the action ${{\triangleright}}$ of $\Omega^1$ on the spinors which preserves grades, we follow [@Ma:rieq] and set $$f e^+{{\triangleright}}y f^- = \alpha\, f y f^+\ ,\ f e^-{{\triangleright}}x f^+ = \beta\, f x f^- , \quad {{\triangleright}}:\Omega^{1,0}{\otimes}{{\mathcal{S}}}_-\to {{\mathcal{S}}}_+,\quad {{\triangleright}}:\Omega^{0,1}{\otimes}{{\mathcal{S}}}_+\to {{\mathcal{S}}}_-$$ and other degree combinations zero, where we have explicitly inserted two constant complex parameters $\alpha,\beta$ (one could absorb one of these in the normalisation of the $f^\pm$). Apart from the constant parameters, this is just the product of the appropriate degree subspaces inside ${{\mathbb{C}}}_q[SU_2]$ as in [@Ma:rieq].
If we write $\pi{{\rm d}}x={{\partial}}_+ x\,e^++{{\partial}}_- x\,e^-$, the Dirac operator ${{ \slashed{D} }}=({{\triangleright}}{\otimes}{{\rm id}})\nabla_{{\mathcal{S}}}$ comes out for $|x|=-1$ and $|y|=1$ as $${{ \slashed{D} }}(x\,f^++y\,f^-) = \alpha q^{-1} {{\partial}}_+ y\, f^+ +\beta q\, {{\partial}}_- x\,f^-$$ which apart from the $\alpha,\beta$ weightings completes our recap of the Dirac operator introduced in [@Ma:rieq]. The grading bimodule map is given by $\gamma=\pm{{\rm id}}$ on ${{\mathcal{S}}}_\pm$.
The new ingredient we need beyond [@Ma:rieq] is ${{\mathcal{J}}}$. We set ${{\mathcal{J}}}(x f^\pm)=\pm\delta^{\pm1}\,x^* f^\mp$ for $\delta$ real, giving ${\epsilon}=-1$ and $\epsilon''=-1$. The connection preserves $j$ since it vanishes on the generators, while using $e^{\pm *}=-q^{\mp 1}e^\mp$ we get $$\begin{aligned}
&({{\triangleright}})\sigma_{{\mathcal{S}}}({{\mathcal{J}}}(x\, f^+){\otimes}(y\, e^{-})^*) = -\,\delta\,q ({{\triangleright}})\sigma_{{\mathcal{S}}}(x^* f^- {\otimes}e^+\,y^*)\cr
=&\ -\,\delta\,q^{-1} ({{\triangleright}})\sigma_{{\mathcal{S}}}(x^* f^- {\otimes}y^*\,e^+) = -\,\delta\,q^{-1} ({{\triangleright}})(x^*\,y^*\, e^{+ }(d{\otimes}a - q b{\otimes}c)f^-)\cr
=&\ -\,\delta\,q^{-2} ({{\triangleright}})(x^*\,y^*(d\, e^{+ }{\otimes}a - q\, b\, e^{+ }{\otimes}c)f^-)
= -\,\alpha\,\delta\,q^{-2} x^*\,y^*\,f^+\ ,\cr
&({{\triangleright}})\sigma_{{\mathcal{S}}}({{\mathcal{J}}}(x\, f^-){\otimes}(y\, e^{+})^*) = \delta^{-1}q^{-1}\,({{\triangleright}})\sigma_{{\mathcal{S}}}(x^* f^+ {\otimes}e^-\,y^*)\cr
=&\ \delta^{-1}q^{1}\,({{\triangleright}})\sigma_{{\mathcal{S}}}(x^* f^+ {\otimes}y^*\,e^-) = \delta^{-1}q^{1}\,({{\triangleright}})(x^*\,y^*\, e^{- }(a{\otimes}d - q^{-1}c{\otimes}b)f^+)\cr
=&\ \delta^{-1}q^{2}\,({{\triangleright}})(x^*\,y^*(a\, e^{- }{\otimes}d - q^{-1}c\, e^{- }{\otimes}b)f^+)
= \beta\, \delta^{-1}q^{2}\,x^*\,y^*\, f^-\ .\end{aligned}$$ Referring back to (\[lapreserves\]) with ${\epsilon}'=1$, we need to compare these results with $$\begin{aligned}
{{\mathcal{J}}}(y\, e^{-}{{\triangleright}}x\, f^+) =&\ {{\mathcal{J}}}(\beta\,y\,x\,f^-) = -\,\delta^{-1}\,\beta^*\, (y\,x)^*\, f^+\ ,\cr
{{\mathcal{J}}}(y\, e^{+}{{\triangleright}}x\, f^-) =&\ {{\mathcal{J}}}(\alpha\,y\,x\,f^+) = \delta\,\alpha^*(y\,x)^*\,f^-\ .\end{aligned}$$ In the case $\epsilon'=1$, (\[lapreserves\]) becomes the condition $$\label{qsphereparam} \delta^2\,\alpha^* = \beta\, q^{2},$$ for $q$ real, which requires that $\beta/\alpha^*$ is real. Assuming the latter, we therefore define $\delta$ as the (say, positive) square root of $\beta q^2/\alpha^*$ and have now satisfied all the algebraic axioms (2)-(6) of a spectral with dimension $n=2$, by Theorem \[sptripres\].
Next we define a positive hermitian inner product ${{\langle}},{{\rangle}}:\overline{ {{\mathcal{S}}}}{\otimes}_A {{\mathcal{S}}}\to A$ by the following, for some $\mu>0$, $${{\langle}}\overline{ x_+\, f^+ + x_-\, f^-}, y_+\, f^+ + y_-\, f^- {{\rangle}}=x_+{}^*\,y_+ + \mu\, x_-{}^*\,y_- \ .$$ So far we have a $A$ valued inner product, but we really need an honest ${{\mathbb{C}}}$ valued inner product for a Dirac operator. We define $${{\langle}}\!{{\langle}},{{\rangle}}\!{{\rangle}}=\frac{\smallint {{\langle}},{{\rangle}}\,e^+\wedge e^-}{\smallint e^+\wedge e^-}\ .$$ where $\smallint$ is the de Rham cohomology class in $H_{dR}^2({{\mathbb{C}}}_q[S^2])\cong {{\mathbb{C}}}$. This gives a hermitian inner product ${{\langle}}\!{{\langle}},{{\rangle}}\!{{\rangle}}:\overline{ {{\mathcal{S}}}}{\otimes}_A {{\mathcal{S}}}\to {{\mathbb{C}}}$. (This is just the Haar integral of the $A$-valued inner product.)
\[prrp1\] For $ {{\langle}}\!{{\langle}}\ ,\ {{\rangle}}\!{{\rangle}}$ defined by the Haar integral and $\mu=q\delta^{-2}$, ${{ \slashed{D} }}$ is hermitian.
**Proof:** We have, using our notations, $$\begin{aligned}
{{\langle}}\overline{{{ \slashed{D} }}(x\,f^+)},y\,f^-{{\rangle}}=&\ \beta^*\, q\, {{\langle}}\overline{ {{\partial}}_- x\,f^- },y\,f^-{{\rangle}}=\
\beta^*\, q\,\mu\, ({{\partial}}_- x)^*\,y \ ,\cr
{{\langle}}\overline{x\,f^+},{{ \slashed{D} }}(y\,f^-){{\rangle}}=&\ \alpha\, q^{-1}\, {{\langle}}\overline{x\,f^+}, {{\partial}}_+ y\, f^+ {{\rangle}}= \alpha\, q^{-1}\, x^*\, {{\partial}}_+ y
\ \end{aligned}$$ for all $x\,f^+$ and $y\,f^-$ of grade zero. So if $ \beta^*\, q\,\mu=\alpha$ we have $$\begin{aligned}
{{\langle}}\overline{x\,f^+},{{ \slashed{D} }}(y\,f^-){{\rangle}}- {{\langle}}\overline{{{ \slashed{D} }}(x\,f^+)},y\,f^-{{\rangle}}= \alpha\,q^{-1}\,(x^*{{\partial}}_+ y
- q\,({{\partial}}_- x){}^*y)\ .\end{aligned}$$ Using $|x|=-1$ and $|y|=1$, with $\pi{{\rm d}}x={{\partial}}_+ x\,e^++{{\partial}}_- x\,e^-$ etc, we also have $$\begin{aligned}
\pi{{\rm d}}(x^*y)=&\ (x^*{{\partial}}_+y - q\, ({{\partial}}_- x){}^*y)\, e^+ + (x^*{{\partial}}_-y - q^3\, ({{\partial}}_+x){}^*y)\, e^-\ ,\cr
\pi{{\rm d}}(x^*y\, e^-)=&\
\pi{{\rm d}}(x^*y)\wedge e^-= (x^*{{\partial}}_+y - q\, ({{\partial}}_- x){}^*y)\, e^+ \wedge e^-\ ,\end{aligned}$$ which gives $$\begin{aligned}
{{\langle}}\!{{\langle}}\overline{x\,f^+},{{ \slashed{D} }}(y\,f^-){{\rangle}}\!{{\rangle}}= {{\langle}}\!{{\langle}}\overline{{{ \slashed{D} }}(x\,f^+)},y\,f^-{{\rangle}}\!{{\rangle}}\ ,\end{aligned}$$ on applying the cohomology class given by the Haar integral. Taking the complex conjugate provides the other equation needed to show that ${{ \slashed{D} }}$ is hermitian. The condition on the parameters here is equivalent to the one stated given that we already assumed (\[qsphereparam\]). $\largesquare$
Proceeding with ${{ \slashed{D} }}$ hermitian by the above proposition, it remains to look at the isometry property of ${{\mathcal{J}}}$. For this we note that the underlying Haar integral on functions, $\int:{{\mathbb{C}}}_q[SU_2]\to {{\mathbb{C}}}$, is well-known to be a twisted trace in that there is an algebra automorphism $\varsigma$ such that $\int xy = \int \varsigma(y)x$ for all $x,y$ in ${{\mathbb{C}}}_q[SU_2]$. Explicitly, $$\varsigma(a^ib^jc^kd^l)=q^{2(l-i)}a^ib^jc^kd^l$$ on monomials from which one can see that $\varsigma$ preserves degree and skew-commutes with $*$ in the sense $\varsigma(x^*)=(\varsigma^{-1}(x))^*$ for all $x\in {{\mathbb{C}}}_q[SU_2]$.
\[prrp2\] For $ {{\langle}}\!{{\langle}}\ ,\ {{\rangle}}\!{{\rangle}}$ defined by the Haar integral and $q\ne 1$, ${{\mathcal{J}}}$ is not an isometry but obeys $${{\langle}}\!{{\langle}}\overline{{{\mathcal{J}}}(x.f^\pm)},{{\mathcal{J}}}(y.f^\pm){{\rangle}}\!{{\rangle}}=q^{\pm 1} {{\langle}}\!{{\langle}}\overline{\varsigma^{-1}(y).f^\pm},x.f^\pm{{\rangle}}\!{{\rangle}},\quad\forall |x|,|y|=\mp1$$ where $\varsigma$ is an algebra automorphism whereby the Haar integral is a twisted trace.
**Proof:** Consider, for $|x|=|y|=\mp1$, $${{\langle}}\overline{{{\mathcal{J}}}(x.f^\pm)},{{\mathcal{J}}}(y.f^\pm){{\rangle}}= {{\langle}}\overline{ \pm\delta^{\pm1}\,x^* f^\mp }, \pm\delta^{\pm1}\,y^* f^\mp {{\rangle}}= \left\{\begin{array}{cc} \delta^2\,\mu\,x\,y^* & \mathrm{upper\ sign} \\ \delta^{-2}\, x\,y^* & \mathrm{lower\ sign}\end{array}\right.$$ and compare this to $${{\langle}}\overline{y.f^\pm},x.f^\pm{{\rangle}}= \left\{\begin{array}{cc}y^*\,x & \mathrm{upper\ sign} \\ \mu\,y^*\,x & \mathrm{lower\ sign}\end{array}\right.$$ Given than $\delta^2\mu=q$ and integrating, the twisted trace property tells us that ${{\mathcal{J}}}$ is some kind of twisted $q$-isometry in the manner stated. It is clear when $q\ne 1$ that we can fun instances proving that ${{\mathcal{J}}}$ is not a usual isometry. $\largesquare$
Note that the Haar integral is not the only choice. If we take an ordinary trace in the form of a linear map $\tau:A\to {{\mathbb{C}}}$ such that $\tau(xy)=\tau(yx)$, we can define ${{\langle}}\!{{\langle}}\ ,\ {{\rangle}}\!{{\rangle}}=\tau{{\langle}}\ ,\ {{\rangle}}$ in the same way, the above proof shows that we do then have ${{\mathcal{J}}}$ an isometry when $\mu=\delta^{-2}$, but we would then lose that ${{ \slashed{D} }}$ is hermitian as this depended on translation invariance in the form of vanishing on a total differential.
To summarise, to finish off the algebraic conditions (2)-(6) we needed $ \alpha = \beta^*\, q^{2}\,\delta^{-2} $ and for ${{ \slashed{D} }}$ to be hermitian we need $\alpha= \beta^*\, q\,\mu$ which, given the first condition is equivalent to $\mu=q\delta^{-2}$. However we cannot in general make ${{\mathcal{J}}}$ an isometry when $q\ne 1$. Finally, by rescaling of the $f^\pm$ while preserving the form of our other constructions (this requires $f^+$ to change by at most a phase), we can without loss of generality set $\alpha=1$ and then have only one free parameter $\beta>0$ in our above construction, with $\delta=\sqrt{\beta} q$ and $\mu=\beta^{-1} q^{-1}$ uniquely determined up to the sign of $\delta$. Thus we have a 1-parameter moduli of Dirac operators under the above construction, with $\beta=1$ recovering the ${{ \slashed{D} }}$ introduced in [@Ma:rieq].
Considering the results on Dirac operators on the non commutative sphere in [@DLPS-sphere], this should not come as a surprise that we cannot obey all the conditions. There the conclusion was to sacrifice the bimodule condition, which is (4) on our list, and replace it by the commutator being a compact operator. However we have kept all the algebraic conditions including the bimodule condition and $\nabla_{{\mathcal{S}}}$ preserving $j$, kept ${{ \slashed{D} }}$ being hermitian and dropped only that ${{\mathcal{J}}}$ is an isometry in favour of some twisted $q$-version of that.
A Dirac operator on the quantum disk {#diskdir}
------------------------------------
There is an algebra of functions on a ‘deformed disk’ $A={{\mathbb{C}}}_q[D]$, generated by $z$ and $\bar z$ with commutation relation $z\bar z=q^{-2}\bar zz-q^{-2}+1$ and involution $z^*=\bar z$ with $q$ real and nonzero (see [@klilesdisk]). The algebra ${{\mathbb{C}}}_q[D]$ is $\mathbb{Z}$ graded, by $|z|=1$ and $|\bar z|=-1$. Putting ${{w}}=1-\bar zz$, we have $z{{w}}=q^{-2}{{w}}z$ and $\bar z{{w}}=q^{2}{{w}}\bar z$, so for any polynomial $p({{w}})$ $$z.p({{w}})=p(q^{-2}{{w}}).z\ ,\quad \bar z.p({{w}})=p(q^{2}{{w}}).\bar z\ .$$
There is a differential calculus given by $$\begin{aligned}
{{\rm d}}z\wedge {{\rm d}}\bar z=-q^{-2}\, {{\rm d}}\bar z\wedge {{\rm d}}z\ &,& z. {{\rm d}}z=q^{-2}\, {{\rm d}}z. z\ ,\quad
z. {{\rm d}}\bar z=q^{-2}\, {{\rm d}}\bar z. z\ ,\cr {{\rm d}}z\wedge {{\rm d}}z={{\rm d}}\bar z\wedge {{\rm d}}\bar z=0 &,&
\bar z. {{\rm d}}z=q^{2}\, {{\rm d}}z. \bar z\ ,\quad \bar z. {{\rm d}}\bar z=q^{2}\, {{\rm d}}\bar
z. \bar z\ .\end{aligned}$$ \[dpolydisk\] A proof by induction on powers of ${{w}}$ gives, for any polynomial $p({{w}})$, $$\begin{aligned}
\label{missedp}
{{\rm d}}p({{w}}) =&\ q^2 \frac{p(q^{-2}{{w}})-p({{w}})}{{{w}}(1-q^{-2})} z\, {{\rm d}}\bar z\ +
\frac{p(q^{2}{{w}})-p({{w}})}{{{w}}(1-q^{2})}\bar z\, {{\rm d}}z\ .\end{aligned}$$
Recall that the $*$-Hopf algebra $U_q(su_{1,1})$ is defined by generators $X_+,X_-$ and an invertible grouplike generator $q^{H\over 2}$ with $$q^{H\over 2}X_\pm q^{-{H\over 2}}=q^{\pm 1}X_\pm,\quad [X_+,X_-]={q^H-q^{-H}\over q-q^{-1}},\quad \Delta X_\pm=X_\pm{\otimes}q^{H\over 2}+
q^{-{H\over 2}}{\otimes}X_\pm$$ and $*$-structure $X_+^*= -X_-$, $(q^{H\over 2})^*=q^{H\over 2}$ (we follow the conventions of [@Ma:book]). There is a left action of $U_q(su_{1,1})$ on ${{\mathbb{C}}}_q[D]$ (similar to that given by [@klilesdisk], but adjusted to be unitary in the sense of [@Ma:book], i.e. $(h {{\triangleright}}a)^*= S(h^*){{\triangleright}}a^* $) given by $$\begin{aligned}
&X_{\pm}{{\triangleright}}1=0\ ,\ q^{H\over 2}{{\triangleright}}1=1\ ,\ q^{H\over 2}{{\triangleright}}z=q^{-1}z\ ,\ q^{H\over 2}{{\triangleright}}\bar z=q\,\bar z\ ,\cr
&X_+{{\triangleright}}z = q^{-1/2} \ ,
X_+{{\triangleright}}\bar z= -q^{-1/2} \bar z^2
\ ,\
X_- {{\triangleright}}\bar z= q^{1/2}
\ ,
X_- {{\triangleright}}z= -q^{1/2} z^2 \ .\end{aligned}$$ This action extends to the calculus by $$\begin{aligned}
&q^{H\over 2}{{\triangleright}}{{\rm d}}z=q^{-1}{{\rm d}}z\ ,\ q^{H\over 2}{{\triangleright}}{{\rm d}}\bar z=q\,{{\rm d}}\bar z\ ,\ X_+{{\triangleright}}{{\rm d}}z=0\ ,\ X_-{{\triangleright}}{{\rm d}}\bar z=0\ ,\cr
&X_+{{\triangleright}}{{\rm d}}\bar z=-q^{-1/2}({{\rm d}}\bar z\,\bar z+\bar z\,{{\rm d}}\bar z)\ ,\ X_-{{\triangleright}}z=-q^{1/2}(z\,{{\rm d}}z+{{\rm d}}z\,z)\ .\end{aligned}$$
Now we consider integration on the deformed disk. In [@klilesdisk] an integral is given which has classical limit the Lebesgue integral on the unit disk, and is subsequently used to examine noncommutative function theory on the disk. However we use another integral with $U_q(su_{1,1})$ invariance. A partially defined map $\int:{{\mathbb{C}}}_q[D]\to {{\mathbb{C}}}$, invariant for the $U_q(su_{1,1})$ action, is defined by $$\int {{w}}^{n+1} =\frac{1}{[n]_{q^{-2}}} \ ,\quad n\ge 1\ ,$$ and $\int$ applied to any monomial of nonzero grade gives zero. (We shall not go into detail over the domain of this integral.) To spell the invariance out explicitly, we require that the following diagram commutes: $$\xymatrix{
U_q(su_{1,1}) {\otimes}{{\mathbb{C}}}_q[D] \ar[r]^{\ \ \ \quad {{\triangleright}}} \ar[d]^{{{\rm id}}{\otimes}\int } & {{\mathbb{C}}}_q[D] \ar[d]^\int \\
U_q(su_{1,1}) {\otimes}{{\mathbb{C}}}\ar[r]^{\ \quad \epsilon{\otimes}{{\rm id}}} & {{\mathbb{C}}}}$$ We take care that there are two conflicting views of what is going on with the quantum disk. As a unital $C^*$ algebra, ${{\mathbb{C}}}_q[D]$ corresponds to a deformation of a compact topological space, the closed unit disk. Classically this is not a manifold, but it is a manifold with boundary. However the $U_q(sl_2)$ action is taking us in quite a different direction - classically it corresponds to the Möbius action on the open disk, and as such its invariants are really related to hyperbolic space, rather than the closed unit disk. The problem with the integral is simply that the classical volume of hyperbolic space, under its usual invariant measure, is infinite. Recalling that ${{w}}=1-\bar zz$, we have $$\begin{aligned}
q^{1/2}X_+{{\triangleright}}{{w}}=&\ q^{-1}\bar z^2z-q^{-1}\bar z= - q^{-1}\bar z{{w}}\ ,\cr
q^{-1/2}X_-{{\triangleright}}{{w}}=&\ - q^{-1}z+q^{-1}\bar zz^2=-q^{-1}{{w}}z\ ,\cr
q^{H\over 2}{{\triangleright}}{{w}}=&\ 1-(q^{H\over 2}{{\triangleright}}\bar z)( q^{H\over 2}{{\triangleright}}z)={{w}}\ .\end{aligned}$$ and induction gives, $$q^{1/2}X_+{{\triangleright}}{{w}}^n=-q^{-1}\bar z[n]_{q^{-2}}{{w}}^n\ ,\quad n\ge 0\ .$$ Then $$\begin{aligned}
q^{1/2}X_+{{\triangleright}}(z{{w}}^n)=&\ {{w}}^n-z\bar z[n]_{q^{-2}}{{w}}^n \cr
=&\ -\ q^{-2} [n-1]_{q^{-2}} {{w}}^n+q^{-2} [n]_{q^{-2}} {{w}}^{n+1} \ .\end{aligned}$$ By invariance, the integral applied to this should give zero, so we get $$\int [n-1]_{q^{-2}} {{w}}^n=\int [n]_{q^{-2}} {{w}}^{n+1} \ ,$$ which, on choosing a normalisation, gives the formula we gave for the integral.
We next show that the integral is a twisted trace, in the sense $$\int a\, b=\int \varsigma(b)\,a\ ,\quad \forall a,b \in {{\mathbb{C}}}_q[D]$$ for the degree algebra automorphism $\varsigma(b)=q^{2|b|}\,b$ on homogeneous elements. This can be shown for $b=z$ by $\int az=0$ unless $a$ has degree $-1$, and in the $-1$ case writing $a=\bar z\,p(w)$ for a polynomial $p$. The result is then checked by explicit calculation. A straightforward inductive argument then extends this result to $b=z^m$ and similarly for negative degrees.
Next we take generators $\{s,\bar s\}$ of a spinor bimodule $\mathcal{S}$, with relations for homogenous $a\in {{\mathbb{C}}}_q[D]$, $$s.a=q^{|a|}\,a.s\ ,\quad \bar s.a=q^{|a|}\,a.\bar s$$ where the power of $q$ in the commutation relations is half that for the relations with ${{\rm d}}z,
\bar{{\rm d}}z$. Suppose that $\nabla_\mathcal{S}(s)=0$ and $\nabla_\mathcal{S}(\bar s)=0$, then define $${{\rm d}}z{{\triangleright}}\bar s=\alpha\,{{w}}\,s\ ,\ {{\rm d}}\bar z{{\triangleright}}s=\beta\,{{w}}\,\bar s\ ,\
{{\rm d}}z{{\triangleright}}s=0\ ,\ {{\rm d}}\bar z{{\triangleright}}\bar s=0$$ for two parameters $\alpha,\beta$. We have $$\begin{aligned}
\sigma_\mathcal{S}(s{\otimes}{{\rm d}}a) =&\ \nabla_\mathcal{S}(s.a)-\nabla_\mathcal{S}(s).a
= q^{|a|} \nabla_\mathcal{S}(a.s)-\nabla_\mathcal{S}(s).a = q^{|a|} {{\rm d}}a{\otimes}s\ ,\cr
\sigma_\mathcal{S}(\bar s{\otimes}{{\rm d}}a) =&\ q^{|a|} {{\rm d}}a{\otimes}\bar s\ .\end{aligned}$$ Then the Dirac operator is, writing ${{\rm d}}a=\tfrac{\partial a}{\partial z} {{\rm d}}z+\tfrac{\partial a}{\partial \bar z} {{\rm d}}\bar z$, $$\begin{aligned}
{{ \slashed{D} }}(a.s)=&\ (\tfrac{\partial a}{\partial z}{{\rm d}}z+\tfrac{\partial a}{\partial \bar z}{{\rm d}}\bar z){{\triangleright}}s=\beta\,(\tfrac{\partial a}{\partial \bar z})\,{{w}}\,\bar s\ ,\cr
{{ \slashed{D} }}(a.\bar s)=&\ (\tfrac{\partial a}{\partial z}{{\rm d}}z+\tfrac{\partial a}{\partial \bar z}{{\rm d}}\bar z){{\triangleright}}\bar s=\alpha\,(\tfrac{\partial a}{\partial z})\,{{w}}\,s\ .\end{aligned}$$ We set $\gamma(s)=s$ and $\gamma(\bar s)=-\bar s$.
Next we set $\mathcal{J}(s)=\delta\bar s$ and $\mathcal{J}(\bar s)=-\delta^{-1} s$ for some real parameter $\delta$, giving ${\epsilon}=-1$ and ${\epsilon}''=-1$. Since $\nabla_S$ on the generators is zero, it is clear that the connection preserves $j$. We compute using the definitions and commutation rules above that $$\begin{aligned}
({{\triangleright}})\sigma_S({{\mathcal{J}}}(a s){\otimes}{{\rm d}}z)=& \delta({{\triangleright}}) \sigma_S(\bar s a^*{\otimes}{{\rm d}}z) = \delta q q^{|a^*|} ({{\triangleright}})(a^*{{\rm d}}z{\otimes}\bar s)=\delta q q^{|a^*|} a^*\alpha{{w}}s\\
{{\mathcal{J}}}({{\rm d}}\bar z{{\triangleright}}a s)=& q^{2|a|}{{\mathcal{J}}}( a{{\rm d}}\bar z{{\triangleright}}s)=q^{2|a|}{{\mathcal{J}}}( \beta a {{w}}\bar s)=\beta^* {{\mathcal{J}}}( {{w}}a \bar s)=\beta^* (-\delta^{-1})s ({{w}}a)^*\\
=&-\beta^*\delta^{-1}q^{|a^*|}a^*{{w}}s\ \end{aligned}$$ for all $a$ of homogeneous degree. Similarly for the other cases. Hence (\[lapreserves\]) holds with ${\epsilon}'=1$ provided $$\label{qdiskparamJ} \delta^2 q\alpha=-\beta^*.$$ We assume this and Theorem \[sptripres\] then tells us that (2)-(6) hold with dimension $n=2$.
Finally, we define a positive hermitian inner product ${{\langle}},{{\rangle}}:\overline{ {{\mathcal{S}}}}{\otimes}_A {{\mathcal{S}}}\to A$ by the following, for some $\mu>0$, $${{\langle}}\overline{ s\,a_++\bar s\, a_-}, s\,b_++ \bar s\,b_- {{\rangle}}=a_+{}^*\,{{w}}\,b_+ + \mu\, a_-{}^*\,{{w}}\,b_- $$ for all $a_\pm,b_\pm\in A$. Now we have, $$\begin{aligned}
{{\langle}}\overline{ {{ \slashed{D} }}(a.\bar s)}, b. s {{\rangle}}=&\ {{\langle}}\overline{ \alpha\,\tfrac{\partial a}{\partial z}\,{{w}}\,s }, b. s {{\rangle}}=\alpha^*q^{-|b|-|\tfrac{\partial a}{\partial z}|}
{{\langle}}\overline{ s\,\tfrac{\partial a}{\partial z}\,{{w}}}, s.b {{\rangle}}\cr
=&\ \alpha^*q^{-|b|-|\tfrac{\partial a}{\partial z}|} {{w}}\, \tfrac{\partial a}{\partial z}{}^*\, {{w}}\, b
= \alpha^*q^{|b|-|\tfrac{\partial a}{\partial z}|} {{w}}\, \tfrac{\partial a}{\partial z}{}^*b\, {{w}}\ ,\cr
{{\langle}}\overline{ a.\bar s}, {{ \slashed{D} }}(b. s) {{\rangle}}=&\ {{\langle}}\overline{ a.\bar s}, \beta\,\tfrac{\partial b}{\partial \bar z}\,{{w}}\,\bar s {{\rangle}}= \beta\, q^{ -|a|-|\frac{\partial b}{\partial \bar z}| }
{{\langle}}\overline{ \bar s.a}, \bar s \, \tfrac{\partial b}{\partial \bar z}\,{{w}}{{\rangle}}\cr
=&\ \mu\, \beta\, q^{ -|a|-|\frac{\partial b}{\partial \bar z}| }a^*\, {{w}}\, \tfrac{\partial b}{\partial \bar z}\,{{w}}= \mu\, \beta\, q^{ |a|-|\frac{\partial b}{\partial \bar z}| } {{w}}\, a^*\, \tfrac{\partial b}{\partial \bar z}\,{{w}}\ .\end{aligned}$$ If we define the complex valued inner product ${{\langle}}\!{{\langle}},{{\rangle}}\!{{\rangle}}$ by the integral of ${{\langle}},{{\rangle}}$, then as the integral vanishes unless the grade is zero we find $$\begin{aligned}
{{\langle}}\!{{\langle}}\overline{ {{ \slashed{D} }}(a.\bar s)}, b. s {{\rangle}}\!{{\rangle}}=&\ \alpha^*\int {{w}}\, \tfrac{\partial a}{\partial z}{}^*b\, {{w}}\ ,\cr
{{\langle}}\!{{\langle}}\overline{ a.\bar s}, {{ \slashed{D} }}(b. s) {{\rangle}}\!{{\rangle}}=&\ \mu\, \beta \int {{w}}\, a^*\, \tfrac{\partial b}{\partial \bar z}\,{{w}}\ .\end{aligned}$$ Now $$\begin{aligned}
{{\rm d}}(a^*b)=&\ (\tfrac{\partial a}{\partial z}\,{{\rm d}}z+\tfrac{\partial a}{\partial \bar z}\, {{\rm d}}\bar z)^*b+a^*(\tfrac{\partial b}{\partial z}\,{{\rm d}}z+\tfrac{\partial b}{\partial \bar z}\,{{\rm d}}\bar z)=
({{\rm d}}\bar z\,\tfrac{\partial a}{\partial z}{}^*+{{\rm d}}z\,\tfrac{\partial a}{\partial \bar z}{}^*)b+a^*(\tfrac{\partial b}{\partial z}\,{{\rm d}}z+\tfrac{\partial b}{\partial \bar z} \, {{\rm d}}\bar z)\ ,\end{aligned}$$ so we get (using the previous notation) $\tfrac{\partial a^*b}{\partial \bar z}=q^{2(|b|-|\tfrac{\partial a}{\partial z}|)}\tfrac{\partial a}{\partial z}{}^*b+a^*\tfrac{\partial b}{\partial \bar z}$, so if $\mu\, \beta=-\alpha^*$ which, given (\[qdiskparamJ\]) is equivalent to $$\label{qdiskparam}\mu =q^{-1}\delta^{-2},$$ we get $${{\langle}}\!{{\langle}}\overline{ {{ \slashed{D} }}(a.\bar s)}, b. s {{\rangle}}\!{{\rangle}}- {{\langle}}\!{{\langle}}\overline{ a.\bar s}, {{ \slashed{D} }}(b. s) {{\rangle}}\!{{\rangle}}=
\alpha^*\int {{w}}\, \tfrac{\partial a^*b}{\partial \bar z}\, {{w}}\ .$$ With this choice, to show that ${{ \slashed{D} }}$ is hermitian all we need to show is that for all $a$ with $|a|=-1$ we have $$\int {{w}}\, \tfrac{\partial a}{\partial \bar z}\, {{w}}=0\ .$$ We set $a=\bar z{{w}}^m$ for some $m\ge 1$, and then $\frac{\partial a}{\partial \bar z}\,{{\rm d}}\bar z={{\rm d}}\bar z\, {{w}}^m+\bar z\,
\frac{\partial ({{w}}^m)}{\partial \bar z}\,{{\rm d}}\bar z$, so $\frac{\partial a}{\partial \bar z}={{w}}^m+\bar z\, \frac{\partial ({{w}}^m)}{\partial \bar z}$. Now from (\[missedp\]), $$\begin{aligned}
\tfrac{\partial a}{\partial \bar z}=&\ {{w}}^m-q^2\,\bar zz\,[m]_{q^2}\,{{w}}^m \cr
=&\ {{w}}^m+q^2\,{{w}}\,[m]_{q^2}\,{{w}}^{m-1}-q^2\,[m]_{q^2}\,{{w}}^{m-1} \cr
=&\ [m+1]_{q^2}{{w}}^m-q^2\,[m]_{q^2}{{w}}^{m-1}\ ,\cr
q^{-2m} \tfrac{\partial a}{\partial \bar z}=&\ [m+1]_{q^{-2}}{{w}}^m-[m]_{q^{-2}}{{w}}^{m-1}\ .\end{aligned}$$ Now the formula for the integral shows that ${{ \slashed{D} }}$ is hermitian (The use of the hermitian property for the inner product means that we have checked this for all cases.). However note that if we were to set $a=\bar z$, then the condition would not be satisfied. There is a condition on the domain of ${{ \slashed{D} }}$ which would classically include functions vanishing on the boundary of the disk – we shall not pursue this matter further here.
The condition that $\mathcal{J}$ is an isometry would require equality under the integral of $${{\langle}}\overline{{{\mathcal{J}}}(a s)}, {{\mathcal{J}}}(bs){{\rangle}}=\mu \delta^2 a {{w}}b^*=q^{-1}a{{w}}b^*,\quad {{\langle}}\overline{bs},as{{\rangle}}=q^{-|a|-|b|}b^* {{w}}a$$ on homogeneous elements. The second expression integrates to zero unless $|a|=|b|$ so gives $\varsigma(b^*){{w}}a$. From this and the above twisted trace property of the integral, we can conclude that $${{\langle}}\!{{\langle}}\overline{{{\mathcal{J}}}(as)},{{\mathcal{J}}}(bs){{\rangle}}\!{{\rangle}}= q^{-1}{{\langle}}\!{{\langle}}\overline{\varsigma^{-1} (b)s},as{{\rangle}}\!{{\rangle}},\quad \forall a,b\in {{\mathbb{C}}}_q[D]$$ much as in the spirit of the $q$-sphere example. The same applies with $s$ replaced by $\bar s$ and $q^{-1}$ by $q$. Finally, we have some freedom to rescale the generators $s,\bar s$ and using this we can without loss of generality assume $\alpha=1$ and $\beta<0$ say when $q>0$. In that case we have a 1-parameter family of Dirac operators by our construction with $\delta=\sqrt{-q^{-1}\beta}, \mu=-\beta^{-1}$.
Holomorphic bimodules and Chern connections
===========================================
An integrable almost complex structure (see [@BegSmiComplex]) on a star algebra $A$ with star differential calculus $(\Omega,{{\rm d}},\wedge)$ is a bimodule map $J:\Omega^1\to \Omega^1$ with $J^2=-{{\rm id}}$ obeying certain conditions. This basically is a decomposition of bimodules $\Omega^n=\oplus_{p+q=n}
\Omega^{p,q}$ with ${{\rm d}}={\partial}+{\overline{\partial}}$ where ${\partial}:\Omega^{p,q}\to \Omega^{p+1,q}$ and ${\overline{\partial}}:\Omega^{p,q}\to \Omega^{p,q+1}$ with ${\partial}^2={\overline{\partial}}{}^2=0$ and $\Omega^{p,q}\wedge\Omega^{p',q'}\subset \Omega^{p+p',q+q'}$ . The direct sum gives projection maps $\pi^{p,q}:\Omega^{p+q}\to\Omega^{p,q}$.
We can then define the notion of a holomorphic bundle $E$ but note that in [@BegSmiComplex] the basic definition of this is given in terms of a left holomorphic section. This awkwardly lends itself to looking at hermitian inner products with the antilinear side on the right, which is opposite to the usual convention for the Dirac operator. However, in the spinor bundle case we can use the antilinear isomorphism $j:
{{\mathcal{S}}}\to\overline{{{\mathcal{S}}}}$ to swap the side of the antilinear part by commutativity of the following diagram: $$\begin{aligned}
\label{sideswitch}
\xymatrix{
\overline{{{\mathcal{S}}}}\otimes_A {{\mathcal{S}}}\ar[r]^{{{\langle}},{{\rangle}}} & A \\
{{\mathcal{S}}}\otimes_A \overline{{{\mathcal{S}}}} \ar[ur]_{{{\langle}},{{\rangle}}} \ar[u]^{j{\otimes}j^{-1}}
}\end{aligned}$$ In general there is no necessity for a holomorphic bundle to have any obvious antilinear isomorphism.
Holomorphic bimodules
---------------------
A left ${\overline{\partial}}$-connection ${\overline{\partial}}_E:E\to \Omega^{0,1}A{\otimes}_A E$ on a left module $E$ is a linear map satisfying the left ${\overline{\partial}}$-Liebniz rule, for $e\in E$ and $a\in A$ $${\overline{\partial}}_E(a.e)={\overline{\partial}}a{\otimes}e+a.{\overline{\partial}}_E(e)\ .$$
[@BegSmiComplex] A holomorphic structure on a left $A$-module $E$ is is a left ${\overline{\partial}}$-connection ${\overline{\partial}}_E:E\to \Omega^{0,1}A{\otimes}_A E$ with vanishing holomorphic curvature, i.e. $({\overline{\partial}}{\otimes}{{\rm id}}-{{\rm id}}\wedge{\overline{\partial}}_E){\overline{\partial}}_E:E\to \Omega^{0,2}A{\otimes}_A E$ vanishes. Then we call $(E,{\overline{\partial}}_E)$ a left holomorphic module. If in addition there is a bimodule map $\sigma_E:E{\otimes}_A\Omega^{0,1}A\to \Omega^{0,1}A{\otimes}_A E$ so that ${\overline{\partial}}_E(e.a)={\overline{\partial}}_E(e).a+\sigma_E(e{\otimes}{\overline{\partial}}a)$ for all $a\in A$ and $e\in E$, we say that $(E,{\overline{\partial}}_E,\sigma_E)$ is a left holomorphic bimodule.
Now suppose that we have a hermitian inner product on a holomorphic left module. Classically the ${\overline{\partial}}$-connection given by the holomorphic structure can be extended to a unique connection $\nabla_E:E\to\Omega^1{\otimes}_A E$ preserving the hermitian inner product and with curvature only mapping to $\Omega^{1,1}{\otimes}_A E$, the Chern connection. We shall show that this is also the case in noncommutative geometry. We give two constructions of the same connection, as both has its advantages. For one we require some material on duals and coevaluations, and for the other we consider Christoffel symbols.
We begin with a hermitian inner product on $E$, ${{\langle}},{{\rangle}}:E{\otimes}_A\overline{E}\to A$, and we say that the left connection $\nabla_E:E\to \Omega^{1}A{\otimes}_A E$ preserves the metric if $$\begin{aligned}
\label{hermcon1}
{{\rm d}}\,{{\langle}},{{\rangle}}=({{\rm id}}{\otimes}{{\langle}},{{\rangle}})(\nabla_E{\otimes}{{\rm id}})+({{\langle}},{{\rangle}}{\otimes}{{\rm id}})({{\rm id}}{\otimes}\tilde\nabla):E{\otimes}\overline{E}\to \Omega^1A\ ,\end{aligned}$$ where $\tilde\nabla:\overline{E}\to \overline{E}{\otimes}_A\Omega^1A$ is the right connection constructed from $\nabla_E$ by $\tilde\nabla(\overline{e})=\overline{f}{\otimes}\kappa^*$, where $\nabla_E(e)=\kappa{\otimes}f$.
Duals and coevaluations
-----------------------
For an $A$-bimodule $E$ we use the notation $E^\circ={}_A{{\rm{Hom}}}(E,A)$, and there is an evaluation bimodule map ${{\rm ev}}_E:E{\otimes}_A E^\circ\to A$. If $E$ is finitely generated projective (fgp for short) as a left module, then there is a coevaluation map ${{\rm coev}}_E:A\to E^\circ{\otimes}_A E$ (written items of a ‘dual basis’) so that $$\begin{aligned}
({{\rm id}}{\otimes}{{\rm ev}}_E)({{\rm coev}}_E{\otimes}{{\rm id}})={{\rm id}}:E^\circ\to E^\circ\ ,\quad ({{\rm ev}}_E{\otimes}{{\rm id}})({{\rm id}}{\otimes}{{\rm coev}}_E)={{\rm id}}:E\to E\ .\end{aligned}$$
Choosing a side, we suppose that ${{\langle}},{{\rangle}}:E{\otimes}_A\overline{E}\to A$ is a hermitian inner product on $E$ (and therefore that it is a bimodule map). This hermitian inner product is called non-degenerate if there is a bimodule isomorphism $G:\overline{E}\to E^\circ$ so that ${{\langle}},{{\rangle}}={{\rm ev}}\circ({{\rm id}}{\otimes}G)$. In this case we can define an ‘inverse inner product’ ${{\langle}},{{\rangle}}^{-1}:A\to \overline{E}{\otimes}_A E$ by ${{\langle}},{{\rangle}}^{-1}=(G^{-1}{\otimes}{{\rm id}}){{\rm coev}}_E$.
\[unichernnog\] Suppose that $(E,{\overline{\partial}}_E)$ is a holomorphic left module, where $E$ is finitely generated projective as a left $A$-module and ${{\langle}},{{\rangle}}:E{\otimes}\overline{E}\to A$ is a non-degenerate hermitian inner product on $E$. Then there is a unique left connection $\nabla_E:E\to\Omega^1 {\otimes}_A E$, preserving the hermitian metric and obeying $(\pi^{0,1}{\otimes}{{\rm id}})\nabla_E={\overline{\partial}}_E$.
The $(\pi^{1,0}{\otimes}{{\rm id}})\nabla_E={\partial}_E$ part is given by the following diagram, where $\tilde{\partial}: \overline{E}\to
\overline{E}{\otimes}_A \Omega^{1,0}$ is the right ${\partial}$-covariant derivative defined by $\tilde{\partial}(\overline{e})=\overline{f}{\otimes}\kappa^*$, where $\kappa{\otimes}f={\overline{\partial}}_E(e)$:
0.5 mm
(180,60)(-10,27) (20,60)[(0,1)[25]{}]{} (29.99,50.5)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (29.95,51)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (29.89,51.49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (29.8,51.98)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (29.69,52.47)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (29.56,52.95)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (29.4,53.42)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (29.21,53.88)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (29.01,54.34)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (28.78,54.78)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (28.53,55.21)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (28.26,55.63)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (27.97,56.04)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (27.66,56.43)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (27.33,56.8)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (26.98,57.16)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (26.62,57.5)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (26.23,57.82)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (25.84,58.12)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (25.43,58.4)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (25,58.66)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (24.56,58.9)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (24.11,59.12)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (23.65,59.31)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (23.18,59.48)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (22.71,59.63)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (22.23,59.75)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (21.74,59.85)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (21.24,59.92)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (20.75,59.97)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (20.25,60)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (19.75,60)[(1,0)[0.5]{}]{} (19.25,59.97)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (18.76,59.92)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (18.26,59.85)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (17.77,59.75)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (17.29,59.63)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (16.82,59.48)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (16.35,59.31)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (15.89,59.12)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (15.44,58.9)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (15,58.66)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (14.57,58.4)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (14.16,58.12)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (13.77,57.82)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (13.38,57.5)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (13.02,57.16)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (12.67,56.8)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (12.34,56.43)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (12.03,56.04)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (11.74,55.63)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (11.47,55.21)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (11.22,54.78)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (10.99,54.34)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (10.79,53.88)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (10.6,53.42)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (10.44,52.95)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (10.31,52.47)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (10.2,51.98)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (10.11,51.49)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (10.05,51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (10.01,50.5)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (10,50)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(60,60)[(0,1)[25]{}]{} (60,60)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (60.01,59.5)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (60.05,59)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (60.11,58.51)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (60.2,58.02)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (60.31,57.53)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (60.44,57.05)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (60.6,56.58)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (60.79,56.12)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (60.99,55.66)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (61.22,55.22)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (61.47,54.79)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (61.74,54.37)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (62.03,53.96)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (62.34,53.57)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (62.67,53.2)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (63.02,52.84)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (63.38,52.5)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (63.77,52.18)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (64.16,51.88)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (64.57,51.6)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (65,51.34)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (65.44,51.1)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (65.89,50.88)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (66.35,50.69)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (66.82,50.52)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (67.29,50.37)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (67.77,50.25)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (68.26,50.15)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (68.76,50.08)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (69.25,50.03)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (69.75,50)[(1,0)[0.5]{}]{} (70.25,50)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (70.75,50.03)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (71.24,50.08)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (71.74,50.15)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (72.23,50.25)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (72.71,50.37)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (73.18,50.52)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (73.65,50.69)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (74.11,50.88)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (74.56,51.1)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (75,51.34)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (75.43,51.6)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (75.84,51.88)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (76.23,52.18)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (76.62,52.5)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (76.98,52.84)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (77.33,53.2)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (77.66,53.57)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (77.97,53.96)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (78.26,54.37)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (78.53,54.79)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (78.78,55.22)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (79.01,55.66)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (79.21,56.12)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (79.4,56.58)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (79.56,57.05)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (79.69,57.53)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (79.8,58.02)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (79.89,58.51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (79.95,59)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (79.99,59.5)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(99.99,60.5)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (99.95,61)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (99.89,61.49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (99.8,61.98)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (99.69,62.47)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (99.56,62.95)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (99.4,63.42)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (99.21,63.88)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (99.01,64.34)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (98.78,64.78)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (98.53,65.21)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (98.26,65.63)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (97.97,66.04)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (97.66,66.43)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (97.33,66.8)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (96.98,67.16)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (96.62,67.5)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (96.23,67.82)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (95.84,68.12)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (95.43,68.4)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (95,68.66)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (94.56,68.9)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (94.11,69.12)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (93.65,69.31)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (93.18,69.48)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (92.71,69.63)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (92.23,69.75)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (91.74,69.85)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (91.24,69.92)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (90.75,69.97)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (90.25,70)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (89.75,70)[(1,0)[0.5]{}]{} (89.25,69.97)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (88.76,69.92)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (88.26,69.85)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (87.77,69.75)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (87.29,69.63)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (86.82,69.48)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (86.35,69.31)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (85.89,69.12)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (85.44,68.9)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (85,68.66)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (84.57,68.4)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (84.16,68.12)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (83.77,67.82)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (83.38,67.5)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (83.02,67.16)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (82.67,66.8)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (82.34,66.43)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (82.03,66.04)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (81.74,65.63)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (81.47,65.21)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (81.22,64.78)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (80.99,64.34)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (80.79,63.88)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (80.6,63.42)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (80.44,62.95)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (80.31,62.47)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (80.2,61.98)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (80.11,61.49)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (80.05,61)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (80.01,60.5)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (80,60)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(70,45)[(0,1)[5]{}]{} (70,25)[(0,1)[10]{}]{} (100,25)[(0,1)[35]{}]{} (130,50)[(0,1)[35]{}]{} (130,50)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (130.01,49.5)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (130.05,49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (130.11,48.51)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (130.2,48.02)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (130.31,47.53)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (130.44,47.05)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (130.6,46.58)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (130.79,46.12)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (130.99,45.66)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (131.22,45.22)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (131.47,44.79)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (131.74,44.37)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (132.03,43.96)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (132.34,43.57)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (132.67,43.2)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (133.02,42.84)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (133.38,42.5)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (133.77,42.18)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (134.16,41.88)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (134.57,41.6)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (135,41.34)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (135.44,41.1)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (135.89,40.88)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (136.35,40.69)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (136.82,40.52)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (137.29,40.37)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (137.77,40.25)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (138.26,40.15)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (138.76,40.08)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (139.25,40.03)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (139.75,40)[(1,0)[0.5]{}]{} (140.25,40)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (140.75,40.03)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (141.24,40.08)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (141.74,40.15)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (142.23,40.25)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (142.71,40.37)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (143.18,40.52)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (143.65,40.69)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (144.11,40.88)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (144.56,41.1)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (145,41.34)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (145.43,41.6)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (145.84,41.88)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (146.23,42.18)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (146.62,42.5)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (146.98,42.84)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (147.33,43.2)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (147.66,43.57)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (147.97,43.96)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (148.26,44.37)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (148.53,44.79)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (148.78,45.22)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (149.01,45.66)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (149.21,46.12)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (149.4,46.58)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (149.56,47.05)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (149.69,47.53)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (149.8,48.02)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (149.89,48.51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (149.95,49)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (149.99,49.5)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(169.99,50.5)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (169.95,51)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (169.89,51.49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (169.8,51.98)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (169.69,52.47)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (169.56,52.95)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (169.4,53.42)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (169.21,53.88)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (169.01,54.34)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (168.78,54.78)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (168.53,55.21)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (168.26,55.63)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (167.97,56.04)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (167.66,56.43)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (167.33,56.8)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (166.98,57.16)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (166.62,57.5)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (166.23,57.82)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (165.84,58.12)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (165.43,58.4)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (165,58.66)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (164.56,58.9)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (164.11,59.12)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (163.65,59.31)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (163.18,59.48)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (162.71,59.63)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (162.23,59.75)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (161.74,59.85)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (161.24,59.92)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (160.75,59.97)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (160.25,60)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (159.75,60)[(1,0)[0.5]{}]{} (159.25,59.97)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (158.76,59.92)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (158.26,59.85)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (157.77,59.75)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (157.29,59.63)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (156.82,59.48)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (156.35,59.31)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (155.89,59.12)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (155.44,58.9)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (155,58.66)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (154.57,58.4)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (154.16,58.12)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (153.77,57.82)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (153.38,57.5)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (153.02,57.16)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (152.67,56.8)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (152.34,56.43)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (152.03,56.04)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (151.74,55.63)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (151.47,55.21)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (151.22,54.78)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (150.99,54.34)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (150.79,53.88)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (150.6,53.42)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (150.44,52.95)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (150.31,52.47)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (150.2,51.98)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (150.11,51.49)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (150.05,51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (150.01,50.5)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (150,50)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(179.99,70.5)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (179.95,71)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (179.89,71.49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (179.8,71.98)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (179.69,72.47)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (179.56,72.95)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (179.4,73.42)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (179.21,73.88)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (179.01,74.34)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (178.78,74.78)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (178.53,75.21)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (178.26,75.63)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (177.97,76.04)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (177.66,76.43)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (177.33,76.8)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (176.98,77.16)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (176.62,77.5)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (176.23,77.82)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (175.84,78.12)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (175.43,78.4)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (175,78.66)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (174.56,78.9)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (174.11,79.12)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (173.65,79.31)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (173.18,79.48)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (172.71,79.63)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (172.23,79.75)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (171.74,79.85)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (171.24,79.92)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (170.75,79.97)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (170.25,80)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (169.75,80)[(1,0)[0.5]{}]{} (169.25,79.97)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (168.76,79.92)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (168.26,79.85)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (167.77,79.75)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (167.29,79.63)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (166.82,79.48)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (166.35,79.31)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (165.89,79.12)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (165.44,78.9)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (165,78.66)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (164.57,78.4)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (164.16,78.12)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (163.77,77.82)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (163.38,77.5)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (163.02,77.16)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (162.67,76.8)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (162.34,76.43)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (162.03,76.04)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (161.74,75.63)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (161.47,75.21)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (161.22,74.78)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (160.99,74.34)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (160.79,73.88)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (160.6,73.42)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (160.44,72.95)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (160.31,72.47)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (160.2,71.98)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (160.11,71.49)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (160.05,71)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (160.01,70.5)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (160,70)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(160,60)[(0,1)[10]{}]{} (170,25)[(0,1)[25]{}]{} (180,25)[(0,1)[45]{}]{} (30,25)[(0,1)[25]{}]{} (10,25)[(0,1)[25]{}]{} (94,77)[(0,0)\[cc\][${{\langle}},{{\rangle}}^{-1}$]{}]{}
(174,86)[(0,0)\[cc\][${{\langle}},{{\rangle}}^{-1}$]{}]{}
(70,40)[(0,0)\[cc\][$\partial$]{}]{}
(14,64)[(0,0)\[cc\][$\partial_E$]{}]{}
(25,82)[(0,0)\[cc\][$E$]{}]{}
(65,85)[(0,0)\[cc\]]{}
(43,55)[(0,0)\[cc\][$=$]{}]{}
(115,55)[(0,0)\[cc\][$-$]{}]{}
(105,64)[(0,0)\[cc\][$E$]{}]{}
(75,65)[(0,0)\[cc\][$\overline{E}$]{}]{}
(154,64)[(0,0)\[cc\][$\tilde\partial$]{}]{}
(140,35)[(0,0)\[cc\][${{\langle}},{{\rangle}}$]{}]{}
(77,47)[(0,0)\[cc\][${{\langle}},{{\rangle}}$]{}]{}
(35,29.5)[(0,0)\[cc\][$E$]{}]{}
(1,30)[(0,0)\[cc\][$\Omega^{1,0}$]{}]{}
Note that the RHS of the diagram only depends on the value ${{\langle}},{{\rangle}}^{-1}\in \overline{E}{\otimes}_A E$ (with the emphasis on ${\otimes}_A$). It is reasonably easy to see that this defines a left ${\partial}$-connection. Then applying ${{\rm id}}{\otimes}{{\langle}}-,\overline{c}{{\rangle}}$ to this shows that $${\partial}{{\langle}}e,\overline{c}{{\rangle}}=({{\rm id}}{\otimes}{{\langle}},{{\rangle}})({\partial}_E e{\otimes}\overline{c})+({{\langle}},{{\rangle}}{\otimes}{{\rm id}})(e{\otimes}\tilde{\partial}\,\overline{c})\ .$$ We now need to check applying the ${\overline{\partial}}$ derivative, which means checking the equation
0.5 mm
(155,55)(-10,23) (10,60)[(0,1)[15]{}]{} (30,60)[(0,1)[15]{}]{} (10,60)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (10.01,59.5)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (10.05,59)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (10.11,58.51)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (10.2,58.02)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (10.31,57.53)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (10.44,57.05)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (10.6,56.58)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (10.79,56.12)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (10.99,55.66)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (11.22,55.22)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (11.47,54.79)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (11.74,54.37)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (12.03,53.96)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (12.34,53.57)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (12.67,53.2)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (13.02,52.84)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (13.38,52.5)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (13.77,52.18)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (14.16,51.88)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (14.57,51.6)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (15,51.34)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (15.44,51.1)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (15.89,50.88)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (16.35,50.69)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (16.82,50.52)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (17.29,50.37)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (17.77,50.25)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (18.26,50.15)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (18.76,50.08)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (19.25,50.03)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (19.75,50)[(1,0)[0.5]{}]{} (20.25,50)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (20.75,50.03)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (21.24,50.08)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (21.74,50.15)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (22.23,50.25)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (22.71,50.37)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (23.18,50.52)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (23.65,50.69)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (24.11,50.88)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (24.56,51.1)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (25,51.34)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (25.43,51.6)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (25.84,51.88)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (26.23,52.18)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (26.62,52.5)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (26.98,52.84)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (27.33,53.2)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (27.66,53.57)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (27.97,53.96)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (28.26,54.37)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (28.53,54.79)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (28.78,55.22)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (29.01,55.66)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (29.21,56.12)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (29.4,56.58)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (29.56,57.05)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (29.69,57.53)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (29.8,58.02)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (29.89,58.51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (29.95,59)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (29.99,59.5)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(20,45)[(0,1)[5]{}]{} (20,25)[(0,1)[10]{}]{} (65,60)[(0,1)[15]{}]{} (74.99,50.5)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (74.95,51)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (74.89,51.49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (74.8,51.98)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (74.69,52.47)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (74.56,52.95)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (74.4,53.42)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (74.21,53.88)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (74.01,54.34)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (73.78,54.78)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (73.53,55.21)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (73.26,55.63)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (72.97,56.04)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (72.66,56.43)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (72.33,56.8)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (71.98,57.16)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (71.62,57.5)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (71.23,57.82)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (70.84,58.12)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (70.43,58.4)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (70,58.66)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (69.56,58.9)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (69.11,59.12)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (68.65,59.31)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (68.18,59.48)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (67.71,59.63)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (67.23,59.75)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (66.74,59.85)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (66.24,59.92)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (65.75,59.97)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (65.25,60)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (64.75,60)[(1,0)[0.5]{}]{} (64.25,59.97)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (63.76,59.92)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (63.26,59.85)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (62.77,59.75)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (62.29,59.63)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (61.82,59.48)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (61.35,59.31)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (60.89,59.12)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (60.44,58.9)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (60,58.66)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (59.57,58.4)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (59.16,58.12)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (58.77,57.82)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (58.38,57.5)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (58.02,57.16)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (57.67,56.8)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (57.34,56.43)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (57.03,56.04)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (56.74,55.63)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (56.47,55.21)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (56.22,54.78)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (55.99,54.34)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (55.79,53.88)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (55.6,53.42)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (55.44,52.95)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (55.31,52.47)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (55.2,51.98)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (55.11,51.49)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (55.05,51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (55.01,50.5)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (55,50)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(95,50)[(0,1)[25]{}]{} (75,50)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (75.01,49.5)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (75.05,49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (75.11,48.51)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (75.2,48.02)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (75.31,47.53)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (75.44,47.05)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (75.6,46.58)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (75.79,46.12)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (75.99,45.66)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (76.22,45.22)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (76.47,44.79)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (76.74,44.37)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (77.03,43.96)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (77.34,43.57)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (77.67,43.2)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (78.02,42.84)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (78.38,42.5)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (78.77,42.18)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (79.16,41.88)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (79.57,41.6)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (80,41.34)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (80.44,41.1)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (80.89,40.88)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (81.35,40.69)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (81.82,40.52)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (82.29,40.37)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (82.77,40.25)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (83.26,40.15)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (83.76,40.08)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (84.25,40.03)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (84.75,40)[(1,0)[0.5]{}]{} (85.25,40)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (85.75,40.03)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (86.24,40.08)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (86.74,40.15)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (87.23,40.25)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (87.71,40.37)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (88.18,40.52)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (88.65,40.69)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (89.11,40.88)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (89.56,41.1)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (90,41.34)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (90.43,41.6)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (90.84,41.88)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (91.23,42.18)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (91.62,42.5)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (91.98,42.84)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (92.33,43.2)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (92.66,43.57)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (92.97,43.96)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (93.26,44.37)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (93.53,44.79)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (93.78,45.22)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (94.01,45.66)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (94.21,46.12)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (94.4,46.58)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (94.56,47.05)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (94.69,47.53)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (94.8,48.02)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (94.89,48.51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (94.95,49)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (94.99,49.5)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(55,25)[(0,1)[25]{}]{} (115,50)[(0,1)[25]{}]{} (115,50)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (115.01,49.5)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (115.05,49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (115.11,48.51)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (115.2,48.02)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (115.31,47.53)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (115.44,47.05)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (115.6,46.58)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (115.79,46.12)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (115.99,45.66)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (116.22,45.22)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (116.47,44.79)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (116.74,44.37)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (117.03,43.96)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (117.34,43.57)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (117.67,43.2)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (118.02,42.84)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (118.38,42.5)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (118.77,42.18)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (119.16,41.88)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (119.57,41.6)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (120,41.34)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (120.44,41.1)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (120.89,40.88)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (121.35,40.69)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (121.82,40.52)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (122.29,40.37)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (122.77,40.25)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (123.26,40.15)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (123.76,40.08)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (124.25,40.03)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (124.75,40)[(1,0)[0.5]{}]{} (125.25,40)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (125.75,40.03)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (126.24,40.08)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (126.74,40.15)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (127.23,40.25)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (127.71,40.37)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (128.18,40.52)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (128.65,40.69)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (129.11,40.88)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (129.56,41.1)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (130,41.34)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (130.43,41.6)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (130.84,41.88)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (131.23,42.18)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (131.62,42.5)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (131.98,42.84)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (132.33,43.2)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (132.66,43.57)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (132.97,43.96)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (133.26,44.37)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (133.53,44.79)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (133.78,45.22)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (134.01,45.66)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (134.21,46.12)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (134.4,46.58)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (134.56,47.05)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (134.69,47.53)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (134.8,48.02)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (134.89,48.51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (134.95,49)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (134.99,49.5)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(154.99,50.5)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.95,51)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.89,51.49)(0.06,-0.49)[1]{}[(0,-1)[0.49]{}]{} (154.8,51.98)(0.09,-0.49)[1]{}[(0,-1)[0.49]{}]{} (154.69,52.47)(0.11,-0.49)[1]{}[(0,-1)[0.49]{}]{} (154.56,52.95)(0.14,-0.48)[1]{}[(0,-1)[0.48]{}]{} (154.4,53.42)(0.16,-0.47)[1]{}[(0,-1)[0.47]{}]{} (154.21,53.88)(0.09,-0.23)[2]{}[(0,-1)[0.23]{}]{} (154.01,54.34)(0.1,-0.23)[2]{}[(0,-1)[0.23]{}]{} (153.78,54.78)(0.11,-0.22)[2]{}[(0,-1)[0.22]{}]{} (153.53,55.21)(0.12,-0.22)[2]{}[(0,-1)[0.22]{}]{} (153.26,55.63)(0.14,-0.21)[2]{}[(0,-1)[0.21]{}]{} (152.97,56.04)(0.15,-0.2)[2]{}[(0,-1)[0.2]{}]{} (152.66,56.43)(0.1,-0.13)[3]{}[(0,-1)[0.13]{}]{} (152.33,56.8)(0.11,-0.12)[3]{}[(0,-1)[0.12]{}]{} (151.98,57.16)(0.12,-0.12)[3]{}[(0,-1)[0.12]{}]{} (151.62,57.5)(0.12,-0.11)[3]{}[(1,0)[0.12]{}]{} (151.23,57.82)(0.13,-0.11)[3]{}[(1,0)[0.13]{}]{} (150.84,58.12)(0.13,-0.1)[3]{}[(1,0)[0.13]{}]{} (150.43,58.4)(0.21,-0.14)[2]{}[(1,0)[0.21]{}]{} (150,58.66)(0.21,-0.13)[2]{}[(1,0)[0.21]{}]{} (149.56,58.9)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (149.11,59.12)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (148.65,59.31)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (148.18,59.48)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (147.71,59.63)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (147.23,59.75)(0.48,-0.12)[1]{}[(1,0)[0.48]{}]{} (146.74,59.85)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (146.24,59.92)(0.49,-0.07)[1]{}[(1,0)[0.49]{}]{} (145.75,59.97)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (145.25,60)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (144.75,60)[(1,0)[0.5]{}]{} (144.25,59.97)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (143.76,59.92)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (143.26,59.85)(0.49,0.07)[1]{}[(1,0)[0.49]{}]{} (142.77,59.75)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (142.29,59.63)(0.48,0.12)[1]{}[(1,0)[0.48]{}]{} (141.82,59.48)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (141.35,59.31)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (140.89,59.12)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (140.44,58.9)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (140,58.66)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (139.57,58.4)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (139.16,58.12)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (138.77,57.82)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (138.38,57.5)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (138.02,57.16)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (137.67,56.8)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (137.34,56.43)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (137.03,56.04)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (136.74,55.63)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (136.47,55.21)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (136.22,54.78)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (135.99,54.34)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (135.79,53.88)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (135.6,53.42)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (135.44,52.95)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (135.31,52.47)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (135.2,51.98)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (135.11,51.49)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (135.05,51)(0.06,0.49)[1]{}[(0,1)[0.49]{}]{} (135.01,50.5)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (135,50)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{}
(145,60)[(0,1)[15]{}]{} (155,25)[(0,1)[25]{}]{} (19,40)[(0,0)\[cc\][${\overline{\partial}}$]{}]{}
(58,65)[(0,0)\[cc\][${\overline{\partial}}_E$]{}]{}
(85,35)[(0,0)\[cc\][${{\langle}},{{\rangle}}$]{}]{}
(125,35)[(0,0)\[cc\][${{\langle}},{{\rangle}}$]{}]{}
(27,47)[(0,0)\[cc\][${{\langle}},{{\rangle}}$]{}]{}
(150,65)[(0,0)\[cc\][$\hat{\partial}$]{}]{}
(15,74.5)[(0,0)\[cc\][$E$]{}]{}
(35,75)[(0,0)\[cc\][$\overline{E}$]{}]{}
(45,50)[(0,0)\[cc\][$=$]{}]{}
(105,50)[(0,0)\[cc\][$+$]{}]{}
where $\hat{\partial}: \overline{E}\to
\overline{E}{\otimes}_A \Omega^{0,1}$ is the right ${\overline{\partial}}$-covariant derivative defined by $\hat{\partial}(\overline{c})=\overline{g}{\otimes}\xi^*$, where $\xi{\otimes}g={\partial}_E(c)$, as defined in the previous picture. We have $$\begin{aligned}
({\overline{\partial}}{{\langle}}e,\overline{c}{{\rangle}})^*=&\ {\partial}{{\langle}}c,\overline{e}{{\rangle}}=({{\rm id}}{\otimes}{{\langle}},{{\rangle}})({\partial}_E c{\otimes}\overline{e})+({{\langle}},{{\rangle}}{\otimes}{{\rm id}})(c{\otimes}\tilde{\partial}(\overline{e}))
= \xi\,{{\langle}}g,\overline{e}{{\rangle}}+{{\langle}}c,\overline{f}{{\rangle}}\, \kappa^*\ ,\end{aligned}$$ so on taking star again, $$\begin{aligned}
{\overline{\partial}}{{\langle}}e,\overline{c}{{\rangle}}=&\ {{\langle}}g,\overline{e}{{\rangle}}^*\,\xi^*+\kappa\,{{\langle}}c,\overline{f}{{\rangle}}^* ={{\langle}}e,\overline{g}{{\rangle}}\,\xi^*+\kappa\,{{\langle}}f,\overline{c}{{\rangle}}\cr
=&\ ({{\langle}},{{\rangle}}{\otimes}{{\rm id}})(e{\otimes}\hat{\partial}(\overline{c}))+({{\langle}},{{\rangle}}{\otimes}{{\rm id}})({\overline{\partial}}_E(e){\otimes}\overline{c})\ .\qquad\largesquare\end{aligned}$$
Suppose that the conditions for Proposition \[unichernnog\] hold, and that $E$ is a bimodule with $(E,{\overline{\partial}}_E,\sigma_E)$ a left bimodule ${\overline{\partial}}$-connection on $E$ for $\sigma_E:E{\otimes}_A\Omega^{0,1}A\to
\Omega^{0,1}A{\otimes}_A E$. Then the connection $\nabla_E$ in Proposition \[unichernnog\] is a bimodule connection, where $\sigma_E:E{\otimes}_A\Omega^{1,0}A\to
\Omega^{1,0}A{\otimes}_A E$ is defined by, for $\eta\in\Omega^{1,0}A$ $$\begin{aligned}
\sigma_E(e{\otimes}\eta) =&\ ({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})\big({{\rm id}}{\otimes}({{\rm id}}{\otimes}\star^{-1})\Upsilon\,\overline{\sigma_E}\, \Upsilon^{-1}(\star{\otimes}{{\rm id}})
{\otimes}{{\rm id}}\big)(e{\otimes}\eta{\otimes}{{\langle}},{{\rangle}}^{-1})\end{aligned}$$ or maybe more obviously as, for ${{\langle}},{{\rangle}}^{-1}=\overline{c}{\otimes}g$, and where $\xi{\otimes}k=\sigma_E(c{\otimes}\eta^*)$ $$\begin{aligned}
\sigma_E(e{\otimes}\eta) =&\ {{\langle}}e,\overline{k}{{\rangle}}\,\xi^*{\otimes}g\ .\end{aligned}$$
From the diagram in Proposition \[unichernnog\], where ${{\langle}},{{\rangle}}^{-1}(1)=\overline{c}{\otimes}g$, $$\begin{aligned}
{\partial}_E(e.a) =&\ {\partial}{{\langle}}e\,a{\otimes}\overline{c}{{\rangle}}{\otimes}g
-({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e\,a{\otimes}\tilde{\partial}(\overline{c}){\otimes}g) \cr
=&\ {\partial}{{\langle}}e{\otimes}a\,\overline{c}{{\rangle}}{\otimes}g
-({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e{\otimes}a\,\tilde {\partial}(\overline{c}){\otimes}g) \cr
=&\ {\partial}{{\langle}}e{\otimes}a\,\overline{c}{{\rangle}}{\otimes}g
-({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e{\otimes}\tilde {\partial}(a\,\overline{c}){\otimes}g) \cr
&\ +({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e{\otimes}\tilde {\partial}(a\, \overline{c}){\otimes}g)
-({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e{\otimes}a\,\tilde {\partial}(\overline{c}){\otimes}g)\ .\end{aligned}$$ Now $a.\overline{c}{\otimes}g=\overline{c}{\otimes}g.a\in \overline{E}{\otimes}_A E$, so we have $$\begin{aligned}
{\partial}_E(e.a)
=&\ {\partial}{{\langle}}e{\otimes}\overline{c}{{\rangle}}{\otimes}g\,a
-({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e{\otimes}\tilde {\partial}(\overline{c}){\otimes}g\,a) \cr
&\ +({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e{\otimes}\tilde {\partial}(a\, \overline{c}){\otimes}g)
-({{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}})(e{\otimes}a\,\tilde {\partial}(\overline{c}){\otimes}g)\ ,\end{aligned}$$ so we have $$\begin{aligned}
{\partial}_E(e.a)-{\partial}_E(e).a =&\
({{\langle}},{{\rangle}}{\otimes}{{\rm id}})\big(e{\otimes}( \tilde {\partial}(\overline{c.a^*})-a\,\tilde {\partial}(\overline{c}))\big){\otimes}g\ .\end{aligned}$$ Now ${\overline{\partial}}_E(c.a^*)={\overline{\partial}}_E(c).a^*+\sigma(c{\otimes}{\overline{\partial}}a^*)$, and if we put $\kappa{\otimes}f={\overline{\partial}}_E(c)$ and $\xi{\otimes}k=\sigma_E(c{\otimes}{\overline{\partial}}a^*)$ then $$\begin{aligned}
\tilde {\partial}(\overline{c.a^*})-a\,\tilde {\partial}(\overline{c}) =&\ \overline{f.a^*}{\otimes}\kappa^*
+ \overline{k}{\otimes}\xi^*-a\,\overline{f}{\otimes}\kappa^*
= \overline{k}{\otimes}\xi^*\ ,\end{aligned}$$ giving $$\begin{aligned}
{\partial}_E(e.a)-{\partial}_E(e).a =&\
{{\langle}}e,\overline{k}{{\rangle}}.\xi^*{\otimes}g\ .\end{aligned}$$ Finally we use $$\begin{aligned}
({{\rm id}}{\otimes}\star^{-1})\Upsilon\,\overline{\sigma_E}\, \Upsilon^{-1}(\star{\otimes}{{\rm id}})({\partial}a{\otimes}\overline{c})
=\overline{k}{\otimes}\xi^*\ ,\end{aligned}$$ which gives the result.
Christoffel symbol approach
---------------------------
We shall use the matrix formalism for finitely generated projective modules and results given in [@BegMa3]. Of course the use of projection matrices for finitely generated projective modules is long established, but the use of matrices for inner products and Christoffel symbols in noncommutative geometry is more recent.
Suppose that $E$ is a left finitely generated projective module, and fix a dual basis $e^i\in E$ and $e_i\in E^\circ$ for $1\le i\le n$, where $E^\circ=
{}_A{{\rm{Hom}}}(E,A)$. Then $P_{ji}=e_i(e^j)={{\rm ev}}(e^j{\otimes}e_i)$ is a matrix with entries in $A$, and $P^2=P$. We can describe a left covariant derivative $\nabla_E$ by Christoffel symbols, defined by $$\begin{aligned}
\label{chrsymbdef}
\Gamma_k^i = -\,({{\rm id}}{\otimes}{{\rm ev}})(\nabla_E e^i {\otimes}e_k)\in\Omega^1 \ ,\end{aligned}$$ so we have $$\begin{aligned}
\label{christoffeldef}
\nabla_E e^i \,=\, -\,\Gamma_k^i{\otimes}e^k\ .\end{aligned}$$ Fit the Christoffel symbols into matrix notation by setting $$\begin{aligned}
\label{chrmatrix}
(\Gamma)_{ij}\,=\,\Gamma^i_j\ .\end{aligned}$$ Then a necessary and sufficient condition that $\Gamma\in M_n(\Omega^1)$ is the Christoffel symbols for a left connection on $E$ is that $$\begin{aligned}
\Gamma\,P\,=\,\Gamma\ ,\quad \Gamma\,=\,P\,\Gamma-{{\rm d}}P.P\ .\end{aligned}$$ The curvature of the connection is given by $$R_E(e^i)=-(({{\rm d}}\Gamma+\Gamma\wedge\Gamma).P)_{ik}{\otimes}e^k\ .$$
Suppose that we set $g^{ij}={{\langle}}e^i,\overline{e^j}{{\rangle}}$, so the hermitian condition gives $g^{ij*}=g^{ji}$. This corresponds to the invertible bimodule map $G:\overline{E}\to E^\circ$ being $G(\overline{e^i})=e_j.g^{ji}$, and we write the inverse as $G^{-1}(e_i)=\overline{g_{ij}.e^j}$, where without loss of generality we can assume that $g_{ij}.{{\rm ev}}(e^j{\otimes}e_k)= g_{ik}$. It is convenient to define matrices $g^\bullet,g_\bullet\in M_n(A)$ by $$\begin{aligned}
\label{matrixform}
(g_\bullet)_{ij}=g_{ij}\ ,\quad (g^\bullet)_{ij}=g^{ij}\ .\end{aligned}$$ and then we have $$\begin{aligned}
\label{matrixform77}
g^{\bullet*}=g^\bullet \ ,\quad g_\bullet^*=g_\bullet\ ,\quad
g^\bullet g_\bullet=P\ ,\quad g_\bullet P=g_\bullet\ ,\quad Pg^\bullet= g^\bullet\ .\end{aligned}$$
\[unichern\] Given a fgp holomorphic left $A$-module $E$ with a hermitian metric ${{\langle}},{{\rangle}}:E{\otimes}\overline{E}\to A$, there is a unique connection $\nabla_E:E\to\Omega^1 A{\otimes}_A E$, called the Chern connection, which preserves the hermitian metric and for which $(\pi^{0,1}{\otimes}{{\rm id}})\nabla_E={\overline{\partial}}_E$, the canonical ${\overline{\partial}}$-operator for $E$. If we write $\Gamma=\Gamma_++\Gamma_-$, with $\Gamma_+\in M_n(\Omega^{1,0})$ and $\Gamma_-\in M_n(\Omega^{0,1})$, then $\Gamma_-$ is determined by ${\overline{\partial}}_E$ and $$\begin{aligned}
-\,\Gamma_+= \partial g^\bullet.g_\bullet+g^\bullet\ (\Gamma_-)^* \, g_\bullet\ .\end{aligned}$$
Take a dual basis $(e^i,e_i)$ for $E$, and set $\nabla_E(e^i)=-\,\Gamma^i_{a}{\otimes}e^a$, using the definition of the Christoffel symbols in (\[chrsymbdef\]). Then the equation (\[hermcon1\]) for preserving the metric evaluated at $e^i{\otimes}\overline{e^j}$ becomes $$\begin{aligned}
{{\rm d}}\,g^{ij} &=& -\,({{\rm id}}{\otimes}{{\langle}},{{\rangle}})(\Gamma^i_{a}{\otimes}e^a{\otimes}\overline{e^j})-({{\langle}},{{\rangle}}{\otimes}{{\rm id}})(e^i{\otimes}\overline{e^a}{\otimes}(\Gamma^j_{a})^*)\cr
&=& -\,\Gamma^i_{a}\,g^{aj}-g^{ia}\,(\Gamma^j_{a})^*\ .\end{aligned}$$ Applying $\pi^{1,0}$ to this gives $$\begin{aligned}
\pi^{1,0}({{\rm d}}g^{ij})
&=& -\,\pi^{1,0}(\Gamma^i_{a})\,g^{aj}-g^{ia}\,\pi^{1,0}((\Gamma^j_{a})^*) \cr
&=& -\,\pi^{1,0}(\Gamma^i_{a})\,g^{aj}-g^{ia}\,(\pi^{0,1}(\Gamma^j_{a}))^* \ ,\end{aligned}$$ which is rearranged to give the answer. $\largesquare$
The $\Omega^{0,2}$ and $\Omega^{2,0}$ components of the curvature of the Chern connection in Proposition \[unichern\] vanish.
First $$\begin{aligned}
R_E(e^i)=&\ ({{\rm d}}{\otimes}{{\rm id}}_E-{{\rm id}}\wedge\nabla_E)\nabla_E(e^i) = -\,{{\rm d}}\Gamma^i{}_{a}{\otimes}e^a + \Gamma^i{}_{a}\wedge\nabla_E e^a \cr
=&\ -\,{{\rm d}}\Gamma^i{}_{a}{\otimes}e^a - \Gamma^i{}_{a}\wedge\Gamma^a{}_{b}{\otimes}e^b = -\,( {{\rm d}}\Gamma^i{}_{a} + \Gamma^i{}_{j}\wedge\Gamma^j{}_{a} )P_{ab}{\otimes}e^b\ .\end{aligned}$$ The $\Omega^{0,2}$ component of the curvature vanishes because ${\overline{\partial}}_E$ has zero holomorphic curvature. From Proposition \[unichern\], $$\begin{aligned}
{\partial}\Gamma_+ =&\ \partial g^\bullet\wedge\partial g_\bullet-\partial g^\bullet\wedge \Gamma_-{}^* \, g_\bullet
-g^\bullet\ \partial (\Gamma_-{}^*) \, g_\bullet + g^\bullet\ \Gamma_-{}^* \wedge \partial g_\bullet\ ,\cr
\Gamma_+ \wedge \Gamma_+ =&\ \partial g^\bullet.g_\bullet\wedge \partial g^\bullet.g_\bullet+
\partial g^\bullet \wedge \Gamma_-{}^* \, g_\bullet
+ g^\bullet\ \Gamma_-{}^* \, g_\bullet \wedge \partial g^\bullet.g_\bullet
+ g^\bullet\ \Gamma_-{}^* \wedge \Gamma_-{}^* \, g_\bullet\ ,\end{aligned}$$ and then $$\begin{aligned}
{\partial}\Gamma_+ + \Gamma_+ \wedge \Gamma_+=&\ ({\partial}g^\bullet+ g^\bullet\, \Gamma_-{}^*)\wedge({\partial}g_\bullet+ g_\bullet.{\partial}g^\bullet. g_\bullet) -g^\bullet\ \partial (\Gamma_-{}^*) \, g_\bullet
+ g^\bullet\ \Gamma_-{}^* \wedge \Gamma_-{}^* \, g_\bullet \cr
=&\ ({\partial}g^\bullet+ g^\bullet\, \Gamma_-{}^*)\wedge({\partial}g_\bullet+ g_\bullet.{\partial}g^\bullet. g_\bullet) - g^\bullet\ ({\overline{\partial}}\Gamma_-+\Gamma_- \wedge \Gamma_-)^* \, g_\bullet\ ,\end{aligned}$$ and the last bracket vanishes as it is just the holomorphic curvature of the holomorphic connection. Then $$\begin{aligned}
({\partial}g_\bullet+ g_\bullet.{\partial}g^\bullet. g_\bullet).P = {\partial}(P^*).g_\bullet\end{aligned}$$ and $$\begin{aligned}
({\partial}g^\bullet+ g^\bullet\, \Gamma_-{}^*).P^* ={\partial}g^\bullet+ g^\bullet\, \Gamma_-{}^*\ ,\end{aligned}$$ so $$\begin{aligned}
({\partial}\Gamma_+ + \Gamma_+ \wedge \Gamma_+).P
=&\ ({\partial}g^\bullet+ g^\bullet\, \Gamma_-{}^*).P^* \wedge {\partial}(P^*).g_\bullet\end{aligned}$$ where $Q=P^*=g_\bullet g^\bullet$ obeys $Q^2=Q$. Differentiating this gives ${\partial}Q.Q=(1-Q).{\partial}Q$, so $Q.{\partial}Q.Q=0$.$\largesquare$
Examples of Chern connections
=============================
The Chern connection on $\Omega^{1,0}$ for $M_2({{\mathbb{C}}})$
----------------------------------------------------------------
For the algebra $A=M_2({{\mathbb{C}}})$, the decomposition $\Omega^1=\Omega^{1,0}\oplus \Omega^{0,1}$ of the differential calculus gives an integrable almost complex structure. The complex differentials are given by the graded commutators ${\partial}=[E_{12}s,-\}$ and ${\overline{\partial}}=[E_{21}t,-\}$. There is a Riemannian structure $${{\langle}}u\oplus v,\overline{x\oplus y}{{\rangle}}= ux^*+vy^*\in A\ ,$$ and this can be converted to a Hilbert space inner product by taking the trace.
Now put a holomorphic structure on the bimodule $E=\Omega^{1,0}$. For the holomorphic connection ${\overline{\partial}}_E$ use $$\Omega^{1,0} \stackrel{{\overline{\partial}}}\longrightarrow \Omega^{1,1} \stackrel{\wedge^{-1}}\longrightarrow
\Omega^{0,1}{\otimes}_{M_2({{\mathbb{C}}})} \Omega^{1,0}\ ,$$ where we have used the fact that $\wedge:\Omega^{0,1}{\otimes}_{M_2({{\mathbb{C}}})} \Omega^{1,0}\to \Omega^{1,1} $ is a bimodule isomorphism. To check that this is a left ${\overline{\partial}}$-connection, for $\xi\in \Omega^{1,0}$, $${\overline{\partial}}_E(a.\xi)=\wedge^{-1}({\overline{\partial}}a\wedge\xi+a.{\overline{\partial}}\xi)={\overline{\partial}}a{\otimes}\xi+a.{\overline{\partial}}_E(\xi)\ .$$ The curvature of this ${\overline{\partial}}$-connection maps to $\Omega^{0,2}{\otimes}_{M_2({{\mathbb{C}}})} \Omega^{1,0}$, and thus must be zero as we set $s^2=t^2=0$, thus we have exhibited a holomorphic structure on $E=\Omega^{1,0}$.
We take the single basis element $s$ on $E=\Omega^{1,0}$. As ${\overline{\partial}}_E(s)=2\,E_{21}t{\otimes}s$ we get the Christoffel symbol $\Gamma_{-1,1}=-2\,E_{21}t$. We take the metric $${{\langle}}b\,s,\overline{a\,s}{{\rangle}}=b\,a^*\ ,$$ so $g^\bullet$ is a 1 by 1 matrix with the single element $g^{1,1}=1$. Then by Proposition \[unichern\] $$\Gamma_{+1,1}=-\,(-2\,E_{21}t)^*=-2\,E_{12}s\ .$$ so we have $$\nabla_E(s)=2\,E_{12}s{\otimes}s + 2\,E_{21}t{\otimes}s \ .$$ We get a bimodule covariant derivative, as $$\nabla_E(s.a)-\nabla_E(s).a=\nabla_E(a.s)-\nabla_E(s).a={{\rm d}}a{\otimes}s+[a,\nabla_E(s)]=-{{\rm d}}a{\otimes}s\ ,$$ which extends to the map $$\sigma_E(a.s{\otimes}\xi)=-a.\xi{\otimes}s\ .$$ This gives a natural bimodule connection on $\Omega^{1,0}$ which classically on a Kahler manifold would be part of the Levi-Civita connection for the hermitian metric. Note that in Section \[SecMatrix\] we have coincidentally taken $S=\Omega^{1,0}\oplus\Omega^{0,1}$ as a bundle but in this example the Chern connection is not the one used there to construct the ${{ \slashed{D} }}$ operator.
Chern connection on the standard quantum sphere
-----------------------------------------------
On $A={{\mathbb{C}}}_q[S^2]$ take the holomorphic connection on ${{\mathcal{S}}}_+$ (generated by $f^+$ as given earlier) given by ${\overline{\partial}}_{{{\mathcal{S}}}_+}:{{\mathcal{S}}}_+\to \Omega^{0,1}{\otimes}_A {{\mathcal{S}}}_+$, where $${\overline{\partial}}_{{{\mathcal{S}}}_+}(x\,f^+)={\overline{\partial}}x.k_1{\otimes}k_2.f^+\ ,$$ where $k=a{\otimes}d-q^{-1}\,c{\otimes}b=k_1{\otimes}k_2$ in a compact notation with summation understood. (Note ${\overline{\partial}}x$ denotes taking the $e^-$ component of $\pi{{\rm d}}$.) This has zero curvature as $\Omega^{0,2}=0$ and we are in the case where the grading operator $\gamma$ splits the spinor bundle into two parts, one of which is holomorphic. Now we use (\[sideswitch\]) to switch the side of the antilinearity on the inner product in Section \[ncHopfDirac\] as follows: $$\begin{aligned}
{{\langle}}x.f^+,\overline{y.f^+}{{\rangle}}=&\ {{\langle}}\overline{{{\mathcal{J}}}( x.f^+) } , {{\mathcal{J}}}^{-1}(y.f^+){{\rangle}}={{\langle}}\delta\,\overline{x^*.f^- } , \delta\,y^*.f^-{{\rangle}}=\delta^2\mu\, x\,y^*\ .\end{aligned}$$ For ${{ \slashed{D} }}$ to be hermitian, Proposition \[prrp1\] gives $\delta^2\mu=q$, but as in fact we will only be interested in the inner product on ${{\mathcal{S}}}_+$, we are free to absorb this $q$ factor in the normalisation of the inner product and hence we omit it in what follows. Then we can write ${{\langle}},{{\rangle}}^{-1}(1)=\overline{k_1^*.f^+}{\otimes}k_2.f^+$. Then the formula in Proposition \[unichernnog\] gives $$\begin{aligned}
{\partial}_{{{\mathcal{S}}}_+}(x\,f^+) =&\ {\partial}{{\langle}}x\,f^+, \overline{k_1^*.f^+}{{\rangle}}{\otimes}k_2.f^+ -
( {{\langle}},{{\rangle}}{\otimes}{{\rm id}}{\otimes}{{\rm id}}) (x\,f^+{\otimes}\tilde{\partial}(\overline{k_1^*.f^+}) {\otimes}k_2.f^+)\end{aligned}$$ Now, where $k_1'{\otimes}k_2'$ is an independent copy of $k$, $${\overline{\partial}}_{{{\mathcal{S}}}_+}(k_1^*.f^+)={\overline{\partial}}k_1^*.k_1'{\otimes}k_2'.f^+$$ so $$\begin{aligned}
{\partial}_{{{\mathcal{S}}}_+}(x\,f^+) =&\ {\partial}{{\langle}}x\,f^+{\otimes}\overline{k_1^*.f^+}{{\rangle}}{\otimes}k_2.f^+ -
{{\langle}}x\,f^+,\overline{k_2'.f^+}{{\rangle}}\,({\overline{\partial}}k_1^*.k_1')^* {\otimes}k_2.f^+ \cr
=&\ {\partial}(x\,k_1){\otimes}k_2.f^+ -
x\,(k_2')^*\,(k_1')^*\,{\partial}k_1 {\otimes}k_2.f^+ = {\partial}x.k_1{\otimes}k_2.f^+ \ .\end{aligned}$$ Thus the Chern connection is just $$\nabla_{{{\mathcal{S}}}_+}(x\,f^+) =\pi{{\rm d}}x.k_1{\otimes}k_2.f^+ \ ,$$ which is just the connection on ${{\mathcal{S}}}_+$ given in Section \[ncHopfDirac\].
We also compute the example $E=\Omega^{1,0}$. We have ${\overline{\partial}}(x\,e^+)={\overline{\partial}}x\wedge e^+$, and using the isomorphism $\wedge:\Omega^{0,1}{\otimes}_A \Omega^{1,0}\to
\Omega^{1,1}$ we write a holomorphic connection on $E$ as, where $k_1'{\otimes}k_2'$ is another copy of $k_1{\otimes}k_2$ (similarly for further primes) $${\overline{\partial}}_E(x\,e^+) = {\overline{\partial}}x.k_1k_1' {\otimes}k_2'k_2\,e^+\ .$$ We take the inner product $${{\langle}}x\,e^+,\overline{y\,e^+}{{\rangle}}_E= x\,y^*\ ,$$ and then ${{\langle}},{{\rangle}}_E^{-1}=\overline{k_1^{\prime*}k_1^*\,e^+} {\otimes}k_2'k_2\,e^+$. Now using $${\overline{\partial}}_E(k_1^{\prime*}k_1^*\,e^+) = {\overline{\partial}}(k_1^{\prime*}k_1^*).k_1''k_1''' {\otimes}k_2'''k_2''\,e^+\ .$$ we get $$\begin{aligned}
(\tilde{\partial}{\otimes}{{\rm id}}){{\langle}},{{\rangle}}_E^{-1} =&\ \overline{ k_2'''k_2''\,e^+ } {\otimes}({\overline{\partial}}(k_1^{\prime*}k_1^*).k_1''k_1''' )^* {\otimes}k_2'k_2\,e^+\ .\end{aligned}$$ We use the diagram formula in Proposition \[unichernnog\] to write $$\begin{aligned}
{\partial}_E(x\,e^+) =&\ {\partial}{{\langle}}x\,e^+, \overline{k_1^{\prime*}k_1^*\,e^+} \ {{\rangle}}_E {\otimes}k_2'k_2\,e^+
- {{\langle}}x\,e^+, \overline{ k_2'''k_2''\,e^+ } {{\rangle}}\, ({\overline{\partial}}(k_1^{\prime*}k_1^*).k_1''k_1''' )^* {\otimes}k_2'k_2\,e^+ \cr
=&\ {\partial}(x\,k_1k_1') {\otimes}k_2'k_2\,e^+
- x\,( k_2'''k_2'')^* \, (k_1''k_1''' )^* ({\overline{\partial}}(k_1^{\prime*}k_1^*) )^* {\otimes}k_2'k_2\,e^+ \cr
=&\ {\partial}(x\,k_1k_1') {\otimes}k_2'k_2\,e^+
- x\, {\partial}(k_1k_1') {\otimes}k_2'k_2\,e^+ = {\partial}(x).k_1k_1'{\otimes}k_2'k_2\,e^+ \ .\end{aligned}$$ Putting these parts together, we get $$\nabla_E(x\,e^+) = \pi{{\rm d}}x.k_1k_1'{\otimes}k_2'k_2\,e^+\ ,$$ which, given that the relevant component of the metric in [@Ma:rieq] can be written as $g_{-+}=e^-k_1k_2'{\otimes}k_2'k_2 e^+$, is the same as the $\Omega^{1,0}$ part of the quantum Levi-Civita connection on the $q$-sphere found there. Thus, both the connection for the Dirac operator and the Levi-Civita connection for the $q$-sphere are obtained from the Chern construction.
A Chern connection on the quantum open disk
-------------------------------------------
The calculus on the quantum disk in Section \[diskdir\] was constructed to be $U_q(sl_2)$ invariant, and also carries a hint of the hyperbolic structure. If we look for a central quantum symmetric metric, we are naturally led to $$g={{w}}^{-2}\,({{\rm d}}z{\otimes}{{\rm d}}\bar z+q^{-2}\, {{\rm d}}\bar z{\otimes}{{\rm d}}z)$$ Of course, this cannot be regarded as being a Riemannian structure for the closed disk, but rather it lives on the open disk, in fact if we let $q\to 1$ we get the classical hyperbolic metric. The inverse of $w$ appearing here indicates that we are dealing with unbounded functions on the disk. Now for $n\ge 1$, $$q^{1/2}X_+{{\triangleright}}({{w}}^n{{w}}^{-n})=0$$ so $$q^{1/2}X_+{{\triangleright}}({{w}}^{-n})=q^{-1}{{w}}^{-n}\bar z[n]_{q^{-2}}=q^{2n-1}\bar z[n]_{q^{-2}}{{w}}^{-n}\ .$$ Now we check that the metric is invariant to the $U_q(sl_2)$ action. This is easy for $q^{H\over 2}$ while for $X_+$ ($X_-$ is similar), $$\begin{aligned}
& q^{1/2}X_+{{\triangleright}}({{w}}^{-2}\,({{\rm d}}z{\otimes}{{\rm d}}\bar z+q^{-2}\, {{\rm d}}\bar z{\otimes}{{\rm d}}z)) \cr
=&\ (q^{1/2}X_+{{\triangleright}}({{w}}^{-2}))\,( q^{H\over 2} {{\triangleright}}({{\rm d}}z{\otimes}{{\rm d}}\bar z+q^{-2}\, {{\rm d}}\bar z{\otimes}{{\rm d}}z)) \cr
& +\ (q^{-H\over 2} {{\triangleright}}({{w}}^{-2}))\,( X_+ {{\triangleright}}({{\rm d}}z{\otimes}{{\rm d}}\bar z+q^{-2}\, {{\rm d}}\bar z{\otimes}{{\rm d}}z)) \cr
=&\ q^{3}\bar z[2]_{q^{-2}}{{w}}^{-2}({{\rm d}}z{\otimes}{{\rm d}}\bar z+q^{-2}\, {{\rm d}}\bar z{\otimes}{{\rm d}}z)\cr
& +\ {{w}}^{-2}\,( q^{-H\over 2} {{\triangleright}}{{\rm d}}z{\otimes}X_+ {{\triangleright}}{{\rm d}}\bar z+q^{-2}\,X_+ {{\triangleright}}{{\rm d}}\bar z{\otimes}q^{H\over 2} {{\triangleright}}{{\rm d}}z)) \cr
=&\ q^{3}\bar z[2]_{q^{-2}}{{w}}^{-2}({{\rm d}}z{\otimes}{{\rm d}}\bar z+q^{-2}\, {{\rm d}}\bar z{\otimes}{{\rm d}}z)\cr
& -\ {{w}}^{-2}\,(1+q^{-2})\,( q\,{{\rm d}}z{\otimes}\bar z. {{\rm d}}\bar z +q^{-3}\, \bar z. {{\rm d}}\bar z{\otimes}{{\rm d}}z)) \cr
=&\ q^{3}\bar z\, {{w}}^{-2}([2]_{q^{-2}} -1-q^{-2} )({{\rm d}}z{\otimes}{{\rm d}}\bar z+q^{-2}\, {{\rm d}}\bar z{\otimes}{{\rm d}}z)=0\ .\end{aligned}$$
An integrable almost complex structure is given by $J({{\rm d}}z)=\mathrm{i}\, {{\rm d}}z$ and $J({{\rm d}}\bar z)=-\mathrm{i}\, {{\rm d}}\bar z$. Now we examine the Chern connection for the holomorphic bundle $E=\Omega^{1,0}$ (we use $E$ as $\Omega^{1,0}$ is a rather cumbersome subscript) on the unit disk ${{\mathbb{C}}}_q[D]$. For the holomorphic connection ${\overline{\partial}}_E$ (and thus the holomorphic structure) on $E=\Omega^{1,0}$ use $$\Omega^{1,0} \stackrel{{\overline{\partial}}}\longrightarrow \Omega^{1,1} \stackrel{\wedge^{-1}}\longrightarrow
\Omega^{0,1}{\otimes}_{{{\mathbb{C}}}_q[D]} \Omega^{1,0}\ ,$$ where we have used the fact that $\wedge:\Omega^{0,1}{\otimes}_{{{\mathbb{C}}}_q[D]} \Omega^{1,0}\to \Omega^{1,1} $ is a bimodule isomorphism. To check that this is a left ${\overline{\partial}}$-connection, for $\xi\in \Omega^{1,0}$, $${\overline{\partial}}_E(a.\xi)=\wedge^{-1}({\overline{\partial}}a\wedge\xi+a.{\overline{\partial}}\xi)={\overline{\partial}}a{\otimes}\xi+a.{\overline{\partial}}_E(\xi)\ .$$ The curvature of this ${\overline{\partial}}$-connection maps to $\Omega^{0,2}{\otimes}_{{{\mathbb{C}}}_q[D]} \Omega^{1,0}$, and thus must be zero.
We take the single basis element ${{\rm d}}z$ on $E=\Omega^{1,0}$. As ${\overline{\partial}}_E({{\rm d}}z)=0$ we get $\Gamma_- =0$. Taking the invariant metric, $g^\bullet$ is a 1 by 1 matrix with the single element $$g^{1,1}={{\langle}}{{\rm d}}z,\overline{{{\rm d}}z}{{\rangle}}={{w}}^2\ .$$ Then $$\Gamma_{+1,1}=-\,{\partial}({{w}}^2)\,{{w}}^{-2}=\bar z\,{{\rm d}}z\,[2]_{q^{-2}}{{w}}^{-1}\ ,$$ so we have the Chern connection $$\nabla_E({{\rm d}}z)=-\bar z\,{{\rm d}}z\,[2]_{q^{-2}}{{w}}^{-1}{\otimes}{{\rm d}}z$$ associated to the above hermitian metric.
Now we shall consider the different bimodule, the sub-bimodule $S_+$ of the spinor bundle from Section \[diskdir\] generated by $s$. We define ${\overline{\partial}}_{{{\mathcal{S}}}_+}(s)=0$ (i.e. $\Gamma_{-1,1}=0$), giving this sub-bimodule a holomorphic structure. We use (\[sideswitch\]) to switch the sides of the previous spinor inner product: $$\begin{aligned}
{{\langle}}a\,s,\overline{b\,s}{{\rangle}}=&\ {{\langle}}\overline{{{\mathcal{J}}}(a\,s)},{{\mathcal{J}}}^{-1}(b\,s){{\rangle}}=\delta^2\,{{\langle}}\overline{\bar s\,a^*},\bar s\,b^*{{\rangle}}=\delta^2\,\mu\,a\,w\,b^*\ .\end{aligned}$$ Following the previous method, we have ${{\langle}}s,\bar s{{\rangle}}=\delta^2\,\mu\,w$, so $g^{11}=\delta^2\,\mu\,w$. Then the Chern connection has its other Christoffel symbol $$\Gamma_{+1,1}=-{\partial}(w)w^{-1}=\bar z.{{\rm d}}z.w^{-1}\ .$$ This is the natural Chern connection here but note that it is not the bimodule connection used for the construction of the Dirac operator.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper gives an explicit method for computing the resultant of any sparse unmixed bivariate system with given support. We construct square matrices whose determinant is exactly the resultant. The matrices constructed are of hybrid Sylvester and Bézout type. The results extend those in [@Khe] by giving a complete combinatorial description of the matrix. Previous work by D’Andrea [@D] gave pure Sylvester type matrices (in any dimension). In the bivariate case, D’Andrea and Emiris [@DE] constructed hybrid matrices with one Bézout row. These matrices are only guaranteed to have determinant some multiple of the resultant. The main contribution of this paper is the addition of new Bézout terms allowing us to achieve exact formulas. We make use of the exterior algebra techniques of Eisenbud, Fl[ø]{}ystad, and Schreyer [@ES; @EFS].'
address: 'Department of Mathematics, UC Berkeley, Berkeley CA, USA'
author:
- 'Amit Khetan [^1]'
bibliography:
- 'jsc02.bib'
title: The Resultant of an Unmixed Bivariate System
---
Introduction
============
Let $f_1, \dots, f_{n+1} \in \mathbb{C}[x_1, x_1^{-1}, \dots, x_n,
x_n^{-1}]$ be Laurent polynomials in $n$ variables with the same Newton polytope $Q \subset \mathbb{R}^n$. Let $A = Q \cap
\mathbb{Z}^n$. So we can write:
$$f_i = \sum_{\alpha \in A} C_{i\alpha}x^{\alpha}.$$
We will assume that $Q$ is actually $n$-dimensional, and furthermore that $A$ affinely spans $\mathbb{Z}^n$.
\[d:ares\] The $A$-[*resultant*]{} ${\rm Res}_A(f_1, \dots, f_{n+1})$ is the irreducible polynomial in the $C_{i\alpha}$, unique up to sign, which vanishes whenever $f_1, \dots, f_{n+1}$ have a common root in the algebraic torus $(\mathbb{C}^\ast)^n$.
The existence, uniqueness, and irreducibility of the $A$-resultant are proved in the book by Gelfand, Kapranov, and Zelevinsky [@GKZ]. The $A$-resultant, also called the sparse resultant, allows one to eliminate $n$ variables from $n+1$ unmixed equations. Hence, resultants can be quite useful in solving systems of polynomial equations [@CLO2]. It is an important problem to find efficiently computable, explicit formulas for the resultant.
When $n=1$, we are in the case of the classical resultant of two polynomials in one variable of the same degree. There are two formulas due to Sylvester and Bézout which represent the resultant as the determinant of an easily computable matrix. Sylvester’s matrix has entries that are either 0 or a coefficient of $f_1$ or $f_2$. The entries in Bézout’s matrix are linear in the coefficients of each of the $f_i$ hence quadratic overall.
Our work deals with the case $n=2$. We give a determinantal formula which is of hybrid Sylvester and Bézout type. A preliminary version of these results appeared in the ISSAC 2002 Proceedings [@Khe]. This paper makes the formula completely explicit and provides complete proofs. Our approach follows work by Jouanolou [@J] and Dickenstein and D’Andrea [@DD] who found formulas for the “dense” resultant, when the polytope $Q$ is a coordinate simplex of some degree. We make heavy use of new techniques by Eisenbud, Fl[ø]{}ystad and Schreyer [@ES; @EFS] relating resultants to complexes over an exterior algebra.
\[thm:blmtrx\] The resultant of a system $(f_1, f_2, f_3) \in \mathbb{C}[x_1, x_2,
x_1^{-1}, x_2^{-1}]$ with common Newton polygon $Q$ is the determinant of the block matrix: $$\label{e:blmtrx}
\begin{pmatrix}
B & L \\ \tilde{L} & 0 \end{pmatrix}.$$ The entries of $L$ and $\tilde{L}$ are linear forms, and the entries of $B$ are cubic forms in the coefficients $C_{i\alpha}$.
The columns of $B$ and $\tilde{L}$ are indexed by the lattice points in $Q$, the rows of $B$ and $L$ are indexed by the interior lattice points in $2 \cdot Q$, the matrix $\tilde{L}$ has three rows indexed by $\{ f_1, f_2, f_3 \}$, and the columns of the matrix $L$ are indexed by pairs $(f_i, a)$ where $i \in \{1,2,3\}$ and $a$ runs over the interior lattice points of $Q$. Each entry of $L$ and $\tilde L$ is either zero or is a coefficient of some $f_i$ and is determined in the following straightforward manner. The entry of $\tilde L$ in row $f_i$ and column $a$ is the coefficient of $x^a$ in $f_i$. The entry of $L$ in row $b$ and column $(f_i, a)$ is the coefficient of $x^{b-a}$ in $f_i$. The entries of the matrix $B$ are linear forms in [*bracket variables*]{}. A bracket variable is defined as
$$[abc] = \det \bmatrix C_{1a} & C_{1b} & C_{1c} \\
C_{2a} & C_{2b} & C_{2c} \\
C_{3a} & C_{3b} & C_{3c} \endbmatrix,$$
where $C_{ia}$ is the coefficient of $x^a$ in $f_i$. An explicit formula for $B$ is described in Section 3 below.
\[example\]
$$\begin{aligned}
f_1 &= C_{11} + C_{12}x + C_{13}y + C_{14}xy + C_{15}x^2y + C_{16}xy^2 \\
f_2 &= C_{21} + C_{22}x + C_{23}y + C_{24}xy + C_{25}x^2y + C_{26}xy^2\\
f_3 &= C_{31} + C_{32}x + C_{33}y + C_{34}xy + C_{35}x^2y + C_{36}xy^2\end{aligned}$$
The system above has Newton polygon as shown in Figure \[f:newt\]. We will show that the resultant of this system is the determinant of the matrix in Table \[tbl:matrix\].
---------- --------------- ---------- --------------- --------------- ---------- ---------- ---------- ----------
0 $[124]$ 0 $[126]-[234]$ $-[235]$ $-[236]$ $c_{11}$ $c_{21}$ $c_{31}$
0 0 0 0 0 0 $c_{12}$ $c_{22}$ $c_{32}$
0 $[126]-[135]$ 0 $[146]-[236]$ $[156]+[345]$ $[346]$ $c_{13}$ $c_{23}$ $c_{33}$
0 $-[145]$ 0 $[156]-[345]$ $[256]$ $[356]$ $c_{14}$ $c_{24}$ $c_{34}$
0 0 0 0 0 0 $c_{15}$ $c_{25}$ $c_{35}$
0 $[156]$ 0 $[356]$ $[456]$ 0 $c_{16}$ $c_{26}$ $c_{36}$
$c_{11}$ $c_{12}$ $c_{13}$ $c_{14}$ $c_{15}$ $c_{16}$ 0 0 0
$c_{21}$ $c_{22}$ $c_{23}$ $c_{24}$ $c_{25}$ $c_{26}$ 0 0 0
$c_{31}$ $c_{32}$ $c_{33}$ $c_{34}$ $c_{35}$ $c_{36}$ 0 0 0
---------- --------------- ---------- --------------- --------------- ---------- ---------- ---------- ----------
: Resultant matrix for Example \[example\][]{data-label="tbl:matrix"}
In Section \[s:toric\] we provide some preliminary results about toric varieties and their homogeneous coordinates which allow us to present our formula in Section \[s:formula\]. Section \[s:tate\] describes the exterior algebra techniques of Eisenbud, Schreyer, and Fl[ø]{}ystad. Section \[s:torictate\] applies these results to the toric setting, while Section \[s:proofs\] goes on to prove our formula. Finally Section \[s:general\] briefly discusses possible generalizations to more variables.
Acknowledgments
---------------
I would like to thank my advisor Bernd Sturmfels for providing direction and support. I thank David Eisenbud for introducing me to exterior algebra methods. I thank David Speyer for the proof of Lemma \[comblemma\].
Toric Varieties {#s:toric}
===============
Let $\mathbb{Q} \subset \mathbb{R}^n$ be a lattice polytope of dimension $n$, and $A = Q \cap \mathbb{Z}^n = \{ \alpha_1, \dots,
\alpha_N \}$. We assume that $A$ affinely spans $\mathbb{Z}^n$. The [*toric variety*]{} $X_A$ is the dimension $n$ variety defined as the Zariski closure of the following set in $\mathbb{P}^{N-1}$:
$$X_A^0 = \{ (x^{\alpha_1}:\cdots : x^{\alpha_N}) \ : \ x = (x_1, \dots, x_n) \in (\mathbb{C}^\ast)^n \}.$$
Now a polynomial system $(f_1, \dots, f_{n+1})$ can be thought of as $n+1$ hyperplane sections of $X_A$ in $\mathbb{P}^{N-1}$. Generically, such a system defines a codimension $n+1$ plane.
For any $n$-dimensional irreducible projective variety $X$, it turns out that the condition on a linear subspace of codimension $n+1$ meeting $X$ is actually a closed condition of codimension 1 (see [@GKZ] for details). Therefore we can make the following definition.
If $X \subset \mathbb{P}^{N-1}$ is a variety of dimension $n$, the codimension $n+1$ planes meeting $X$ define a hypersurface in the Grassmannian $G(n+1, N)$. The equation of this hypersurface is called the [*Chow form*]{} of $X$.
In particular, the $A$-resultant is the Chow form of $X_A$. As a consequence we have the following strengthening of Definition \[d:ares\].
${\rm Res}_A(f_1, \dots, f_{n+1}) = 0$ if and only if the $f_i$ have a common root on $X_A$.
Returning to the defining polytope $Q$, let $d_1, \dots, d_s$ denote the facets of $Q$. Let $\eta_i$ be the first lattice vector along the inner normal to facet $d_i$. The [*normal fan*]{} of $Q$ is the set of cones, one for each vertex, spanned by the $\eta_i$ corresponding to facets incident to that vertex. The next proposition can be found in Fulton’s book [@Ful].
The $\eta_i$ are in 1-1 correspondence with the $T$-invariant prime Weil divisors on $X_A$. Let $D_i$ denote the divisor corresponding to $\eta_i$.
The polytope $Q$ can be characterized completely in terms of the rays in its normal fan as follows:
$$Q = \{ m \in \mathbb{R}^n \: \ \langle m, \eta_i \rangle \geq -a_i, \ i = 1, \dots, s \}.$$
The very ample divisor corresponding to the embedding of $X_A$ into $\mathbb{P}^{N-1}$ corresponding to $Q$ is just $D = \sum a_i D_i$. We can now define the [*homogeneous coordinate ring*]{} of $X_A$. This was introduced by Cox [@Cox] and the propositions below follow from this paper.
Let $S = \mathbb{C}[y_1, \dots, y_s]$ be the polynomial ring with one variable for each $\eta_i$. Consider the short exact sequence of abelian groups:
$$\begin{CD} 0 @>>> \mathbb{Z}^{n} @>\phi>> \mathbb{Z}^s @>\pi>> G
@>>> 0. \end{CD}$$
Here $\phi(m) = (\langle m, \eta_1 \rangle, \dots, \langle m, \eta_s
\rangle)$, and $G$ is the cokernel of $\phi$.
Define a $G$-grading on $S$ as follows. Given $y^\alpha \in S$, let $\deg(y^\alpha) = \pi(\alpha) \in G$.
Now we will identify the lattice points in $Q$ with a graded piece of $S$.
Let $\alpha \in Q \cap \mathbb{Z}^n$. Define $\alpha_i = \langle
\alpha, \eta_i \rangle + a_i$ for $i = 1, \dots, s$ and the $a_i$ are the defining data for $Q$ as above. The $Q$-[*homogenization*]{} of $x^\alpha$ is $\prod_{i=1}^s y_i^{\alpha_i}$. We will write this as $y^{\alpha}$ and use the letter $\alpha$ to denote both a vector $\alpha \in \mathbb{Z}^n$ and its homogenization $(\alpha_1, \dots
\alpha_s)$, where the meaning will be clear from the context.
\[p:sq\] Let $a = (a_1, \dots, a_s)$ be the defining data for $Q$. The monomials in the $\pi(a)$ graded piece of $S$ are in 1-1 correspondence with the lattice points in $Q$. Denote this graded piece by $S_Q$. Moreover, $H^0(X_A, \mathcal{O}(D)) \cong S_Q$.
There is a similar characterization of the interior lattice points of $Q$.
\[p:sintq\] Let $\omega_0 = (1, 1, \dots, 1) \in \mathbb{Z}^s$. The monomials in the $\pi(a - \omega_0)$ graded piece of $S$ are in 1-1 correspondence with the interior lattice points of $Q$. Denote this graded piece $S_{{\rm
int}(Q)}$. We have $H^0(X, \mathcal{O}(D- \sum_{i=1}^s D_i)) \cong
S_{{\rm int}(Q)}$.
Formula for $B$ {#s:formula}
===============
We now return to case of two variables. So $(f_1, f_2, f_3) \in
\mathbb{C}[x_1, x_2, x_1^{-1}, x_2^{-1}]$ have common Newton polygon $Q \subset \mathbb{R}^2$. The rays in the normal fan of $Q$ are $\{
\eta_1, \dots, \eta_s \}$, assumed to be in counterclockwise order. We pick out the distinguished cone spanned by $\{\eta_1, \eta_2\}$ and partition the vectors in the fan as follows:
$$\begin{aligned}
\label{eqn:Rsets}
R_1 &= \{i \ | \ \eta_i = c_1 \eta_1 + c_2 \eta_2 \ {\rm with} \ c_1 \geq 0 \ {\rm and} \ c_2 \leq 0 \} \nonumber \\
R_2 &= \{i \ | \ \eta_i = c_1 \eta_1 + c_2 \eta_2 \ {\rm with} \ c_1 \leq 0 \ {\rm and} \ c_2 \geq 0 \} \\
R_3 &= \{i \ | \ \eta_i = c_1 \eta_1 + c_2 \eta_2 \ {\rm with} \ c_1 < 0 \ {\rm and} \ c_2 < 0 \}. \nonumber \end{aligned}$$
It is possible that $R_3$ as defined is empty. If that is the case we need to [*refine*]{} the fan, by adding in one new vector, say $\eta_{s+1} = -\eta_1 - \eta_2$. This new vector $\eta_{s+1}$ lies in the interior of some cone spanned by $\eta_i$ and $\eta_{j}$, hence can be written as $c_1 \eta_i + c_2 \eta_j$ for some positive $c_1,
c_2$. Define $a_{s+1} = c_1 a_i + c_2 a_j$. As above, given $\alpha
\in Q$ we denote by $\alpha_{s+1}$ the quantity $\langle \alpha,
\eta_{s+1} \rangle + a_{s+1}$.
In fact, if there is a single fan vector $\eta_i$ such that $-\eta_i$ is not a ray in the fan, then we can choose our distinguished cone to be the one containing $-\eta_i$, and $R_3$ is guaranteed not to be empty. However, for polytopes such that every edge has a corresponding parallel edge, this is not the case.
A good way to think about these sets is that we choose a distinguished vertex $p$ of $Q$ having normal cone spanned by $\{\eta_1,
\eta_2\}$. The set $R_3$ consists of all edges of $Q$ such that the corresponding inner normals are maximized at $v$. If there is no such edge, then our refinement adds in a “length 0” edge whose inner normal is maximized at $p$. $R_1$ is the set of the remaining edges clockwise from $v$, while $R_2$ is the set of remaining edges counterclockwise from $v$.
This partition is illustrated in Figure \[f:fan\] for Example \[example\] with the choice of the vertex $p$. Edge $4$ has the only normal maximized at $p$, thus is the only element in $R_3$. The edges in $R_1$ and $R_2$ are $\{1,5\}$ and $\{2, 3\}$ respectively.
We can now state an explicit formula for the matrix $B$ appearing in the Theorem \[thm:blmtrx\].
\[thm:main\]
The matrix $B$ from Theorem \[thm:blmtrx\] is the matrix of the linear map $\Delta_Q \ : \ (S_Q)^\ast \to S_{{\rm int}(2Q)}$ defined as follows:
$$\Delta_Q((y^\alpha)^\ast) = \sum_{(u,v,w) \in F_{\alpha} \subset
A^3} [uvw] y^{u+v+w - \alpha - \omega_0}.$$
Here $\omega_0 = (1, 1, \dots, 1)$, and $F_\alpha$ is the set of all triples $(u,v,w) \in A^3$ satisfying the following Boolean combination of inequalities: $$\begin{aligned}
\label{eqn:ineq}
\forall i \in R_1 \quad u_i + v_i + w_i >& \alpha_i \nonumber \\
\exists i \in R_1 \quad v_i + w_i \leq& \alpha_i \nonumber \\
\forall j \in R_2 \quad v_j + w_j >& \alpha_j \\
\exists j \in R_2 \quad w_j \leq& \alpha_j \nonumber \\
\forall k \in R_3 \quad w_k >& \alpha_k \nonumber,\end{aligned}$$ where the $R_i$ are as described in (\[eqn:Rsets\]).
Let’s see how this works for Example \[example\]. Specifically, consider the point $\alpha = (1,1)$ corresponding to the monomial $xy$. The homogenization is $y_1y_2y_3y_4y_5$. If the monomials are numbered $1, \dots, 6$ as in the equations, then the only solutions to the inequalities above are: $$(u,v,w) = \{ (2,6,1), (4, 6, 1), (5, 6, 1), (2,4,3), (5,4,3), (2,6,3),
(5,6,3) \}.$$
It follows that $$\begin{aligned}
\Delta_Q((y_1y_2y_3y_4y_5)^\ast) &= ([261]+[243])y_3y_4^3y_5 + ([461] + [263])y_2y_3^2y_4^2 \\
&+ ([561] + [543])y_1y_2y_3y_4y_5 + [563]y_1y_2^2y_3^2, \end{aligned}$$ which corresponds to the fourth column of the matrix in Table \[tbl:matrix\].
Tate Resolution {#s:tate}
===============
In this section we describe a complex used by Eisenbud and Schreyer [@ES; @EFS] to compute Chow forms of projective varieties. This begins as a complex of free modules over an exterior algebra, however there is a functor which transforms it into a complex of vector bundles on the Grassmannian. The determinant of this new complex will be the Chow form.
Suppose $X \subset \mathbb{P}^{N-1}$ is an irreducible variety of dimension $ n$. The ambient projective space $\mathbf{P} =
\mathbb{P}^{N-1}$ has the graded coordinate ring $R = \mathbb{C}[X_1,
\dots, X_N]$. If we let $W$ be the $\mathbb{C}$ vector space spanned by the $X_i$, identified with the degree 1 part of $R$, then $\mathbf{P}$ is the projectivization $\mathbb{P}(W)$. The ring $R$ can also be identified with the symmetric algebra $Sym(W)$.
Now let $V = W^\ast$, the dual vector space, with a corresponding dual basis $ e_1, \dots e_N$. We will consider the [*exterior algebra*]{} $E = \bigwedge V$, which is also graded where the $e_i$ have degree $
-1$. We will use the standard notation $E(k)$ to refer the rank 1 free $E$-module generated in degree $-k$.
Now given any coherent sheaf $\mathcal{F}$ on $\mathbf{P}$, there is an associated exact complex of graded free $E$-modules, called the [*Tate resolution*]{}, denoted $T(\mathcal{F})$. The terms of $T(\mathcal{F})$ can be written in terms of the sheaf cohomology of twists of $\mathcal{F}$. Namely, we have:
$$\label{e:tate}
T^e(\mathcal{F}) = \oplus_{j=0}^{N-1} [H^j(\mathcal{F}(e-j))
\otimes_{\mathbb{C}} E(j-e)]$$
for all $e \in \mathbb{Z}$. See Eisenbud-Fl[ø]{}ystad-Schreyer [@EFS].
Suppose further that $\mathcal{F}$ is chosen to be supported on $X$. Recall that the Chow form of $X$ is the defining equation of the set of codimension $n+1$-planes meeting $X$. Such a plane is specified by an $n+1$ dimensional subspace $W_f = \mathbb{C} \{ f_1, \dots,
f_{n+1} \} \subset W$. Let $\mathbf{G_{n+1}}$ be the Grasmannian of codimension $n+1$-planes on $\mathbf{P}$. Let $\mathcal{T}$ be the [*tautological bundle*]{} on $\mathbf{G_{n+1}}$, that is to say the fiber at the point corresponding to $f$ is just $W_f$.
The following proposition is a consequence of Theorem 0.1 in [@EFS].
There is an additive functor $U_{n+1}$ from graded free modules over $E$ to vector bundles on $\mathbf{G_{n+1}}$, such that $U_{n+1}(E(p))
= \wedge^p \mathcal{T}$. Furthermore, if $\mathcal{F}$ is a sheaf of rank $k$ supported on a variety $X \subset \mathbb{P}(V)$ of dimension $n$, $U_{n+1}(T(\mathcal{F}))$ is a complex of vector bundles whose determinant is the $k$-th power of the Chow form of $X$.
The determinant of a complex of vector bundles on $\mathbf{G_{n+1}}$ is a homogeneous polynomial function on $\mathbf{G_{n+1}}$ whose value at a particular point is the corresponding determinant of the complex of [*vector spaces*]{} over that point. The determinant of a complex of vector spaces is defined in [@GKZ Appendix A].
So, in particular if we could choose $\mathcal{F}$ so that enough cohomology vanishes, this new complex $U_{n+1}(T(\mathcal{F}))$ may have only two terms and a single non trivial map $\Psi_{\mathcal{F}}$. Such sheaves are called [*weakly Ulrich*]{}, see [@ES Section 2]. In this case, to compute the Chow form we need only compute the determinant of $\Psi_{\mathcal{F}}$. This is exactly what we do in the next section. But first we need to describe the maps in the Tate resolution, and also how the functor $U_{n+1}$ acts.
The maps in the Tate resolution are composed of maps $H^j(\mathcal{F}(e-j)) \otimes E(j-e) \to H^k(\mathcal{F}(e+1-k))
\otimes E(k-e-1)$. All such maps for $k > j$ must be 0 by degree considerations.
When $k = j$ we have a [*linear map*]{} $H^j(\mathcal{F}(e-j)) \otimes
E(j-e) \to H^j(\mathcal{F}(e+1-j)) \otimes E(j-e-1)$ which is canonical and completely well understood. Explicitly we consider the graded $R$-module $M^j = \oplus_{l > 0} H^j(\mathcal{F}(l))$. The Bernstein-Gel’fand-Gel’fand correspondence [@EFS Section 2] applied to $M^j$ results in a map $M^j_{e-j} \otimes E(j-e)
\to M^j_{e-j+1} \otimes E(j-e-1)$ which is just multiplication by the element $m = \sum X_i \otimes e_i$. By [@EFS Theorem 4.1] these are exactly the linear maps in the Tate Resolution.
Much more mysterious are the nonlinear [*diagonal maps*]{} corresponding to $k < j$. Indeed one of the major contributions of this paper is an explicit formula for one of these diagonal maps in the case of a toric surface. Eisenbud and Schreyer [@ES] outline a general procedure for computing the Tate resolution, and therefore the diagonal maps, however it requires computing a free resolution and is not an explicit formulation.
Before moving on to the toric setting let us complete the description of the functor $U_{n+1}$ by describing how it acts on morphisms. The functoriality and other useful properties of the construction below are in Proposition 1.1 of [@ES] .
Given a map $E(q) \to E(q-p)$ we need to construct a map $\bigwedge^q
\mathcal{T} \to \bigwedge^{q-p} \mathcal{T}$. Any map $E(q) \to E(q-p)$ is defined by a single element $a \in \bigwedge^p V$. This also defines a map $\bigwedge^p W \to \mathbb{C}$. As $\mathcal{T}$ is a subbundle of $W \otimes \mathcal{O}_{\mathbf{G_{n+1}}}$, there is an induced map $a \ :\ \bigwedge^p \mathcal{T} \to
\mathcal{O}_{\mathbf{G_{n+1}}}$. Finally, to construct the map $U_{n+1}(a) \ : \ \bigwedge^q \mathcal{T} \to \bigwedge^{q-p}
\mathcal{T}$, start with the standard diagonal map $\Delta \ : \
\bigwedge^q \mathcal{T} \to \bigwedge^{q-p} \mathcal{T} \otimes
\bigwedge^{p} \mathcal{T}$ and compose with the map $1 \otimes a$.
We will need to use a more explicit description of the map, in terms of our chosen bases. Recall that a fiber of $\mathcal{T}$ is a subspace $W_f = \mathbb{C}\{f_1, \dots, f_{n+1} \}$. We can write the $f_i$ as:
$$f_i = \sum_{j=1}^N C_{ij} X_j.$$
The coefficients form a $(n+1) \times N$ matrix $C$. Given ordered subsets $I = \{i_1, \dots, i_p \} \subset \{1, \dots, n+1 \}$ and $J =
\{j_1, \dots, j_p \} \subset \{1, \dots, N \}$, of the same size $p$, let $C_{I,J}$ denote the determinant of the submatrix of $C$ with rows from $I$ and columns from $J$. We will also use the notation $f_I =
(-1)^I \bigwedge_{i \in I} f_i$ and $e_J = \bigwedge_{j
\in J} e_j$. Note the sign factor added to the $f$ part only in order to simplify the signs in the next proposition:
\[l:map\]
Let $J \subset \{1, \dots, N \}$ with $|J| = p$. We view $e_J$ as a map from $E(q)$ to $E(q-p)$. In that case for any $I \subset \{1,
\dots, n+1\}$ with $|I| = q$:
$$(U_{n+1}(e_J))(f_I) = \sum_{I_1 \subset I, \ |I_1| = p} C_{I_1, J}
f_{I \setminus I_1}$$
This is a direct translation of the above description applied to our particular choice of bases. The diagonal map splits up $f_I$ into a sum of pieces corresponding to a choice of $I_1$ and its complement. The action of $e_J$ on the piece corresponding to $I_1$ is exactly the determinant of the specified minor. The only thing to check is that the sign works out.
Toric Tate Resolution {#s:torictate}
=====================
We return to the case in question, where $X_A$ is a toric surface with corresponding polytope $Q$. As we saw earlier, the sections of the corresponding very ample divisor are just the elements of the vector space $S_Q$. Therefore, we will apply the exterior algebra construction with $W = S_Q$ and $V = S_Q^\ast$. The corresponding projective space is $\mathbf{P} = \mathbb{P}(W) \cong
\mathbf{P}^{N-1}$, and the exterior algebra is $E = \bigwedge V$.
Any Weil divisor on the toric surface $X_A$ yields a rank one reflexive sheaf which can be extended to a sheaf on $\mathbf{P}$ under the given embedding. We will consider the particular divisor corresponding to ${\rm int}(2Q)$ i.e. $2D - \sum D_i$. Let $\mathcal{F}$ be the corresponding sheaf $\mathcal{O}_{X_A}({\rm int}(2Q))$ extended to a sheaf of $\mathbf{P}$.
\[p:cohom\] $$\begin{aligned}
\label{eqn:H0} H^0 (\mathcal{F}(k)) &\cong S_{{\rm int} ((2+k)Q)} \\
\label{eqn:H1} H^1(\mathcal{F}(k)) &\cong 0 \\
\label{eqn:H2} H^2 (\mathcal{F}(k)) &\cong S_{(-2-k)Q}^\ast\end{aligned}$$
for all $k \in \mathbb{Z}$.
First of all, since all sheaves are supported on $X_A$, it is equivalent to compute cohomology on $X_A$. By construction, $X_A$ is normal and thus Cohen Macaulay by Hochster’s theorem. The dualizing sheaf is $\mathcal{O}(\omega) = \mathcal{O}(-\sum D_i)$. Also, twisting by $1$ on $\mathbf{P}$ is the same as twisting by $D$ on $X_A$. Therefore, $\mathcal{F}(k) = \mathcal{O}((k+2)D - \omega)$.
Now (\[eqn:H0\]) follows from Proposition \[p:sintq\]. For $k > -2$, $\mathcal{F}(k)$ is an ample divisor minus the canonical divisor. Therefore, the higher cohomology, $H^1$ and $H^2$ must be zero by Mustata’s vanishing result, [@Mus Theorem 2.4 (ii)].
Furthermore, $\mathcal{O}(D)$ is very ample, hence locally free, so Serre duality tells us $H^i(\mathcal{O}((k+2)D - \omega)) \cong
H^{2-i} (\mathcal{O}((-2-k)D))^\ast$. In particular, applying Proposition \[p:sq\] to $i=2$ gives us statement (\[eqn:H2\]) in the proposition. For $k \leq -2$, $\mathcal{O}((-2-k)D)$ is generated by its sections and so all higher cohomology, in particular $H^1$ vanishes, completing the proof of (\[eqn:H1\]).
The Tate resolution $T(\mathcal{F})$ has terms:
\[thm:tate\] $$\begin{aligned}
T^e(\mathcal{F}) =& S^\ast_{-eQ} \otimes E(2-e) \quad {\rm for} \ e < -1 \\
T^{-1}(\mathcal{F}) =& S^\ast_Q \otimes E(3) \oplus S_{{\rm int}(Q)} \otimes E(1) \\
T^{0}(\mathcal{F}) =& S^\ast_0 \otimes E(2) \oplus S_{{\rm int}(2Q)} \otimes E(0) \\
T^e(\mathcal{F}) =& S_{{\rm int}(eQ)} \otimes E(-e) \quad {\rm for} \ e >0,\end{aligned}$$ with maps as follows:
& \^[i\_m]{} & (S\_[2Q]{})\^E(4) & \^[i\_m]{} & (S\_Q)\^E(3) & \^[i\_m]{} & (S\_0)\^E(2) & & 0 &\
&& & \^[\_[2Q]{}]{} & & \^[\_Q]{}& & \^[\_0]{} & &\
&& 0 & & S\_[[int]{}(Q)]{} E(1) & \^[m]{} & S\_[[int]{}(2Q)]{} E & \^[m]{} & S\_[[int]{}(3Q)]{} E(-1) & \^[m]{} & .\
The horizontal maps $\wedge m$ and $i_m$ are all multiplication by the element $m = \sum y^\alpha \otimes e_\alpha$ where $\alpha$ ranges over the lattice points in $Q$, and $e_\alpha$ is the corresponding dual vector in $E$.
We simply plug in our known cohomology from \[p:cohom\] into (\[e:tate\]) to obtain the terms. The horizontal maps are indeed multiplication by $m$, as per our discussion in the previous section, noting only that the Serre duality respects the $S$-module structure of the cohomology.
Now we apply the functor $U_3$ to $T(\mathcal{F})$. Once again let $\mathcal{T}$ denote the tautological bundle on the Grassmannian of codimension 3 planes in $\mathbb{P}^{N-1}$. Note that $\bigwedge^p
\mathcal{T} = 0$ for $p > 3$ or $p < 0$. Therefore $U_3(T(\mathcal{F}))$ is the two term complex below:
& & (S\_Q)\^\^3 & \^ & (S\_0)\^\^2 & &\
0 & & & \^[\_Q]{} & & & & 0.\
& & S\_[[int]{}(Q)]{} \^1 & \^ & S\_[[int]{}(2Q)]{} \^0 &\
Since $\mathcal{F}$ is of rank 1, the resultant is up to a constant the determinant of the matrix of the nontrivial map $(\widehat{i_m} +
\widehat{\Delta}_Q) \oplus \widehat{\wedge m}$. However, we can of course normalize the maps in the Tate resolution so that we have the resultant up to sign. From here on we assume that such a normalization has been made.
All that is left to do is describe the maps $\widehat{\wedge m},
\widehat{\Delta}_Q$, and $\widehat{i_m}$. It is enough to define these maps on each fiber, that is, for each choice of $(f_1, f_2, f_3)$.
To describe the maps $\widehat{\wedge m}$ and $\widehat{i_m}$ we introduce the [*Sylvester map*]{} $\Psi_t \ : \ S_t \otimes
\mathbb{C}^3 \to S_{t+Q}$ which sends $(g_1, g_2, g_3)$ to $f_1g_1 +
f_2g_2 + f_3g_3$.
\[p:sylvmaps\] The map $\widehat{\wedge m}$ is $\Psi_{{\rm int}(Q)}$, and the map $\widehat{i_m}$ is $(\Psi_0)^\ast$ on each fiber over the Grassmannian.
First consider $\wedge m$. We pass to $\bigwedge^1 \mathcal{T}$, which has a basis at each fiber indexed by $f_1, f_2, f_3$. By Lemma \[l:map\], on the factor corresponding to $f_i$ we must replace each $e_{\alpha}$ in $m$ by the corresponding coefficient $C_{i\alpha}$. So on the factor corresponding to $f_i$, multiplication by $m =
\sum_{\alpha \in A} y^\alpha \otimes e_{\alpha}$ becomes multiplication by $\sum_{\alpha \in A} C_{i\alpha}y^\alpha = f_i$ . This is exactly the Sylvester map.
On the other hand $i_m$ is the map sending $(y^\alpha)^\ast$ to $e_\alpha$. To apply the functor $U_3$ we pick the basis $(f_1 \wedge
f_2 \wedge f_3)$ on $\bigwedge^3 \mathcal{T}$ and $(f_2 \wedge f_3, -f_1
\wedge f_3, f_1 \wedge f_2)$ on $\bigwedge^2 \mathcal{T}$. Another application of Lemma \[l:map\] shows that $e_\alpha$ is replaced by the vector $(C_{1\alpha}, C_{2\alpha}, C_{3\alpha})$ in terms of this second basis. This is exactly the dual Sylvester map $(\Psi_0)^\ast$.
Computing $\widehat{\Delta_Q}$ from $\Delta_Q$ is straightforward.
\[p:bezmap\] Write
$$\Delta_Q((y^\alpha)^\ast) = \sum_\beta \sum_{u,v,w} c_{uvw} (e_u \wedge e_v \wedge e_w) y^{\beta},$$
then for each fiber $(f_1, f_2, f_3)$ on the Grassmannian:
$$\widehat{\Delta_Q}((y^\alpha)^\ast) = \sum_\beta \sum_{u,v,w} c_{uvw} [uvw] y^{\beta}.$$
Here, both $\bigwedge^3 \mathcal{T}$ and $\bigwedge^0 \mathcal{T}$ are 1 dimensional vector spaces. Lemma \[l:map\] tells us to replace $e_u \wedge e_v \wedge e_w$ by the determinant of the maximal minor with columns $u, v, w$ of the coefficient matrix of the $f_i$, i.e the bracket $[uvw]$.
Putting it all together we have a proof of Theorem \[thm:blmtrx\].
The Chow form is the determinant, up to sign, of the map $(\widehat{i_m} +
\widehat{\Delta}_Q) \oplus \widehat{\wedge m}$. However, the blocks of the matrix corresponding to $\wedge m$ and $i_m$ are just Sylvester maps, by Proposition \[p:sylvmaps\], whose matrices are $L$ and $\tilde L$ respectively. The matrix of $\widehat{\Delta_Q}$ has entries which are linear forms in the bracket variables by Proposition \[p:bezmap\] above.
As a corollary we note that the matrix must be square. That is, $3 +
\#{\rm int}(2Q) = 3\cdot\#{\rm int}(Q) + \#Q$. This identity also arises from the simple fact that the third difference of the quadratic Erhart polynomial of $Q$ is 0.
All that is left is to prove our formula for $\widehat{\Delta_Q}$ in Theorem \[thm:main\], for which, by the above, we need to prove the corresponding formula for $\Delta_Q$. It turns out that it is easy to compute $\Delta_0$, and we can verify a formula for $\Delta_Q$ by making sure it lifts $\Delta_0$. This is described below.
The Map $\Delta_Q$ {#s:proofs}
==================
The map $\Delta_0$ is closely related to the [*toric Jacobian*]{} [@Cox2]. The toric Jacobian is usually constructed as the determinant of a matrix of partial derivatives. Cattani, Cox, and Dickenstein [@CCD] construct a different element, which they call $\Delta_{\sigma}$, referring to the choice $\sigma$ of a cone in the fan, which is a constant times the Jacobian modulo the ideal $I = (f_1,
f_2, f_3)$. Moreover, while the Jacobian of three forms supported on $Q$ has [*toric residue*]{} [@CCD] equal to the normalized area of $Q$, this new element has residue 1. Therefore, we will call this the [*normalized Jacobian*]{} and it is unique modulo $I$.
Let $y_1, y_2$ be edge variables such that the corresponding edges meet at a vertex $p$. Let $y_3, \dots, y_s$ be the remaining edge variables of the homogeneous coordinate ring $S$. A monomial $m$ in $S_Q$ is divisible by $y_i$ if and only if the corresponding lattice point in $Q$ is not on the corresponding edge.
Therefore, we can define a partition of the monomials in $S_Q$ into three sets $\mu_1, \mu_2, \mu_3$, where $\mu_1$ is defined to be the set of all monomials divisible by $y_1$, $\mu_2$ is the set of monomials divisible by $y_2$ but not divisible by $y_1$, and $\mu_3$ divisible by $y_3\cdots y_s$ but not by either $y_1$ or $y_2$.
Note that $\mu_1$ corresponds to points not on edge 1, $\mu_2$ is the points on edge 1, but not edge 2, and $\mu_3$ is the unique point, the vertex $p$, on both edges 1 and 2.
Set $M_i = \sum_{s \in \mu_i} s \otimes e_s \in
S_Q \otimes E$. Define $J_0 = \frac{M_1}{y_1} \wedge \frac{M_2}{y_2}
\wedge \frac{M_3}{y_3\cdots y_s}$, an element of $S_{{\rm int}(3Q)}
\otimes E_{-3}$. A choice for the map $\Delta_0 \ : \ (S_0)^\ast
\otimes E(2) \to S_{int(3Q)} \otimes E(-1)$ is $1 \otimes 1 \mapsto
-J_0$.
The element $J_0$ is chosen so that $U_3(J_0)$ is the normalized toric Jacobian as constructed in [@CCD].
First note that the map $\Delta_0$ is determined by the image of $1
\otimes 1 \in (S_0)^\ast \otimes E_0$. By abuse of notation we denote $\Delta_0(1 \otimes 1)$ by just $\Delta_0$. By exactness, $\Delta_0$ is in the kernel of $\wedge m$ and not in the image of the previous map $\wedge m$. Furthermore, $\Delta_0$ is unique with respect to this property, up to a constant and modulo the image of $\wedge m$. Thus we need to check that our choice $J_0$ is also in the kernel of the horizontal map $\wedge m$, but not in the image of the previous map $\wedge m$. Finally, we argue that if we choose the constant -1, the determinant of the complex will be exactly the resultant (up to sign).
To start with we notice $m = M_1 + M_2 + M_3$, and so $J_0 \wedge m =
\frac{M_1}{y_1} \wedge \frac{M_2}{y_2} \wedge \frac{M_3}{y_3\cdots
y_s} \wedge (M_1 + M_2 + M_3) = 0$. So $J_0$ is indeed in the kernel of $\wedge m$.
To show that $J_0$ is not in the image of the previous map, we twist the whole Tate resolution by 1, so that the map $\Delta_0$ goes from $(S_0)^\ast \otimes E(3)$ to $S_{{\rm int}(3Q)} \otimes E$, and then apply the functor $U_3$. This also gives a complex whose determinant is the resultant (Theorem 0.1, in [@ES]), in particular it is exact when the resultant is non-zero. In this situation the image of the lower map is just the ${\rm int}(3Q)$ graded piece of the ideal $I
= (f_1, f_2, f_3)$, and the normalized toric Jacobian is known to be a nonzero element modulo this ideal(see [@CCD; @Cox2]). Therefore, $J_0$, which specializes to the Jacobian, cannot be in the image of the map $\wedge m$.
Finally, the specialized complex above, with the normalized toric Jacobian as the diagonal map, appears in [@DE] where the authors show that the determinant of the complex is exactly the resultant up to sign. Therefore, the map $1\otimes 1 \mapsto -J_0$ above is a valid choice, up to sign, for the map $\Delta_0$ in Theorem \[thm:tate\].
Now let’s take the degree $-3$ part of the Tate resolution to get:
$$\begin{diagram}
0 & \rTo &
(S_Q)^\ast & \rTo^{i_m} & (S_0)^\ast \otimes \bigwedge^1 V
& \rTo & 0 \\
\oplus & & \oplus &
\rdTo^{\Delta_Q}& \oplus & \rdTo^{\Delta_0} & \oplus\\
0 & \rTo & S_{{\rm int}(Q)} \otimes \bigwedge^2 V & \rTo^{\wedge m} &
S_{{\rm int}(2Q)} \otimes \bigwedge^3 V & \rTo^{\wedge m}
& S_{{\rm int}(3Q)} \otimes \bigwedge^4 V. \\
\end{diagram}$$
Let $\{n_\alpha\}$ be the basis of $(S_Q)^\ast$ dual to the monomial basis $\{y^\alpha\}$ of $S_Q$. The map on the top row sends $n_\alpha$ to $e_\alpha$. Because these maps form a complex we have the relation $\Delta_Q(n_\alpha) \wedge m = -\Delta_0(e_\alpha) =
J_0 \wedge e_\alpha$.
The map $\Delta_Q$ is not canonically defined, even after picking $\Delta_0$. In fact the next proposition shows that [*any*]{} map satisfying the above relation will do.
Define $\Delta_Q(n_\alpha)$ to be [*any*]{} element $d_\alpha$, homogeneous of degree -3, such that $d_\alpha \wedge m = J_0
\wedge e_\alpha$. This defines a valid choice for $\Delta_Q$.
The map $i_m$ in the top row sending $n_\alpha$ to $e_\alpha$ for each $\alpha \in Q$ is clearly injective (in fact an isomorphism of vector spaces). We will use this to show that the bottom row is exact at the term $S_{{\rm int}(2Q)} \otimes \bigwedge^3 V$. So pick an element $k$ in the kernel of $\wedge m \ : S_{{\rm int}(2Q)} \otimes \bigwedge^3 V
\to S_{{\rm int}(3Q)} \otimes \bigwedge^4 V$. Now $(0,k)$ is in the kernel of the whole complex. Therefore, by exactness there exists an element $(a, b) \in (S_Q)^\ast \oplus (S_{{\rm int}(Q)} \otimes
\bigwedge^2 V)$ mapping on to it. But now $i_m(a) = 0$, therefore $a=0$. So $b \wedge m = k$ as desired.
Now suppose the Tate resolution is fixed with $\Delta_0$ defined as in Proposition 6.1. Let $\tilde{\Delta}_Q$ be any map satisfying the above relation. Therefore, for any $n_{\alpha}$, $\Delta_Q(n_{\alpha})
\wedge m = - {\Delta}_0(e_{\alpha}) = \tilde{\Delta}_Q(n_{\alpha})
\wedge m $. So, $\Delta_Q$ and $\tilde{\Delta}_Q$ differ by an element of the kernel of $\wedge m$. By the argument in the previous paragraph, this is the same as differing by an element of the image of the previous $\wedge m$. Therefore, replacing $\Delta_Q$ by $\tilde{\Delta}_Q$ does not change exactness at this step of the Tate resolution. As the Tate resolution is a minimal free resolution, this new choice can always be extended ad infinitum, and so $\tilde{\Delta}_Q$ is itself a valid map.
So we need only find for every lattice point $\alpha$ in $Q$, an element $d_\alpha$ such that $d_\alpha \wedge m = J_0 \wedge
e_\alpha$. In [@Khe] it was shown how to reduce this to a problem in linear algebra. In this paper, we show instead that the explicit, combinatorial formula from Theorem \[thm:main\] does the trick. We restate Theorem \[thm:main\] below using the language of exterior algebras developed above. Recall the definitions of the sets $R_i$ from \[eqn:Rsets\]. The fan has possibly been refined as described earlier to guarantee that $R_3$ is non-empty.
\[thm:main2\] The map $\Delta_Q \ : \ (S_Q)^\ast \otimes E \to S_{{\rm int}(2Q)} \otimes
E(-3)$ can be defined as follows:
$$\Delta_Q(n_\alpha) = \sum_{(u,v,w) \in F_{\alpha} \subset A^3}
y^{u+v+w - \alpha - \omega_0} \otimes e_u \wedge e_v \wedge e_w.$$
Here $\omega_0 = (1, 1, \dots, 1)$, and $F_\alpha$ is the set of all triples $(u,v,w) \in A^3$ satisfying the Boolean combination of inequalities in (\[eqn:ineq\])
The next lemma will rewrite $J_0 \wedge e_{\alpha}$ in a form more convenient for our purposes.
\[l:del0alpha\] $$J_0 \wedge e_\alpha = \sum_{t,u,v,w} y^{t+u+v+w - \alpha - \omega_0} \otimes e_u \wedge e_v \wedge e_w \wedge e_t,$$
where $t, u, v, w$ satisfy:
$$\begin{aligned}
\label{ineq1} \forall i \in R_1 \quad t_i + u_i + v_i + w_i >& \alpha_i\\
\label{ineq2} \exists i \in R_1 \quad t_i + v_i + w_i \leq& \alpha_i \\
\label{ineq3} \forall j \in R_2 \quad t_j + v_j + w_j >& \alpha_j \\
\label{ineq4} \exists j \in R_2 \quad t_j + w_j \leq& \alpha_j \\
\label{ineq5} \forall k \in R_3 \quad t_k + w_k >& \alpha_k \\
\label{ineq6} \exists k \in R_3 \quad t_k \leq & \alpha_k.\end{aligned}$$
First note that if $\exists k \in R_3$ such that $w_k \leq \alpha_k$, then both $e_u \wedge e_v \wedge e_w \wedge e_t$ and $e_u \wedge e_v
\wedge e_t \wedge e_w$, with the same power of $y$, appear in the sum and cancel out. So condition (\[ineq5\]) can be replaced by the stronger condition
$$\forall k \in R_3 \quad w_k > \alpha_k. \tag{\ref{ineq5}'}$$
We will show that every term in $J_0 \wedge e_\alpha$ satisfies these conditions, and conversely every tuple $(t,u,v,w)$ satisfying the conditions corresponds to a term in $\Delta_0 \wedge e_\alpha$.
The element $J_0$ can be rewritten as $y^{u+v+w - \omega_0} \otimes
\sum e_u \wedge e_v \wedge e_w $ where $u_1 > 0$, $v_1 = 0$ but $v_2 >
0$, and $w_1 = w_2 = 0$. Wedge this with $e_{\alpha}$, and we show that the terms $e_u \wedge e_v \wedge e_w \wedge e_{\alpha}$ all appear on the right hand side. So choose $t = \alpha$ then $t_1 + v_1
+ w_1 = \alpha_1$, $t_2 + w_2 = \alpha_2$ and $t_k = \alpha_k$ for all k, thus conditions (\[ineq2\]), (\[ineq4\]), and (\[ineq6\]) are satisfied. On the other hand, $w_i > 0$ for all $i \neq 1,2$. This, combined with $v_2 > 0$ implies condition (\[ineq3\]), while $u_1 >
0$ implies condition (\[ineq1\]). Now, the set $R_3$ is constructed so that $w$, the vertex where edges 1 and 2 meet, satisfies condition (\[ineq5\]’) for all $\alpha$ [*except*]{} when $\alpha = w$, in which case $J_0 \wedge e_\alpha = 0$. Thus all the terms in $J_0
\wedge e_\alpha$ appear in the desired sum.
Conversely, pick any tuple $(t,u,v,w)$ satisfying (\[ineq1\]), (\[ineq2\]), (\[ineq3\]), (\[ineq4\]), (\[ineq6\]) , and the modified (\[ineq5\]’). Define $\gamma = \alpha - t$. So, in our notation, $\alpha_i - t_i = \langle \eta_i, \gamma \rangle$.
By conditions (\[ineq2\]), (\[ineq4\]), (\[ineq6\]) there exists $i_0$, $j_0$, $k_0$ in $R_1$, $R_2$, $R_3$ respectively such that $\langle \eta_{i_0}, \gamma \rangle \geq 0$, $\langle \eta_{j_0},
\gamma \rangle \geq 0$ and $\langle \eta_{k_0}, \gamma \rangle \geq
0$. Since the region $R_3$ is between $R_1$ and $R_2$, we must either have $\eta_{k_0}$ a positive linear combination of $\eta_{i_0}$ and $\eta_{j_0}$, or $\langle \eta_{i_0}, \gamma \rangle = \langle
\eta_{j_0}, \gamma \rangle = 0$.
However, we also have $w_{i_0} \leq \alpha_{i_0}$ and $w_{j_0} \leq
\alpha_{j_0}$, but $w_{k_0} > \alpha_{k_0}$, which rules out the first case. Thus $t_{i_0} = \alpha_{i_0}$ and $t_{j_0} = \alpha_{j_0}$. By conditions (\[ineq2\]) and (\[ineq4\]) we must have $w_{i_0} =
w_{j_0} = 0$. This is possible only if the facets corresponding to $\eta_{i_0}$ and $\eta_{j_0}$ meet at a vertex. The only vertex where the sets $R_1$ and $R_2$ meet is the vertex $p$ when $w_1 = w_2 =
0$. But now, $\gamma$ must be 0, since $\eta_1$ and $\eta_2$ are linearly independent. Thus $t = \alpha$. So, by condition (\[ineq2\]), $v_1 = 0$, by condition (\[ineq3\]) $v_2 > 0$, and by condition (\[ineq1\]), $u_1 > 0$. Hence, every term in the right hand sum also appears in $J_0 \wedge e_\alpha$.
We must show that if $\Delta_Q$ is defined as above, then $\Delta_Q(n_\alpha) \wedge m = J_0 \wedge e_\alpha$. The left hand side is the sum
$$\sum_{(u,v,w, t)} y^{u+v+w+t - \alpha - \omega_0} \otimes e_u \wedge e_v \wedge e_w \wedge e_t,$$
where $(u,v,w)$ satisfy (\[eqn:ineq\]) and $t$ is unconstrained.
On the other hand, by Lemma \[l:del0alpha\], the right hand side is
$$\sum_{t,u,v,w} y^{t+u+v+w - \alpha - \omega_0} \otimes e_u \wedge e_v \wedge e_w \wedge e_t,$$
where $(u,v,w,t)$ satisfy the inequalities (\[ineq1\])-(\[ineq6\]).
So, it is enough to show for any fixed 4 tuple $(u,v,w,t)$ the sum of all signed permutations satisfying (\[eqn:ineq\]), is equal to the sum of all signed permutations satisfying (\[ineq1\])-(\[ineq6\]).
We consider the poset corresponding to the power set of $P =
\{u,v,w,t\}$. This is a four-dimensional cube whose vertices are the 16 subsets of $P$, and two subsets $p$ and $q$ are connected by a directed edge from $p$ to $q$ if $p$ is the union of $q$ with a single element of $P$. A maximum oriented path (of length 5) in this poset corresponds to a permutation of $(u,v,w,t)$. Given a permutation $(u,v, w, t)$, the path starts at $\emptyset$, has first vertex $\{t\}$, second vertex $\{w, t\}$ and so on. Define the [*sign*]{} of this path to be the sign of the corresponding permutation. We will consider formal sums of signed paths, remembering that if the same path occurs twice in the sum with opposite signs, then the contribution from that path is 0.
Let $A_i$ be a condition on a vertex $p$ which evaluates to true if $\sum_{v \in p} v_k > \alpha_k$ holds for all indices $k$ in $R_i$. Note that if $p$ satisfies $A_i$ and $q \supset p$ then $q$ satisfies $A_i$. Label a vertex $B_i$ if it satisfies condition $A_i, \dots,
A_3$ but fails to satisfy conditions $A_1, \dots, A_{i-1}$. With this notation the permutations $(u,v,w,t)$ satisfying (\[eqn:ineq\]) are oriented paths through the cube labeled $(B_4, B_3, B_2, B_1,
B_1)$. The permutations, this time ordered $(t, u, v, w)$, satisfying (\[ineq1\])-(\[ineq6\]) are paths of the form $(B_4, B_4, B_3,
B_2, B_1)$. Note that this introduces a sign of $(-1)^3$ into our formula.
So, to complete the proof it is enough to show the following lemma that was proved by David Speyer in a personal communication.
\[comblemma\] The sum of oriented paths in the cube of the form $(B_4, \dots, B_i,\\
B_i, B_{i-1}, \dots, B_1)$ is $(-1)^{i-1}$ times the sum of paths of the form $(B_4, B_3, B_2, \\ B_1, B_1)$.
In particular when $i=4$ we have our desired result.
By induction it is enough to show that the sum of paths of the form $(B_4, \dots, B_i, B_i, B_{i-1}, \dots, B_1)$ is negative the sum of paths of the form $(B_4, \dots, \\ B_{i-1}, B_{i-1}, \dots, B_1)$. Let $S_1$ denote the first sum and $S_2$ the second.
For the moment, consider any two vertices $p$ and $q$ of the cube, labeled $B_i$ and $B_{i-1}$ respectively, joined by an oriented path of length 2. There are exactly two such paths passing through intermediate vertices $a$ and $b$ respectively. As $a$ contains $p$ and is contained in $q$, by the definition of the labels $a$ satisfies $A_i, \dots, A_3$ but fails to satisfy $A_1, \dots, A_{i-2}$. If $a$ obeys $A_{i-1}$ then it has label $B_{i-1}$, otherwise it has label $B_i$. The case for $b$ is identical.
Returning to the claim consider two disjoint paths of vertices $v_4,
\dots, v_i$ and $v_{i-1}, \dots v_1$ where $v_j$ has label $B_j$ and it is possible to join these paths by adding a single vertex between them. As above, there are two possibilities for this new vertex, $a$ and $b$, each of which has label $B_i$ or $B_{i-1}$. The permutations associated to the two ways of completing the path differ by a single exchange, hence have opposite signs. If $a$ and $b$ have the same label they cancel in the sum $S_1$ or $S_2$. If they have opposite labels than one contributes positively to one of the sums, and the other contributes negatively to the other sum. Therefore, the two sums are negative of each other.
Future Work {#s:general}
===========
This paper is, in the author’s opinion, just the tip of the iceberg in the application of exterior algebra methods to sparse resultants. I am actively working on several more general results and have ideas on many more.
In this paper we investigated the sheaf $\mathcal{O}({\rm int}(2Q))$ on a toric surface. One of the important properties was the vanishing of all “middle” cohomology. Other sheaves also have this property and give rise to different formulas for the resultant of a surface. We can also consider sheaves that do have middle cohomology, although it seems more difficult to make the maps explicit. In the special case of products of projective spaces, this is hinted at in Section 6 of the paper by Dickenstein and Emiris [@DiE].
It is of course of great interest to consider toric varieties of higher dimension, that is more than 3 equations. I know of a sheaf giving rise, via the Tate resolution, to a determinantal formula for the Chow form of any toric threefold. The sticking point is finding an explicit formula, analogous to Theorem \[thm:main\]. Hopefully, this will be worked out in a future publication.
For four dimensions or higher, it appears the best we can hope for is matrices whose determinant is a nontrivial multiple of the resultant. In this situation it should be possible to identify the extraneous factor with a minor of the matrix. See [@DD; @D].
An important generalization would be to [*mixed*]{} resultants, i.e. equations with different supports. Tate resolutions do not obviously apply, but there may be an appropriate extension.
Finally, returning to the specific formula presented here, there are several places where choice is involved. An interesting question would be to classify all possible formulas, for all the different choices. Another issue is to investigate the efficiency, both in theory and for an implementation. It may be possible to speed up the computation of the Bézout map $\Delta_Q$.
[^1]: E-mail: akhetan@math.berkeley.edu
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'An automorphism $\theta$ of a spherical building $\Delta$ is called *capped* if it satisfies the following property: if there exist both type $J_1$ and $J_2$ simplices of $\Delta$ mapped onto opposite simplices by $\theta$ then there exists a type $J_1\cup J_2$ simplex of $\Delta$ mapped onto an opposite simplex by $\theta$. In previous work we showed that if $\Delta$ is a thick irreducible spherical building of rank at least $3$ with no Fano plane residues then every automorphism of $\Delta$ is capped. In the present work we consider the spherical buildings with Fano plane residues (the *small buildings*). We show that uncapped automorphisms exist in these buildings and develop an enhanced notion of “opposition diagrams” to capture the structure of these automorphisms. Moreover we provide applications to the theory of “domesticity” in spherical buildings, including the complete classification of domestic automorphisms of small buildings of types $\sF_4$ and $\sE_6$.'
author:
- James Parkinson
- Hendrik Van Maldeghem
title: Opposition diagrams for automorphisms of small spherical buildings
---
Introduction {#introduction .unnumbered}
============
Let $\theta$ be an automorphism of a thick irreducible spherical building $\Delta$ of type $(W,S)$. The *opposite geometry* of $\theta$ is the set $\operatorname{\mathrm{Opp}}(\theta)$ of all simplices $\sigma$ of $\Delta$ such that $\sigma$ and $\sigma^{\theta}$ are opposite in $\Delta$. This geometry forms a natural counterpart to the more familiar fixed element geometry $\mathrm{Fix}(\theta)$, however by comparison very little is known about $\operatorname{\mathrm{Opp}}(\theta)$.
This paper is the continuation of [@PVM:17a], where we initiated a systematic study of $\operatorname{\mathrm{Opp}}(\theta)$ for automorphisms of spherical buildings. In particular in [@PVM:17a] we showed that if $\Delta$ is a thick irreducible spherical building of rank at least $3$ containing no Fano plane residues then $\operatorname{\mathrm{Opp}}(\theta)$ has the following weak closure property: if there exist both type $J_1$ and $J_2$ simplices in $\operatorname{\mathrm{Opp}}(\theta)$ then there exists a type $J_1\cup J_2$ simplex in $\operatorname{\mathrm{Opp}}(\theta)$. Automorphisms with this property are called *capped*, and the thick irreducible spherical buildings of rank at least $3$ with no Fano plane residues are called *large buildings*. Thus every automorphism of a large building is capped.
In the present paper we investigate $\operatorname{\mathrm{Opp}}(\theta)$ for the thick irreducible spherical buildings of rank at least $3$ containing a Fano plane residue. These are called the *small buildings*. In particular we show that, in contrast to the case of large buildings, uncapped automorphisms exist for all small buildings (with the possible exception of $\sE_8(2)$ where we provide conjectural examples).
A key tool in [@PVM:17a] was the notion of the *opposition diagram* of an automorphism $\theta$, consisting of the triple $(\Gamma,J,\pi)$, where $\Gamma$ is the Coxeter graph of $(W,S)$, $J$ is the union of all $J'\subseteq S$ such that there exists a type $J'$ simplex in $\operatorname{\mathrm{Opp}}(\theta)$, and $\pi$ is the automorphism of $\Gamma$ induced by $\theta$ (less formally, the opposition diagram is drawn by encircling the nodes $J$ of $\Gamma$). If $\theta$ is capped then this diagram turns out to encode a lot of information about the automorphism, essentially because it completely determines the partially ordered set $\mathcal{T}(\theta)$ of all types of simplices mapped onto opposite simplices by $\theta$. However for an uncapped automorphism the opposition diagram does not necessarily determine $\cT(\theta)$. For example in the polar space $\Delta=\sB_3(2)$ there are collineations $\theta_1$, $\theta_2$ and $\theta_3$ each with opposition diagram
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; (-1.5,0)–(-0.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east);
(that is, each $\theta_i$ maps a vertex of each type to an opposite vertex) whose partially ordered sets $\mathcal{T}(\theta_i)$, for $i=1,2,3$, are the following (see Theorem \[thm:existenceBn(2)\] for explicit examples):
at (0,0.3) ; at (-2,-1) (1) [$\{1\}$]{}; at (0,-1) (2) [$\{3\}$]{}; at (2,-1) (3) [$\{2\}$]{}; at (-2,0) (4) [$\{1,3\}$]{}; at (0,0) (5) [$\{1,2\}$]{}; at (2,0) (6) [$\{2,3\}$]{}; at (0,1) (7) [$\{1,2,3\}$]{}; (1)–(4); (1)–(5); (2)–(4); (2)–(6); (3)–(5); (3)–(6); (4)–(7); (5)–(7); (6)–(7);
at (0,0.3) ; at (-2,-1) (1) [$\{1\}$]{}; at (0,-1) (2) [$\{3\}$]{}; at (2,-1) (3) [$\{2\}$]{}; at (-2,0) (4) [$\{1,3\}$]{}; at (0,0) (5) [$\{1,2\}$]{}; at (2,0) (6) [$\{2,3\}$]{}; (1)–(4); (1)–(5); (2)–(4); (2)–(6); (3)–(5); (3)–(6);
at (0,0.3) ; at (-2,-1) (1) [$\{1\}$]{}; at (0,-1) (2) [$\{3\}$]{}; at (2,-1) (3) [$\{2\}$]{}; at (-2,0) (4) [$\{1,3\}$]{}; at (2,0) (6) [$\{2,3\}$]{}; (1)–(4); (2)–(4); (2)–(6); (3)–(6);
Note that only $\theta_1$ is capped (hence, in particular, analogues of $\theta_2$ and $\theta_3$ cannot exist for polar spaces $\sB_3(\mathbb{F})$ with $|\mathbb{F}|>2$ by the main result of [@PVM:17a]).
Thus the opposition diagram of an uncapped automorphism needs to be enhanced to properly understand these automorphisms. We achieve this by defining the *decorated opposition diagram* of an uncapped automorphism.
The full definition is given in Section \[sec:1\], however for the purpose of this introduction consider the following simplified situation. Suppose that $\theta$ is an automorphism with the property that the induced automorphism $\pi$ of the Coxeter graph $\Gamma$ is the opposition automorphism $w_0$. Then the *decorated opposition diagram* of $\theta$ is the quadruple $(\Gamma,J,K,\pi)$ where $(\Gamma,J,\pi)$ is the opposition diagram, and
$K=\{j\in J\mid \text{there exists a type $J\backslash\{j\}$ simplex mapped onto an opposite simplex by $\theta$}\}$.
Less formally, the decorated opposition diagram is drawn by encircling the nodes of $J$, and then shading those nodes of $K$. Thus, for example, the decorated opposition diagrams of the two uncapped automorphisms of $\sB_3(2)$ given above are
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (-1.5,0)–(-0.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{};
and
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (-1.5,0)–(-0.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{};
. At an intuitive level, the more encircled nodes that are shaded on the decorated opposition diagram of an uncapped automorphism, the “closer” the automorphism is to being capped.
The main theorem of this paper is Theorem \[thm:main\*\] below. Part (a) of the theorem shows that the decorated opposition diagram of an uncapped automorphism lies in a small list of diagrams, hence severely restricting the structure of uncapped automorphisms. Part (b) deals with the existence of uncapped automorphisms, showing that the list provided in part (a) has no redundancies, with only the $\sE_8(2)$ case remaining open due to the size of the building rendering our computational techniques inadequate. We strongly believe that the two $\sE_8(2)$ diagrams are indeed realised as opposition diagrams; see Conjecture \[conj:2\] for details.
\[thm:main\*\]
Let $\theta$ be an uncapped automorphism of a thick irreducible spherical building $\Delta$ of rank at least $3$. Then the decorated opposition diagram of $\theta$ appears in Table \[table:1\] or Table \[table:2\].
Let $\Delta$ be a small building. Each diagram appearing in the respective row of Table \[table:1\] or Table \[table:2\] can be realised as the decorated opposition diagram of some uncapped automorphism of $\Delta$, with the exception perhaps of the two $\sE_8(2)$ diagrams.
[|c|l|]{}
----------
$\Delta$
----------
&*Diagrams*\
------------
$\sA_n(2)$
------------
&
[l]{}
at (0,0.5) ; at (0,-0.5) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; (-1,0)–(1,0); (-4.north west) rectangle (-4.south east); (-2.north west) rectangle (-2.south east); (2.north west) rectangle (2.south east); (4.north west) rectangle (4.south east); (-3.north west) rectangle (-3.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (-5.north west) rectangle (-5.south east); (5.north west) rectangle (5.south east); (-5,0)–(-1,0); (1,0)–(5,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
\
-------------------------------
$\sB_n(2)$ *or* $\sB_n(2,4)$,
$(3\leq j\leq n)$
-------------------------------
&
[l]{}
at (0,0.5) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; at (2,-0.7) [$j$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (-5,0)–(-2,0); (0,0)–(4,0); (4,0.07)–(5,0.07); (4,-0.07)–(5,-0.07); (-2,0)–(0,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
at (0,0.3) ; at (0,-0.5) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (5.north west) rectangle (5.south east); (-5,0)–(-2,0); (0,0)–(4,0); (4,0.07)–(5,0.07); (4,-0.07)–(5,-0.07); (-2,0)–(0,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
\
--------------------------------
$\sD_{n}(2)$, $n\geq 4$ *even*
$(4\leq 2j\leq n-2)$
--------------------------------
&
[l]{}
at (0,0.8) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); at (2,-0.25) [$2j$]{}; (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to (5,0.5); (4,0) to (5,-0.5); (-2,0)–(0,0); at (2,0) [$\bullet$]{}; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
at (0,0.8) ; at (0,-0.8) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-4.north west) rectangle (-4.south east); (-2.north west) rectangle (-2.south east); (2.north west) rectangle (2.south east); (4.north west) rectangle (4.south east); (5a.north west) rectangle (5a.south east); (-5.north west) rectangle (-5.south east); (-3.north west) rectangle (-3.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (5b.north west) rectangle (5b.south east); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to (5,0.5); (4,0) to (5,-0.5); (-2,0)–(0,0);
\
\
-------------------------------
$\sD_{n}(2)$, $n\geq 4$ *odd*
$(4\leq 2j\leq n-3)$
-------------------------------
&
[l]{}
at (0,0.8) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); at (1,-0.25) [$2j$]{}; (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to \[bend left\] (5,0.5); (4,0) to \[bend right=45\] (5,-0.5); (-2,0)–(0,0); at (1,0) [$\bullet$]{}; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
at (0,0.8) ; at (0,-0.8) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-4.north west) rectangle (-4.south east); (-2.north west) rectangle (-2.south east); (2.north west) rectangle (2.south east); (4.north west) rectangle (4.south east); (-5.north west) rectangle (-5.south east); (-3.north west) rectangle (-3.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (5a.north west) rectangle (5b.south east); (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to \[bend left\] (5,0.5); (4,0) to \[bend right=45\] (5,-0.5); (-2,0)–(0,0); at (5,0.5) [$\bullet$]{}; at (5,-0.5) [$\bullet$]{}; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
\
--------------------------------
$\sD_{n}(2)$, $n\geq 4$ *even*
$(3\leq 2j+1\leq n-3)$
--------------------------------
&
[l]{}
at (0,0.8) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); at (1,-0.25) [$2j+1$]{}; (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to \[bend left\] (5,0.5); (4,0) to \[bend right=45\] (5,-0.5); (-2,0)–(0,0); at (1,0) [$\bullet$]{}; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (5a.north west) rectangle (5b.south east); at (0,-0.8) ; (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to \[bend left\] (5,0.5); (4,0) to \[bend right=45\] (5,-0.5); (-2,0)–(0,0); at (5,0.5) [$\bullet$]{}; at (5,-0.5) [$\bullet$]{}; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
\
-------------------------------
$\sD_{n}(2)$, $n\geq 4$ *odd*
$(3\leq 2j+1\leq n-2)$
-------------------------------
&
[l]{}
at (0,0.8) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); at (2,-0.25) [$2j+1$]{}; (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to (5,0.5); (4,0) to (5,-0.5); (-2,0)–(0,0); at (2,0) [$\bullet$]{}; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (5a.north west) rectangle (5a.south east); (5b.north west) rectangle (5b.south east); at (0,-0.8) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (0,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-5,0)–(-2,0); (0,0)–(4,0); (4,0) to (5,0.5); (4,0) to (5,-0.5); (-2,0)–(0,0);
\
\[table:1\]
[|c|l|]{}
----------
$\Delta$
----------
&*Diagrams*\
------------
$\sE_6(2)$
------------
&
[l]{}
at (0,0.3) ; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{}; (2.north west) rectangle (2.south east); (4.north west) rectangle (4.south east); (3.north west) rectangle (5.south east); (1.north west) rectangle (6.south east); (-2,0)–(-1,0); (-1,0) to \[bend left=45\] (0,0.5); (-1,0) to \[bend right=45\] (0,-0.5); (0,0.5)–(1,0.5); (0,-0.5)–(1,-0.5); at (0,-0.5) [$\bullet$]{}; at (0,0.5) [$\bullet$]{}; at (1,-0.5) [$\bullet$]{}; at (1,0.5) [$\bullet$]{}; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{};
at (0,0.3) ; at (0,-1.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (5.north west) rectangle (5.south east); (6.north west) rectangle (6.south east); (-2,0)–(2,0); (0,0)–(0,-1); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{};
\
------------
$\sE_7(2)$
------------
&
[l]{}
at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (6.north west) rectangle (6.south east); at (0,0) [$\bullet$]{}; at (2,0) [$\bullet$]{}; at (0,-1.3) ; at (0,0.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (-2,0)–(3,0); (0,0)–(0,-1);
at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (6.north west) rectangle (6.south east); (2.north west) rectangle (2.south east); (5.north west) rectangle (5.south east); (7.north west) rectangle (7.south east); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (-2,0)–(3,0); (0,0)–(0,-1);
\
------------
$\sE_8(2)$
------------
&
[l]{}
at (0,0.3) ; at (0,-1.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (6.north west) rectangle (6.south east); (7.north west) rectangle (7.south east); (8.north west) rectangle (8.south east); (-2,0)–(4,0); (0,0)–(0,-1); at (-2,0) [$\bullet$]{}; at (2,0) [$\bullet$]{}; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{};
at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (8.north west) rectangle (8.south east); (7.north west) rectangle (7.south east); (6.north west) rectangle (6.south east); (5.north west) rectangle (5.south east); (4.north west) rectangle (4.south east); (3.north west) rectangle (3.south east); (2.north west) rectangle (2.south east); (1.north west) rectangle (1.south east); (-2,0)–(4,0); (0,0)–(0,-1); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; at (0,0.3) ; at (0,-1.3) ;
\
------------
$\sF_4(2)$
------------
&
[l]{}
at (0,0.5) ; at (0,-0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
at (0,0.5) ; at (0,-0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
at (0,0.5) ; at (0,-0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
\
--------------
$\sF_4(2,4)$
--------------
&
[l]{}
at (0,0.5) ; at (0,-0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (-0.15,0.3) – (0.08,0) – (-0.15,-0.3);at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
\
\[table:2\]
Let us briefly describe corollaries to Theorem \[thm:main\*\](a) (see Section \[sec:applications\] for details and precise statements). Recall that the *displacement* $\operatorname{\mathrm{disp}}(\theta)$ of an automorphism $\theta$ is the maximum length of $\delta(C,C^{\theta})$, with $C$ a chamber.
\[cor:cor1\] Let $\theta$ be an automorphism of a thick irreducible spherical building $\Delta$.
If $\theta$ is an involution then $\theta$ is capped.
If $\theta$ is uncapped then $\mathcal{T}(\theta)$ is determined by the decorated opposition diagram of $\theta$.
If $\theta$ is uncapped then $\operatorname{\mathrm{disp}}(\theta)$ is determined by the decorated opposition diagram of $\theta$.
In particular, if $\Delta$ has type $(W,S)$ and $J=\operatorname{\mathrm{Typ}}(\theta)$ then Corollary \[cor:cor1\](c) implies that (see Corollary \[cor:disp\]) $$\operatorname{\mathrm{disp}}(\theta)=\begin{cases}
\mathrm{diam}(W)-\mathrm{diam}(W_{S\backslash J})&\text{if $\theta$ is capped}\\
\mathrm{diam}(W)-\mathrm{diam}(W_{S\backslash J})-1&\text{if $\theta$ is uncapped.}
\end{cases}$$ To illustrate this in an example, it follows that if $\theta$ is a nontrivial automorphism of a thick $\sE_8$ building then $\mathrm{disp}(\theta)\in\{57,90,107,108,119,120\}$, which is a surprisingly restricted list of possibilities (see Remark \[rem:disp\]). Moreover, displacements of $107$ or $119$ can only occur for uncapped automorphisms of the small building $\sE_8(2)$. We also provide applications of Theorem \[thm:main\*\](a) to the study of *domesticity* in spherical buildings (recall that an automorphism is called *domestic* if it maps no chamber to an opposite chamber). These automorphisms have recently enjoyed extensive investigation, including the series [@TTM:11; @TTM:12; @TTM:12b] where domesticity in projective spaces, polar spaces, and generalised quadrangles is studied, [@HVM:12] where symplectic polarities of large $\sE_6$ buildings are classified in terms of domesticity, [@HVM:13] where domestic trialities of $\sD_4$ buildings are classified, and [@PTM:15] where domesticity in generalised polygons is studied.
To give one example of our applications to domesticity, suppose that $\Delta$ is a simply laced spherical building, and that $\theta$ is a domestic automorphism inducing opposition on the type set with the property that $\theta$ maps at least one vertex of each type onto an opposite vertex (such automorphisms are called “exceptional domestic”). Then we show that in fact $\theta$ maps simplices of each type $J\subsetneq S$ onto opposite simplices (such automorphisms are called “strongly exceptional domestic”). In particular, this implies that $\operatorname{\mathrm{disp}}(\theta)=\mathrm{diam}(\Delta)-1$ for exceptional domestic automorphisms. Theorem \[thm:main\*\](b) provides the first known examples of exceptional domestic automorphisms of spherical buildings of rank at least $3$ (examples were previously only known for generalised polygons; see [@PTM:15]). In fact Theorem \[thm:main\*\](b) shows that, with the possible exception of $\sE_8(2)$, every small building admits a strongly exceptional domestic automorphism.
The proof of Theorem \[thm:main\*\](b) for the small buildings of exceptional type involves computations using $\mathsf{MAGMA}$ [@MAGMA], and in particular the Groups of Lie Type Package [@CMT:04]. In fact for the small buildings of type $\sF_4$ and $\sE_6$ we are able to prove a much stronger result and completely classify the domestic automorphisms of these buildings. To perform these calculations we implemented the minimal faithful permutation representations of the $\mathbb{ATLAS}$ groups $\sF_4(2)$, $\sF_4(2).2$, $\sE_6(2)$, $\sE_6(2).2$, ${^2}\sE_6(2^2)$, and ${^2}\sE_6(2^2).2$ (respective permutation degrees $69615$, $139230$, $139503$, $279006$, $3968055$ and $3968055$) into the $\mathsf{MAGMA}$ system. At the time of writing these representations were not readily available in either $\mathsf{MAGMA}$ or $\mathsf{GAP}$, and therefore they are provided on the first author’s webpage. We conclude this introduction with an outline of the structure of the paper. In Section \[sec:1\] we provide definitions and background. The proofs of Theorem \[thm:main\*\](a) and its corollaries are contained in Section \[sec:2\]. The proof of Theorem \[thm:main\*\](b) is divided across Section \[sec:classical\] for the classical types and Section \[sec:exceptional\] for the exceptional types. Moreover, Section \[sec:exceptional\] contains the complete classification of domestic automorphisms of the small buildings of types $\sF_4$ and $\sE_6$.
Definitions and background {#sec:1}
==========================
We refer to [@AB:08] for the general theory of buildings. In this section we will briefly recall some notation, mainly from [@PVM:17a Section 1]. Let $\Delta$ be a spherical building of type $(W,S)$, typically considered as a simplicial complex with type map $\operatorname{\tau}:\Delta\to 2^S$. Let $\cC$ be the set of chambers (maximal simplices) of $\Delta$, and let $\delta:\cC\times \cC\to W$ be the Weyl distance function.
Chambers $C$ and $D$ of $\Delta$ are *opposite* if and only if they are at maximal distance in the chamber graph (with adjacency given by the union of the $s$-adjacency relations: $C\sim_s D$ if and only $\delta(C,D)=s$). Equivalently, chambers $C,D\in\cC$ are opposite if and only if $
\delta(C,D)=w_0
$ where $w_0$ is the longest element of $W$.
If $J\subseteq S$ we write $J^{\mathrm{op}}=J^{w_0}=w_0^{-1}Jw_0$ (the ‘opposite type’ to $J$). The definition of opposition for chambers extends naturally to arbitrary simplices as follows (see [@AB:08 Lemma 5.107]).
Simplices $\alpha,\beta$ of $\Delta$ are *opposite* if $\tau(\beta)=\tau(\alpha)^{\mathrm{op}}$ and there exists a chamber $A$ containing $\alpha$ and a chamber $B$ containing $\beta$ such that $A$ and $B$ are opposite.
An *automorphism* of $\Delta$ is a simplicial complex automorphism $\theta:\Delta\to\Delta$. Note that $\theta$ does not necessarily preserve types. Indeed each automorphism $\theta:\Delta\to\Delta$ induces a permutation $\pi_{\theta}$ of the type set $S$, given by $\delta(C,D)=s$ if and only if $\delta(C^{\theta},D^{\theta})=s^{\pi_{\theta}}$, and this permutation is a diagram automorphism of the Coxeter graph $\Gamma$ of $(W,S)$. If $\Delta$ is irreducible, then from the classification of irreducible spherical Coxeter systems we see that $\pi_{\theta}:S\to S$ is either:
the identity, in which case $\theta$ is called a *collineation* (or *type preserving*),
has order $2$, in which case $\theta$ is called a *duality*, or
has order $3$, in which case $\theta$ is called a *triality*; this case only occurs in type $\sD_4$.
Automorphisms $\theta:\Delta\to\Delta$ that induce opposition on the type set (that is, $\pi_{\theta}=w_0$, where $w_0$ is the diagram automorphism given by $s^{w_0}=w_0^{-1}sw_0$) are called *oppomorphisms*. For example, oppomorphisms of an $\sE_6$ building are dualities, and oppomorphisms of an $\sE_7$ building are collineations (see, for example, [@AB:08 Section 5.7.4]).
Let $\theta$ be an automorphism of $\Delta$. The *opposite geometry* of $\theta$ is $$\mathrm{Opp}(\theta)=\{\sigma\in\Delta\mid \sigma\text{ is opposite }\sigma^{\theta}\}.$$ A fundamental result of Leeb [@Lee:00 Section 5] and Abramenko and Brown [@AB:09 Proposition 4.2] states that if $\theta$ is a nontrivial automorphism of a thick spherical building then $\mathrm{Opp}(\theta)$ is necessarily nonempty (this result has been generalised to the setting of twin buildings; see [@DPM:13]).
The *type* $\operatorname{\mathrm{Typ}}(\theta)$ of an automorphism $\theta$ is the union of all subsets $J\subseteq S$ such that there exists a type $J$ simplex in $\operatorname{\mathrm{Opp}}(\theta)$. The *opposition diagram* of $\theta$ is the triple $(\Gamma,\operatorname{\mathrm{Typ}}(\theta),\pi_{\theta})$. Less formally, the opposition diagram of $\theta$ is depicted by drawing $\Gamma$ and encircling the nodes of $\operatorname{\mathrm{Typ}}(\theta)$, where we encircle nodes in minimal subsets invariant under $w_0\circ \pi_{\theta}$. We draw the diagram ‘bent’ (in the standard way) if $w_0\circ\pi_{\theta}\neq 1$. For example, consider the diagrams
at (-4,0) [(a)]{}; at (0,0.3) ; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{}; (-2,0)–(-1,0); (-1,0) to \[bend left=45\] (0,0.5); (-1,0) to \[bend right=45\] (0,-0.5); (0,0.5)–(1,0.5); (0,-0.5)–(1,-0.5); (2.north west) rectangle (2.south east); (1.north west) rectangle (6.south east);
at (-4,-0.5) [(b)]{}; at (0,0.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (-2,0)–(2,0); (0,0)–(0,-1); (1.north west) rectangle (1.south east); (6.north west) rectangle (6.south east);
Diagram (a) represents a collineation $\theta$ of an $\sE_6$ building with $\operatorname{\mathrm{Typ}}(\theta)=\{1,2,6\}$, and diagram (b) represents a duality $\theta$ of an $\sE_6$ building with $\operatorname{\mathrm{Typ}}(\theta)=\{1,6\}$.
We call an opposition diagram *empty* if no nodes are encircled (that is, $\operatorname{\mathrm{Typ}}(\theta)=\emptyset$), and *full* if all nodes are encircled (that is, $\operatorname{\mathrm{Typ}}(\theta)=S$).
Let $\Delta$ be a spherical building of type $(W,S)$. Let $\theta$ be a nontrivial automorphism of $\Delta$, and let $J\subseteq S$. Then $\theta$ is called:
*capped* if there exists a type $\operatorname{\mathrm{Typ}}(\theta)$ simplex in $\operatorname{\mathrm{Opp}}(\theta)$, and *uncapped* otherwise.
*domestic* if $\operatorname{\mathrm{Opp}}(\theta)$ contains no chamber.
*$J$-domestic* if $\operatorname{\mathrm{Opp}}(\theta)$ contains no type $J$ simplex (this terminology is reserved for subsets $J$ which are stable under $w_0\circ\pi_{\theta}$).
*exceptional domestic* if $\theta$ is domestic with full opposition diagram.
*strongly exceptional domestic* if $\theta$ is domestic, but not $J$-domestic for any strict subset $J$ of $S$ invariant under $w_0\circ\pi_{\theta}$.
Note that if $\theta$ is a domestic automorphism with $w_0\circ\pi_{\theta}=1$ then $\theta$ is exceptional domestic if and only if there exists a vertex of each type mapped to an opposite vertex, and $\theta$ is strongly exceptional domestic if and only if there exists a panel of each cotype mapped to an opposite panel (recall that a *panel* is a codimension $1$ simplex).
To study uncapped automorphisms $\theta$ we introduce the decorated opposition diagram. Let $\mathcal{J}_{\theta}$ denote the set of subsets $I\subseteq S$ which are minimal with respect to the condition $I^{\pi_{\theta} w_0}=I$. For example, if $\theta$ induces opposition on $\Gamma$ then $\mathcal{J}_{\theta}=\{\{s\}\mid s\in S\}$ is the set of all singleton subsets of $S$.
The *decorated opposition diagram* of an uncapped automorphism $\theta$ is the quadruple $(\Gamma,J,K_{\theta},\pi_{\theta})$ where $J=\operatorname{\mathrm{Typ}}(\theta)$ and $K_{\theta}\subseteq J$ is the union of all $J'\in\mathcal{J}_{\theta}$ such that there exists a type $J\backslash J'$ simplex mapped onto an opposite simplex.
Less formally, the decorated opposition diagram is drawn by shading the nodes of $K_{\theta}$ on the opposition diagram. For example, consider the following.
at (-4,0) [(a)]{}; at (0,0.3) ; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{}; (2.north west) rectangle (2.south east); (4.north west) rectangle (4.south east); (3.north west) rectangle (5.south east); (1.north west) rectangle (6.south east); (-2,0)–(-1,0); (-1,0) to \[bend left=45\] (0,0.5); (-1,0) to \[bend right=45\] (0,-0.5); (0,0.5)–(1,0.5); (0,-0.5)–(1,-0.5); at (0,-0.5) [$\bullet$]{}; at (0,0.5) [$\bullet$]{}; at (1,-0.5) [$\bullet$]{}; at (1,0.5) [$\bullet$]{}; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{};
at (-4,-0.5) [(b)]{}; at (0,0.3) ; at (0,-1.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (5.north west) rectangle (5.south east); (6.north west) rectangle (6.south east); (-2,0)–(2,0); (0,0)–(0,-1); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{};
The decorated opposition diagram (a) represents an uncapped collineation of $\sE_6(2)$ with the property that there are simplices of types $S\backslash\{2\}$ and $S\backslash\{4\}$ mapped onto opposite simplices, and no simplices of types $S\backslash\{3,5\}$ nor $S\backslash\{1,6\}$ mapped onto opposite simplices – this automorphism is exceptional domestic, but not strongly exceptional domestic. The diagram (b) represents an uncapped duality of $\sE_6(2)$ with the property that there are panels of each cotype mapped onto opposite panels – this automorphism is strongly exceptional domestic.
Residue arguments are used extensively in the proof of Theorem \[thm:main\*\](a), and so we conclude this section with a summary of the techniques. We first briefly define residues and projections (see [@AB:08] for details). The *residue* $\operatorname{\mathrm{Res}}(\alpha)$ of a simplex $\alpha\in\Delta$ is the set of all simplices of $\Delta$ which contain $\alpha$, together with the order relation induced by that of $\Delta$. Then $\operatorname{\mathrm{Res}}(\alpha)$ is a building whose diagram is obtained from the diagram of $\Delta$ by removing all nodes which belong to $\tau(\alpha)$. The *projection onto $\alpha$* is the map $\operatorname{\mathrm{proj}}_{\alpha}:\Delta\to\operatorname{\mathrm{Res}}(\alpha)$ defined as follows. Firstly, if $B$ is a chamber of $\Delta$ then there is a unique chamber $A\in \operatorname{\mathrm{Res}}(\alpha)$ such that $\ell(\delta(A,B))<\ell(\delta(A',B))$ for all chambers $A'\in \operatorname{\mathrm{Res}}(\alpha)$ with $A'\neq A$, and we define $\operatorname{\mathrm{proj}}_{\alpha}(B)=A$. In other words, $\operatorname{\mathrm{proj}}_{\alpha}(B)$ is the unique chamber $A$ of $\operatorname{\mathrm{Res}}(\alpha)$ with the property that every minimal length gallery from $B$ to $\operatorname{\mathrm{Res}}(\alpha)$ ends with the chamber $A$. Now, if $\beta$ is an arbitrary simplex we define $$\operatorname{\mathrm{proj}}_{\alpha}(\beta)=\bigcap_{B}\,\operatorname{\mathrm{proj}}_{\alpha}(B)$$ where the intersection is over all chambers $B$ in $\operatorname{\mathrm{Res}}(\beta)$. In other words, $\operatorname{\mathrm{proj}}_{\alpha}(\beta)$ is the unique simplex $\gamma$ of $\operatorname{\mathrm{Res}}(\alpha)$ which is maximal subject to the property that every minimal length gallery from a chamber of $\operatorname{\mathrm{Res}}(\beta)$ to $\operatorname{\mathrm{Res}}(\alpha)$ ends in a chamber containing $\gamma$.
Let $\theta$ be an automorphism of $\Delta$, and suppose that $\sigma\in\operatorname{\mathrm{Opp}}(\theta)$. It follows from [@Tit:74 Theorem 3.28] that the projection map $\operatorname{\mathrm{proj}}_{\sigma}:\operatorname{\mathrm{Res}}(\sigma^{\theta})\to\operatorname{\mathrm{Res}}(\sigma)$ is an isomorphism. Define $$\theta_{\sigma}:\operatorname{\mathrm{Res}}(\sigma)\xrightarrow{\sim} \operatorname{\mathrm{Res}}(\sigma)\quad \text{by}\quad \theta_{\sigma}=\operatorname{\mathrm{proj}}_{\sigma}\circ\,\theta.$$ The type map induced by $\theta_{\sigma}$ is as follows.
\[prop:typemap\] Let $\theta$ be an automorphism of a spherical building $\Delta$ of type $(W,S)$. Suppose that $\sigma\in\operatorname{\mathrm{Opp}}(\theta)$ and let $J=\tau(\sigma)$. Then the type map on $S\backslash J$ induced by $\theta_{\sigma}$ is $w_{S\backslash J}\circ w_0\circ \pi_{\theta}$.
This follows easily from [@AB:08 Corollary 5.116].
We will use Proposition \[prop:typemap\] many times in our residue arguments. For example, consider a duality $\theta$ of an $\sD_n$ building, and suppose that $v\in\operatorname{\mathrm{Opp}}(\theta)$ is a type $i$ vertex, with $i\leq n-2$. The residue of $v$ is a building of type $\sA_{i-1}\times \sD_{n-i}$, and the induced automorphism $\theta_v$ of $\operatorname{\mathrm{Res}}(v)$ is a duality on the $\sA_{i-1}$ component, and a duality (respectively collineation) on the $\sD_{n-i}$ component if $i$ is even (respectively odd).
From [@Tit:74 Proposition 3.29] we have:
\[prop:proj\] Let $\theta$ be an automorphism of a spherical building $\Delta$ and let $\alpha\in\operatorname{\mathrm{Opp}}(\theta)$. If $\beta\in\operatorname{\mathrm{Res}}(\alpha)$ then $\beta$ is opposite $\beta^{\theta}$ in the building $\Delta$ if and only if $\beta$ is opposite $\beta^{\theta_{\alpha}}$ in the building $\operatorname{\mathrm{Res}}(\alpha)$.
The following corollary facilitates inductive residue arguments.
\[cor:proj\] Let $\theta:\Delta\to\Delta$ be a domestic automorphism and let $\sigma\in\operatorname{\mathrm{Opp}}(\theta)$. Then $\theta_{\sigma}:\operatorname{\mathrm{Res}}(\sigma)\to \operatorname{\mathrm{Res}}(\sigma)$ is a domestic automorphism of the building $\operatorname{\mathrm{Res}}(\sigma)$.
Let $J=\tau(\sigma)$. If $\theta_{\sigma}$ is not domestic then there is a chamber $\sigma'$ of $\operatorname{\mathrm{Res}}(\sigma)$ mapped onto an opposite chamber by $\theta_{\sigma}$. Then $\sigma\cup\sigma'$ is a chamber of $\Delta$, and from Proposition \[prop:proj\] this chamber is mapped onto an opposite chamber, a contradiction.
Theorem \[thm:main\*\](a) and its corollaries {#sec:2}
=============================================
In this section we prove Theorem \[thm:main\*\](a) and give applications to determining the partially ordered set $\mathcal{T}(\theta)$, domesticity, cappedness of involutions, and calculating displacement.
Proof of Theorem \[thm:main\*\](a)
----------------------------------
By [@PVM:17a Theorem 1] if $\theta$ is an uncapped automorphism of a thick irreducible spherical building $\Delta$ of rank at least $3$ then $\Delta$ is a small building. These are precisely the buildings listed in the first column of Tables \[table:1\] and \[table:2\]. Moreover, the following proposition from [@PVM:17a] explains why collineations of $\sA_n$, trialities of $\sD_4$, and dualities of $\sF_4$ do not appear in Tables \[table:1\] and \[table:2\].
\[prop:1.1\] Every collineation of a thick $\sA_n$ building is capped, every triality of a thick $\sD_4$ building is capped, and every duality of a thick $\sF_4$ building is capped.
See [@PVM:17a Corollary 3.9, Theorem 3.17, Lemma 4.1].
Buildings of type $\sA_n$ play an important role in our proof techniques owing to their prevalence as residues of spherical buildings of arbitrary type. Every thick building of type $\sA_n$ with $n>2$ is a projective space $\mathsf{PG}(n,\KK)$ over a division ring $\KK$, where the type $i$ vertices of the building are the $(i-1)$-spaces of the projective space. Thus points have type $1$, lines have type $2$, and so on.
Let $\mathbb{F}$ be a field. A duality of $\sA_{2n-1}(\mathbb{F})$ with $
U^{\theta}=\{v\mid (u,v)=0\text{ for all $u\in U$}\}
$ for some nondegenerate symplectic form $(\cdot,\cdot)$ on $\FF^{2n}$ is called a *symplectic polarity*.
Let us recall some useful facts concerning dualities of type $\sA$ buildings.
\[lem:An\] If the projective space $\Delta=\mathsf{PG}(n,\KK)$ admits a duality $\theta$ for which all points are absolute (equivalently no type $1$ vertex is mapped to an opposite), then $n$ is odd, $\KK$ is a field, and $\theta$ is a symplectic polarity.
\[lem:sp\] If $\theta$ is a symplectic polarity of an $\sA_{2n-1}$ building $\Delta$ then $\theta$ is $\{i\}$-domestic for each odd $i$, and each vertex mapped to an opposite vertex is contained in a type $\{2,4,\ldots,2n-2\}$ simplex mapped to an opposite simplex. In particular, symplectic polarities are capped.
\[thm:Asmall\] Let $\theta$ be a domestic duality of the small building $\Delta=\sA_n(2)$ with $n\geq 2$. Then either $\theta$ is a strongly exceptional domestic duality or $n$ is odd and $\theta$ is a symplectic polarity.
The following proposition shows that the diagrams for uncapped dualities of $\sA_n$ buildings are as claimed in the first row of Table \[table:1\].
\[prop:1.2\] Every uncapped duality of $\sA_n(2)$ is a strongly exceptional domestic duality.
If $\theta$ is uncapped then necessarily $\theta$ is domestic, and so by Theorem \[thm:Asmall\] $\theta$ is either a symplectic polarity or is strongly exceptional domestic. The first case is eliminated by Lemma \[lem:sp\].
We now consider the small buildings of types $\sB_n$ and $\sD_n$. We first require some preliminary results. It is convenient at times to use terminology like “$x$ is domestic for $\theta$” and “$x$ is non-domestic for $\theta$” as short hand for “$\theta$ does not map $x$ to an opposite” and “$x$ is mapped to an opposite by $\theta$”. If the automorphism $\theta$ is clear from context we will simply say “$x$ is domestic” or “$x$ is non-domestic”.
\[lem:type1\] Let $n\geq 4$ and let $\Delta$ be a building of type $\sB_n$ or $\sD_{n+2}$ with thick projective plane residues. Let $\theta$ be an automorphism and let $J=\operatorname{\mathrm{Typ}}(\theta)$. If there exists $j\in J$ odd with $j\leq n$, then $\{1,2,\ldots,j\}\subseteq J$.
Let $v$ be a non-domestic type $j$ vertex. Then $\theta_v$ acts as a duality on the $\sA_{j-1}$ component of the residue of $v$ (by Proposition \[prop:typemap\]). Since $j$ is odd, this duality is either non-domestic or is exceptional domestic (see Theorem \[thm:Asmall\]), and in either case $1,2,\ldots,j-1\in J$, and hence the result.
\[lem:1\] Let $\Delta$ be a building of type $\sB_n$ or $\sD_{n+2}$ with $n\geq 4$ and thick projective plane residues, and let $\theta$ be a collineation. Let $J=\operatorname{\mathrm{Typ}}(\theta)$. Suppose that $3\leq j<n$, and that $\{j-1,j\}\subseteq J$ and $j+1\notin J$. Then there exists a type $\{1,j\}$-simplex mapped onto an opposite simplex by $\theta$.
We first show that $\theta$ is not $\{j-1,j\}$-domestic. For if $\theta$ is $\{j-1,j\}$-domestic, then since $\theta$ is also $\{j-1,j+1\}$-domestic it follows from [@PVM:17a Lemma 3.25] that either $\theta$ is $\{j-1\}$-domestic or $\{j\}$-domestic, a contradiction. Thus there exists a type $\{j-1,j\}$ simplex $\sigma$ mapped onto an opposite. If $v$ is the type $j$ vertex of this simplex then $\theta_v$ acts as a duality on the $\sA_{j-1}$ component (Proposition \[prop:typemap\]) mapping a hyperplane to an opposite (by Proposition \[prop:proj\]). Thus $\theta_v$ is either non-domestic or strongly exceptional domestic on the $\sA_{j-1}$ component, and in either case there exists a non-domestic type $\{1,j\}$ simplex (note that $j-1\geq 2$).
\[lem:2\] Let $\Delta$ be a small building of type $\sB_n$ or $\sD_{n+1}$, and let $j<n$. Suppose that $\theta$ is an uncapped collineation of type $J=\{1,2,3,\ldots,j\}$. Then $\theta$ is $\{1,2,3,\ldots,j-1\}$-domestic.
Suppose that there is a non-domestic type $\{1,2,\ldots,j-1\}$ simplex, and let $v$ be the type $j-1$ vertex this simplex. If $\theta$ is uncapped then necessarily $\theta_v$ acts as the identity on the “upper” residue of type $\sB_{n-j+1}$ or $\sD_{n-j+2}$ (by Proposition \[prop:proj\]). Thus [@PVM:17a Lemma 3.28] with $i=j-2$ and $\ell=j-3$ (note the index shift due to the fact that we used projective dimension in [@PVM:17a]) implies that every $(j-1)$-space in the polar space of $\Delta$ has a fixed point. Thus no type $j$ vertex of $\Delta$ is mapped onto an opposite vertex, contradicting the fact that $j\in J$.
We can now complete the proof of Theorem \[thm:main\*\](a) for buildings of type $\sB_n$. We allow the additional generality of thin cotype $n$ panels in the following proposition in order to facilitate our later arguments for type $\sD_n$.
\[prop:B\] Let $\Delta$ be a (possibly non-thick) building of type $\sB_n$ with Fano plane residues and $n\geq 3$, and let $\theta$ be a collineation of $\Delta$. If $\theta$ is uncapped, then the decorated opposition diagram of $\theta$ is one of the diagrams in Table \[table:1\].
Suppose that $\theta$ is uncapped. Let $J=\operatorname{\mathrm{Typ}}(\theta)$, and let $j=\max J$. Then $j\geq 3$, for if $j=1$ then $\theta$ is capped, and if $j=2$ then either $J=\{2\}$ and $\theta$ is capped, or $J=\{1,2\}$ in which case [@PVM:17a Fact 3.21] implies that $\theta$ is capped.
We claim that $J$ contains an odd element. For if every element of $J$ is even then for each non-domestic type $j$-vertex $v$ the induced automorphism $\theta_v$ is a point domestic duality of an $\sA_{j-1}$ building (by Propositions \[prop:typemap\] and \[prop:proj\]). Thus $\theta_v$ is a symplectic polarity (Lemma \[lem:An\]), and so there exists a type $\{2,4,\ldots,j-2\}$ simplex of the residue mapped to an opposite (Lemma \[lem:sp\]). Hence by Proposition \[prop:proj\] there is a type $\{2,4,\ldots,j-2,j\}=J$ simplex of $\Delta$ mapped onto an opposite and so $\theta$ is capped, a contradiction.
Let $k\in J$ be the maximal odd node. By Lemma \[lem:type1\] we have $\{1,2,\ldots,k\}\subseteq J$. Consider the following cases.
If $j=n$ then by [@PVM:17a Proposition 3.12(2)] there is a non-domestic type $\{1,n\}$ simplex. In the $\sA_{n-1}$ residue of the type $n$ vertex of this simplex we have a strongly exceptional domestic duality of $\sA_{n-1}$ (since it is domestic and maps a point to an opposite), and hence there are panels of each cotype $1,2,\ldots,n-1$ mapped onto opposites in $\Delta$. Thus $\theta$ has either the first diagram listed in Table \[table:1\] (with $j=n$) or the second diagram listed in Table \[table:1\] (strongly exceptional domestic).
If $k=j<n$ then $J=\{1,2,\ldots,j\}$, and by Lemma \[lem:1\] there exists a non-domestic type $\{1,j\}$ simplex. Considering the type $\sA_{j-1}$ residue of the type $j$ vertex of this simplex, and noting that $j-1$ is even, we see that in $\Delta$ there are non-domestic simplices of each type $J\backslash\{j'\}$ with $j'=1,2,\ldots,j-1$ (using Theorem \[thm:Asmall\]), and hence the diagram of $\theta$ is either $$\begin{aligned}
\label{eq:diags}
\begin{tikzpicture}[scale=0.5,baseline=-0.5ex]
\node at (0,0.3) {};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-5,0) (-5) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-4,0) (-4) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-3,0) (-3) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-2,0) (-2) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (0,0) (-1) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (1,0) (1) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (2,0) (2) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (3,0) (3) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (4,0) (4) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (5,0) (5) {$\bullet$};
\node at (2,-0.7) {$j$};
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-5.north west) rectangle (-5.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-4.north west) rectangle (-4.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-3.north west) rectangle (-3.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-2.north west) rectangle (-2.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-1.north west) rectangle (-1.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (1.north west) rectangle (1.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (2.north west) rectangle (2.south east);
\draw (-5,0)--(-2,0);
\draw (0,0)--(4,0);
\draw (4,0.07)--(5,0.07);
\draw (4,-0.07)--(5,-0.07);
\draw [style=dashed] (-2,0)--(0,0);
\node [inner sep=0.8pt,outer sep=0.8pt] at (-5,0) (-5) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-4,0) (-4) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-3,0) (-3) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-2,0) (-2) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (0,0) (-1) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (1,0) (1) {$\bullet$};
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\node [inner sep=0.8pt,outer sep=0.8pt] at (5,0) (5) {$\bullet$};
\end{tikzpicture}\quad\text{or}\quad
\begin{tikzpicture}[scale=0.5,baseline=-0.5ex]
\node at (0,0.3) {};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-5,0) (-5) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-4,0) (-4) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-3,0) (-3) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-2,0) (-2) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (0,0) (-1) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (1,0) (1) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (2,0) (2) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (3,0) (3) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (4,0) (4) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (5,0) (5) {$\bullet$};
\node at (2,-0.7) {$j$};
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-5.north west) rectangle (-5.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-4.north west) rectangle (-4.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-3.north west) rectangle (-3.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-2.north west) rectangle (-2.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (-1.north west) rectangle (-1.south east);
\draw [line width=0.5pt,line cap=round,rounded corners,fill=ggrey] (1.north west) rectangle (1.south east);
\draw [line width=0.5pt,line cap=round,rounded corners] (2.north west) rectangle (2.south east);
\draw (-5,0)--(-2,0);
\draw (0,0)--(4,0);
\draw (4,0.07)--(5,0.07);
\draw (4,-0.07)--(5,-0.07);
\draw [style=dashed] (-2,0)--(0,0);
\node [inner sep=0.8pt,outer sep=0.8pt] at (-5,0) (-5) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-4,0) (-4) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-3,0) (-3) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (-2,0) (-2) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (0,0) (-1) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (1,0) (1) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (2,0) (2) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (3,0) (3) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (4,0) (4) {$\bullet$};
\node [inner sep=0.8pt,outer sep=0.8pt] at (5,0) (5) {$\bullet$};
\end{tikzpicture}\end{aligned}$$ The first digram is eliminated by Lemma \[lem:2\].
If $k<j<n$ then $j$ is even, and as above we have $\{2,4,\ldots,j-2,j\}\subseteq J$. In particular $\{k,k+1\}\subseteq J$ and $k+2\notin J$ (as $k$ is maximum odd node of $J$, and note that $k+2\leq n$). Lemma \[lem:1\] implies that there is a non-domestic type $\{1,k+1\}$ simplex. If $k+1=j$ then as above we have the diagrams (\[eq:diags\]) and Lemma \[lem:2\] eliminates the first of the diagrams. If $k+1<j$ then $k+3\leq j<n$. If $\theta$ is $\{1,k+3\}$-domestic, then since $\theta$ is not $\{k+3\}$-domestic, [@PVM:17a Lemma 3.29] implies that $\theta$ is $\{1,k+1\}$-domestic, a contradiction. Hence there exists a type $\{1,k+3\}$ simplex mapped onto an opposite. However, considering the $\sA_{k+2}$ residue of the type $k+3$ vertex of this simplex we see that $\theta$ is not $\{k+2\}$-domestic, contradicting the maximality of $k$.
Hence the result.
\[cor:B1i\] Let $\Delta$ be a building of type $\sB_n$ with thick projective spaces, and let $\theta$ be a collineation and $n\geq i\geq 3$. If $\theta$ is $\{1,i\}$-domestic then $\theta$ is either $\{1\}$-domestic or $\{i\}$-domestic.
If $\theta$ is capped then the result is true by definition. If $\theta$ is uncapped then the result follows directly from the classification of uncapped diagrams given above.
The assumption $i\geq 3$ cannot be removed from Corollary \[cor:B1i\]. For example, consider the exceptional domestic collineation of the generalised quadrangle $\sB_2(2)$ (see [@TTM:12b Section 4]) . More generally, for each $n\geq 2$ there exists an uncapped collineation of $\sB_n(2)$ with $\operatorname{\mathrm{Typ}}(\theta)=\{1,2\}$ (see Theorem \[thm:existenceBn(2)\]).
We now continue with the analysis of buildings of type $\sD_n$. Recall that each building of type $\sD_n$ can be realised as the oriflamme geometry of the space $\mathbb{F}^{2n}$ equipped with an orthogonal form of Witt index $n$, for some field $\mathbb{F}$. The vertices of type $j$ for $j\in\{1,\ldots,n-2\}$ are the totally isotropic spaces of dimension $j$, and the vertices of type $n-1$ and $n$ are the totally isotropic subspaces of dimension $n$ (corresponding to the orbits of the action of the associated simple orthogonal group). To each such building $\Delta$ of type $\sD_n$ there is an associated (non-thick) building $\Delta'$ of type $\sB_n$. The type $j$ vertices of $\Delta'$, for $1\leq j\leq n$, are the totally isotropic subspaces of dimension $j$. Each type $n-1$ vertex of $\Delta'$ determines a type $\{n-1,n\}$ simplex of $\Delta$, and vice versa, as follows. A type $n-1$ vertex of $\Delta'$ is an $(n-1)$-dimensional totally isotropic space $W$, and there are precisely two totally isotropic $n$-dimensional subspaces $U,V$ containing $W$ and $(U,V)$ is an $\{n-1,n\}$-simplex of $\Delta$. Conversely, if $(U,V)$ is a type $\{n-1,n\}$ simplex of $\Delta$ then $W=U\cap V$ is a type $n-1$ vertex of $\Delta'$.
We first recall two facts from [@PVM:17a].
\[lem:containedB\] Let $\Delta$ be a thick building of type $\sD_n$ with $n$ odd, and let $\Delta'$ be the associated non-thick $\sB_n$ building. A collineation $\theta$ maps a type $\{n-1,n\}$ simplex of $\Delta$ to an opposite simplex if and only if it maps the associated type $n-1$ vertex of $\Delta'$ to an opposite vertex.
\[lem:Ddual\] No duality of a thick building of type $\sD_n$ is $\{1\}$-domestic.
\[lem:1nDodd\] Let $\Delta$ be a thick building of type $\sD_n$ with $n\geq 5$ odd, and let $\theta$ be a collineation. If $\theta$ is $\{1,n-1,n\}$-domestic then $\theta$ is either $\{1\}$-domestic or $\{n-1,n\}$-domestic.
Suppose that $\theta$ is neither $\{1\}$-domestic nor $\{n-1,n\}$-domestic. Since $\theta$ maps a type $\{n-1,n\}$-simplex to an opposite, by familiar residue arguments there are vertices of types $2,4,\ldots,n-3$ mapped onto opposite vertices. These vertex types are therefore also mapped onto opposites in the associated non-thick $\sB_n$ building $\Delta'$. If there are no type $n-2$ or $n-1$ vertices of $\Delta'$ mapped onto opposite vertices, then $\theta$ is $\{n-3,n-2\}$-domestic and $\{n-3,n-1\}$-domestic (on $\Delta'$) and thus since $\theta$ is not $\{n-3\}$-domestic it follows from [@PVM:17a Lemma 3.25] that every space of vector space dimension at least $n-2$ contains a fixed point. However by Lemma \[lem:containedB\] there are $n-1$ dimensional spaces mapped onto opposites, a contradiction. Thus either (i) $\theta$ is not $\{n-3,n-2\}$-domestic, or (ii) $\theta$ is not $\{n-3,n-1\}$-domestic (on $\Delta'$).
Consider case (i). Let $v$ be the type $n-2$ vertex of a non-domestic type $\{n-3,n-2\}$ simplex. Then $\theta_v$ acts on the upper type $\sA_1\times\sA_1$ residue by permuting the components, and thus $\theta_v$ is non-domestic on this upper residue (see [@PVM:17a Lemma 3.7]). Moreover $\theta_v$ is a duality on the lower type $\sA_{n-3}$ residue mapping a hyperplane (a type $n-3$ vertex) of this residue onto an opposite, and thus $\theta_v$ also maps a point (a type $1$ vertex) to an opposite. Thus $\theta$ maps a type $\{1,n-1,n\}$ simplex to an opposite, a contradiction.
Consider case (ii). Since $\theta$ is neither $\{1\}$-domestic nor $\{n-1\}$-domestic on $\Delta'$, and since $n-1\leq 4$, Corollary \[cor:B1i\] implies that there exists a type $\{1,n-1\}$ simplex of $\Delta'$ mapped to an opposite. Now Lemma \[lem:containedB\] implies that $\theta$ is not $\{1,n-1,n\}$-domestic on $\Delta$. This contradiction establishes the result.
\[prop:D1\] Let $\Delta$ be the building $\sD_n(2)$, $n\geq 4$, and let $\theta$ be a collineation of $\Delta$. If $\theta$ is uncapped then the decorated opposition diagram of $\theta$ is contained in Table \[table:1\].
Let $\theta$ be an uncapped collineation of $\sD_n(2)$, and let $J=\operatorname{\mathrm{Typ}}(\theta)$. Let $j=\max J$.
*Case 1: $j\in\{n-1,n\}$ with $n$ odd*. Then necessarily $\{n-1,n\}\subseteq J$. If $J\backslash\{n-1,n\}$ contains no odd types, then the induced automorphism in every residue of a non-domestic $\{n-1,n\}$-simplex is a symplectic polarity, and hence $\theta$ is capped, a contradiction. Thus $J\backslash \{n-1,n\}$ contains an odd node, and so by Lemma \[lem:type1\] we have $1\in J$. Thus by Lemma \[lem:1nDodd\] there exists a type $\{1,n-1,n\}$ simplex mapped onto an opposite simplex, and it easily follows that $\theta$ maps simplices of each type $S\backslash\{i\}$ with $i=1,2,\ldots,n-2$ to opposite. Hence the claimed diagram.
*Case 2: $j\in\{n-1,n\}$ with $n$ even*. By duality symmetry we may assume that $j=n$. If $n-1\in J$, then by [@PVM:17a Proposition 3.12(3)(b)] there is a type $\{n-1,n\}$-simplex mapped onto an opposite, and then considering the type $\sA_{n-2}$ residue we easily deduce that there are simplices of each cotype $S\backslash \{i\}$ with $i=1,2,\ldots,n-2$ mapped onto opposites. It then easily follows that there are also simplices of each type $S\backslash \{n-1\}$ and $S\backslash \{n\}$ mapped onto opposite. So suppose that $n-1\notin J$. If $J\backslash\{n-1,n\}$ contains no odd indices, then as above we deduce that $\theta$ is capped. Thus $J\backslash\{n-1,n\}$ contains an odd node, and so $1\in J$ by Lemma \[lem:type1\], and by [@PVM:17a Proposition 3.12(3)(a)] there is a type $\{1,n\}$ simplex mapped onto an opposite. It now easily follows that $\theta$ is strongly exceptional domestic.
*Case 3: $j\notin\{n-1,n\}$.* If $j$ is odd, then considering the upper residue of a type $j$ non-domestic we obtain a duality of a $\sD_{n-j}$, and since every duality of a $\sD_{n-j}$ maps a point to an opposite point (Lemma \[lem:Ddual\]) we have $j+1\in J$, a contradiction. Thus $j$ is even. If $j=2$ then $\theta$ is capped (see [@PVM:17a Fact 3.22]). So $j\geq 4$ (and hence $n\geq 6$). If $J$ has only even types then clearly $\theta$ is capped. Thus $J$ contains an odd node, and hence by Lemma \[lem:type1\] we have $1\in J$. Applying Corollary \[cor:B1i\] in the non-thick $\sB_n$ building it follows that there is a type $\{1,j\}$-simplex mapped onto an opposite, and the result easily follows, using Lemma \[lem:2\] to show that the last node is not shaded.
\[prop:D2\] Let $\theta$ be a duality of the $\sD_n(2)$ building. If $\theta$ is uncapped then the decorated opposition diagram of $\theta$ is contained in Table \[table:1\].
Let $\theta$ be an uncapped duality of $\sD_n(2)$, and let $J=\operatorname{\mathrm{Typ}}(\theta)$. Let $j=\max J$.
*Case 1: $j\in\{n-1,n\}$ with $n$ even*. Then necessarily $\{n-1,n\}\subseteq J$. In the residue of such a simplex we have an exceptional domestic duality of $\sA_{n-2}(2)$, and and the result easily follows.
*Case 2: $j\in\{n-1,n\}$ with $n$ odd*. In the residue of a non-domestic type $j$ vertex we obtain an exceptional domestic duality of $\sA_{n-1}(2)$, and again the result easily follows.
*Case 3: $j\notin\{n-1,n\}$.* If $j$ is even, then considering the upper residue of a non-domestic type $j$ vertex we obtain a duality of $\sD_{n-j}(2)$, and since every duality of $\sD_{n-j}(2)$ maps a point to an opposite point we have $j+1\in J$, a contradiction. Thus $j$ is odd. If $j=1$ then $\theta$ is obviously capped. So $j\geq 3$ (and hence $n\geq 5$). In the lower residue of a non-domestic type $j$ vertex we obtain an exceptional domestic duality of $\sA_{j-1}(2)$, and hence the result, using Lemma \[lem:2\] to see that the last node is not shaded.
Propositions \[prop:D1\] and \[prop:D2\] establish Theorem \[thm:main\*\](a) for buildings of type $\sD_n$. We now consider the exceptional types.
\[lem:F41234\] Let $\Delta$ be the building $\sF_4(2)$, and let $\theta$ be a collineation. If $\operatorname{\mathrm{Typ}}(\theta)=\{1,2,3,4\}$ then there exists either a non-domestic type $\{1,2\}$ simplex, or a non-domestic type $\{3,4\}$ simplex.
This follows from the classification given in Theorem \[thm:F4\]. We note that no circular logic is introduced by postponing the proof until Section \[sec:exceptional\].
We are now ready to prove Theorem \[thm:main\*\](a) for the small exceptional buildings. Before doing so we would like to correct [@HVM:12 Main Result 2.2], where it is asserted that every domestic duality of an $\sE_6$ building is a symplectic polarity. In fact this result only holds for large $\sE_6$ buildings. The oversight in the proof of [@HVM:12 Main Result 2.2] is in the proof of [@HVM:12 Lemma 5.2], where the existence of exceptional domestic automorphisms of $\sA_4(2)$ is overlooked.
\[prop:exceptional\] If $\theta$ is an uncapped automorphism of a building of exceptional type then the decorated opposition diagram of $\theta$ is contained in Table \[table:2\].
\(1) Let $\theta$ be an uncapped collineation of $\sE_6(2)$ and let $J=\operatorname{\mathrm{Typ}}(\theta)$. Suppose that $J=S$, and so the opposition diagram has the subsets $\{2\}$, $\{4\}$, $\{3,5\}$ and $\{1,6\}$ encircled. Let $\sigma$ be a non-domestic type $\{3,5\}$ simplex. Then $\theta_{\sigma}$ is an automorphism of an $\sA_2\times\sA_1\times\sA_1$ building acting as a duality on the $\sA_2$ component and interchanging the two $\sA_1$ components (by Proposition \[prop:typemap\]). Thus $\theta_{\sigma}$ is not domestic on the $\sA_1\times\sA_1$ component (see [@PVM:17a Lemma 3.7]) and must be exceptional domestic on the $\sA_2$ component (for otherwise $\theta$ is capped). Hence there are non-domestic simplices of types $S\backslash \{2\}$ and $S\backslash \{4\}$, and so the encircled nodes $2$ and $4$ are shaded. Suppose that there is a non-domestic simplex $\sigma'$ either of type $S\backslash\{3,5\}$ or $S\backslash\{1,6\}$. Then $\theta_{\sigma'}$ is an automorphism of an $\sA_1\times\sA_1$ building interchanging the two components (again by Proposition \[prop:typemap\]), and hence is not domestic, and hence $\theta$ is capped, a contradiction. Thus the encircled subsets $\{3,5\}$ and $\{1,6\}$ are not shaded.
Suppose that $J\neq S$. Then the first argument of the previous paragraph shows that $\{3,5\}\cap J=\emptyset$. A similar argument shows that $4\notin J$. Thus if $J\neq S$ we have $\{3,4,5\}\cap J=\emptyset$. If $\{1,6\}\subseteq J$ then $2\in J$ (for in the residue of a non-domestic type $\{1,6\}$ simplex we obtain a duality of $\sD_4$, and no duality of $\sD_n$ is point domestic; see [@PVM:17a Proposition 3.16]), and $\theta$ is capped. If $J=\{2\}$ then $\theta$ is obviously capped. Thus there are no uncapped collineations of $\sE_6$ with $\operatorname{\mathrm{Typ}}(\theta)\neq S$.
\(2) Let $\theta$ be an uncapped duality of an $\sE_6$ building and let $J=\operatorname{\mathrm{Typ}}(\theta)$. We claim that $J=S$. If $1\in J$ then $6\in J$, and vice versa (since no duality of $\sD_n$ is point domestic), and this argument shows that if $J=\{1,6\}$ then $\theta$ is capped, a contradiction. So $\{2,3,4,5\}\cap J\neq \emptyset $. If $3\in J$ then $\{2,3,4,5,6\}\subseteq J$ (considering the $\sA_4$ component of the residue of a non-domestic type $3$ vertex) and similarly if $5\in J$ then $\{1,2,3,4,5\}\subseteq J$. Thus if either $3\in J$ or $5\in J$ then $J=S$. If $2\in J$ then $\{2,3,5\}\subseteq J$ (considering the $\sA_5$ residue of a non-domestic type $2$ vertex), and thus again $J=S$. If $4\in J$ then $\{1,3,4,5,6\}\subseteq J$ (considering the $\sA_2\times\sA_2$ component of the residue of a non-domestic type $4$ vertex), and so once more $J=S$.
Thus all nodes are encircled. We claim that $\theta$ is strongly exceptional domestic, and so all nodes are shaded. To prove that there exists a cotype $j$ panels mapped onto opposite panels for each $j\in\{1,3,4,5,6\}$, note first that there exists a non-domestic type $\{2,4\}$ simplex (by considering the $\sA_4$ component of the residue of a non-domestic type $3$ vertex). If $v$ is the type $2$ vertex of such a simplex, then $\theta_v$ is a domestic duality of $\sA_5$ mapping a plane of this projective space onto an opposite, and thus $\theta_v$ is strongly exceptional domestic, and hence the result. Finally, to see that there is a non-domestic cotype $2$ panel, let $v$ be the type $1$ vertex of a non-domestic cotype $4$ panel. Using the classification of uncapped $\sD_5$ diagrams we see that $\theta_v$ is strongly exceptional domestic, and it follows that there exists a cotype $2$ panel of $\sE_6$ mapped onto an opposite.
\(3) Let $\theta$ be an uncapped collineation of an $\sE_7$ building and let $J=\operatorname{\mathrm{Typ}}(\theta)$. If $J=S$ then $\theta$ is strongly exceptional domestic (considering the $\sA_6$ residue of a non-domestic type $2$ vertex shows that $\theta$ maps simplices of each type $S\backslash\{j\}$ onto opposites for $j=1,3,4,5,6,7$, and considering the $\sE_6$ residue of the type $7$ vertex of a non-domestic type $\{2,7\}$ simplex, and using (2), shows that there is a simplex of type $S\backslash\{2\}$ mapped onto an opposite).
Suppose that $J\neq S$. Then $2\notin J$ (for otherwise the induced duality of the $\sA_6$ residue is strongly exceptional domestic) and $5\notin J$ (for otherwise the induced dualities of the $\sA_4$ and $\sA_2$ residues are both strongly exceptional domestic). We note the following: If $3\in J$ then $\{3,4,6\}\subseteq J$ (considering the $\sA_5$ component of the residue) and if $4\in J$ then $\{1,3,4,6\}\subseteq J$ (considering the $\sA_2$ and $\sA_3$ components of the residue). Thus if either $3\in J$ or $4\in J$ then $\{1,3,4,6\}\subseteq J$. If $6\in J$ then $\{1,6\}\subseteq J$ (since no duality of the $\sD_5$ component of the residue is point domestic). If $7\in J$ then $\{1,6,7\}\subseteq J$ (since every duality of $\sE_6$ maps both type $1$ and type $6$ vertices to opposites). It follows that either $J=\{1\}$, $J=\{1,6\}$, $J=\{1,6,7\}$, $J=\{1,3,4,6\}$, or $J=\{1,3,4,6,7\}$. In the first, second, and third cases it is clear using the above arguments that $\theta$ is capped, a contradiction. We claim that $J=\{1,3,4,6,7\}$ is impossible (for any collineation, capped or uncapped). For if $J=\{1,3,4,6,7\}$ then by [@PVM:17a Proposition 4.3(2)] there exists a type $\{3,7\}$ simplex $\sigma$ mapped to an opposite simplex, and if $v$ is the type $7$ vertex of $\sigma$ then $\theta_v$ is a duality of an $\sE_6$ building mapping a type $3$ vertex to an opposite, thus forcing $2,5\in J$, a contradiction.
The previous paragraph shows that if $\theta$ is uncapped and $J\neq S$ then $J=\{1,3,4,6\}$. Considering the $\sA_2\times\sA_3$ component of the residue of a non-domestic type $4$ vertex shows that there are simplices of types $\{3,4,6\}$ and $\{1,4,6\}$ mapped onto opposites, thus the nodes $1$ and $3$ are shaded. If there exist either type $\{1,3,6\}$ or $\{1,3,4\}$ simplices mapped onto opposite simplices then considering the residue of the type $1$ vertex of such a simplex we deduce that $\theta$ is capped, a contradiction. Thus the nodes $4$ and $6$ are not shaded.
\(4) Let $\theta$ be an uncapped (hence nontrivial) collineation of an $\sE_8$ building and let $J=\operatorname{\mathrm{Typ}}(\theta)$. If $J=S$ then easy residue arguments show that $\theta$ is strongly exceptional domestic.
We claim that if $J\neq S$ then $J\subseteq \{1,6,7,8\}$. To see this, note that if $2\in J$ then $\{3,5,7\}\in J$ (considering an $\sA_7$ residue), if $3\in J$ then $\{2,4,5,6,7,8\}\subseteq J$ (considering the $\sA_6$ component of the residue), if $4\in J$ then $\{1,3,5,6,7,8\}\subseteq J$ (considering the $\sA_2\times\sA_4$ component of the residue), and if $5\in J$ then $\{1,2,3,4,7\}\subseteq J$ (considering the $\sA_4\times\sA_3$ residue). Combining these statements it follows that if $\{2,3,4,5\}\cap J\neq \emptyset$ then $J=S$, and hence the claim.
Suppose that $J\neq S$, and so $J\subseteq \{1,6,7,8\}$. We claim that $J=\{1,6,7,8\}$. For if $1\in J$ then $8\in J$ (since no duality of $\sD_7$ is point domestic), if $6\in J$ then $J=\{1,6,7,8\}$ (considering the $\sD_5\times \sA_2$ residue and recalling that no duality of $\sD_5$ is point domestic), and if $7\in J$ then $6\in J$ (considering the duality of $\sE_6$ and using (2) above) and so again $J=\{1,6,7,8\}$. Thus $J=\{8\},\{1,8\}$ or $\{1,6,7,8\}$. The first two cases are clearly capped, hence the claim. Now considering the residue of a type $6$ non-domestic vertex we see that there are simplices of types $\{1,6,7\}$ and $\{1,6,8\}$ mapped onto opposite simplices (hence the nodes $7$ and $8$ are shaded). If there exists a simplex of type $\{6,7,8\}$ or $\{1,7,8\}$ mapped onto an opposite then considering the $\sD_5$ residue we deduce that $\theta$ is capped, and so the nodes $1$ and $6$ are not shaded.
\(5) Let $\theta$ be an uncapped collineation of an $\sF_4$ building and let $J=\operatorname{\mathrm{Typ}}(\theta)$. If $2\in J$ then $3,4\in J$ (by the duality in the $\sA_2$ component of the residue) and similarly if $3\in J$ then $1,2\in J$. Thus either $J=\{1\}$, $J=\{4\}$, $J=\{1,4\}$, or $J=\{1,2,3,4\}$. The first and second cases are trivially capped. The third case is capped by [@PVM:17a Lemma 4.5]. Thus $J=\{1,2,3,4\}$.
If $\Delta=\sF_4(2)$ then by Lemma \[lem:F41234\] there is either a type $\{1,2\}$ or $\{3,4\}$ simplex mapped onto an opposite simplex. In the first case, by considering the residue of the type $2$ vertex, we see that there are panels of cotype $3$ and $4$ mapped onto opposites, and hence the nodes $3$ and $4$ are shaded. The second case is symmetric, with the nodes $1$ and $2$ shaded. Of course both cases may occur simultaneously, and then all nodes are shaded. Finally, note that if either nodes $1$ or $2$ are shaded then both are shaded (if the $i$ node is shaded and $i\in\{1,2\}$ then consider the residue of the type $3$ vertex of a non-domestic cotype $i$ panel). Similarly, if either nodes $3$ or $4$ are shaded then both are shaded. Hence the result for $\sF_4(2)$.
If $\Delta=\sF_4(2,4)$ then considering the $\sA_2(4)$ component of a type $2$ non-domestic vertex we deduce that there are simplices of type $\{2,3,4\}$ mapped onto opposites. Then considering the $\sA_2(2)$ residue of a type $\{3,4\}$ non-domestic simplex we deduce that there are also simplices of type $\{1,3,4\}$ mapped onto opposites. Thus the nodes $1,2$ are shaded. If there exists a simplex of type $\{1,2,4\}$ or $\{1,2,3\}$ mapped onto an opposite, then considering the type $\sA_2(4)$ residue of the $\{1,2\}$ subsimplex we deduce that $\theta$ is non-domestic, and hence capped, a contradiction. Thus the nodes $3$ and $4$ are not shaded.
Theorem \[thm:main\*\](a) now follows from Propositions \[prop:1.1\], \[prop:1.2\], \[prop:B\], \[prop:D1\], \[prop:D2\], and \[prop:exceptional\].
Applications {#sec:applications}
------------
This section contains applications and corollaries of Theorem \[thm:main\*\](a).
\[cor:app1\] Let $\theta$ be a an exceptional domestic automorphism of a thick irreducible spherical building $\Delta$.
If $\theta$ is an oppomorphism and $\Delta$ is simply laced, then $\theta$ is strongly exceptional domestic.
If $\theta$ is not an oppomorphism then $\theta$ is not strongly exceptional domestic.
The first statement follows by noting that in Tables \[table:1\] and \[table:2\], if $\theta$ is an oppomorphism and $\Delta$ is simply laced, then whenever all nodes are encircled they are all shaded (see the first, third, sixth rows of Table \[table:1\] and the first, second, and third rows of Table \[table:2\]). The second statement follows by inspecting the third and fourth rows of Table \[table:1\] and the first row of Table \[table:2\].
The following lemma is in preparation for our next corollary to Theorem \[thm:main\*\](a).
\[lem:orderproj\] Let $\theta$ be an involution of a thick spherical building, and suppose that the simplex $\sigma$ is mapped onto an opposite simplex. Then the induced automorphism $\theta_{\sigma}$ of $\operatorname{\mathrm{Res}}(\sigma)$ is either the identity or it is an involution.
Let $\alpha$ be a simplex of $\mathrm{Res}(\sigma)$. If $\alpha^{\theta}=\operatorname{\mathrm{proj}}_{\sigma^{\theta}}(\alpha)$ then $\alpha^{\theta_{\sigma}}=\alpha$ (because the projection maps $\mathrm{proj}_{\sigma}:\mathrm{Res}(\sigma^{\theta})\to\mathrm{Res}(\sigma)$ and $\mathrm{proj}_{\sigma^{\theta}}:\mathrm{Res}(\sigma)\to\mathrm{Res}(\sigma^{\theta})$ are mutually inverse bijections). If $\alpha^{\theta}=\mathrm{proj}_{\sigma^{\theta}}(\alpha)$ then $\alpha^{\theta_\sigma}=\alpha$. If $\alpha^{\theta}\neq \operatorname{\mathrm{proj}}_{\sigma^{\theta}}(\alpha)$ then, since $\theta$ maps $\alpha^{\theta}$ onto $\alpha$, the projection $\mathrm{proj}_{\sigma}(\alpha^{\theta})$ is mapped onto $\mathrm{proj}_{\sigma^{\theta}}(\alpha)$. Thus $\theta_{\sigma}^2=1$.
\[cor:involutions\] Every involution of a thick irreducible spherical building is capped.
The result is of course true for large buildings of rank at least 3 (where all automorphisms are capped by [@PVM:17a]), and thus it remains to show that involutions of small buildings and of arbitrary generalised polygons are capped. Let us begin with the former. We use the decorated opposition diagrams in Tables \[table:1\] and \[table:2\] to show that every uncapped automorphism has order strictly greater than $2$. Consider type $\sA_n$, and let $\theta$ be uncapped. By Theorem \[thm:main\*\](a) there exists a non-domestic type $\{3,4,\ldots,n\}$ simplex $\sigma$. Then $\theta_{\sigma}$ is a domestic duality of the Fano plane. However by [@PTM:15] the only domestic duality of the Fano plane is the unique exceptional domestic duality, and this has order $8$. Thus, by Lemma \[lem:orderproj\] $\theta$ has order strictly greater than $2$.
The arguments are similar for all other uncapped diagrams. The key fact is that in some residue one finds a domestic duality of the Fano plane. For example, in the first $\sE_6(2)$ diagram in Table \[table:2\] we have a non-domestic type $\{1,3,5,6\}$ simplex $\sigma$ (because, for example, the node $2$ is shaded), and $\theta_{\sigma}$ is a domestic duality of the Fano plane residue.
We now show that every involution of an arbitrary generalised $m$-gon, $m\geq 2$, is capped. Recall that a generalised $m$-gon $\Delta$ is a bipartite graph with diameter $m$ and girth $2m$. A chamber is a pair of vertices connected by an edge. If $\{x,y\}$ is a chamber we write $x\sim y$ and call $x$ and $y$ adjacent. In particular, if $x\sim y$ then the vertices $x$ and $y$ have different types. Vertices $x$ and $y$ of $\Delta$ are opposite if and only if the distance between them is $m$, and this in turn is equivalent to the existence of a path $x=x_0\sim x_1\sim\cdots \sim x_{m}=y$ with $x_j\neq x_{j+2}$ for all $j=0,\ldots,m-2$. If the distance between vertices $x,y$ is $k<m$ then there is a unique geodesic from $x$ to $y$. In this case, writing $x=z_0\sim z_1\sim\cdots \sim z_k=y$ the vertex $z_1$ (respectively $z_{k-1}$) is the projection of $y$ onto $x$ (respectively $x$ onto $y$).
*Claim 1:* Every involutary collineation of a thick generalised $2n$-gon $\Delta$, $n\geq 1$, is capped.
*Proof of Claim 1:* The case $n=1$ is trivial, and so suppose that $\theta$ is an uncapped involutary collineation of a generalised $2n$-gon with $n\geq 2$. Thus $\theta$ is domestic (on chambers), and maps at least one vertex of each type onto an opposite vertex. Let $\Delta'$ denote the fixed elements of $\theta$. Let $x_0$ be a type $1$ vertex mapped onto an opposite vertex $x_{2n}=x_0^{\theta}$, and consider any geodesic path $x_0\sim x_1\sim \cdots \sim x_{2n-1}\sim x_{2n}$. If $x_1^\theta\neq x_{2n-1}$ then the chamber $\{x_0,x_1\}$ is mapped onto an opposite chamber and $\theta$ is capped. Hence $x_1^\theta=x_{2n-1}$, and it follows that $x_i^\theta=x_{2n-i}$, for all $i\in\{0,1,2,\ldots,2n\}$. In particular $x_n^\theta=x_n$ is fixed. Consider another geodesic $x_0\sim y_1\sim\cdots \sim y_{2n-1}\sim x_{2n}$ with $y_1\neq x_1$. Then $y_n^{\theta}=y_n$. By considering the path from $x_n$ to $x_0$ to $y_n$ we see that $x_n$ and $y_n$ are opposite, and thus there is a pair of opposite vertices $x_n,y_n\in\Delta'$.
Similarly, by considering a type 2 vertex $x_0'$ that is mapped onto an opposite vertex we deduce the existence of a pair of opposite vertices $x_n',y_n'\in\Delta'$. Since the vertices $x_n',y_n'$ have different type to the vertices $x_n,y_n$ we conclude that for each type $j\in\{1,2\}$ there are pairs of opposite vertices of type $j$ in $\Delta'$. It follows that $\Delta'$ is a sub-$2n$-gon (because the fixed structure of an collineation of a $2n$-gon is either empty, consists of pairwise opposite elements, is a tree of diameter at most $2n$, or is a sub-$2n$-gon, and the first three options are impossible from the above considerations).
Now, the distance from $x_n'$ to $x_n$ is at most $2n-1$ (by types and diameter) and hence the unique geodesic from $x_n'$ to $x_n$ is fixed by $\theta$. In particular the chamber $\{z,x_n\}$ is fixed, where $z\sim x_n$ is the projection of $x_n'$ onto $x_n$. Note that $z\neq x_{n-1},x_{n+1}$ because $x_{n-1}^{\theta}=x_{n+1}$ is not fixed. We claim that every vertex $z_1\sim z$ is fixed. With $y_j$ as above, note that $z$ and $y_{n-1}$ are opposite (consider the path from $z$ to $x_0$ to $y_{n-1}$). Hence the distance from $z_1$ to $y_{n-1}$ is $2n-1$, and so there is a unique geodesic $z_1\sim z_2\sim\cdots z_{2n-1}=y_{n-1}$. If $z_1^{\theta}\neq z_1$ then $z_n$ and $z_n^{\theta}$ are opposite (consider the path from $z_n$ to $z_0$ to $z_n^{\theta}$). Similarly, since $y_{n-1}^{\theta}=y_{n+1}$ we have $y_{n-1}\neq y_{n-1}^{\theta}$ and so $z_{n+1}$ and $z_{n+1}^{\theta}$ are opposite. Hence the chamber $\{z_n,z_{n+1}\}$ is mapped onto an opposite chamber, a contradiction.
It now follows from [@HVM:98 Proposition 1.8.1] that the sub-$2n$-gon $\Delta'$ has the property that whenever $x\in \Delta'$ has the same type as $z$, then all neighbours of $x$ are fixed (and hence are in $\Delta'$). But $x_n'$ has the same type as $z$, contradicting the fact that the projection of $x_0'$ onto $x_n'$ is mapped onto the projection of $x_0'^\theta$ onto $x_n'$ and that these projections are distinct. This contradiction completes the proof of Claim 1.
*Claim 2:* Every involutary duality of a thick generalised $(2n-1)$-gon $\Delta$, $n\geq 2$, is capped.
*Proof of Claim 2:* Let $\theta$ be a polarity of a generalised $(2n-1)$-gon and suppose that $\theta$ maps some element $x_0$ to an opposite element $x_{2n-1}$. Suppose that $\theta$ is not capped, i.e., $\theta$ does not map any chamber to an opposite chamber. Let $x_1\sim x_0$ be arbitrary. Consider the path $x_0\sim x_1\sim\cdots\sim x_{2n-1}$. In a similar way to the previous proof we deduce that $x_i^\theta=x_{2n-1-i}$ for all $i\in\{0,1,2,\ldots,2n-1\}$. Hence $x_n^\theta=x_{n-1}$. Consider a second path $x_0\sim y_1\sim\cdots \sim y_{2n-2}\sim x_{2n-1}$ with $y_1\neq x_1$. Then also $y_{n-1}^\theta=y_n$. Let $z_0\sim x_n$ be arbitrary but distinct from $x_{n-1}$ and $x_{n+1}$ (using thickness). There is a unique path $z_0\sim z_1\sim\cdots \sim z_{2n-2}=y_{n-1}$ from $z_0$ to $y_{n-1}$. By considering the path $z_{n-2}\sim \cdots \sim z_0\sim x_n\sim x_n^{\theta}\sim z_0^{\theta}\sim\cdots \sim z_{n-2}^{\theta}$ we see that $z_{n-2}$ is mapped onto an opposite vertex. Similarly, since $y_{n-1}^{\theta}=y_n$ we see that $z_{n-1}$ is mapped onto an opposite vertex (consider the path $z_{n-1}\sim\cdots \sim y_{n-1}\sim y_{n-1}^{\theta}\sim \cdots \sim z_{n-1}^{\theta}$). Hence the chamber $\{z_{n-2},z_{n-1}\}$ is mapped onto an opposite chamber, a contradiction. This completes the proof of Claim $2$.
Finally, we note that no duality of a thick generalised $2n$-gon is domestic and no collineation of a thick generalised $(2n-1)$-gon is domestic (see [@PTM:15 Lemmas 3.1 and 3.2]), completing the proof of the corollary.
Corollary \[cor:involutions\] shows that every uncapped automorphism has order at least $3$. Since every known example of an uncapped automorphism has order at least $4$ (see the examples in Sections \[sec:classical\] and \[sec:exceptional\], and also the rank $2$ classification in [@PTM:15]) we are led to make the following conjecture.
If $\theta$ is an automorphism of a thick irreducible spherical building, and if $\theta$ has order $3$, then $\theta$ is capped.
Note that if we remove the shading from the diagrams in Tables \[table:1\] and \[table:2\] then the diagrams we obtain are contained in [@PVM:17a Tables 1–5]. Thus Theorem \[thm:main\*\](a) has the following immediate corollary.
The (undecorated) opposition diagram of any automorphism of a thick irreducible spherical building is contained in *[@PVM:17a Tables 1–5]*.
We now use Theorem \[thm:main\*\](a) to determine the partially ordered set $\mathcal{T}(\theta)$ for all automorphisms $\theta$. We first note that, by the proposition below, it is sufficient to determine the maximal elements of $\mathcal{T}(\theta)$.
\[prop:redd\] Let $\mathcal{M}(\theta)$ be the set of maximal elements of $\mathcal{T}(\theta)$. Then $$\mathcal{T}(\theta)=\{J\subseteq S\mid J^{\pi_{\theta} w_0}=J\text{ and }J\subseteq M\text{ for some $M\in\mathcal{M}(\theta)$}\}.$$
This follows immediately from the facts that if $\sigma$ is a non-domestic type $K$ simplex then (i) $K$ is preserved by $w_0\circ\pi_{\theta}$, and (ii) if $J\subseteq K$ is preserved under $w_0\circ \pi_{\theta}$ then the type $J$ subsimplex of $\sigma$ is also non-domestic (see [@PVM:17a Lemma 1.3]).
Thus it remains to compute the set $\mathcal{M}(\theta)$ of maximal elements of $\mathcal{T}(\theta)$. We do this in the corollary below. Recall that if $\theta$ is uncapped then the decorated opposition diagram of $\theta$ is $(\Gamma,\operatorname{\mathrm{Typ}}(\theta),K_{\theta},\pi_{\theta})$ where, in particular, $K_{\theta}$ is the set of shaded nodes.
\[cor:maximal\] Let $\theta$ be an automorphism of a spherical building $\Delta$.
If $\theta$ is capped then $\mathcal{M}(\theta)=\{\operatorname{\mathrm{Typ}}(\theta)\}$.
If $\theta$ is uncapped then $\mathcal{M}(\theta)=\{\operatorname{\mathrm{Typ}}(\theta)\backslash\{k\}\mid k\in K_{\theta}\}$.
The first statement is obvious, so consider the second statement. Let $(\Gamma,J,K,\pi)$ be the decorated opposition diagram, and so $J=\operatorname{\mathrm{Typ}}(\theta)$. If $J=K$ then there are non-domestic simplices of each type $\operatorname{\mathrm{Typ}}(\theta)\backslash\{k\}$ with $k\in J$, and these are clearly the maximal types mapped to opposite (otherwise $\theta$ is capped). Suppose now that $J\backslash K$ consists of a single minimal $w_0\circ\pi$ invariant subset $J'$ (thus $J'$ is either a singleton, or $J'$ consists of a pair, as in the second $\sD_{2n}(2)$ diagram in Table \[table:1\]). In this case the only $w_0\circ \pi$ stable strict subset of $J$ that is not contained in an element of $\{J\backslash\{k\}\mid k\in K\}$ is $J\backslash J'$, and since $J'$ is not shaded all simplicies of this type are domestic. Hence the result in this case.
By Theorem \[thm:main\*\](a) the only remaining cases are the $6$ diagrams where $J\backslash K$ consists of precisely $2$ minimal $w_0\circ\pi$ invariant sets. Specifically, these examples are the $\sE_6(2)$ collineation diagram, the first $\sE_7(2)$ and $\sE_8(2)$ diagrams, the first two $\sF_4(2)$ diagrams (these are dual to one another), and the $\sF_4(2,4)$ diagram. In these cases the result is implied by the following claim.
*Claim:* Suppose that the decorated opposition diagram of $\theta$ is one of the $6$ diagrams listed above. Then $\theta$ is $\{i,j\}$-domestic where $i$ and $j$ are the two shaded nodes.
*Proof of Claim:* Consider the $\sE_6$ diagram. If there is a non-domestic type $\{2,4\}$ simplex then with $v$ the type $4$ vertex of this simplex the map $\theta_v$ acts on the $\sA_2\times\sA_2$ component of the residue swapping the components (by Proposition \[prop:typemap\]). It follows that $\theta$ is not domestic, a contradiction. Similar arguments apply for $\sE_7$ and $\sE_8$, using an $\sA_5$ and $\sE_6$ residue respectively. For the first $\sF_4(2)$ diagram, suppose there is a non-domestic type $\{1,2\}$ simplex $\sigma$. Then $\theta_{\sigma}$ is a domestic duality of $\sA_2(2)$, and hence is the exceptional domestic duality of the Fano plane. It follows that there is non-domestic type $\{1,2,3\}$ simplex, contradicting the node $4$ being unshaded. A dual argument applies to the second $\sF_4(2)$ diagram. The $\sF_4(2,4)$ diagram is similar. Hence the proof of the claim is complete, and the corollary follows.
Suppose that $\theta$ has the $\sE_6(2)$ collineation diagram in Table \[table:2\]. Then the partially ordered set $\mathcal{T}(\theta)$ is (using Proposition \[prop:redd\] and Corollary \[cor:maximal\]):
at (0,0.3) ; at (-3,-1) (1) [$\{2\}$]{}; at (-1,-1) (2) [$\{3,5\}$]{}; at (1,-1) (3) [$\{1,6\}$]{}; at (3,-1) (4) [$\{4\}$]{}; at (-4,0) (5) [$\{2,3,5\}$]{}; at (-2,0) (6) [$\{1,2,6\}$]{}; at (0,0) (7) [$\{1,3,5,6\}$]{}; at (2,0) (8) [$\{3,4,5\}$]{}; at (4,0) (9) [$\{1,4,6\}$]{}; at (-2,1) (10) [$\{1,2,3,5,6\}$]{}; at (2,1) (11) [$\{1,3,4,5,6\}$]{}; (1)–(5); (1)–(6); (2)–(5); (2)–(7); (2)–(8); (3)–(6); (3)–(7); (3)–(9); (4)–(8); (4)–(9); (5)–(10); (6)–(10); (7)–(10); (7)–(11); (8)–(11); (9)–(11);
As a final application we will compute the displacement of an arbitrary automorphism $\theta$ in Corollary \[cor:disp\] below. Recall that, by definition, $\operatorname{\mathrm{disp}}(\theta)=\max\{d(C,C^{\theta})\mid C\in\mathcal{C}\}$, where $\mathcal{C}$ is the set of chambers of $\Delta$, and $d(C,D)=\ell(\delta(C,D))$ for chambers $C,D\in\mathcal{C}$.
\[prop:disp\] Let $\theta$ be any automorphism of a thick irreducible spherical building of type $(W,S)$. Then $$\operatorname{\mathrm{disp}}(\theta)=\operatorname{\mathrm{diam}}(W)-\min \{\mathrm{diam}(W_{S\backslash J})\mid J\in\mathcal{M}(\theta)\}.$$
Let $N=\min \{\mathrm{diam}(W_{S\backslash J})\mid J\in\mathcal{M}(\theta)\}$. We note first that $$\begin{aligned}
\label{eq:ineq}
N=\min\{\operatorname{\mathrm{diam}}(W_{S\backslash J})\mid \text{there exists a type $J$ simplex in $\operatorname{\mathrm{Opp}}(\theta)$}\}\end{aligned}$$ because the minimum is obviously attained at a maximal element of $\mathcal{T}(\theta)$.
Let $J\subseteq \operatorname{\mathrm{Typ}}(\theta)$ be any subset for which there exists a non-domestic type $J$ simplex. Then for all chambers $C$ containing this simplex we have $\delta(C,C^{\theta})\in W_{S\backslash J}w_0$ (see [@PVM:17a Lemma 2.5]) and thus $$\operatorname{\mathrm{disp}}(\theta)\geq \ell(\delta(C,C^{\theta}))\geq \ell(w_0)-\ell(w_{S\backslash J})=\mathrm{diam}(W)-\mathrm{diam}(W_{S\backslash J}).$$ Since this inequality holds for all $J$ such that there exists a type $J$ simplex in $\operatorname{\mathrm{Opp}}(\theta)$ the formula (\[eq:ineq\]) gives $\operatorname{\mathrm{disp}}(\theta)\geq \operatorname{\mathrm{diam}}(W)-N$.
On the other hand, let $C$ be any chamber with $\ell(\delta(C,C^{\theta}))$ maximal. By the arguments of [@AB:09 Lemma 2.4 and Theorem 4.2] we have $\delta(C,C^{\theta})=w_{I}w_0$ for some $I\subseteq S$ with $I^{\pi_{\theta}}=I^{w_0}$. Hence the type $J=S\backslash I$ simplex of $C$ is mapped onto an opposite simplex. Thus $$\mathrm{disp}(\theta)=\ell(\delta(C,C^{\theta}))=\ell(w_0)-\ell(w_{S\backslash J})=\operatorname{\mathrm{diam}}(W)-\operatorname{\mathrm{diam}}(W_{S\backslash J})\leq \operatorname{\mathrm{diam}}(W)-N,$$ hence the result.
\[cor:disp\] Let $\theta$ be an automorphism of a thick irreducible spherical building and let $J=\operatorname{\mathrm{Typ}}(\theta)$. Then $$\operatorname{\mathrm{disp}}(\theta)=\begin{cases}
\mathrm{diam}(W)-\mathrm{diam}(W_{S\backslash J})&\text{if $\theta$ is capped}\\
\mathrm{diam}(W)-\mathrm{diam}(W_{S\backslash J})-1&\text{if $\theta$ is uncapped.}
\end{cases}$$ In particular, if $\theta$ is exceptional domestic then $\operatorname{\mathrm{disp}}(\theta)=\operatorname{\mathrm{diam}}(\Delta)-1$.
The case of capped automorphisms is [@PVM:17a Theorem 5]. In the case of an uncapped automorphism we note that by Corollary \[cor:maximal\] the maximal elements of $\mathcal{T}(\theta)$ are of the form $\operatorname{\mathrm{Typ}}(\theta)\backslash\{j\}$ for some $j\in\operatorname{\mathrm{Typ}}(\theta)$, and then the result follows from Proposition \[prop:disp\].
\[rem:disp\] Corollary \[cor:disp\] shows that the set of possible displacements is extremely restricted. For example, consider an $\sE_8$ building $\Delta$, where a priori there are $\ell(w_0)=120$ potential displacements. However, by Corollary \[cor:disp\], [@PVM:17a Theorem 3], and Theorem \[thm:main\*\](a) the only possible displacements are: $$\begin{aligned}
0&=\operatorname{\mathrm{diam}}(\sE_8)-\operatorname{\mathrm{diam}}(\sE_8) &&\text{for the trivial (hence capped) automorphism}\\
57&=\operatorname{\mathrm{diam}}(\sE_8)-\operatorname{\mathrm{diam}}(\sE_7)&&\text{for capped automorphisms with $\operatorname{\mathrm{Typ}}(\theta)=\{8\}$}\\
90&=\operatorname{\mathrm{diam}}(\sE_8)-\operatorname{\mathrm{diam}}(\sD_6)&&\text{for capped automorphisms with $\operatorname{\mathrm{Typ}}(\theta)=\{1,8\}$}\\
107&=\operatorname{\mathrm{diam}}(\sE_8)-\operatorname{\mathrm{diam}}(\sD_4)-1&&\text{for uncapped automorphisms with $\operatorname{\mathrm{Typ}}(\theta)=\{1,6,7,8\}$}\\
108&= \operatorname{\mathrm{diam}}(\sE_8)-\operatorname{\mathrm{diam}}(\sD_4)&&\text{for capped automorphisms with $\operatorname{\mathrm{Typ}}(\theta)=\{1,6,7,8\}$}\\
119&=\operatorname{\mathrm{diam}}(\sE_8)-1&&\text{for uncapped automorphisms with $\operatorname{\mathrm{Typ}}(\theta)=S$}\\
120&= \operatorname{\mathrm{diam}}(\sE_8)&&\text{for non-domestic (hence capped) automorphisms}.\end{aligned}$$ In particular, note that for $\sE_8$ buildings the displacement determines the (decorated) opposition diagram of the automorphism. This phenomenon is not true for all types; for example in $\sB_7(\mathbb{F})$ displacement $45$ is obtained by both capped automorphisms with $\operatorname{\mathrm{Typ}}(\theta)=\{1,2,3,4,5\}$ and capped automorphisms with $\operatorname{\mathrm{Typ}}(\theta)=\{2,4,6\}$.
Uncapped automorphisms for classical types {#sec:classical}
==========================================
In this section we prove Theorem \[thm:main\*\](b) for classical types. Thus our aim is to construct uncapped automorphisms with each of the diagrams listed in Tables \[table:1\] and \[table:2\] for the buildings $\sA_n(2)$, $\sB_n(2)$, $\sB_n(2,4)$, and $\sD_n(2)$.
The buildings $\sA_n(2)$
------------------------
In this section we work with the concrete model $\sA_n(2)=\mathsf{PG}(n,\FF_2)$ for the small building of type $\sA_n$. Thus an $i$-space of $\sA_n(2)$ means a subspace of $\mathbb{F}_2^n$ of (projective) dimension $i$, and this corresponds to a type $i+1$ vertex of the building. Let $\theta$ be a duality of $\sA_n(2)$. Recall that a point $p$ of $\sA_n(2)$ is called *absolute* with respect to $\theta$ if $p\in p^{\theta}$ (that is, $p$ is not mapped to an opposite hyperplane). Dually, a hyperplane $\pi$ is absolute if $\pi^{\theta}\in \pi$ (that is, $\pi$ is not mapped to an opposite point).
\[lem:dimension\] Let $\theta$ be a duality of a projective space. Suppose that $U$ is an $m$-space consisting of absolute points of $\theta$, and let $k=\dim(U\cap U^{\theta})$. Then $m-k$ is even.
The hyperplanes through $\langle U^{\theta},U\rangle$ form a dual space of (projective) dimension $k$, and the inverse image is a $k$-space contained in $U$. Choose a complementary $(m-k-1)$-space $H$ in $U$, and so $H$ intersects neither $U^{\theta}$ nor $U^{\theta^{-1}}$. Then for each $x\in H$ we have that $x^{\theta}\cap H$ is a hyperplane of $H$ through $x$, and hence is absolute. Thus $\theta$ is a symplectic polarity on $H$, and so $m-k$ is even (see Lemma \[lem:An\]).
\[thm:existenceAn(2)\] For each $n\geq 2$ there exists a unique duality $\theta$ of $\sA_n(2)$ (up to conjugation) with the property that the set of absolute points of $\theta$ is the union of two distinct hyperplanes. This duality is strongly exceptional domestic, with order $8$ if $n$ is even and $4$ if $n$ is odd.
We first demonstrate the existence of a duality whose absolute points form the union of two hyperplanes. Let $J_1$, $J_2$, and $J_3$ be the matrices $$\begin{aligned}
J_1&=\begin{bmatrix}
0&1\\
1&0\end{bmatrix},& J_2&=\begin{bmatrix}
0&0&1\\
1&0&1\\
1&1&0
\end{bmatrix}, & J_3&=\begin{bmatrix}
0&0&1&1\\
1&0&0&1\\
1&0&0&0\\
1&1&0&0
\end{bmatrix}\end{aligned}$$ and let $A$ be the $(n+1)\times (n+1)$ matrix in block diagonal form $$A=\mathrm{diag}(J,J_1,J_1,\ldots,J_1)\quad\text{with $J=J_2$ for even $n$ and $J=J_3$ for odd $n$}.$$ Let $\theta$ be the duality of $\sA_n(2)$ with matrix $A$. That is, $X^{\theta}=(AX)^{\perp}$ where $X$ is written as a column vector. Then $X$ is absolute if and only if $X\in (AX)^{\perp}$, and hence by direct calculation $X$ is absolute if and only if $X_0X_1=0$. The matrix for the collineation $\theta^2$ is given by $A^{-t}A$, and it follows by calculation that $\theta$ has order $8$ if $n$ is even, and order $4$ if $n$ is odd.
We now prove that there is at most one duality $\theta$ up to conjugation with the given property, and that such a duality is necessarily strongly exceptional domestic. We proceed by induction on $n$, the case $n=2$ being contained in [@PTM:15].
So let $\theta$ be a duality of $\sA_n(2)$ such that $\alpha_1\cup\alpha_2$ is the set of absolute points for $\theta$ with $\alpha_1\neq\alpha_2$ two hyperplanes of $\sA_n(2)$. Let $\beta$ be the hyperplane containing $\alpha_1\cap\alpha_2$ and different from both $\alpha_1$ and $\alpha_2$. Note that $\alpha_1\cup \alpha_2\cup\beta$ is the entire point set. Let $p_i=\alpha_i^\theta$, $i=1,2$ and $q=\beta^\theta$; then $L=\{p_1,p_2,q\}$ is a line.
Note that $q$ is absolute (for if $q\in\beta$ we have $q^{\theta}\ni\beta^{\theta}=q$). Thus $q\in \alpha_1\cup \alpha_2$. In fact we claim that $q\in\alpha_1\cap \alpha_2$. For if not we have $\beta^{\theta}=q\notin\beta$ and so $\beta$ is not absolute, contradicting the fact that $\beta=q^{\theta^{-1}}$ is absolute (since $q$ is absolute).
Since $L=\{p_1,p_2,q\}$ is a line and $q\in \alpha_1\cap \alpha_2$ we either have $p_1,p_2\in\beta\backslash(\alpha_1\cup\alpha_2)$ or $p_1,p_2\in\alpha_1\cup\alpha_2$. We treat these two cases below. Before doing this, we observe that in the first case $n$ is necessarily even, and in the second case $n$ is necessarily odd. To see this, note that if $p_1,p_2\in\beta\backslash(\alpha_1\cup\alpha_2)$ then the point $p_1$ is non-absolute and the mapping $\rho_1:z\mapsto z^\theta\cap\alpha_1$, $z\in\alpha_1$, is a duality on $\alpha_1$ every point of which is absolute, forcing $n$ to be even (see Lemma \[lem:An\]). On the other hand, if $p_1,p_2\in\alpha_1\cup\alpha_2$ then we have $(\alpha_1\cap\alpha_2)^\theta=\<p_1,p_2\>\subseteq \alpha_1\cap\alpha_2$ and so Lemma \[lem:dimension\] implies $(n-2)-1=n-3$ is even, and so $n$ is odd. We also observe that since $\alpha_1$ and $\alpha_2$ are the only two hyperplanes all of whose points are absolute, every even power of $\theta$ preserves the set $\{\alpha_1,\alpha_2\}$, and hence also the set $\{p_1,p_2\}$. It follows that $p_i^\theta\in\{\alpha_1,\alpha_2\}$ for $i=1,2$.
*Case 1:* $p_1,p_2\in\beta\backslash(\alpha_1\cup\alpha_2)$. As noted above $n$ is even, and so we may assume $n\geq 4$. Let $\sigma=\{x,\xi\}$ be any non-domestic (point-hyperplane)-flag for $\theta$ (that is, a non-domestic type $\{1,n\}$-simplex of the building). We note that such simplices exist, and indeed they obviously all arise as follows: Since the absolute hyperplanes for $\theta$ are precisely the hyperplanes through one of the points $p_1$ or $p_2$, if we select any point $x\in\beta\setminus(\alpha_1\cup\alpha_2)$ and any hyperplane $\xi$ through $x$ not containing $p_1$ or $p_2$, then $\sigma=\{x,\xi\}$ is non-domestic.
We claim that the mapping $\theta_\sigma:z\mapsto z^\theta\cap \xi\cap x^\theta$ for $z\in\xi\cap x^\theta$ has exactly two hyperplanes consisting entirely of absolute points. Note that $q\in\xi$ and also $q\in x^\theta$. Note also that, since $p_i^\theta$ contains the absolute point $q_i:=\<p_i,x\>\cap(\alpha_1\cap\alpha_2)$, also $x^\theta$ contains $q_i$, $i=1,2$. Since $\xi$ does not contain $p_i$, but it does contain $x$, it does not contain $q_i$, $i=1,2$. Consequently $x^\theta\cap\alpha_1\cap\alpha_2$ is not contained in $\xi$ and the claim follows.
Thus for every non-domestic (point-hyperplane)-pair $\sigma=\{x,\xi\}$ the induced duality $\theta_{\sigma}$ on the $\sA_{n-2}(2)$ residue has precisely two hyperplanes of absolute points. Since $n-2$ is even this duality again satisfies the condition of Case 1, and so by induction $\theta$ is domestic. Since $\theta$ has non-domestic points necessarily $\theta$ is strongly exceptional domestic by Theorem \[thm:Asmall\].
We now show that $\theta$ is unique, up to a projectivity (and under the assumptions of Case 1). Let $\rho_1$ be the symplectic polarity on $\alpha_1$ introduced in the paragraph before Case 1. Noting that $q^{\rho_1}=\alpha_1\cap\alpha_2$, we see that the data $\alpha_1,\alpha_2$ and $\rho_1$ are projectively unique. This determines $q$. All choices of $p_1$ outside $\alpha_1\cup\alpha_2$ are projectively equivalent, and then $p_2$ is the third point on the line determined by $p_1$ and $q$. We then know the image of an arbitrary point $x_1$ of $\alpha_1\setminus\alpha_2$, as $x_1^\theta=\<x^{\rho_1}, p_1\>$. This determines the images of all points of $\alpha_1$. Since $p_1^\theta=\alpha_1$, we know the images of a basis, which suffices to determine the whole duality.
*Case 2:* $p_1,p_2\in\alpha_1\cup\alpha_2$. As noted above, $n$ is odd. Take an arbitrary point $z\in\beta\setminus(\alpha_1\cup\alpha_2)$ and set $H:= z^\theta$. Then $\varphi:x\mapsto x^\theta\cap H$ is a duality in the $(n-1)$-dimensional projective space $H$ such that its absolute points form two hyperplanes $H\cap\alpha_i$, $i=1,2$. Hence by the previous case is domestic, and since $z$ was arbitrary amongst the non-domestic points for $\theta$ we conclude that $\theta$ is domestic. Thus by Theorem \[thm:Asmall\] $\theta$ is strongly exceptional domestic.
It remains to show that $\theta$ is unique up to conjugation with a projectivity. Let $D_i=H\cap \alpha_i$, $i=1,2$. Set $\{i,j\}=\{1,2\}$ and $D_i^{\varphi^{-1}}=p_i'$. Then $\{q,p_1',p_2'\}$ is a line in $H\cap\beta$ (since ${p_i'}^{\varphi}=D_i$ it suffices to see that $q^{\varphi}=\beta\cap H$, and this follows from the definition of $\varphi$ as $\beta=q^{\theta}$). It also follows that $D_i^{\theta^{-1}}=\<p_i',z\>$. Since $D_i\subseteq\alpha_i$, we conclude $\alpha_i^{\theta^{-1}}\in\< p_i',z\>$. But $\alpha_i^{\theta^{-1}}\in\{p_1,p_2\}$. We claim that $\alpha_i^{\theta^{-1}}=p_i$. Suppose not. Then $\alpha_i^{\theta^{-1}}=p_j$. Now from $z^\theta=H$ and $p_i^\theta=\alpha_j$ follows that $t_i^\theta=\<D_j,z\>$, with $\{t_i,p_i,z\}$ a line. But $p_j'^\theta$ is a hyperplane through $D_j$ distinct from $\alpha_j$ and $H$ (as $p_j\in H$ and is not absolute); hence $p_j'^\theta=\<D_j,z\>$ and so $t_i=p'_j$. Now $p_j'^{\theta^{-1}}=\<D_i,z\>$ and $p_i^{\theta^{-1}}=\alpha_i$. It follows that $z^{\theta^{-1}}=H$. Hence $z^{\theta^2}=z$, for all $z\in\beta\setminus (\alpha_1\cup\alpha_2)$. It follows that $p_i^{\theta^2}=p_i$, contradicting $p_i^{\theta^2}=\alpha_j^\theta=p_j$. Our claim follows.
But now, just like in the proof of our previous claim, we have that $\{p_i,p_i',z\}$ is a line and $p_i'^\theta=\<D_j,z\>$. It follows that $p_i^{\theta^2}=p_j$ and so $z^{\theta^2}=z'$, with $\{z,z',q\}$ a line.
Now, $\alpha_1,\alpha_2,H,z$ and $\varphi$ are unique up to conjugation with a projectivity. But then, given $z^\theta=H$, the duality $\theta$ is completely determined, since $q$ is determined and hence also $z'$ (with the above notation). This determines the image $x^\theta$ of an arbitrary point in $H$ as $x^\theta=\<x^\varphi,z'\>$. Furthermore, we also have $z^\theta=H$, and so $\theta$ is determined.
The buildings $\sB_n(2)$, $\sB_n(2,4)$, and $\sD_n(2)$
------------------------------------------------------
It will be more convenient for us to regard $\sB_n(2)\cong \sC_n(2)$ as a symplectic polar space. We begin by recalling the standard models of the $\sC_n(2)$, $\sD_n(2)$, and $\sB_{n-1}(2,4)$ buildings in the ambient projective space $\mathsf{PG}(2n-1,2)$. Let $V=\FF_2^{2n}$, and let $(\cdot,\cdot)$ be the (symplectic and symmetric) bilinear form on $V=\FF_2^{2n}$ given by $$\begin{aligned}
\label{eq:form}
(X,Y)=X_1Y_{2n}+X_2Y_{2n-1}+\cdots+X_{2n}Y_{1}.\end{aligned}$$ The points of the polar space $\mathsf{C}_n(2)$ are the $0$-spaces of $\mathsf{PG}(2n-1,2)$, and points $p=\langle X\rangle$ and $q=\langle Y\rangle$ are collinear (including the case $p=q$) if and only if $(X,Y)=0$. A subspace $U$ of $V$ is *totally isotropic* if $(X,Y)=0$ for all $X,Y\in U$. The totally isotropic subspaces of maximal dimension have projective dimension $n-1$, and for each $0\leq k\leq n-1$ the $k$-spaces of the polar space $\mathsf{C}_n(2)$ are the totally isotropic subspaces of $V$ with projective dimension $k$. To obtain the building of $\mathsf{C}_n(2)$ as a labelled simplicial complex one takes the totally isotropic $(k-1)$-spaces to be the type $k$ vertices of the building for $1\leq k\leq n$, with incidence of vertices given by symmetrised containment of the corresponding spaces. The full collineation group of $\mathsf{C}_n(2)$ is the symplectic group $\mathsf{Sp}_{2n}(2)$ consisting of all matrices $g\in\mathsf{GL}_{2n}(2)$ satisfying $g^TJg=J$, where $J$ is the matrix of the symplectic form $(\cdot,\cdot)$ (see [@Tit:74 Corollary 5.9]).
Let $F^{+}$ and $F^-$ be quadratic forms on $V$ with Witt indices $n$ and $n-1$ respectively. We will fix the specific choices $$\begin{aligned}
F^+(X)&=X_1X_{2n}+X_2X_{2n-1}+\cdots+X_{n}X_{n+1} \\
F^-(X)&=X_1X_{2n}+X_2X_{2n-1}+\cdots+X_{n}X_{n+1}+X_{n}^2+X_{n+1}^2. \end{aligned}$$ For $\epsilon\in\{-,+\}$, a subspace $U\subseteq V$ is *singular* with respect to $F^{\epsilon}$ if $F^{\epsilon}(X)=0$ for all $X\in U$. The maximal dimensional singular subspaces of $V$ with respect to $F^{\epsilon}$ have vector space dimension equal to the Witt index of $F^{\epsilon}$. The points of $\mathsf{D}_n(2)$, respectively the polar space $\mathsf{B}_{n-1}(2,4)$, are those points of $\mathsf{PG}(2n-1,2)$ that are singular with respect to $F^+$, respectively $F^-$. In both cases points $p=\langle X\rangle$ and $q=\langle Y\rangle$ are collinear (including the case $p=q$) if and only if $(X,Y)=0$, where $(\cdot,\cdot)$ is as in (\[eq:form\]).
Let $\mathsf{GO}^{\epsilon}_{2n}(2)$ be the group of all matrices of $\mathsf{GL}_{2n}(2)$ preserving the quadratic form $F^{\epsilon}$, and let $\mathsf{O}_{2n}^{\epsilon}(2)$ be the corresponding index $2$ simple subgroup of $\mathsf{GO}^{\epsilon}_{2n}(2)$ (c.f. [@ATLAS §2.4]). Since $\mathsf{GO}_{2n}^{\epsilon}(2)$ preserves collinearity, the group $\mathsf{GO}_{2n}^{+}(2)$ acts on $\mathsf{D}_n(2)$ and the group $\mathsf{GO}_{2n}^-(2)$ acts on $\mathsf{B}_{n-1}(2,4)$. In fact the group $\mathsf{GO}_{2n}^-(2)$ is the full automorphism group of $\mathsf{B}_{n-1}(2,4)$ (see [@Tit:74]). In the case of $\mathsf{D}_n(2)$ the maximal singular subspaces are partitioned into two sets of equal cardinality by the action of $\mathsf{O}_{2n}^+(2)$, and an automorphism $\theta$ of $\mathsf{D}_n(2)$ mapping points to points is called a *collineation* if this partition of maximal singular subspaces is preserved by $\theta$, and a *duality* otherwise. Then $\mathsf{O}^+_{2n}(2)$ is the group of all collineations of $\mathsf{D}_n(2)$, and $\mathsf{GO}_{2n}^+(2)\backslash\mathsf{O}_{2n}^+(2)$ is the set of all dualities of $\mathsf{D}_n(2)$ (see [@Tit:74]).
To obtain the building of $\mathsf{B}_{n-1}(2,4)$ as a labelled simplicial complex one takes the singular $(k-1)$-spaces to be the type $k$ vertices of the building for $1\leq k\leq n-1$, with incidence of vertices given by symmetrised containment of the corresponding spaces. The situation for $\mathsf{D}_n(2)$ is slightly different: For $1\leq k\leq n-2$ the singular $(k-1)$-spaces are taken to be the type $k$ vertices of the building, and the singular $(n-1)$-spaces in one part of the partition mentioned above are taken to be the type $n-1$ vertices of the building, and those in the other part of the partition are taken to be the type $n$ vertices of the building. A type $n-1$ vertex is declared to be incident with a type $n$ vertex if the corresponding $(n-1)$-spaces meet in an $(n-2)$-space. For all other types incidence is given by symmetrised containment of the corresponding spaces.
Note the index shifts that occur (for example an $\{n\}$-domestic collineation of a $\mathsf{C}_n(2)$ building is a collineation that is domestic on the totally isotropic $(n-1)$-spaces). A point $p$ of a polar space is an *absolute point* of an automorphism $\theta$ if $p^{\theta}$ is collinear with $p$ (including $p^{\theta}=p$).
\[lem:Cn(2)\] Let $\theta$ be a collineation of $\mathsf{C}_n(2)$.
If $\theta$ fixes a subspace of $\mathsf{PG}(2n-1,2)$ of projective dimension $k\geq n$ then $\theta$ is $\{j\}$-domestic for each $2n-k\leq j\leq n$.
If the set of absolute points of $\theta$ strictly contains the union of two distinct hyperplanes of $\mathsf{PG}(2n-1,2)$ then $\theta$ is $\{1\}$-domestic.
\(a) By considering dimensions, each $(j-1)$-space of $\mathsf{PG}(2n-1,2)$ with $j\geq 2n-k$ intersects the subspace of fixed points. In particular, no totally isotropic $(j-1)$-space is mapped onto an opposite and so $\theta$ is $\{j\}$-domestic for all $2n-k\leq j\leq n$.
\(b) A point $X$ is an absolute point of $\theta\in\mathsf{Sp}_4(2)$ if and only if $(X,\theta X)=X^TJ\theta X=0$, where $J$ is the matrix of the symplectic form $(\cdot,\cdot)$. Thus the set of absolute points of $\theta$ is a quadric, and so if it strictly contains the union of two distinct hyperplanes then all points are absolute.
In the following proofs we use the standard notations $p\perp q$ if points $p$ and $q$ are collinear (including the case $p=q$), and $p^{\perp}$ for the set of all points collinear to $p$.
\[lem:fixed-subspace\] Let $\Delta=\sC_n(2)$ with $n\geq 2$ and let $\theta$ be a collineation.
If the fixed points of $\theta$ form a $(2n-3)$-space $W$, then the absolute points form a subspace containing $W$.
If the fixed points of $\theta$ form a $(2n-2)$-space $W$, then every absolute point is fixed.
\(a) Let $p$ be a point not contained in $W$ and suppose $p$ is absolute. Let $q\in\<W,p\>\setminus W$. We claim that $q$ is absolute. Indeed, let $r:=\<p,q\>\cap W$. If $p\perp q$, then the plane $\pi=\<p,q,p^\theta\>$ contains the triangle $\{p,p^\theta,r\}$ of points collinear in $\sC_n(2)$ and so $
q\perp q^\theta$, as both points belong to $\pi$. If $p\notin q^\perp$, then $\pi$ contains the line $\<p,p^\perp\>$, which belongs to $\sC_n(2)$, but also contains the line $\<p,r\>$, which does not belong to $\sC_n(2)$. Also $\<p^\theta, r\>$ does not belong to $\sC_n(2)$, and it follows that the line $\<r,s\>$, where $\{p,p^\theta,s\}$ is the line of $\sC_n(2)$ through $p$ and $p^\theta$, belongs to $\sC_n(2)$. Hence also the line $\{s,q,q^\theta\}$ belongs to $\sC_n(2)$, which proves our claim.
So, if there are no absolute points besides those in $W$, then $(i)$ holds. If some absolute point $p\notin W$ exists, then there are three possibilities. Either exactly one hyperplane through $W$ consists of absolute points (and then $(i)$ holds), or all three hyperplanes through $W$ consist of absolute points (and then, again, $(i)$ holds), or exactly two hyperplanes $H_1$ and $H_2$ through $W$ consist of absolute points. In this final case, let $H$ be the third hyperplane through $W$. Let $t,t_1,t_2$ be points such that $t^\perp = H$ and $t_i^\perp= H_i$, $i=1,2$. Then, since $\theta$ fixes $H$, we have $t\in W$. Since $t_i\in t_i^\perp= H_i$, $i=1,2$, we deduce $t_i\in W$, $i=1,2$. Hence $\theta$ induces collineations in $H,H_1,H_2$ having a hyperplane $W$ as fixed points. Consequently, these collineations are central involutions. Since all points of $W$ are fixed, all subspaces through $\{t,t_1,t_2\}$ are fixed. Hence the centres of the above collineations are $t,t_1,t_2$. Since the collineations in $H_i$, $i=1,2$, map points to a collinear point, the centers are $t_i$. But then the centre of the collineation in $H$ is $t$ and hence it also maps points to collinear points, a contradiction. This shows (a).
\(b) If the fixed points of $\theta$ form a $(2n-2)$-space $W$, then $\theta$ is a central elation in $\PG(2n-1,2)$, and the centre is necessarily $W^\perp$ since every point of $W$ is fixed, and hence every hyperplane through $W^\perp$ is fixed. No line through $W^\perp$ not contained in $W$ is a line of $\sC_n(2)$, whence (b).
\[lem:CBase\] A collineation $\theta$ of the generalised quadrangle $\mathsf{C}_2(2)$ is exceptional domestic if and only if the set of absolute points of $\theta$ equals the union of two distinct hyperplanes in $\mathsf{PG}(3,2)$.
It is known that $\sC_2(2)$ admits a unique exceptional domestic collineation (see [@TTM:12b]), and direct inspection shows that the set of absolute points of this collineation forms the union of two distinct hyperplanes in $\mathsf{PG}(3,2)$. It remains to show that no other collineation of $\sC_2(2)$ has such a structure of absolute points. This can be done, for example, using the character tables in the $\mathbb{ATLAS}$, see [@ATLAS p.5]. We omit the details.
\[lem:CBase3\] Let $\Delta=\sC_n(2)$ with $n\geq 3$ and let $\theta$ be a collineation. If the absolute points of $\theta$ lie on a union of two hyperplanes, and if the fixed points of $\theta$ form a $(2n-4)$-space $W$, then $\theta$ has decorated opposition diagram
at (0,0.3) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-5,0)–(-0.5,0); (0.5,0)–(4,0); (4,0.07)–(5,0.07); (4,-0.07)–(5,-0.07); (-1,0)–(1,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
The hypothesis implies that every $3$-space contains a fixed point, and thus $\theta$ is $\{i\}$-domestic for all $4\leq i\leq n$.
By the hypothesis on the structure of the absolute points of $\theta$ there exist points in $\mathrm{Opp}(\theta)$. Let $p$ be an arbitrary point in $\mathrm{Opp}(\theta)$. We will show below that the induced collineation $\theta_p$ of $\mathsf{C}_{n-1}(2)$ is $\{2\}$-domestic (in the inherited labelling). Hence $\theta$ is $\{1,2\}$-domestic. So if $\theta$ is capped then $\theta$ is $\{2\}$-domestic, however by [@TTM:12 Theorem 5.1] every such collineation fixes a geometric hyperplane pointwise, contrary to our hypothesis that the fixed points form a $(2n-4)$-space. Thus $\theta$ is uncapped, and then by Theorem \[thm:main\*\](a) the decorated opposition diagram of $\theta$ is forced to be as claimed.
Therefore it only remains to show that $\theta_p$ is $\{2\}$-domestic (that is, point-domestic on $\sC_{n-1}(2)$). We fix some notation. Let $H_i$, $i=1,2$, be the two hyperplanes all points of which are absolute. Set $S=H_1\cap H_2$ and let $H$ be the hyperplane distinct from $H_i$, $i=1,2$, and containing $S$. Note that all points of $\mathrm{Opp}(\theta)$ are contained in $H$ (more precisely they form the set $H\setminus S$).
First we claim that any line in $\mathrm{Opp}(\theta)$ incident to $p$ must necessarily be contained in the hyperplane $H$. Suppose the such a line $L$ is not contained in $H$. Then $L=\{p,q_1,q_2\}$, with $q_i\in H_i$ and hence $q_i^\theta\perp q_i$, $i=1,2$. Since $p$ is not collinear to $p^\theta$, it must be collinear to $q_i^\theta$ for some $i\in\{1,2\}$. But then $q_i^\theta$ is collinear to all points of $L$, and so the line $L^\theta\ni q_i^\theta$ is not opposite the line $L$. Hence the claim.
Consider the subspace $\xi:= p^\perp\cap (p^\theta)^\perp$ of dimension $2n-3$. Then clearly $\xi$ contains the subspace $p^\perp\cap W$. We claim that $\dim(p^\perp\cap W)=2n-5$. Indeed, if not, then $W$ is a hyperplane of $\xi$. By Lemma \[lem:fixed-subspace\](b) and our previous claim, all lines of $\sC_n(2)$ through $p$ are contained in $H$, implying $p^\perp=H$. But since $H$ is fixed by $\theta$ we deduce that $p\in W$, a contradiction. Our claim follows.
Hence $\dim(p^\perp\cap W)=2n-5$. It follows that $\dim(\xi\cap W)=2n-5$ as well, since $p^\perp\cap W=(p^\theta)^\perp\cap W$. Now let $q\in\xi\setminus W$. Suppose $q\notin H$. Then the line $\<p,q\>$ is not mapped to an opposite, as we showed above. Suppose $q\in S\setminus W$. Then $q^\theta\perp q$, and since $p^\theta\perp q$, we deduce that $q$ is collinear to $\<p,q\>^\theta$, implying that $\<p,q\>\notin\mathrm{Opp}(\theta)$. Hence, if $\theta_p$ is not $\{2\}$-domestic, then $\xi\cap(H\setminus S)\neq\emptyset$. Under that conditon, if $\xi$ is not contained in $H$, then $\xi\cap H_i$ is a hyperplane of $\xi$, $i=1,2$, and this contradicts Lemma \[lem:fixed-subspace\](a).
Hence we deduce that if $\theta_p$ is not $\{2\}$-domestic, then $\xi\subseteq H$. In this case, since both $p$ and $p^\theta$ are in $H$, we have $p^\perp=\<p,\xi\>=H$ and $(p^{\theta})^{\perp}=\langle p^{\theta},\xi\rangle=H$. However $\perp$ is a symplectic polarity and so $p^\perp=H=(p^{\theta})^{\perp}$ forces $p=p^{\theta}$, a contradiction. The lemma is proved.
\[thm:existenceBn(2)\] Let $\theta$ be a collineation of $\mathsf{C}_n(2)$. Suppose that the set of absolute points of $\theta$ equals the union of two distinct hyperplanes of $\mathsf{PG}(2n-1,2)$. Then $\theta$ is domestic. Moreover, if $k$ is the projective dimension of the subspace of points of $\mathsf{PG}(2n-1,2)$ fixed by $\theta$, then
if $k=n-2$ then $\theta$ is strongly exceptional domestic, and
if $k=n-1+j$ for some $0\leq j\leq n-3$ then $\theta$ is uncapped with decorated opposition diagram
at (0,0.3) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; at (2,-0.7) [$n-j$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (-5,0)–(-0.5,0); (0.5,0)–(4,0); (4,0.07)–(5,0.07); (4,-0.07)–(5,-0.07); (-1,0)–(1,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
Moreover examples exist for each $n-2\leq k\leq 2n-4$.
Suppose that $\theta$ is a collineation of $\mathsf{C}_n(2)$ such that the set of absolute points of $\theta$ is the union of two distinct hyperplanes $H_1$ and $H_2$ of $\mathsf{PG}(2n-1,2)$. We show by induction on $n-j$ that $\theta$ is domestic, with Lemma \[lem:CBase\] providing the base case $n-j=3$.
Let $p$ be any point not in $H_1\cup H_2$. Thus $p$ is mapped to an opposite point by $\theta$. Let $\mathrm{Res}(p)$ be the set of totally isotropic subspaces containing $p$. Thus $\mathrm{Res}(p)$ is a $\mathsf{C}_{n-1}(2)$ building, whose points are the lines through $p$, lines are the planes through $p$, and so forth. Let $\theta_p=\mathrm{proj}_{\mathrm{Res}(p)}\circ \theta$, regarded as a collineation of $\mathsf{C}_{n-1}(2)$. Since $p^{\perp}$ and $(p^{\theta})^{\perp}$ are hyperplanes of $\mathsf{PG}(2n-1,2)$ the spaces $H_i'=p^{\perp}\cap(p^{\theta})^{\perp}\cap H_i$ are $(2n-4)$-spaces for $i=1,2$ (as in the proof of Lemma \[lem:CBase\]). Let $q\in p^{\perp}\cap (p^{\theta})^{\perp}\cap (H_1\cup H_1)$, and let $L=\langle p,q\rangle$. Similar arguments as those in Lemma \[lem:CBase\] show that
if $q$ is fixed by $\theta$, then $L$ is fixed by $\theta_p$, and
if $q$ is mapped to a distinct collinear point by $\theta$ then $L$ is either fixed by $\theta_p$, or is mapped to a distinct coplanar line by $\theta_p$.
Thus for all non-domestic points $p$ the induced collineation $\theta_p$ of the $\mathsf{C}_{n-1}(2)$ building $\mathrm{Res}(p)$ has the property that the set of points mapped to collinear points (including fixed points) contains the union of two distinct hyperplanes in $\mathsf{PG}(2n-3,2)$. Thus by Lemma \[lem:Cn(2)\] and the induction hypothesis the collineation $\theta_p$ is domestic, and hence $\theta$ is domestic.
Now suppose that the absolute points of $\theta$ form a union of two hyperplanes, and that the fixed point set $F$ of $\theta$ is an $(n-2)$-space of $\mathsf{PG}(2n-1,2)$. We prove by induction on $n$ that $\theta$ is strongly exceptional domestic, with Lemma \[lem:CBase\] providing the base case. The above argument shows that $\theta$ is necessarily domestic, and so it remains to show that there are non-domestic panels of each cotype $1,2,\ldots,n$. We claim that for $n\geq 3$ there exists a non-domestic point $p$ such that the hyperplane $p^{\perp}$ intersects $F$ in an $(n-3)$-space $F'$. To see this it suffices to show that there is a point $p$ with $p\notin H_1\cup H_2$ and $p\notin F^{\perp}$. The number of points in $H_1\cup H_2$ is $3\cdot 2^{2n-2}-1$ and the number of points in $F^{\perp}$ is $2^{n+1}-1$. Thus for $n\geq 3$ there is a point $p\notin H_1\cup H_2$ and $p\notin F^{\perp}$. By the induction hypothesis, there are panels of cotypes $2,3,\ldots,n$ of $\mathrm{Res}(p)$ mapped to an opposite panels by $\theta_p$, and thus there are panels of each cotype $2,3,\ldots,n$ of $\mathsf{C}_n(2)$ mapped to an opposite by $\theta$. It is then easy to see that there is also a non-domestic cotype $1$ panel (by a residue argument) and hence $\theta$ is strongly exceptional domestic. Now suppose that the absolute points of $\theta$ form a union of two hyperplanes, and that the fixed point set $F$ of $\theta$ is a $k$-space with $k=n-1+j$ for some $0\leq j\leq n-3$. An argument as in the previous paragraph shows that there is a non-domestic point $p$ such that $p^{\perp}$ intersects $F$ in an $(n-2+j)$-space. By induction, with Lemma \[lem:CBase3\] as the base case, the collineation $\theta_p$ of the $\sC_{n-1}(2)$ building $\mathrm{Res}(p)$ has diagram
at (0,0.3) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; at (2,-0.75) [$n-j$]{}; at (-5,-0.75) [$2$]{}; at (-4,-0.75) [$3$]{}; at (5,-0.75) [$n$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (-5,0)–(-0.5,0); (0.5,0)–(4,0); (4,0.07)–(5,0.07); (4,-0.07)–(5,-0.07); (-1,0)–(1,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
Moreover, for any other non-domestic point $p$ we have that either $\theta_p$ has the above diagram, or $\theta_p$ is domestic on type $n-1-j$ vertices. Thus no simplex $\mathsf{C}_n(2)$ of type $\{1,2,\ldots,n-j-1\}$ is mapped to an opposite by $\theta$, hence the result.
To conclude we prove existence of collineations with each diagram. Recursively define elements $g_n\in\mathsf{Sp}_{2n}(2)$, for $n\geq 2$, by $$g_2=\begin{bmatrix}
0&1&0&0\\
1&0&0&0\\
1&0&0&1\\
0&0&1&0
\end{bmatrix},\quad g_3=\begin{bmatrix}
0&0&1&0&0&0\\
0&1&0&0&0&0\\
1&0&0&0&0&0\\
0&1&0&0&0&1\\
1&0&0&0&1&0\\
0&0&0&1&0&0
\end{bmatrix}, \quad g_n=\begin{bmatrix}
1&0&0&0&0\\
1&1&0&0&0\\
0&0&g_{n-2}&0&0\\
0&0&0&1&0\\
0&0&0&1&1
\end{bmatrix}.$$ Moreover, for each $j\geq 0$ define $g_n^{(j)}\in\mathsf{Sp}_{2n}(2)$ by $$g_n^{(j)}=\begin{bmatrix}
I_j&0&0\\
0&g_{n-j}&0\\
0&0&I_j
\end{bmatrix}.$$ By direct calculation, the absolute points of $g_{2n}$ and $g_{2n}^{(j)}$ are given by $X_{2n-1}X_{2n}=0$ and the collinear points of $g_{2n+1}$ and $g^{(j)}_{2n+1}$ are given by $X_{n-1}(X_{n-2}+X_{n})=0$. Moreover, the fixed points of $g_n$ form an $(n-2)$-space of $\mathsf{PG}(2n-1,2)$, and the fixed points of $g_n^{(j)}$ form an $(n-2+j)$-space of $\mathsf{PG}(2n-1,2)$. Thus, by the arguments above, $g_n$ is a strongly exceptional domestic collineation of $\mathsf{C}_n(2)$ for each $n\geq 2$, and $g_n^{(j+1)}$ diagram as in (b).
Similar theorems hold, with similar proofs, for the $\sB_n(2,4)$ and $\sD_n(2)$ buildings. We will only sketch the details below. Consider first the case $\sB_n(2,4)$. The following lemmas are similar to the $\sC_n(2)$ case.
\[lem:BBase1\] A collineation $\theta$ of the generalised quadrangle $\mathsf{B}_2(2,4)$ is exceptional domestic if and only if the set of absolute points of $\theta$ is the set of points of $\sB_2(2,4)$ lying on the union of two distinct hyperplanes in $\mathsf{PG}(5,2)$.
\[lem:BBase3\] Let $\Delta=\sB_n(2,4)$ with $n\geq 3$ and let $\theta$ be a collineation. If the absolute points of $\theta$ lie on a union of two hyperplanes, and if the fixed points of $\theta$ are the isotropic points of a $(2n-3)$-space in $\mathsf{PG}(2n+1,2)$, then $\theta$ has decorated opposition diagram
at (0,0.3) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-5,0)–(-0.5,0); (0.5,0)–(4,0); (4,0.07)–(5,0.07); (4,-0.07)–(5,-0.07); (-1,0)–(1,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
\[thm:existenceBn(2,4)\] Let $\theta$ be a collineation of $\mathsf{B}_{n}(2,4)$. Suppose that the set of absolute points of $\theta$ is the set of isotropic points lying on the union of two hyperplanes of $\mathsf{PG}(2n+1,2)$. Let $k$ be the projective dimension of the subspace of points of $\mathsf{PG}(2n+1,2)$ fixed by $\theta$. Then $\theta$ is domestic, and
if $k=n$ then $\theta$ is strongly exceptional domestic, and
if $k=n+1+j$ for some $0\leq j\leq n-3$ then $\theta$ is uncapped with decorated diagram
at (0,0.3) ; at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{}; at (2,-0.7) [$n-j$]{}; (-5.north west) rectangle (-5.south east); (-4.north west) rectangle (-4.south east); (-3.north west) rectangle (-3.south east); (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (-5,0)–(-0.5,0); (0.5,0)–(4,0); (4,0.07)–(5,0.07); (4,-0.07)–(5,-0.07); (-1,0)–(1,0); at (-5,0) (-5) [$\bullet$]{}; at (-4,0) (-4) [$\bullet$]{}; at (-3,0) (-3) [$\bullet$]{}; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0) (5) [$\bullet$]{};
Moreover examples exist for each $n\leq k\leq 2n-2$.
The proofs are very similar to Theorem \[thm:existenceBn(2)\], with the base cases given by Lemma \[lem:BBase1\] and \[lem:BBase3\], and we omit the details. Thus it only remains to exhibit the existence of collineations of $\mathsf{B}_n(2,4)$ with the desired properties. To this end, define matrices $g_n$, $n\geq 3$ by $$\begin{aligned}
g_2=\begin{bmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&1\\
0&0&1&0&0&0\\
0&0&0&1&0&0\\
1&0&0&0&0&0\\
0&0&0&0&1&0
\end{bmatrix},\quad
g_3=\begin{bmatrix}
0&0&1&0&0&0&0&0\\
0&1&1&0&0&0&0&0\\
1&0&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0\\
0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&0&1\\
0&0&0&0&0&0&1&0\\
0&0&0&0&0&1&1&0
\end{bmatrix},\quad g_n=\begin{bmatrix}
1&0&0&0&0\\
1&1&0&0&0\\
0&0&g_{n-2}&0&0\\
0&0&0&1&0\\
0&0&0&1&1
\end{bmatrix}.\end{aligned}$$ Moreover, for each $j\geq 1$ define $g_n^{(j)}$ by $$g_n^{(j)}=\begin{bmatrix}
I_j&0&0\\
0&g_{n-j}&0\\
0&0&I_j
\end{bmatrix}.$$ Since $g_n,g_n^{(j)}\in \mathsf{GO}_{2n+2}^-(2)$ these matrices induce collineations of $\sB_n(2,4)$. It is straightforward to check that $g_n$ satisfies the conditions (a) and $g_n^{(j+1)}$ satisfies the conditions (b).
Consider now the case $\sD_n(2)$.
\[thm:existenceDn(2)\] Let $\theta$ be an automorphism of $\mathsf{D}_n(2)$. Suppose that the set of absolute points of $\theta$ is the set of points of $\mathsf{D}_n(2)$ lying on the union of two hyperplanes of $\mathsf{PG}(2n-1,2)$. Let $k$ be the projective dimension of the subspace of points of $\mathsf{PG}(2n-1,2)$ fixed by $\theta$. Then $\theta$ is domestic, and
if $k=n-1$ and $\theta$ is an oppomorphism then $\theta$ is strongly exceptional domestic, and
if $k=n-1+j$ for some $1\leq j\leq n-3$ and $\theta$ is a non-oppomorphism (for odd $j$) and an oppomorphism (for even $j$) then $\theta$ has diagram
at (0,0.8) ; at (0,-0.8) ; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-2.north west) rectangle (-2.south east); (2.north west) rectangle (2.south east); (4.north west) rectangle (4.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (5a.north west) rectangle (5b.south east); (-2,0)–(-0.5,0); (0.5,0)–(4,0); (4,0) to \[bend left\] (5,0.5); (4,0) to \[bend right=45\] (5,-0.5); (-1,0)–(1,0); at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
*(if $j=1$)*
at (0,0.8) ; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); at (2,-0.25) [$n-j$]{}; (-2,0)–(-0.5,0); (0.5,0)–(4,0); (4,0) to (5,0.5); (4,0) to (5,-0.5); (-1,0)–(1,0); at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
*(if $j$ is even)*
at (0,0.8) ; at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{}; (-2.north west) rectangle (-2.south east); (-1.north west) rectangle (-1.south east); (1.north west) rectangle (1.south east); at (1,-0.25) [$n-j$]{}; (-2,0)–(-0.5,0); (0.5,0)–(4,0); (4,0) to \[bend left\] (5,0.5); (4,0) to \[bend right=45\] (5,-0.5); (-1,0)–(1,0); at (-2,0) (-2) [$\bullet$]{}; at (-1,0) (-1) [$\bullet$]{}; at (1,0) (1) [$\bullet$]{}; at (2,0) (2) [$\bullet$]{}; at (3,0) (3) [$\bullet$]{}; at (4,0) (4) [$\bullet$]{}; at (5,0.5) (5a) [$\bullet$]{}; at (5,-0.5) (5b) [$\bullet$]{};
*(if $j>1$ is odd)*
Moreover examples exist for all $n-1\leq k\leq 2n-4$.
The proofs of statements (a) and (b) are again analogous to those in Theorem \[thm:existenceBn(2)\], with an appropriate start to the induction. We omit the details.
To prove existence, note that the matrices $g_{n-1}$, $n\geq 3$, from the proof of Theorem \[thm:existenceBn(2,4)\] are also elements of $\mathsf{GO}_{2n}^+(2)$. Let $h_3=g_2$ and $h_4=g_3$. Then $h_3$ induces a duality of $\sD_3(2)$ and $h_4$ induces a collineation of $\sD_4(2)$. Let $h_n=g_{n-1}$, and for each $1\leq j\leq n-3$ let $h_n^{(j)}=g_{n-1}^{(j)}$. It is easy to check that $h_n$ satisfies conditions (a), and $h_n^{(j)}$ satisfies conditions (b).
Uncapped automorphisms for exceptional types {#sec:exceptional}
============================================
In this section we prove Theorem \[thm:main\*\](b) for the small buildings of exceptional type. Moreover we completely classify the domestic automorphisms of the buildings $\sF_4(2)$, $\sF_4(2,4)$, and $\sE_6(2)$. We begin, in Section \[sec:detect\], by developing a (computationally feasible) method of detecting whether a given automorphism is domestic. In Section \[sec:minimal\] we briefly describe the implementation of the minimal faithful permutation representations of the relevant $\mathbb{ATLAS}$ groups into $\mathsf{MAGMA}$, and then in Section \[sec:EF\] we give the classification of domestic automorphisms of the buildings $\sF_4(2)$, $\sF_4(2,4)$, and $\sE_6(2)$ making use of these permutation representations. We provide examples of uncapped automorphisms in $\sE_7(2)$, and give conjectures for $\sE_8(2)$ in Section \[sec:E7E8\].
Throughout this section we will use standard notation for Chevalley groups and twisted Chevalley groups $G$, and we refer to Carter [@Car:89] for details. In particular, the symbols $B$, $H$, $N$, $U$, $W$, $S$, $R$, $x_{\alpha}(a)$, $n_{\alpha}(a)$, etc, have their usual meanings. However we note that in the twisted case we use these symbols for the objects in the twisted group (rather than the untwisted group). Then the quadruple $(B,N,W,S)$ forms a Tits system in $G$, and thus $(\Delta,\delta)$ is a building of type $(W,S)$ where $\Delta=G/B$ and $\delta(gB,hB)=w$ if and only if $g^{-1}h\in BwB$. In the case of graph automorphisms of a simply laced Dynkin diagram we assume that the Chevalley generators are chosen so that [@Car:89 Proposition 12.2.3] holds (in particular $x_{\alpha}(a)^{\sigma}=x_{\sigma(\alpha)}(\pm a)$).
Detecting domesticity {#sec:detect}
---------------------
The following lemma shows that under certain hypotheses, to verify domesticity it is sufficient to show that no chamber opposite a given chamber is mapped onto an opposite. As we see in the remark after the lemma, the hypotheses cannot be removed.
\[lem:red2\] Let $\theta$ be an automorphism of a thick spherical building $\Delta$, and let $L=\operatorname{\mathrm{disp}}(\theta)$. Let $C$ be any chamber. Suppose that either
each panel of $\Delta$ has at least $4$ chambers, or
$\theta$ is an involution, or
$\theta$ induces opposition and $L=\ell(w_0)$.
Then there exists a chamber $D$ with $\delta(C,D)=w_0$ and $\ell(\delta(D,D^{\theta}))=L$.
Let $E$ be a chamber with $\ell(\delta(E,E^{\theta}))=L$, and write $v=\delta(E,E^{\theta})$. Let $w=\delta(C,E)$, and suppose that $w\neq w_0$. Then there exists $s\in S$ with $\ell(ws)>\ell(w)$. We show that there is a chamber $D$ with $\delta(E,D)=s$ such that $\ell(\delta(D,D^{\theta}))=L$. Consider each case.
$\ell(sv)<\ell(v)$. Then either:
$\ell(svs^{\theta})=\ell(v)$, in which case we choose the unique $D$ with $\delta(E,D)=s$ such that $\delta(D,E^{\theta})=sv$. Since $\delta(E^{\theta},D^{\theta})=s^{\theta}$ and $\ell(svs^{\theta})>\ell(sv)$ we have $\delta(D,D^{\theta})=svs^{\theta}$ and so $\ell(\delta(D,D^{\theta}))=L$.
$\ell(svs^{\theta})<\ell(v)$, in which case necessarily $\ell(vs^{\theta})<\ell(v)$, and it follows that there exists a reduced expression for $v$ starting with $s$ and ending with $s^{\theta}$. Thus there exists a minimal length gallery $E=E_0\sim_{s_1}E_1\sim_{s_2}\cdots\sim_{s_{\ell-1}}E_{\ell-1}\sim_{s_{\ell}}E_{\ell}=E^{\theta}$ with $s_1=s$ and $s_{\ell}=s^{\theta}$.
If every panel of $\Delta$ has at least $4$ chambers then there exists a chamber $D$ with $\delta(E,D)=s$ such that $D\notin \{E_1,E_{\ell-1}^{\theta^{-1}}\}$. Then there is a gallery $D\sim_{s_1}E_1\sim_{s_2}\cdots\sim_{s_{\ell-1}}E_{\ell-1}\sim_{s_{\ell}}D^{\theta}$, and hence $\delta(D,D^{\theta})=v$ has length $L$.
If $\theta$ is an involution then $\theta$ maps every minimal length gallery from $E$ to $E^{\theta}$ to a minimal length gallery from $E^{\theta}$ to $E$, and it follows by considering types of first and last steps that $E_1^{\theta}=E_{\ell-1}$. Thus for any $D$ with $\delta(E,D)=s$ and $D\neq E_1$ we again have $\delta(D,D^{\theta})=v$.
If $\theta$ induces opposition and $L=\ell(w_0)$ then $v=w_0$, and $svs^{\theta}=sw_0s^{\theta}=w_0s^{\theta}s^{\theta}=w_0$, and so case (1)(b) cannot occur.
$\ell(sv)>\ell(v)$. Then either:
$\ell(svs^{\theta})>\ell(v)$, in which case every chamber $D$ with $\delta(E,D)=s$ has $\delta(D,D^{\theta})=svs^{\theta}$, contradicting $\ell(v)=\operatorname{\mathrm{disp}}(\theta)$. Thus this case cannot occur.
$\ell(svs^{\theta})=\ell(v)$, in which case we choose $D$ to be any chamber with $\delta(E,D)=s$. Then $\delta(D,E^{\theta})=sv$ (since $\ell(sv)>\ell(v)$), and thus $\delta(D,D^{\theta})=sv$ or $\delta(D,D^{\theta})=svs^{\theta}$. The first case is impossible by the definition of displacement, and thus $\delta(D,D^{\theta})=svs^{\theta}$ has length $L$.
Hence the result.
\[rem:counter\] The following examples illustrate that the conclusion of Lemma \[lem:red2\] may fail if the hypotheses of the lemma are not satisfied.
The collineation $\theta$ of the Fano plane given by the upper triangular $3\times 3$ matrix with all upper triangular entries equal to $1$ maps no chamber opposite the base chamber $C=(\langle e_1\rangle,\langle e_1+e_2\rangle)$ to an opposite chamber. However this collineation has displacement $\ell(w_0)=3$, since no nontrivial collineation of a projective plane is domestic.
The exceptional domestic collineation of the generalised quadrangle $\mathsf{GQ}(2)=\sC_2(2)$ is given by $\theta=x_1(1)x_2(1)$ in Chevalley generators. The chambers opposite the base chamber $B$ of $G/B$ are mapped to distances $s_1s_2$ or $s_2s_1$, however $\theta$ has displacement $3$ (by both $s_1s_2s_1$ and $s_2s_1s_2$).
Minimal faithful permutation representations {#sec:minimal}
--------------------------------------------
Let $\mathcal{G}$ be the following set of $\mathbb{ATLAS}$ groups: $$\mathcal{G}=\{\sF_4(2),\sF_4(2).2,{^2}\sE_6(2^2),{^2}\sE_6(2^2).2,\sE_6(2),\sE_6(2).2\}.$$ These groups are, respectively, the collineation group of $\sF_4(2)$, the full automorphism group of $\sF_4(2)$ (including dualities), the “inner” automorphism group of $\sF_4(2,4)$, the full automorphism group of $\sF_4(2,4)$, the collineation group of $\sE_6(2)$, and the full automorphism group of $\sE_6(2)$. In the following section we will need an explicit set of conjugacy class representatives for the groups in $\mathcal{G}$. With the exception of perhaps $\sF_4(2)$, these groups appear to be too large for the standard conjugacy class algorithms in $\mathsf{MAGMA}$ (or $\mathsf{GAP}$) when input as matrix groups using the standard adjoint representation (for example $\sE_6(2).2$ has order $429683151044011150540800$, and in any case it is not an entirely trivial task to construct such extensions as a matrix group). However the available algorithms in both $\mathsf{MAGMA}$ and $\mathsf{GAP}$ for permutation groups turn out to be considerably more efficient, and therefore we require faithful permutation representations of the groups in $\mathcal{G}$.
The degrees $\deg(G)$ of the minimal faithful permutation representations of the groups in $\mathcal{G}$ are well known (see for example [@Vas:96; @Vas:97; @Vas:98]): $\deg(\sF_4(2))=69615$, $\deg(\sF_4(2).2)=139230$, $\deg({^2}\sE_6(2^2))=\deg({^2}\sE_6(2^2).2)=3968055$, $\deg(\sE_6(2))=139503$, and $\deg(\sE_6(2).2)=279006$. In each case the permutation representation can naturally be realised by the action of $G$ on certain maximal parabolic coset spaces (equivalently, on certain vertices of the building). For example, for $G=\sE_6(2).2$ we consider the action on $G/P_1\cup G/P_6$ (the set of type $1$ and type $6$ vertices of the $\sE_6(2)$ building), and for $G={^2}\sE_6(2^2).2$ we consider the action on ${^2}\sE_6(2^2)/P_1$ (the set of type $1$ vertices of the $\sF_4(2,4)$ building), where $P_i$ denotes the maximal parabolic subgroup of type $S\backslash \{s_i\}$.
To our knowledge, at the time of writing these minimal faithful permutation representations were not available in either $\mathsf{GAP}$ or $\mathsf{MAGMA}$. Therefore we have implemented these permutation representations using the above action on vertices of the building and the The Groups of Lie Type package [@CMT:04]. The resulting permutation representations are available on the first author’s webpage, where we also provide lists of conjugacy class representatives and code relevant to the computations in the following sections. We would like to thank Bill Unger from the $\mathsf{MAGMA}$ team at Sydney University for helping us generate the conjugacy class representatives from the permutation representations.
Domestic automorphism of small buildings of types $\sF_4$ and $\sE_6$ {#sec:EF}
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In this section we classify the domestic automorphisms of the buildings $\sF_4(2)$, $\sF_4(2,4)$, and $\sE_6(2)$. This requires two main steps. We first exhibit a list of $n$ examples of pairwise non-conjugate domestic automorphisms for each building (for some integer $n$). Next, using an explicit set of conjugacy class representatives, we show that all but $n$ of these representatives map some chamber to an opposite and are hence non-domestic. Thus we conclude that our list of $n$ examples is complete.
We make frequent use of both commutator relations, and the formula $$\begin{aligned}
\label{eq:n}
n_{\alpha}(a)=x_{\alpha}(a)x_{-\alpha}(-a^{-1})x_{\alpha}(a).\end{aligned}$$ We will also use the following observation: For the buildings $\sE_n(2)$, $n=6,7,8$, the displacement of an automorphism $\theta$ determines the (decorated) opposition diagram of $\theta$ (c.f. Remark \[rem:disp\]). For the buildings $\sF_4(2)$ and $\sF_4(2,4)$ the (capped) automorphisms with types $\{1\}$ and $\{4\}$ are not distinguished by displacement, and furthermore in $\sF_4(2)$ the three uncapped diagrams all have displacement $23$.
Before beginning we outline of a useful technique. Suppose that $\theta\in G$ induces an automorphism of $\Delta=G/B$ such that the hypothesis of Lemma \[lem:red2\] holds. Then there exists $gB\in Bw_0B/B$ such that $\operatorname{\mathrm{disp}}(\theta)=\ell(\delta(gB,\theta gB))$. Each $gB\in Bw_0B/B$ can be written as $gB=uw_0B$ with $u\in U$, and $\delta(gB,\theta gB)$ is the unique $w\in W$ such that $$\begin{aligned}
\label{eq:calcw}
w_0^{-1}u^{-1}\theta uw_0\in BwB.\end{aligned}$$ Thus to determine $\operatorname{\mathrm{disp}}(\theta)$ it is sufficient to analyse the terms $w_0^{-1}u^{-1}\theta uw_0$ with $u\in U$. However $|U|=|\mathbb{F}|^{\ell(w_0)}$, and so even for relatively small buildings it is not computationally feasible practical to check each $u\in U$ (for example, in $\sE_6(2)$ we have $|U|=2^{36}$).
The following idea often provides considerable efficiency. Note that each $u\in U$ can be written as $\prod_{\alpha\in R^+}x_{\alpha}(a_{\alpha})$ with $a_{\alpha}\in\mathbb{F}$ and the product taken in any order (see [@St:16 Lemma 17]; of course the $a_{\alpha}$ depend on the order chosen). Writing $A=\{\alpha\in R^+\mid x_{\alpha}(a)\theta\neq \theta x_{\alpha}(a)\text{ for all $a\in\mathbb{F}$}\}
$ we can write $u=u_A'u_{A}$ where $u_{A}$ is a product over terms $\alpha\in A$, and $u_A'$ is a product over the remaining positive roots. Then $u_A'$ commutes with $\theta$, and so $$\begin{aligned}
\label{eq:calcw2}
w_0^{-1}u^{-1}\theta uw_0=w_0^{-1}u_A^{-1}\theta u_Aw_0.\end{aligned}$$ There are $|\mathbb{F}|^{|A|}$ such elements, and so the technique works best if a conjugacy class representative for $\theta$ is chosen with the property that it commutes with as many elements $x_{\alpha}(a)$, $\alpha\in R^+$, as possible.
The residue of the type $J$ simplex of the chamber $gB$ is the coset $gP_{S\backslash J}$, and this residue is non-domestic for $\theta$ if and only if $
g^{-1}\theta g\in P_{S\backslash J}w_0P_{S\backslash J}$, and thus if and only if $$\begin{aligned}
\label{eq:para}
\text{$g^{-1}\theta g\in BwB$ for some $w\in w_0W_{S\backslash J}$}\end{aligned}$$ In the following we write $g_1\sim g_2$ to mean that $g_1$ and $g_2$ are conjugate in $G$.
\[thm:F4\] Let $G=\sF_4(2)$, and let $\Delta=G/B$ be the associated building. Let $\varphi=(2342)$ and $\varphi'=(1232)$ be the highest root and highest short root (respectively) of the $\sF_4$ root system. There are precisely $6$ conjugacy classes of domestic collineations of $\Delta$, as follows: $$
[|l|l|l|l|l|]{} $\theta$&*capped*&*diagram*&*fixed type $1/4$ vertices*&$\mathbb{ATLAS}$\
$\theta_1=x_{\varphi}(1)$&*yes*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (1.north west) rectangle (1.south east);
&$2287/5103$&$2B$\
$\theta_2=x_{\varphi'}(1)$&*yes*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (4.north west) rectangle (4.south east);
&$5103/2287$&$2A$\
$\theta_3=x_{\varphi}(1)x_{\varphi'}(1)$&*yes*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (1.north west) rectangle (1.south east); (4.north west) rectangle (4.south east);
&$1263/1263$&$2C$\
$\theta_4=x_1(1)x_2(1)$&*no*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
&$127/399$&$4D$\
$\theta_5=x_4(1)x_3(1)$&*no*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
&$399/127$&$4C$\
$\theta_6=x_2(1)x_3(1)$&*no*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
&$151/151$&$4E$\
$$ Moreover, $\theta_{3+i}^2\sim\theta_i$ for $i=1,2,3$, and $\theta_2=\sigma(\theta_1)$, $\theta_3=\sigma(\theta_3)$, $\theta_5=\sigma(\theta_4)$, and $\theta_6=\sigma(\theta_6)$.
We first show that the automorphisms have the claimed diagrams. Note that $\theta_1$, $\theta_2$, and $\theta_3$ are involutions, and hence the hypothesis of Lemma \[lem:red2\] applies. Consider $\theta_1$. Following the strategy of (\[eq:calcw\]) we notice that $\theta_1=x_{\varphi}(1)$ is central in $U$ (by the commutator formulae), and hence, for all $u\in U$, using (\[eq:n\]) we have $$w_0^{-1}u^{-1}\theta_1 uw_0=w_0^{-1}x_{\varphi}(1)w_0=x_{-\varphi}(1)=x_{\varphi}(1)n_{\varphi}(1)x_{\varphi}(1)\in Bs_{\varphi}B.$$ Thus $\delta(gB,\theta_1 gB)=s_{\varphi}$ for all $gB\in Bw_0B/B$, and so $\operatorname{\mathrm{disp}}(\theta)=\ell(s_{\varphi})=15$ (using Lemma \[lem:red2\]). Moreover, note that $s_{\varphi}=w_0w_{\{2,3,4\}}$ (for example, by computing inversion sets), and so there exists a non-domestic type $1$ vertex. All type $2$ or $3$ vertices are domestic, for if, for example, there is a non-domestic type $2$ vertex then there is $g\in G$ with $\delta(gB,\theta gB)\in w_0W_{\{1,3,4\}}$ and hence $\operatorname{\mathrm{disp}}(\theta)\geq 24-4>15$. If there exists a non-domestic type $4$ vertex then by [@PVM:17a Lemma 4.5] there exists a non-domestic type $\{1,4\}$ simplex, which again contradicts the displacement calculation. Thus the diagram for $\theta_1$ is as claimed, and since $\theta_2=\sigma(\theta_1)$ (with $\sigma$ the graph automorphism) the result for $\theta_2$ also follows.
Consider $\theta_3$. Since $x_{\varphi'}(1)$ is also central in $U$ (this special feature of characteristic $2$ follows from the commutator relations) we see that $\theta_3$ is central in $U$. Thus, using commutator relations and (\[eq:n\]) we have $$\begin{aligned}
w_0^{-1}u^{-1}\theta_3 uw_0&=x_{-\varphi'}(1)x_{-\varphi}(1)\\
&=x_{-\varphi'}(1)x_{\varphi}(1)n_{\varphi}(1)x_{\varphi}(1)\\
&=x_{\varphi}(1)x_{(1110)}(1)x_{-(0122)}(1)x_{-\varphi'}(1)n_{\varphi}(1)x_{\varphi}(1)\\
&\in Bx_{-(0122)}(1)x_{-\varphi'}(1)s_{\varphi}B\\
&= Bs_{\varphi}x_{-(0122)}(1)x_{(1110)}(1)B\\
&=Bs_{\varphi}s_{(0122)}B.\end{aligned}$$ We have $s_{\varphi}s_{(0122)}=w_0w_{\{2,3\}}$ (for example, by computing the inversion sets), and hence there exists a non-domestic type $\{1,4\}$ simplex (see (\[eq:para\])). By Lemma \[lem:red2\] the above calculation also shows that $\operatorname{\mathrm{disp}}(\theta)=\ell(w_0w_{\{2,3\}})=20$, and the diagram of $\theta_3$ follows.
Consider $\theta_4$. We first show that $\theta_4$ is domestic. We will work with the conjugate $$\theta_4'=x_{(1220)}(1)x_{1122}(1)=w^{-1}\theta_4 w\quad\text{where}\quad w=s_{(0110)}s_{(1242)}$$ because this representative commutes with more elements $x_{\alpha}(1)$ with $\alpha\in R^+$, making (\[eq:calcw\]) more effective. Indeed $\theta_4'$ commutes with all $x_{\alpha}(1)$ with $\alpha\in R^+\backslash A$, where $$A=\{(0100),(0001),(0110),(0011),(0120),(1220),(0122),(1122)\}.$$ Then, as in (\[eq:calcw2\]), we have $
w_0^{-1}u^{-1}\theta_4'uw_0=w_0^{-1}u_A^{-1}\theta_4'u_Aw_0.
$ There are $2^8$ distinct elements $u_A$, and using the Groups of Lie Type package in $\mathsf{MAGMA}$ we can easily verify that $w_0^{-1}u_A^{-1}\theta_4'u_Aw_0\notin Bw_0B$ for all $u_A$ (see the first author’s webpage for the code). This implies that $\theta_4'$ is domestic, for if $\theta_4'$ were not domestic then the third hypothesis of Lemma \[lem:red2\] holds and hence there exists an element $u_A$ with $w_0^{-1}u_A^{-1}\theta_4'u_Aw_0\in Bw_0B$.
One may see that $\theta_4'$ maps panels of cotypes $1$ and $2$ to opposites by simply exhibiting such panels (the Groups of Lie Type package is helpful here). Checking that there are no cotype $3$ or $4$ panels mapped to opposite panels is more complicated, and we have resorted to exhaustively verifying this by computation. However some efficiencies must be found to make the search feasible. Firstly, it is sufficient to check that there are no non-domestic type $\{1,2\}$ simplices (by a simple residue argument). Writing $P=P_{\{3,4\}}$, the (residues of the) type $\{1,2\}$ simplices of $\Delta$ are the cosets $gP$, $g\in G$. Let $T\subseteq W$ denote a transversal of minimal length representatives for cosets in $W/W_{\{3,4\}}$. A complete set of representatives for $P$ cosets in $G$ (and hence type $\{1,2\}$ simplices in $\Delta$) is $$\{u_w(a)w \mid w\in T,\,a\in\mathbb{F}_2^{\ell(w)}\}\quad\text{where}\quad u_w(a)=x_{\beta_1}(a_1)\cdots x_{\beta_k}(a_k),$$ where $R(w)=\{\beta_1,\ldots,\beta_k\}$ is the inversion set of $w$. Thus, using (\[eq:para\]), it is sufficient to check that $\delta(g,\theta_4'g)\notin w_0W_{\{3,4\}}$ for all $g=u_w(a)w$ with $w\in T$. However there are $4385745$ such elements $g$ (the cardinality of $G/P$) and this would be computationally expensive. Considerable efficiency can be gained by using the fact that the product $u_w(a)$ can be taken in any order (again, see [@St:16 Lemma 17]). Thus, applying the technique (\[eq:calcw2\]), we only need to consider terms $u_w'(a)=x_{\gamma_1}(a_1)\cdots x_{\gamma_{\ell}}(a_{\ell})$ with $\{\gamma_1,\ldots,\gamma_{\ell}\}=R(w)\cap A$. This drastically reduces the number of cases needing checking. In fact it turns out that there are only $3885$ elements to check, and these are very quickly checked by the computer.
Since $\theta_5=\sigma(\theta_4)$ the result for $\theta_5$ follows.
Consider $\theta_6$. Again we use a different conjugate $
\theta_6\sim \theta_6'=x_{(1110)}(1)x_{(0122)}(1).
$ This element commutes with all $x_{\alpha}(1)$ with $\alpha\in R^+\backslash A$, where $$A=\{(0001),(0011),(0122),(0111),(0121),(1120),(1220),(1110),(1100),(1000)\}.$$ A similar argument to before, this time checking $2^{10}$ cases, verifies that $\theta_6'$ (and hence $\theta_6)$ is domestic. It is then straightforward to provide panels of each cotype mapped onto opposites, and hence $\theta_6$ has the claimed diagram.
There are $95$ conjugacy classes in the group $\sF_4(2)$ (computed using the permutation representation), and for $88$ of these classes a quick search finds non-domestic chambers. The $7$ remaining classes must therefore be domestic, because the $6$ examples given above are clearly non-conjugate (they have distinct decorated opposition diagrams), and the identity is also trivially domestic.
The number of fixed type $1$ vertices for each example is easily computed using the permutation representation, and the number of fixed type $4$ vertices is obtained by considering the dual. Finally the $\mathbb{ATLAS}$ classes can be determined by the orders and fixed structures.
Since no duality of a thick $\sF_4$ building is domestic the classification of domestic automorphisms of $\sF_4(2)$ is complete (see [@PVM:17a Lemma 4.1]). We also note that Lemma \[lem:F41234\] follows from the above classification.
We now consider the building $\sF_4(2,4)$. The full automorphism group of this building is ${^2}\sE_6(2^2).2$ (that is, ${^2}\sE_6(2^2)$ extended by the diagram automorphism $\sigma$ of $\sE_6$; see [@Tit:74 Section 10.4] and [@ATLAS page 191]). We write $x_{\alpha}(a)$ for the Chevalley generators in the twisted group ${^2}\sE_6(2^2)$. Thus $a\in\mathbb{F}_2$ (respectively $a\in\mathbb{F}_4$) if $\alpha$ is a long root (respectively short root) of the twisted root system.
\[thm:F424\] Let $G= {^2\sE}_6(2^2)$, and let $\Delta=G/B$ be the associated building of type $\sF_4(2,4)$. Let $\varphi$ (respectively $\varphi'$) be the highest root (respectively highest short root) of the $\sF_4$ root system. There are precisely $4$ classes of nontrivial domestic collineations, as follows (where $\sigma$ is the graph automorphism of $\sE_6$): $$
[|l|l|l|l|l|]{} $\theta$&*capped*&*diagram*&*fixed points*&$\mathbb{ATLAS}$\
$\theta_1=x_{\varphi}(1)$&*yes*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (-0.15,0.3) – (0.08,0) – (-0.15,-0.3);(1.north west) rectangle (1.south east);
&$46135$&$2A$\
$\theta_2=x_{\varphi'}(1)$&*yes*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (-0.15,0.3) – (0.08,0) – (-0.15,-0.3);(1.north west) rectangle (1.south east); (4.north west) rectangle (4.south east);
&$20279$&$2B$\
$\theta_3=\sigma$&*yes*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (-0.15,0.3) – (0.08,0) – (-0.15,-0.3); (4.north west) rectangle (4.south east);
&$69615$&$2E$\
$\theta_4=x_1(1)x_2(1)$&*no*&
at (0,0.3) ; at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (-1.5,0)–(-0.5,0); (0.5,0)–(1.5,0); (-0.5,0.07)–(0.5,0.07); (-0.5,-0.07)–(0.5,-0.07); (-0.15,0.3) – (0.08,0) – (-0.15,-0.3);at (-1.5,0) (1) [$\bullet$]{}; at (-0.5,0) (2) [$\bullet$]{}; at (0.5,0) (3) [$\bullet$]{}; at (1.5,0) (4) [$\bullet$]{};
&$855$&$4A$\
$$ Here $x_{\alpha}(a)$ denote the Chevalley generators in the twisted group. Moreover, $\theta_4^2\sim\theta_1$.
The analysis for $\theta_1$ is similar to the analysis of $\theta_1$ for $\sF_4(2)$. Specifically, this element commutes with all terms $x_{\alpha}(a)$, and the result easily follows.
Consider $\theta_2$. This element commutes with all terms $x_{\alpha}(a)$ with $\alpha\in R^+$ except for $x_{(0010)}(a)$, $x_{(0110)}(a)$ and $x_{(1110)}(a)$ with $a\in\{\xi,\xi^2\}$ (where $\xi$ is a generator of $\FF_4^*$). By commutator relations, if $a\in\{\xi,\xi^2\}$ we have $$\begin{aligned}
x_{(0010)}(-a)\theta_2x_{(0010)}(a)&=\theta_2x_{\varphi-\alpha_1-\alpha_2}(1)\\
x_{(0110)}(-a)\theta_2x_{(0110)}(a)&=\theta_2x_{\varphi-\alpha_1}(1)\\
x_{(1110)}(-a)\theta_2x_{(1110)}(a)&=\theta_2x_{\varphi}(1),\end{aligned}$$ and it follows that for all $u\in U$ we have $$\begin{aligned}
w_0^{-1}u^{-1}\theta_2 uw_0=x_{-\varphi'}(1)x_{-\varphi+\alpha_1+\alpha_2}(a_1)x_{-\varphi+\alpha_1}(a_2)x_{-\varphi}(a_3)\quad\text{with}\quad a_1,a_2,a_3\in\{0,1\}.\end{aligned}$$ Considering each of the $8$ possibilities for the triple $(a_1,a_2,a_3)\in\FF_2^3$ we see that the maximum length of $w=\delta(uw_0B,\theta_2uw_0B)$ is $20$ with $w=s_{\varphi}s_{(0122)}$, and the result follows.
Consider $\theta_4$. This element is conjugate to $\theta_4'=x_{(1220)}(1)x_{(1122)}(1)$, and then an analysis very similar to the case of $\theta_4$ for $\sF_4(2)$ applies. In particular, with $A$ as in the $\sF_4(2)$ case, we need to check each of the elements $\delta(u_Aw_0B,\theta_4'u_Aw_0B)$. This time there are $2048=4^3\times 2^5$ elements $u_A$ to check (since there are $3$ roots in $A$ whose root subgroup is isomorphic to $\mathbb{F}_4$ and the remaining $5$ root subgroups are isomorphic to $\mathbb{F}_2$). A quick check with the computer shows that the maximum length of $\delta(u_Aw_0B,\theta_4'u_Aw_0B)$ is $23$, and hence $\theta_4'\sim \theta_4$ is domestic. Then necessarily $\theta_4$ maps no panels of cotypes $3$ or $4$ to opposite (by a simple residue argument), and then since $\operatorname{\mathrm{disp}}(\theta_4)=23$ it is forced that there are panels of cotypes both $1$ and $2$ mapped onto opposites.
Consider $\theta_3=\sigma$. This element acts on the untwisted group $\sE_6(4)$ as a symplectic polarity, and thus is $\{i\}$-domestic for $i\in\{2,3,4,5\}$ (see [@HVM:12]). It follows that $\sigma$ is $\{i\}$-domestic for $i\in\{1,2,3\}$ on the building $\sF_4(2,4)$, hence the result.
Thus the diagrams of the four automorphisms are as claimed. Next, as in the $\sF_4(2)$ example, we use the permutation representation of ${^2}\sE_6(2^2).2$ to compute a complete list of conjugacy class representatives of this group. It turns out that there are $189$ conjugacy classes, and for $184$ of these classes one can exhibit a chamber mapped onto an opposite chamber. Thus there are at most $4$ classes of nontrivial domestic collineations, and since the examples exhibited above are pairwise non-conjugate (by decorated opposition diagrams) the list is complete.
Finally, the calculation of the numbers of fixed points is immediate from the permutation representation, and the $\mathbb{ATLAS}$ classes can be determined by the orders and fixed structures.
Let $G=\sE_6(2).2$, and let $\Delta=\sE_6(2)/B$ be the associated building of type $\sE_6(2)$. There are precisely $3$ classes of domestic dualities (up to conjugation in the full automorphism group), as follows: $$
[|l|l|l|l|]{} $\theta$&*capped*&*diagram*&*order*\
$\theta_1=\sigma$&*yes*&
at (0,0.3) ; at (0,-1.2) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (-2,0)–(2,0); (0,0)–(0,-1); (1.north west) rectangle (1.south east); (6.north west) rectangle (6.south east);
&$2$\
$\theta_2=x_1(1)\sigma$&*no*&
at (0,0.3) ; at (0,-1.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (5.north west) rectangle (5.south east); (6.north west) rectangle (6.south east); (-2,0)–(2,0); (0,0)–(0,-1); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{};
&$4$\
$\theta_3=x_1(1)x_3(1)\sigma$&*no*&
at (0,0.3) ; at (0,-1.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (2.north west) rectangle (2.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (5.north west) rectangle (5.south east); (6.north west) rectangle (6.south east); (-2,0)–(2,0); (0,0)–(0,-1); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{};
&$8$\
$$
As noted in Theorem \[thm:F424\], the element $\theta_1=\sigma$ acts as a symplectic polarity on $\sE_6(2)$, and thus has the diagram claimed (see [@HVM:12]). For the remaining cases $\theta_2$ and $\theta_3$ we note that it is easy to find vertices of each type mapped onto opposite vertices. Thus it remains to show that these dualities are domestic. The working here is slightly more complicated than the case of collineations of the $\sF_4$ buildings. Writing $\theta=\tilde{\theta}\sigma$ with $\tilde{\theta}\in G$, we need to show that $w_0^{-1}u^{-1}\tilde{\theta}u^{\sigma}w_0\notin Bw_0B$ for all $u\in U$ (here we are applying Lemma \[lem:red2\]).
Consider $\theta_2$. We use the conjugate $\theta_2'=x_{\beta}(1)\sigma$ with $\beta=(111221)$. It turns out, by commutator relations, that if $u\in U$ is arbitrary then $u^{-1}x_{\beta}(1)u^{\sigma}$ can be written in the following form (where we use $\mathsf{MAGMA}$’s built-in lexicographic order on the positive roots $\alpha_1,\ldots,\alpha_{32}$): $$\begin{aligned}
&x_1(a_1)x_7(a_2)x_{12}(a_3)x_{18}(a_4)x_{23}(0)x_{17}(a_5)x_{22}(a_6)x_{27}(0)x_{26}(a_7)x_{30}(0)x_{29}(a_8)x_{32}(a_9)\\
&\quad x_{33}(a_9+1)x_{34}(a_{10})x_{35}(a_{11})x_{36}(a_{12})x_3(a_{13})x_9(a_{14})x_{13}(a_{15})x_{15}(0)x_{19}(0)x_{21}(a_4)x_{25}(a_6)\\
&\quad x_{24}(0)x_{28}(a_7)x_{31}(a_{16})x_4(0)x_{10}(a_{14})x_8(0)x_{14}(a_{15})x_{16}(a_3)x_{20}(a_5)x_5(a_{13})x_{11}(a_2)x_2(0)x_6(a_1),\end{aligned}$$ where $a_1,\ldots,a_{16}\in\mathbb{F}_2$. The point is that there are only $2^{16}$ such terms, rather than $2^{36}=|U|$ terms. It is then a quick check on the computer to verify that $\theta_2$ is domestic (and hence strongly exceptional domestic by Corollary \[cor:app1\]).
The analysis of $\theta_3$ is slightly more challenging. Using the conjugate $\theta_3'=x_{\beta}(1)x_{\beta'}(1)\sigma$ with $\beta=(010111)$ and $\beta'=(001111)$ we see that $u^{-1}x_{\beta}(1)x_{\beta'}(1)u^{\sigma}$ can be written in a similar way to the $\theta_2$ case above, this time with $2^{22}$ degrees of freedom. The verification is $\theta_3$ is domestic is then a long search with the computer. The details are on the first author’s webpage.
To verify that our list of domestic examples is complete we again use explicit conjugacy class representatives computed from the minimal faithful permutation representation, as in the previous theorems. See the first author’s webpage for the relevant code. Note that the character table of $\sE_6(2)$ is not printed in $\mathbb{ATLAS}$, and therefore it is not possible to provide the $\mathbb{ATLAS}$ conjugacy class names.
Let $G=\sE_6(2)$, and let $\Delta=G/B$ be the associated building of type $\sE_6(2)$. There are precisely $3$ classes of domestic collineations, as follows: $$
[|l|l|l|l|l|]{} $\theta$&*capped*&*diagram*&*fixed points*&*order*\
$\theta_1=x_1(1)$&*yes*&
at (0,0.3) ; at (0,0.7) ; at (0,-0.7) ; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{}; (-2,0)–(-1,0); (-1,0) to \[bend left=45\] (0,0.5); (-1,0) to \[bend right=45\] (0,-0.5); (0,0.5)–(1,0.5); (0,-0.5)–(1,-0.5); (2.north west) rectangle (2.south east);
&$10479$&$2$\
$\theta_2=x_1(1)x_2(1)$&*yes*&
at (0,0.3) ; at (0,0.7) ; at (0,-0.7) ; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{}; (-2,0)–(-1,0); (-1,0) to \[bend left=45\] (0,0.5); (-1,0) to \[bend right=45\] (0,-0.5); (0,0.5)–(1,0.5); (0,-0.5)–(1,-0.5); (2.north west) rectangle (2.south east); (1.north west) rectangle (6.south east);
&$2543$&$2$\
$\theta_3=x_1(1)x_3(1)$&*no*&
at (0,0.3) ; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{}; (2.north west) rectangle (2.south east); (4.north west) rectangle (4.south east); (3.north west) rectangle (5.south east); (1.north west) rectangle (6.south east); (-2,0)–(-1,0); (-1,0) to \[bend left=45\] (0,0.5); (-1,0) to \[bend right=45\] (0,-0.5); (0,0.5)–(1,0.5); (0,-0.5)–(1,-0.5); at (0,-0.5) [$\bullet$]{}; at (0,0.5) [$\bullet$]{}; at (1,-0.5) [$\bullet$]{}; at (1,0.5) [$\bullet$]{}; at (-2,0) (2) [$\bullet$]{}; at (-1,0) (4) [$\bullet$]{}; at (0,-0.5) (5) [$\bullet$]{}; at (0,0.5) (3) [$\bullet$]{}; at (1,-0.5) (6) [$\bullet$]{}; at (1,0.5) (1) [$\bullet$]{};
&$847$&$4$\
$$
To analyse $\theta_1$ we work with the conjugate $\theta_1\sim x_{\varphi}(1)$, where $\varphi$ is the highest root. Then an analysis very similar to the $\sF_4(2)$ case shows that $\theta_1$ has the diagram claimed.
The analysis for $\theta_2$ can be done by hand. We work with the conjugate $\theta_2'=x_{\varphi}(1)x_{\varphi'}(1)$ where $\varphi$ is the highest root and $\varphi'=(101111)$ is the highest root of the $\sA_5$ subsystem. Let $u\in U$. By commutator relations and a simple induction we see that $u^{-1}\theta_2'u$ is a product of terms $x_{\alpha}(a)$ with $\alpha\geq \varphi'$ (with $\geq$ being the natural dominance order). In particular, each such $\alpha$ is in $R^+\backslash\sD_5$, where $\sD_5$ is the subsystem generated by $\alpha_2,\ldots,\alpha_6$. Let $v=w_0w_{\sD_5}$, where $w_{\sD_5}$ is the longest element of the parabolic subgroup $\langle s_2,\ldots,s_6\rangle$. Then $R^+\backslash\sD_5=\{\alpha\in R^+\mid v^{-1}\alpha\in-R^+\}$. It follows that $v^{-1}(w_0^{-1}u^{-1}\theta_2'uw_0)v\in B$ for all $u\in U$, and therefore $$w_0^{-1}u^{-1}\theta_2'uw_0\in vBv^{-1}\subseteq BvB\cdot Bv^{-1}B.$$ Hence $w_0^{-1}u^{-1}\theta_2'uw_0\in BwB$ for some $w$ with $\ell(w)\leq 2\ell(v)=2(\ell(w_0)-\ell(w_{\sD_5}))=32$ (in fact we necessarily have strict inequality here by double coset combinatorics). Thus $\operatorname{\mathrm{disp}}(\theta)\leq 32$, and it then follows from the classification of diagrams (and hence of possible displacements) that $\operatorname{\mathrm{disp}}(\theta)\leq 30$. On the other hand, a quick calculation shows that $w_0^{-1}\theta_2'w_0\in Bs_{\varphi}s_{\varphi'}B$, and by computing inversion sets we have $s_{\varphi}s_{\varphi'}=w_0w_{\sA_3}$ (where $\sA_3$ is the subsystem generated by $\alpha_3,\alpha_4,\alpha_5$). Thus $\theta_2'$ maps the type $\{1,2,6\}$ simplex of the chamber $w_0B$ to an opposite simplex, hence the result. The working for $\theta_3$ is more involved. Here Lemma \[lem:red2\] cannot be applied, and it is not practical to directly check every chamber for domesticity (there are $3126356394525$ of them). Instead we argue in a similar fashion as we did for the collineation $\theta_4$ in Theorem \[thm:F4\]. First replace $\theta_3$ by the conjugate $\theta_3\sim \theta_3'=x_{(111210)}(1)x_{(011111)}(1)$. Then $\theta_3'$ commutes with all $x_{\alpha}(a)$ with $\alpha\in R^+\backslash A$ where $$A=\{\alpha_1,\alpha_3,\alpha_4,\alpha_6,(000110),(000011),(101100),(101110),(001111),(0111111),(111210)\}.$$ By a residue argument it is sufficient to show that there are no non-domestic type $\{2,4\}$ simplices (see the claim in the proof of Corollary \[cor:maximal\]). Again one cannot feasibly check all type $\{2,4\}$ simplices (there are $7089243525$ of them). However, as in Theorem \[thm:F4\], with $T$ a transversal of minimal length representatives for the cosets in $W/W_{\{1,3,5,6\}}$, it is sufficient to check that $\delta(g,\theta_3'g)\notin w_0W_{\{1,3,5,6\}}$ for all $g=u_w'(a)w$ with $w\in T$ and $u_w'(a)=x_{\gamma_1}(a_1)\cdots x_{\gamma_{\ell}}(a_{\ell})$ with $\{\gamma_1,\ldots,\gamma_{\ell}\}=R(w)\cap A$. It turns out that there are only $64158$ such elements $g$, and they are readily checked by computer in under an hour.
Automorphisms of small buildings of types $\sE_7$ and $\sE_8$ {#sec:E7E8}
-------------------------------------------------------------
Consider the $\sE_7$ root system $R$. Fix the ordering $\alpha_1,\ldots,\alpha_{63}$ of the positive roots according to increasing height, using the natural lexicographic order for roots of the same height (for example, $(1122100)<(1112110)$). Note that this is the inbuilt order in $\mathsf{MAGMA}$. With this order, the roots $\alpha_{44}=(1112111)$, $\alpha_{45}=(0112211)$, and $\alpha_{46}=(1122210)$ play an special role below.
\[thm:E72\] Let $\theta_1=x_{44}(1)x_{46}(1)$ and $\theta_2=x_{44}(1)x_{45}(1)x_{46}(1)$ in $\sE_7(2)$. Then $\theta_1$ and $\theta_2$ are uncapped with respective decorated opposition diagrams
at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (6.north west) rectangle (6.south east); at (0,0) [$\bullet$]{}; at (2,0) [$\bullet$]{}; at (0,-1.3) ; at (0,0.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (-2,0)–(3,0); (0,0)–(0,-1);
at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (3.north west) rectangle (3.south east); (4.north west) rectangle (4.south east); (6.north west) rectangle (6.south east); (2.north west) rectangle (2.south east); (5.north west) rectangle (5.south east); (7.north west) rectangle (7.south east); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (-2,0)–(3,0); (0,0)–(0,-1);
Moreover $\theta_1^2=\theta_2^2=x_{\varphi}(1)$ where $\varphi$ is the highest root, and hence $\theta_1$ and $\theta_2$ have order $4$.
Consider $\theta_2$ first. We show that $\theta_2$ is domestic using Lemma \[lem:red2\]. Applying (\[eq:calcw2\]) verbatim requires us to check $2^{26}$ elements. The following modification of the theme is more efficient. It follows from commutator relations that $$\begin{aligned}
w_0^{-1}u^{-1}\theta_2 uw_0&=\prod_{\beta\in B}x_{-\beta}(a_{\beta}),\end{aligned}$$ where $B=\{\beta\in R^+\mid \beta\geq \alpha_{44}\text{ or }\beta\geq \alpha_{45}\text{ or }\beta\geq \alpha_{46}\}$ (where here $\alpha\geq \beta$ if and only if $\alpha-\beta$ is a nonnegative combination of simple roots). There are $20$ roots in $B$. Moreover $a_{44}=a_{45}=a_{46}=1$ (by commutator relations), and so there remain only $2^{17}$ elements to consider. It is then readily checked by computer that $\theta_2$ is domestic, and we easily find vertices of each type mapped onto opposite vertices. Finally, commutator relations show that $\theta_2^2=x_{\varphi}(1)$.
For $\theta_1$ we do a similar search to the above to show that $\theta_1$ is domestic. The remaining difficultly is showing that $\theta_1$ is $\{1,3\}$-domestic. Arguing as we did for $\theta_4$ in Theorem \[thm:F4\] it turns out that there are $1141419$ elements to check, and this can be done in an overnight run on the computer.
Thus the proof of Theorem \[thm:main\*\](b) is complete. Our computational techniques are not efficient enough to handle the two diagrams for $\sE_8(2)$ due to the formidable size of the group. Thus for these diagrams we provide conjectural examples. For each of these conjectures we have randomly selected $10^5$ chambers and verified that restricted to this subset of the chamber set the structure of the automorphism is as claimed.
Fix the ordering $\alpha_1,\ldots,\alpha_{120}$ of the positive roots of $\sE_8$ according to increasing height, using the natural lexicographic order for roots of the same height. Then the roots $\alpha_{88}=(11232221)$, $\alpha_{89}=(12243210)$ and $\alpha_{90}=(12233211)$ play a special role below.
\[conj:2\] Let $\theta_1=x_{88}(1)x_{90}(1)$ and $\theta_2=x_{88}(1)x_{89}(1)x_{90}(1)$ in $\sE_8(2)$. Then $\theta_1$ and $\theta_2$ are uncapped with respective decorated opposition diagrams
at (0,0.3) ; at (0,-1.3) ; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (1.north west) rectangle (1.south east); (6.north west) rectangle (6.south east); (7.north west) rectangle (7.south east); (8.north west) rectangle (8.south east); (-2,0)–(4,0); (0,0)–(0,-1); at (-2,0) [$\bullet$]{}; at (2,0) [$\bullet$]{}; at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{};
at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; (8.north west) rectangle (8.south east); (7.north west) rectangle (7.south east); (6.north west) rectangle (6.south east); (5.north west) rectangle (5.south east); (4.north west) rectangle (4.south east); (3.north west) rectangle (3.south east); (2.north west) rectangle (2.south east); (1.north west) rectangle (1.south east); (-2,0)–(4,0); (0,0)–(0,-1); at (-2,0) (1) [$\bullet$]{}; at (-1,0) (3) [$\bullet$]{}; at (0,0) (4) [$\bullet$]{}; at (1,0) (5) [$\bullet$]{}; at (2,0) (6) [$\bullet$]{}; at (3,0) (7) [$\bullet$]{}; at (4,0) (8) [$\bullet$]{}; at (0,-1) (2) [$\bullet$]{}; at (0,0.3) ; at (0,-1.3) ;
We note that $\theta_1^2=\theta_2^2=x_{\varphi}(1)$ where $\varphi$ is the highest root, and hence $\theta_1$ and $\theta_2$ have order $4$. It is not difficult to verify that $\operatorname{\mathrm{Typ}}(\theta_1)=\{1,6,7,8\}$ and $\operatorname{\mathrm{Typ}}(\theta_2)=\{1,2,3,4,5,6,7,8\}$. Thus the difficulty in the above conjecture is to show that $\theta_1$ is $\{7,8\}$-domestic, and that $\theta_2$ is domestic. In principle the approach taken for $\sE_7(2)$ is applicable, however in practice the enormous size of the group $\sE_8(2)$ makes the search impractical. For example, applying the technique of Theorem \[thm:E72\] to $\theta_2$ amounts to checking $2^{30}=1073741824$ elements. Each of these checks requires a sequence of commutator relations in the group $\sE_8(2)$, and while $\mathsf{MAGMA}$ has remarkably efficient algorithms implemented for this, the number of cases renders this computational approach unfeasible.
The examples of uncapped automorphisms that we have constructed thus far fix a chamber of the building. This is clear for the examples in exceptional types because the representatives are either in the Borel subgroup $B$, or are a composition of an element of $B$ with a standard graph automorphism. For the examples constructed in classical types we note that all examples have either order $4$ or $8$. It follows that they lie in a Sylow $2$-group of the automorphism group, and hence are conjugate to an element of $B$ (or $\langle B,\sigma\rangle$ in the case of an order $2$ graph automorphism). However there do exist uncapped automorphisms that do not fix a chamber. For example, in $\sC_3(2)=\mathsf{Sp}_{6}(2)$ the element $$\theta=x_2(1)x_3(1)n_2=E_{11}+E_{23}+E_{24}+E_{25}+E_{32}+E_{33}+E_{45}+E_{54}+E_{55}+E_{66}$$ is exceptional domestic (in fact strongly exceptional domestic), with order $6$. Thus $\theta$ does not lie in any conjugate of $B$, and hence $\theta$ fixes no chamber of $\sC_3(2)$. In fact the fixed structure of $\theta$ consists of three points $p_1$, $p_2$, $p_3$, a line $L$, and three planes $\pi_1$, $\pi_2$, and $\pi_3$ such that $\pi_1$, $\pi_2$ and $\pi_3$ intersect in $L$, $p_i\in\pi_i$ for $i=1,2,3$, and $p_i\notin L$ for $i=1,2,3$.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A set $X \subseteq {{\mathbb R}}$ is strongly meager if for every measure zero set $H$, $X+H \neq {{\mathbb R}}$. Let ${{\mathcal{SM}}}$ denote the collection of strongly meager sets. We show that assuming ${\operatorname{\mathsf {CH}}}$, ${{\mathcal{SM}}}$ is not an ideal.'
address:
- |
Department of Mathematics and Computer Science\
Boise State University\
Boise, Idaho 83725 U.S.A.
- |
Department of Mathematics\
Hebrew University\
Jerusalem, Israel
author:
- Tomek Bartoszynski
- Saharon Shelah
title: Strongly meager sets do not form an ideal
---
[^1]
[^2]
Introduction
============
In 1919 Borel wrote the paper [@Borel] in which he attempted to classify all measure zero subsets of the real line. In this paper he introduced a class of measure zero sets, which are now called strong measure zero sets. In 70’s Galvin, Mycielski and Solovay found a characterization of strong measure zero sets that was formulated using only the concept of a first category set and of a translation. That allowed, after replacing first category with measure zero, to define a dual notion of a strongly meager set. It was expected that the global properties of both families of sets will be similar. Several results listed below support this expectation. Nevertheless additive properties of both families of sets are different. It is well known that the family of strong measure zero sets forms an ideal, i.e. is closed under finite unions. The result of this paper is that, assuming continuum hypothesis, the collection of strongly meager sets is not closed under finite unions.
In this paper we work exclusively in the space $2^\omega $ equipped with the standard product measure denoted as $\mu$. Let ${{\mathcal N}}$ and ${{\mathcal M}}$ denote the ideal of all $\mu$–measure zero sets, and meager subsets of $2^\omega $, respectively. For $x,y \in 2^\omega$, $x+y \in 2^\omega $ is defined as $(x+y)(n) = x(n)+y(n) \pmod 2$. In particular, $(2^\omega ,
\operatorname{+})$ is a group and $\mu$ is an invariant measure.
A set $X$ of real numbers or more generally, a metric space, is strong measure zero if, for each sequence $\{\varepsilon_n: n \in \omega\}$ of positive real numbers there is a sequence $\{X_n: n \in \omega\}$ of subsets of $X$ whose union is $X$, and for each $n$ the diameter of $X_n$ is less than $\varepsilon_n$.
The family of strong measure zero subsets of $2^\omega $ is denoted by ${{\mathcal {SN}}}$.
The following characterization of strong measure zero is the starting point for our considerations.
\[solo\] The following are equivalent:
1. $X \in \mathcal {SN}$,
2. for every set $F \in {{\mathcal M}}$, $X+F \neq 2^\omega$. ${\hspace{0.1in} \square \vspace{0.1in}}$
This theorem indicates that the notion of strong measure zero should have its category analog. Indeed, we define after Prikry:
\[defstrmea\] Suppose that $X \subseteq 2^\omega $.
We say that $X$ is strongly meager if for every $H \in
{{\mathcal N}}$, $X+H \neq 2^\omega $. Let ${{\mathcal{SM}}}$ denote the collection of strongly meager sets.
Observe that if $z \not\in X+F=\{x+f: x \in X, f\in F\}$ then $X \cap
(F+z) = \emptyset$. In particular, a strong measure zero set can be covered by a translation of any dense $G_\delta $ set, and every strongly meager set can be covered by a translation of any measure one set.
If $X \subseteq 2^\omega $ is a group then the concepts of strong measure zero and strongly meager connect to the classical construction of a nonmeasurable set by Vitali (a selector of ${{\mathbb R}}/{{\mathbb Q}}$).
Suppose that $X \subseteq 2^\omega $ is a dense subgroup of $(2^\omega,+)$. Then
1. $X \in {{\mathcal{SM}}}$ if and only if every selector from $2^\omega / X$ is nonmeasurable.
2. $X \in {{\mathcal {SN}}}$ if and only if every selector from $2^\omega / X$ does not have the Baire property.
[[Proof]{}. ]{}The proof below requires the group $X$ to be infinite and the set $2^\omega
/ X$ to be infinite. A dense group will have these properties.
We will show only (1), the proof of (2) is analogous. Note that if $X$ is a selector from $2^\omega / X$ and $X$ is as above then $X$ is nonmeasurable if and only if $X$ does not have measure zero.
$ \rightarrow $ Suppose that $X \in {{\mathcal{SM}}}$ and $H \in {{\mathcal N}}$. Let $x \not \in X+H$. It follows that $[x]_X \cap H =\emptyset$, hence no selector is contained in $H$.
$ \leftarrow$ Suppose that $X \not \in {{\mathcal{SM}}}$ and let $H \in {{\mathcal N}}$ be such that $X+H=2^\omega $. For each $x \in 2^\omega $, $[x]_X \cap H \neq
\emptyset$. It follows that we can choose a selector contained in $H$. ${\hspace{0.1in} \square \vspace{0.1in}}$
Note that $X \not \in {{\mathcal {SN}}}$ if there exists a meager set $F$ such that the family $\{F+x: x \in X\}$ covers $2^\omega $. Instead of the assignment $x {\mapsto}F+x$ we can consider a more general mapping $x {\mapsto}(H)_x$, where $H \subseteq 2^\omega \times
2^\omega $ is a Borel set such that $(H)_x = \{y:{\langle}x,y{\rangle}\in H\}\in {{\mathcal M}}$ for all $x \in 2^\omega $.
$X \in {\operatorname{\mathsf {COV}}}({{\mathcal M}})$ if for every Borel set $H \subseteq 2^\omega \times
2^\omega$ such that $(H)_x \in {{\mathcal M}}$ for all $x \in 2^\omega $, $$\bigcup_{x \in X} (H)_x \neq
2^\omega.$$ Similarly, $X \in {\operatorname{\mathsf {COV}}}({{\mathcal N}})$ if for every Borel set $H \subseteq 2^\omega \times
2^\omega$ such that $(H)_x \in {{\mathcal N}}$ for all $x \in 2^\omega $, $$\bigcup_{x \in X} (H)_x \neq
2^\omega.$$
Note that
${\operatorname{\mathsf {COV}}}({{\mathcal N}}) \subseteq {{\mathcal{SM}}}$ and ${\operatorname{\mathsf {COV}}}({{\mathcal M}}) \subseteq {{\mathcal {SN}}}$.
[[Proof]{}. ]{}Given $F \in {{\mathcal M}}$ let $H=\{(x,y): y \in F+x\}$. It is clear that, $\bigcup_{x \in X} (H)_x = F+X$. ${\hspace{0.1in} \square \vspace{0.1in}}$
Families ${{\mathcal {SN}}}$ and ${{\mathcal{SM}}}$ as well as ${\operatorname{\mathsf {COV}}}({{\mathcal M}})$ and ${\operatorname{\mathsf {COV}}}({{\mathcal N}})$ are dual to each other and we are interested to what extent the properties of one family are shared by the dual one.
Below we present several results of that kind. The proofs of these results as well as quite a lot of additional material can be found in [@BJbook].
Let Borel Conjecture (${\operatorname{\mathsf {BC}}}$) be the assertion that there are no uncountable strong measure zero sets, and Dual Borel Conjecture (${\operatorname{\mathsf {DBC}}}$) be the assertion that there are no uncountable strongly meager sets.
Sierpinski showed that Borel Conjecture contradicts ${\operatorname{\mathsf {CH}}}$. His proof essentially yields the following:
Assume $ {\bf MA} $. Both ${\operatorname{\mathsf {COV}}}({{\mathcal M}})$ and ${\operatorname{\mathsf {COV}}}({{\mathcal N}})$ contain sets of size $
2^{\boldsymbol\aleph_0} $. In particular, both Borel Conjectures are false.
There are many weaker assumptions than $ {\bf MA} $ that contradict ${\operatorname{\mathsf {BC}}}$ or ${\operatorname{\mathsf {DBC}}}$. Nevertheless we have the following:
Borel Conjecture is consistent with ${{\operatorname{\mathsf {ZFC}}}}$.
Dual Borel Conjecture is consistent with ${{\operatorname{\mathsf {ZFC}}}}$.
An uncountable set $X \subseteq 2^\omega $ is a Luzin set if $X \cap F$ is countable for $F \in {{\mathcal M}}$, and is a Sierpinski set if $X \cap G$ is countable for $G \in {{\mathcal N}}$.
Sierpinski showed that every Luzin set is in ${{\mathcal {SN}}}$. In addition we have the following:
Every Luzin set is in ${\operatorname{\mathsf {COV}}}({{\mathcal M}})$.
Every Sierpinski set is in ${\operatorname{\mathsf {COV}}}({{\mathcal N}})$ (and so in ${{\mathcal{SM}}}$).
Results presented above indicate that we have certain degree of symmetry between the notions of strongly meager and strong measure zero. The main objective of this paper is to show that as far as additive properties of both families are concerned it is not the case.
Sierpinski showed that ${{\mathcal {SN}}}$ is a $ \sigma $-ideal. In fact, we have the following:
Assume ${\bf MA} $. Then the additivity of ${{\mathcal {SN}}}$ is $2^{\boldsymbol\aleph_0}$.
Similarly,
1. ${\operatorname{\mathsf {COV}}}({{\mathcal M}})$ is a $ \sigma $-ideal,
2. Assume ${\bf MA} $. Then the additivity of ${\operatorname{\mathsf {COV}}}({{\mathcal M}})$ is $2^{\boldsymbol\aleph_0}$.
Surprisingly the dual results are not true.
It is consistent that ${\operatorname{\mathsf {COV}}}({{\mathcal N}})$ is not a $ \sigma $-ideal.
[[Proof]{}. ]{}It is an immediate consequence of the following theorem of Shelah:
It is consistent that ${\operatorname{\mathsf {cov}}}({{\mathcal N}})=\boldsymbol\aleph_\omega$.
Recall that $${\operatorname{\mathsf {cov}}}({{\mathcal N}})=\min\left\{|{\mathcal A}|: {\mathcal A} \subseteq
{\mathcal N} \ \&\ \bigcup {\mathcal A} =2^\omega \right\}.$$
Suppose that ${\operatorname{\mathsf {cov}}}({{\mathcal N}})=\boldsymbol\aleph_\omega$ and let a family ${\mathcal A}
\subseteq {{\mathcal N}}$ witness that. Let $H \subseteq 2^\omega \times 2^\omega $ be an Borel set with null vertical sections and such that $$\forall G \in {{\mathcal N}}\ \exists x \in 2^\omega \ G \subseteq (H)_x.$$ Such a set can be easily constructed from a universal set.
For each $G \in {\mathcal A} $ choose $x_G \in 2^\omega $ such that $G \subseteq (H)_{x_G}$. It follows that $X=\{x_G : G \in {\mathcal
A}\} \not\in {\operatorname{\mathsf {COV}}}({{\mathcal N}})$. On the other hand, every set of size $<{\operatorname{\mathsf {cov}}}({{\mathcal N}})$ belongs to ${\operatorname{\mathsf {COV}}}({{\mathcal N}})$ and $X$ is a countable union of such sets. ${\hspace{0.1in} \square \vspace{0.1in}}$
The purpose of this paper is to show that
\[biggie\] Assume ${\operatorname{\mathsf {CH}}}$. Then ${{\mathcal{SM}}}$ is not an ideal.
Framework {#out}
=========
The proof of Theorem \[biggie\] occupies the rest of the paper. The construction is motivated by the tools and methods developed in [@RoSh470]. We should note here that by using the forcing notion defined in this paper we can also show that the statement “${{\mathcal{SM}}}$ is not an ideal” is not equivalent to ${\operatorname{\mathsf {CH}}}$. However, since the main result is of interest outside of set theory we present a version of the proof that does not contain any metamathematical references.
The structure of the proof is as follows:
- In section \[out\] we show that in order to show that ${{\mathcal{SM}}}$ is not an ideal it suffices to find certain partial ordering ${\mathcal P} $ (Theorem \[newtrick\]).
- The definition of $ {\mathcal P} $ involves construction of a measure zero set $H$ with some special properties. All results needed to define $H$ are proved in section \[three\], and $H$ together with other parameters is defined in section \[four\].
- ${\mathcal P} $ is defined in section \[seven\]. The proof that $ {\mathcal P} $ has the required properties is a consequence of Theorem \[crucialgeneralmore\], which is the main result of section \[five\], and Theorems \[main\] and \[main1\], which are proved in section \[six\].
We will show that in order to prove \[biggie\] it is enough to construct a partial ordering satisfying several general conditions. Here is the first of them.
Suppose that $ ({\mathcal P}, \geq)$ is a partial ordering. We say that ${\mathcal P} $ has the fusion property if there exists a sequence of binary relations $\{ \geq_n : n \in \omega\}$ (not necessarily transitive) such that
1. If $p \geq_{n} q$ then $p \geq q$,
2. if $p \geq_{n+1} q$ and $r \geq_{n+1} p$ then $r \geq_n q$,
3. if $\{p_n: n \in \omega\}$ is a sequence such that $p_{n+1}
\geq_{n+1} p_n$ for each $n$ then there exists $p_\omega $ such that $p_\omega \geq_n p_n$ for each $n$.
From now on we will work in $2^\omega $ with the set of rationals defined as $${{\mathbb Q}}=\{x
\in 2^\omega : \forall^\infty n \ x(n)=0\}.$$
Let ${\operatorname{\mathsf {Perf}}}$ be the collection of perfect subsets of $2^\omega $. For $p, q \in {\operatorname{\mathsf {Perf}}}$ let $p \geq q$ if $p \subseteq q$.
We will be interested in subsets of ${\operatorname{\mathsf {Perf}}}\times {\operatorname{\mathsf {Perf}}}$. Elements of ${\operatorname{\mathsf {Perf}}}\times {\operatorname{\mathsf {Perf}}}$ will be denoted by boldface letters and if ${\mathbf p} \in {\operatorname{\mathsf {Perf}}}\times {\operatorname{\mathsf {Perf}}}$ then ${\mathbf p}=(p_1, p_2)$. Moreover, for ${\mathbf p}, {\mathbf q} \in {\operatorname{\mathsf {Perf}}}\times {\operatorname{\mathsf {Perf}}}$, ${\mathbf p}\geq {\mathbf q}$ if $p_1 \subseteq q_1$ and $p_2
\subseteq q_2$.
\[newtrick\] Assume ${\operatorname{\mathsf {CH}}}$, fix a measure zero set $H \subseteq 2^\omega $, and suppose that there exists a family $ {\mathcal P} \subseteq {\operatorname{\mathsf {Perf}}}\times {\operatorname{\mathsf {Perf}}}$ such that:
1. $ {\mathcal P} $ has the fusion property,
2. For every $ {\mathbf p} \in
{\mathcal P}$, $n \in \omega $ and $z \in 2^\omega $ there exists ${\mathbf q} \geq_n {\mathbf p}$ such that $q_1 \subseteq H+z$ or $q_2 \subseteq H+z$,
3. for every ${\mathbf p} \in {\mathcal P} $, $ n \in \omega
$, $X \in
[2^\omega]^{\leq \boldsymbol\aleph_0} $, $i=1,2$ and ${{\mathbf t}}\in {\operatorname{\mathsf {Perf}}}$ such that $\mu({{\mathbf t}})>0$, $$\mu{\mathopen\ifcase2{}\oo\or
\big(\or\Big(\else\oo\fi}{\mathopen\ifcase1{}\oo\or
\big\{\or\Big\{\else\oo\fi}z\in 2^\omega : \exists {{\mathbf q}}\geq_n {{\mathbf p}}\
X \cup (q_i +
{{\mathbb Q}})
\subseteq {{\mathbf t}}+{{\mathbb Q}}+z{\mathclose\ifcase1{}\oo\or
\big\}\or\Big\}\else\oo\fi}{\mathclose\ifcase2{}\oo\or
\big)\or\Big)\else\oo\fi}=1.$$
Then ${{\mathcal{SM}}}$ is not an ideal.
[[Proof]{}. ]{} We intend to build by induction sets $X_1,X_2 \in {{\mathcal{SM}}}$ in such a way that $H$ witnesses that $X_1 \cup X_2$ is not strongly meager, that is, $(X_1\cup X_2)+H=2^\omega $. By induction we will define an $\omega_1$-tree of members of ${\mathcal P} $ and then take the selector from the elements of this tree. This is a refinement of the method invented by Todorcevic (see [@GalMil84Gam]), who used an Aronszajn tree of perfect sets to construct a set of reals with some special properties. More examples can be found in [@BarInv].
For each $ \alpha< \omega_1 $, $\mathfrak
T_\alpha $ will denote the $\alpha $’th level of an Aronszajn tree of elements of $
{\mathcal P} $. More precisely, we will define ${{\operatorname{\mathsf {succ}}}}({\mathbf p},\alpha)
\subseteq
{\mathcal P}$ – the collection of all successors of ${\mathbf p}$ on level $ \alpha $. We will require that:
1. $\mathfrak T_0= \{2^\omega \times 2^\omega\} $,
2. ${{\operatorname{\mathsf {succ}}}}({\mathbf p}, \alpha ) $ is countable (so levels of the tree are countable),
3. if ${\mathbf q} \in {{\operatorname{\mathsf {succ}}}}({\mathbf p},\alpha)$ then ${\mathbf q}
\geq {\mathbf p}$,
4. if ${{\operatorname{\mathsf {succ}}}}({\mathbf p},\alpha)$ is defined then for each $n \in \omega $ there is ${\mathbf q} \in {{\operatorname{\mathsf {succ}}}}({\mathbf p},\alpha)$ such that ${\mathbf q} \geq_n {\mathbf p}$.
Note that the tree constructed in this way will be an Aronszajn tree since an uncountable branch would produce an uncountable descending sequence of closed sets. For an arbitrary $
{\mathcal P} $ with fusion property the conditions above will guarantee that we build an $ \omega_1$-tree with countable levels. This suffices for the constructions we are interested in.
Let ${\mathfrak T}=\bigcup_{\alpha<\omega_1} {\mathfrak T}_\alpha $ where ${\mathfrak T}_\alpha = {{\operatorname{\mathsf {succ}}}}(2^\omega \times 2^\omega , \alpha)$. For each $ {\mathbf p} \in {\mathfrak T}_\alpha $ choose $x_{\mathbf
p}^1 \in
p_1 $ and $x_{\mathbf p}^2 \in p_2 $. We will show that we can arrange this construction in such a way that $X_1=\{x^1_{\mathbf p}: {\mathbf p}\in \mathfrak T\}$ and $X_2=\{x^2_{\mathbf p}: {\mathbf p}\in \mathfrak T\}$ are the sets we are looking for.
Let $\{({{\mathbf t}}_\alpha,i_\alpha) : \alpha < \omega_1\}$ be an enumeration of pairs $({{\mathbf t}},i)\in {\operatorname{\mathsf {Perf}}}\times \{1,2\}$ such that $\mu({{\mathbf t}})>0$. Let $\{z_\alpha: \alpha <
\omega_1\}$ be an enumeration of $2^\omega $.
[Successor step]{}.
Suppose that $\mathfrak T_\alpha $ is already constructed. Denote $X^\alpha =
\left\{x^1_{\mathbf p},x^2_{\mathbf p}: {\mathbf p} \in \bigcup_{\beta
\leq \alpha } \mathfrak T_\beta\right\}$.
For each ${{\mathbf p}}\in {\mathfrak T}_\alpha $ and $ n \in \omega $, let $$Z^n_{{{\mathbf p}}}=\left\{z\in 2^\omega : \exists {{\mathbf q}}\geq_n {{\mathbf p}}\ X^\alpha
\cup (q_{i_\alpha} + {{\mathbb Q}})
\subseteq {{\mathbf t}}_\alpha +{{\mathbb Q}}+z\right\}.$$ Note that by A2, each set $Z^n_{{{\mathbf p}}}$ has measure one. Fix $$y_\alpha \in \bigcap_{{{\mathbf p}}\in {\mathfrak T}_\alpha} \bigcap_{n \in
\omega} Z^n_{{{\mathbf p}}}.$$ For each ${{\mathbf p}}\in {\mathfrak T}_\alpha $ choose $\{{\mathbf p}^n: n \in \omega\}$ such that
1. ${\mathbf p}^n \geq_{n+1} {\mathbf p}$ for each $n$,
2. $X^\alpha \cup
(p^n_{i_\alpha} + {{\mathbb Q}})
\subseteq {{\mathbf t}}_\alpha +{{\mathbb Q}}+y_\alpha$.
Next apply A1 to get sets $\{{\mathbf q}^n: n \in \omega\}$ such that for all $n$,
1. ${\mathbf q}^n \geq_{n+1} {\mathbf p}^n$,
2. $q^n_1 \subseteq H+z_\alpha $ or $q^n_2 \subseteq H+z_\alpha $.
Define ${{\operatorname{\mathsf {succ}}}}({\mathbf p},\alpha+1)=\{{\mathbf q}^n: n \in \omega\}$. Note that for each $n \in \omega $ there is ${\mathbf q} \in {{\operatorname{\mathsf {succ}}}}({\mathbf p},\alpha)$ such that ${\mathbf q} \geq_n {\mathbf p}$. For completeness, if $\mathbf p \in \bigcup_{\beta<\alpha } \mathfrak
T_\beta $ then put $${{\operatorname{\mathsf {succ}}}}({\mathbf p}, \alpha+1)=\bigcup \{{{\operatorname{\mathsf {succ}}}}({\mathbf q},
\alpha+1): {\mathbf q} \in {{\operatorname{\mathsf {succ}}}}({\mathbf p}, \alpha)\}.$$
[Limit step]{}.
Suppose that $ \alpha $ is a limit ordinal and $\mathfrak T_\beta $ are already constructed for $ \beta < \alpha $. Suppose that ${\mathbf p}_0 \in \mathfrak T_{\alpha_0}$, $\alpha_0<\alpha$. Find an increasing sequence $\{\alpha_n: n \in \omega\}$ with $\sup_n \alpha_n=\alpha $, and for $k \in \omega $, let $\{{\mathbf p}_n^k : n
\in \omega\}$ be such that
1. ${\mathbf p}_n^k \in \mathfrak T_{\alpha_n}$,
2. ${\mathbf p}_{n+1}^k \geq_{n+k+1} {\mathbf p}_n^k$ for each $k,n\in
\omega $.
Let ${\mathbf p}_\omega^k$ be such that ${\mathbf p}_\omega^k
\geq_{n+k} {\mathbf p}_n^k$. Define ${{\operatorname{\mathsf {succ}}}}({\mathbf p}_0,\alpha)=\{{\mathbf p}^k_\omega : k \in
\omega\}$. This concludes the construction of $\mathfrak T$ and $X_1,X_2$.
$X_1, X_2 \in {{\mathcal{SM}}}$.
We will show that $X_1 \in {{\mathcal{SM}}}$. The proof that $X_2 \in
{{\mathcal{SM}}}$ is the same.
Let $G \subseteq 2^\omega $ be a measure zero set. Find $ \alpha < \omega_1$ such that $G \cap ({{\mathbf t}}_\alpha +
{{\mathbb Q}})=\emptyset$ and $i_\alpha=1$. It follows that, $$X_1 \subseteq X^\alpha \cup \bigcup_{{\mathbf p}
\in {\mathfrak T}_{\alpha+1} }
p_1 \subseteq {{\mathbf t}}_\alpha +{{\mathbb Q}}+y_\alpha \subseteq (2^\omega
\setminus G)+ y_\alpha .$$ Thus $X_1 + y_\alpha \subseteq 2^\omega
\setminus G$ and therefore $ y_\alpha \not \in X_1+G$, which finishes the proof. ${\hspace{0.1in} \square \vspace{0.1in}}$
$X_1 \cup X_2 \not\in {{\mathcal{SM}}}$.
[[Proof]{}. ]{} Let $H$ be the set used in A1. We will show that $(X_1 \cup X_2) + H=2^\omega $. Suppose that $z \in 2^\omega $ and let $\alpha<\omega_1$ be such that $z=z_\alpha $. By our construction, for any $\mathbf p \in \mathfrak T_{\alpha+1} $, $x^1_{\mathbf p}
\in z +H$ or $x^2_{\mathbf p} \in z
+H$. Thus $z \in (X_1\cup X_2)+H$, which ends the proof.
This shows that the sets $X_1, X_2$ and $H$ have the required properties. The proof of \[newtrick\] is finished. ${\hspace{0.1in} \square \vspace{0.1in}}$
Therefore the problem of showing that ${{\mathcal{SM}}}$ is not an ideal reduces to the construction of an appropriate set $ {\mathcal P} $. We will do that in the following sections.
Measure zero set {#three}
================
In this section we will develop tools to define a measure zero set $H$ that will be used in the construction of $ {\mathcal P} $ and will witness that the union of two strongly meager sets $X_1, X_2$ defined in the proof of \[newtrick\] is not strongly meager. The set $H$ will be defined at the end of the next section.
We will need several definitions.
Suppose that $I \subseteq \omega $ is a finite set. Let ${\operatorname{\mathsf {F}}}^I$ be the collection of all functions $f: {{\operatorname{\mathsf {dom}}}}(f)
\longrightarrow 2$, with ${{\operatorname{\mathsf {dom}}}}(f) \subseteq
2^I$. For $f \in {\operatorname{\mathsf {F}}}^I$, let $m_f^0=|\{s: f(s)=0\}|$ and $m_f^1=|\{s:
f(s)=1\}|$.
For a set $B \subseteq 2^I$ let $(B)^1 = 2^I \setminus B$ and $(B)^0=B$.
We will work in the space $(2^I,+)$ with addition mod $2$. For a function $f \in
{\operatorname{\mathsf {F}}}^I$ let $$(B)^f = \bigcap_{s \in {{\operatorname{\mathsf {dom}}}}(f)} (B+s)^{f(s)}.$$ In addition let $(B)^\emptyset = 2^I$.
For $f\in {\operatorname{\mathsf {F}}}^I$ and $ k \in \omega $, let $${\operatorname{\mathsf {F}}}^I_{f,k}=\left\{g\in {\operatorname{\mathsf {F}}}^I: f \subseteq g \ \&\ |{{\operatorname{\mathsf {dom}}}}(g) \setminus
{{\operatorname{\mathsf {dom}}}}(f)| \leq k\right\}.$$
The set $H$ will be defined using an infinite sequence of finite sets. The following theorem describes how to construct one term of this sequence.
\[choosec\] Suppose that $m \in \omega $ and $0<\delta < \varepsilon < 1$ are given. There exists $n \in \omega $ such that for every finite set $I \in [\omega]^{>n}$ there exists a set $C \subseteq 2^I$ such that $1-\varepsilon + \delta \geq |C|\cdot 2^{-|I|} \geq 1-
\varepsilon -\delta$ and for every $f \in
{\operatorname{\mathsf {F}}}^I_{\emptyset, m}$, $$\left| \frac{|(C)^f|}{|(C)^\emptyset|} - (1-\varepsilon)^{m^0_f}
\varepsilon^{m^1_f}\right| < \delta .$$
Note that the theorem says that we can choose $C$ is such a way that for any sequences $s_1, \dots, s_{m} \in 2^I$ the sets $s_1+C, \dots,
s_{m}+C$ are probabilistically independent with error $\delta $. Thus, we want $ \delta $ to be much smaller than $ \varepsilon^m$. In order to prove this theorem it is enough to verify the following:
\[choosec1\] Suppose that $m \in \omega $ and $0< \delta < \varepsilon < 1$ are given. There exists $n \in \omega $ such that for every finite set $I \in [\omega]^{>n}$ there exists a set $C \subseteq 2^I$ such that $1-\varepsilon + \delta \geq |C|\cdot 2^{-|I|} \geq 1-
\varepsilon - \delta $ and for every set $X
\subseteq 2^I$, $|X| \leq m$ $$\left| \frac{|\bigcap_{s\in X} (C+s)|}{2^{|I|}} -
(1-\varepsilon)^{|X|}\right| <
\delta.$$
[[Proof]{}. ]{}Note first that \[choosec1\] suffices to prove \[choosec\]. Indeed, if for every $X \in [2^I]^{\leq m}$, $$\left| \frac{|\bigcap_{s\in X} (C+s)|}{2^{|I|}} -
(1-\varepsilon)^{|X|}\right| <
\delta$$ then we show by induction on $m^1_f$ that for every $f \in {\operatorname{\mathsf {F}}}^I_{\emptyset, m}$, $$\left| \frac{|(C)^f|}{|(C)^\emptyset|} - (1-\varepsilon)^{m^0_f}
\varepsilon^{m^1_f}\right| < 2^m\delta.$$
Fix $ m, \delta$ and $ \varepsilon$, and choose the set $C
\subseteq 2^I$ randomly (for the moment $I$ is arbitrary). For each $s
\in 2^I$ decisions whether $s \in
C$ are made independently with the probability of $s \in C$ equal to $1-
\varepsilon $. Thus the set $C$ is a result of a sequence of Bernoulli trials. Note that by the Chebyshev’s inequality, the probability that $1-\varepsilon + \delta \geq |C|\cdot 2^{-|I|} \geq 1-
\varepsilon -\delta $ approaches $1$ as $|I|$ goes to infinity.
Let $S_n$ be the number of successes in $n$ independent Bernoulli trials with probability of success $p$. We will need the following well–known fact that we will prove here for completeness.
\[cor1\] For every $ \delta >0$, $$P\left(\left|\frac{S_n}{n}-p\right| \geq \delta \right) \leq
2 e^{-n\delta^2/4}.$$
[[Proof]{}. ]{}We will show that $$P\left(\frac{S_n}{n} \geq p+\delta \right) \leq
e^{-n\delta^2/4}.$$ The proof that $$P\left(\frac{S_n}{n} \leq p-\delta \right) \leq
e^{-n \delta^2/4}$$ is the same. Let $q = 1-p$. Then for each $x \geq 0$ we have $$\begin{gathered}
P\left(\frac{S_n}{n} \geq p+\delta \right) \leq \sum_{k \geq
n(p+\delta)}^n {n \choose k} p^k q^{n-k} \leq \\
\sum_{k \geq
n(p+\delta)}^n e^{-x(n(p+\delta) -k)}\cdot {n \choose k}
p^k q^{n-k} \leq \\
e^{-xn\delta} \cdot \sum_{k \geq
n(p+\delta)} {n \choose k} (p e^{xq})^k (q e^{-xp})^{n-k} \leq\\
e^{-xn\delta} \cdot \sum_{k=0}^n {n \choose k} (p e^{xq})^k (q
e^{-xp})^{n-k} =
e^{-xn\delta} \left(p e^{xq} + q e^{-xp}\right)^n \leq \\
e^{-xn\delta} \left(p
e^{x^2q^2} + q e^{x^2p^2}\right)^n \leq
e^{-xn\delta} \left(p e^{x^2} + q e^{x^2}\right)^n =
e^{-xn\delta} e^{nx^2} = e^{n(x^2 - \delta x)}.\end{gathered}$$ The inequality $p e^{xq} + q e^{-xp} \leq p
e^{x^2q^2} + q e^{x^2p^2}$ follows from the fact that $e^x \leq
e^{x^2}+x$, for every $x$. The expression $e^{n(x^2 - \delta x)}$ attains its minimal value at $x = \delta/2$, which yields the desired inequality. ${\hspace{0.1in} \square \vspace{0.1in}}$
Consider an arbitrary set $X \subseteq 2^I$. To simplify the notation denote $V = 2^I \setminus C$ and note that $\bigcap_{s \in X} (C+s) = 2^I \setminus (V+X)$. For a point $t \in 2^I$, $t \not\in X+V$ is equivalent to $(t+X)\cap
V=\emptyset$. Thus the probability that $t \not\in X+V$ is equal to $ (1-\varepsilon)^{|X|}$.
Let $G(X)$ be a subgroup of $(2^I, +)$ generated by $X$. Since every element of $2^I$ has order $2$, it follows that $|G(X)|\leq 2^{|X|}$.
There are sets $\left\{U_j : j \leq {|G(X)|}\right\}$ such that:
1. $ \forall j \ \forall s, t \in U_j \ {\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}s \neq t
\rightarrow s+t \not\in G(X){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$,
2. $ \forall j \leq {|G(X)|} \ |U_j|=2^{|I|}/|G(X)|$,
3. $ \forall i \neq j \ U_i \cap U_j = \emptyset$,
4. $\bigcup_{j \leq |G(X)|} U_j = 2^I$.
[[Proof]{}. ]{}Choose $U_j$’s to be disjoint selectors from the cosets $2^I/G(X)$. ${\hspace{0.1in} \square \vspace{0.1in}}$
Note that if $t_1, t_2 \in U_j$ then the events $t_1 \in X+V$ and $t_2
\in X+V$ are independent since sets $t_1+X$ and $t_2+X$ are disjoint. Consider the sets $X_j=
U_j \cap \bigcap_{s \in X} (C+s)$ for $j
\leq |G(X)|$. The expected value of the size of this set is $ (1-\varepsilon)^{|X|} \cdot
2^{|I|}/|G(X)|$. By \[cor1\] for each $j \leq |G(X)|$, $$P\left(\left|\frac{|X_j|}{2^{|I|}/|G(X)|} - (1-\varepsilon)^{|X|}\right|
\geq \delta \right)
\leq 2
e^{-2^{|I|-2}\delta^2/|G(X)|}.$$ It follows that for every $X \subseteq 2^I$ the probability that $$(1-\varepsilon)^{|X|} - \delta \leq
\frac{\left|\bigcap_{s \in X} (C+s)\right|}{2^{|I|}}
\leq (1-\varepsilon)^{|X|}+\delta$$ is at least $$1-
2|G(X)|e^{-2^{|I|-2} \delta^2/|G(X)|} \geq 1-2^{|X|+1} e^{-2^{|I|-|X|-2}\delta^2}.$$ The probability that it happens for every $X$ of size $\leq m$ is at least $$1-2^{|I|\cdot (m+1)^2}\cdot
e^{- 2^{|I|-m-2}\delta^2} .$$ If $m$ and $ \delta $ are fixed then this expression approaches $1$ as $|I|$ goes to infinity, since $\lim_{x \rightarrow \infty} P(x)e^{-x} = 0$ for any polynomial $P(x)$. It follows that for sufficiently large $|I|$ the probability that the “random” set $C$ has the required properties is $>0$. Thus there exists an actual $C$ with these properties as well. ${\hspace{0.1in} \square \vspace{0.1in}}$
Parameters of the construction {#four}
==============================
We will define now all the parameters of the construction. The actual relations (P1–P7 below) between these parameters make sense only in the context of the computations in which they are used, and are tailored to simplify the calculations in the following sections. The reason why we collected these definitions here is that there are many of them and the order in which they are defined is quite important. Nevertheless this section serves only as a reference.
The following notation will be used in the sequel.
Suppose that $s :
\omega \times \omega \longrightarrow \omega $.
Let $s^{(0)}(i,j)=i$ and $s^{(n+1)}(i,j)=s(s^{(n)}(i,j),j)$. Given $N \in \omega+1$, $n \in \omega $ and $f \in \omega^N$ let $$s^{(n)}(f)=\left\{{\mathopen\ifcase2{}\oo\or
\big(\or\Big(\else\oo\fi}i,s^{(n)}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}f(i),i{\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}{\mathclose\ifcase2{}\oo\or
\big)\or\Big)\else\oo\fi}: i <N\right\}.$$ We will write $ s(f)$ instead of $ s^{(1)}(f) $.
We define real sequences $\{\varepsilon_i, \delta_i, \epsilon_i: i \in \omega \}$, intervals $\{I_i: i
\in \omega \}$, sets $\{C_i: i\in \omega \}$ and integers $\{m_i: i
\in \omega \}$. In addition we will define functions $\bar{{{\operatorname{\mathbf {s}}}}},\tilde{{{\operatorname{\mathbf {s}}}}},{{\operatorname{\mathbf {s}}}}: \omega \times
\omega \longrightarrow \omega $. The sequence $\{\varepsilon_i: i \in \omega \}$ is defined first. We require that
1. $0<\varepsilon_{i+1}< \varepsilon_i$ for $ i \in \omega $,
2. $\sum_{i \in \omega } \varepsilon_i< 1/2$.
Set $ \epsilon_0=\delta_0=1$, $I_0=C_0=\emptyset, m_0=0$ and $\bar{{{\operatorname{\mathbf {s}}}}}(n,0)=\tilde{{{\operatorname{\mathbf {s}}}}}(n,0)= {{\operatorname{\mathbf {s}}}}(n,0)=0$ for all $n \in \omega $. Suppose that $\{\delta_i,\epsilon_i, I_i, C_i,m_i: i<N\}$ are defined. Also assume that $\bar{{{\operatorname{\mathbf {s}}}}}(n,i),\tilde{{{\operatorname{\mathbf {s}}}}}(n,i)$ and $ {{\operatorname{\mathbf {s}}}}(n,i)$ are defined for $i<N$ and $ n \in
\omega $.
Put $v_N=\left|\prod_{k<N}
2^{I_k}\right|$, $l_N=\prod_{k<N} v_k$ and define $\epsilon_N$ such that that
1. $0<v_N \cdot \epsilon_N \leq
\varepsilon_N$,
2. $2^{l_N +N+2} \cdot\epsilon_N < \epsilon_{N-1}$.
Given $ \varepsilon_N$ and $\epsilon_N$ we will define for $k \in
\omega $ $$\bar{{{\operatorname{\mathbf {s}}}}}(k,N)=\left\{
\begin{array}{ll}
\max\left\{l: \dfrac{k}{l+1} \epsilon_N^2
\varepsilon_N^{l} > 4\right\} & \text{if }
k\epsilon_N^2 > 4\\
0 & \text{otherwise}
\end{array}\right. .$$
Next let $\tilde{{{\operatorname{\mathbf {s}}}}}(k,N)=\bar{{{\operatorname{\mathbf {s}}}}}^{(2u_N)}(k,N)$, where $u_N $ is the smallest integer $\geq \log_2(8/\epsilon_N^2)$. Finally define $${{\operatorname{\mathbf {s}}}}(k,N)=\tilde{{{\operatorname{\mathbf {s}}}}}^{(2v_N+1)}(k,N).$$ Note that the functions $\bar{{{\operatorname{\mathbf {s}}}}}(\cdot,N),\tilde{{{\operatorname{\mathbf {s}}}}}(\cdot,N)$, and ${{\operatorname{\mathbf {s}}}}(\cdot,N)$ are nondecreasing and unbounded.
Define
1. $m_{N}=\min\left\{m: {{\operatorname{\mathbf {s}}}}^{(N\cdot
l_N)}(m,N)>0\right\}$,
2. $ \delta_N=2^{-N-2}\cdot \varepsilon_N^{m_N}$.
Finally use \[choosec\] to define $I_N$ and $C_N \subseteq
2^{I_N}$ for $ \delta=\delta_N$, $\varepsilon=\varepsilon_N$ and $m=m_N$.
In addition we require that
1. $I_i$ are pairwise disjoint.
The set $H$ that will witness that ${{\mathcal{SM}}}$ is not an ideal is defined as $$H=\{x \in 2^\omega : \exists^\infty k \ x {{\mathord{\restriction}}}I_k \not\in C_k\}.$$ Note that $$\mu(H)\leq \mu\left(\bigcap_{n} \bigcup_{k>n} \{x \in 2^\omega : x
{{\mathord{\restriction}}}I_k \not\in C_k\}\right) \leq \sum_{k>n} \varepsilon_k + \delta_k
\stackrel{n \rightarrow
\infty}{\longrightarrow} 0.$$
More combinatorics {#five}
==================
This section contains the core of the proof of \[newtrick\]. This is Theorem \[crucialgeneral\] which is in the realm of finite combinatorics and concerns properties of the counting measure on finite product spaces. We will use the following notation:
Suppose that $N_0<N \leq\omega $. Define ${\operatorname{\mathsf {F}}}^N$ to be the collection of all sequences ${\operatorname{\mathbf {F}}}={\langle}f_i: i<N{\rangle}$ such that $f_i \in {\operatorname{\mathsf {F}}}^{I_i}$ for $i < N$. For ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^N$ and $h \in \omega^N$, let $${\operatorname{\mathsf {F}}}^N_{{\operatorname{\mathbf {F}}}, h}=\left\{{\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^N: \forall i < N \ {\operatorname{\mathbf {G}}}(i) \in
{\operatorname{\mathsf {F}}}^{I_i}_{{\operatorname{\mathbf {F}}}(i),h(i)}\right\}.$$ Similarly, $${\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}, h}=\left\{{\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^N_{{\operatorname{\mathbf {F}}}, h}: {\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N_0 =
{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N_0\right\}.$$
We always require that for all $i < N$, $$\left|{{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{\operatorname{\mathbf {F}}}(i){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}\right|+h(i) \leq m_i.$$
Let ${{\operatorname{\mathbf{ C}}}}={\langle}C_i:i < \omega {\rangle}$ be the sequence of sets defined earlier. For $N_0 <N$ and ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^N$ let $$({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}=\prod_{N_0\leq i< N} (C_i)^{{\operatorname{\mathbf {F}}}(i)} = \left\{s
\in 2^{I_{N_0} \cup \dots \cup I_{N-1}}: \forall i \in [N_0, N)
\ s {{\mathord{\restriction}}}I_i \in (C_i)^{{\operatorname{\mathbf {F}}}(i)}\right\}.$$ We will write $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}$ instead of $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_0$ and $(C_{N-1})^{{\operatorname{\mathbf {F}}}(N-1)}$ instead of $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N-1}$.
Suppose that $X$ is a finite set. A distribution is a function $m: X
\longrightarrow {{\mathbb R}}$ such that $$0 \leq m(x) \leq \frac{1}{|X|}.$$ Define $\alpha_m$ to be the largest number $\alpha $ such that $m'=\alpha \cdot m$ is a distribution, and put ${\overline{m}}=\sum_{x \in X} m(x)$ and $\overline{\overline{m}}=\alpha_m \cdot {\overline{m}}$.
Suppose that a distribution $m$ on $X$ is given and $Y \subseteq X$. Define $m_Y: Y \longrightarrow {{\mathbb R}}^+$ as $$m_Y(x) = \frac{|X|}{|Y|} \cdot m(x).$$
Note that $$\alpha_m=\frac{1}{|X|\cdot \max\{m(x): x\in X\}}.$$ Observe also that $(m_Y)_Z=m_Z$ if $Z \subseteq Y \subseteq X$.
A prototypical example of a distribution is defined as follows. Suppose that $p \subseteq 2^\omega $ is a closed (or just measurable) set and $n \in \omega $. Let $m$ be defined on $2^n$ as $$m(s)=\mu(p \cap [s]) \text{ for } s \in 2^n.$$ Note that ${\overline{m}}=\mu(p)$.
The following lemmas list some easy observations concerning these notions.
\[lem1\] Suppose that $N \in \omega $, $k^0+k_0 \leq m_N$, $f \in {\operatorname{\mathsf {F}}}^{I_N}_{\emptyset, k^0}$ and $m$ is a distribution on $2^{I_N}$. There exist $f_0, f_1 \in
{\operatorname{\mathsf {F}}}^{I_N}_{f, k_0}$ such that $|f_0 \setminus f|=|f_1\setminus
f|=k_0$ and $${\overline{m_{(C_N)^{f_0}}}} \leq {\overline{m_{(C_N)^{f}}}} \leq {\overline{m_{(C_N)^{f_1}}}}.$$
[[Proof]{}. ]{} For each $x \in 2^{I_N}$ and $h \in {\operatorname{\mathsf {F}}}^{I_N}_{\emptyset, k}$, let $h^0_x = h \cup \{(x,0)\}$ and $h^1_x = h \cup \{(x,1)\}$. Note that there is $i \in \{0,1\}$ such that $${\overline{m_{(C_N)^{h^i_x}}}} \leq {\overline{m_{(C_N)^{h}}}} \leq
{\overline{m_{(C_N)^{h^{1-i}_x}}}}.$$ Iteration of this procedure $k_0$ times will produce the required examples. ${\hspace{0.1in} \square \vspace{0.1in}}$
\[easy1\] Suppose that $N_0 \leq N$ are natural numbers, $h^0,h_0 \in
\prod_{i<N} m_i$ satisfy $h_0(i)+h^0(i) \leq m_i$ for $i <N$, ${\operatorname{\mathbf {F}}}\in
{\operatorname{\mathsf {F}}}^N_{\emptyset, h^0}$ and $m$ is a distribution on $2^{I_0 \cup \dots \cup I_{N-1}}$. Suppose that for every ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}, h_0}$, $a \leq {\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}} \leq b$. Let ${\operatorname{\mathbf {G}}}^\star \in
{\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}, h_0}$ be such that $|{{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{\operatorname{\mathbf {G}}}^\star(i){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi} \setminus {{\operatorname{\mathsf {dom}}}}({\operatorname{\mathbf {F}}}(i))|=h_1(i)<h_0(i)$ for $N_0 \leq i < N$. Then $$\forall {\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {G}}}^\star, h_0-h_1}\
a \leq {\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}} \leq b.$$
[[Proof]{}. ]{}Since ${\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {G}}}^\star, h_0-h_1} \subseteq
{\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}, h_0}$, the lemma is obvious. ${\hspace{0.1in} \square \vspace{0.1in}}$
The following theorem is a good approximation of the combinatorial result that we require for the proof of \[newtrick\]. The proof of it will give us a slightly stronger but more technical result \[crucialgeneralmore\], which is precisely what we need.
\[crucialgeneral\] Suppose that $N_0 < N$ are natural numbers, $h^0,h_0 \in
\prod_{i<N} m_i$ satisfy $h_0(i)+h^0(i) \leq m_i$ for $i <N$, ${\operatorname{\mathbf {F}}}\in
{\operatorname{\mathsf {F}}}^N_{\emptyset, h^0}$ and $m$ is a distribution on $2^{I_0 \cup \dots \cup I_{N-1}}$ such that $${\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}} \geq \frac{2\cdot \sum_{i=N_0}^N
\epsilon_i}{\prod_{i=N_0}^N (1-8\epsilon_i)}.$$ There exists ${\operatorname{\mathbf {F}}}^\star \in
{\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}, h_0-{{\operatorname{\mathbf {s}}}}(h_0)}$ such that $$\forall {\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}^\star, {{\operatorname{\mathbf {s}}}}(h_0)}\
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}} \geq {\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot\prod_{i=N_0}^{N-1} (1-8\epsilon_i)^2 -
\sum_{i=N_0}^{N-2} \epsilon_i.$$
[**Remark**]{}. It is worth noticing that the complicated formulas appearing in the statement of this theorem are chosen to simplify the inductive proof. Putting them aside, the theorem can be formulated as follows: if ${\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}$ is sufficiently big (where big means only slightly larger than zero), then there exists ${\operatorname{\mathbf {F}}}^\star \in
{\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}, h_0-{{\operatorname{\mathbf {s}}}}(h_0)}$ such that for all ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}^\star, {{\operatorname{\mathbf {s}}}}(h_0)}$ the value of $\dfrac{{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}}}{{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}}$ cannot be significantly smaller than $1$.
The proof of \[crucialgeneral\] will proceed by induction on $N \geq N_0$, and the following theorem corresponds to the single induction step.
Suppose that $N \in \omega $ is fixed.
\[comb\] If $k^0+k_0 \leq m_N$, $m$ is a distribution on $2^{I_N}$ and $f \in {\operatorname{\mathsf {F}}}^{I_N}_{\emptyset, k^0}$ is such that $\overline{\overline{{m}_{({C_N})^{f}}}} \geq
2\epsilon_N$ then there exists $f^\star \in
{\operatorname{\mathsf {F}}}^{I_N}_{f, k_0-\tilde{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$ such that $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{f^\star, \tilde{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\
{\overline{{m}_{({C_N})^{f^\star}}}}\cdot (1+2\epsilon_N) \geq {\overline{{m}_{({C_N})^{g}}}} \geq {\overline{{m}_{({C_N})^{f^\star}}}}\cdot
(1- 2\epsilon_N),$$ and $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{f^\star, \tilde{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\
{\overline{{m}_{({C_N})^{g}}}} \geq {\overline{{m}_{({C_N})^{f}}}}\cdot
(1- 2\epsilon_N).$$
[[Proof]{}. ]{}We start with the following observation:
\[small\] Suppose that ${\overline{{m}_{({C_N})^f}}} \geq \epsilon_N$. There exists $\tilde{f} \in
{\operatorname{\mathsf {F}}}^{I_N}_{f, k_0-\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$ such that $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}, \bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\
{\overline{{m}_{({C_N})^g}}} \geq {\overline{{m}_{({C_N})^f}}}\cdot (1- \epsilon_N).$$ Similarly, there exists $\tilde{f} \in
{\operatorname{\mathsf {F}}}^{I_N}_{f, k_0-\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$ such that $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}, \bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\
{\overline{{m}_{({C_N})^g}}} \leq {\overline{{m}_{({C_N})^{f}}}}\cdot(1+ \epsilon_N).$$
[[Proof]{}. ]{}We will show only the first part, the second part is proved in the same way. If $\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)=0$ then the lemma follows readily from \[lem1\]. Thus, suppose that $\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)>0$ and let $m_{({C_N})^f}$ be a distribution satisfying the requirements of the lemma.
Construct, by induction, a sequence $\{f_n : n < n^\star \}$ such that
1. $f_0=f$,
2. $f_{n+1} \in {\operatorname{\mathsf {F}}}^{I_N}_{f_n,\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$,
3. ${\overline{{m}_{({C_N})^{f_n}}}} \geq
{\overline{{m}_{({C_N})^{f}}}}\cdot\left(1+
\dfrac{n}{2}\epsilon_N\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\right).$
First notice that $\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)$ was defined in such a way that $$\begin{gathered}
{\overline{{m}_{({C_N})^{f}}}}\cdot\left(1+
\frac{1}{2}\left(\frac{k_0}{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}-2\right)
\epsilon_N\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\right) \geq \\
\epsilon_N\left(1+
\frac{1}{2}\left(\frac{k_0}{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}-2\right)
\epsilon_N\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\right)
\geq \\
\frac{1}{2}\frac{k_0}{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\epsilon_N^2\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}
>1.\end{gathered}$$ Therefore, after fewer than $\dfrac{k_0}{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}-2 $ steps the construction has to terminate (otherwise ${\overline{{m}_{({C_N})^{g}}}}
>1 $ for some $g$, which is impossible).
Suppose that $f_n$ has been constructed.
[Case 1]{}. $\forall h \in {\operatorname{\mathsf {F}}}^{I_N}_{f_n, \bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} \
{\overline{{m}_{({C_N})^{h}}}} \geq {\overline{{m}_{({C_N})^{f}}}} \cdot(1-\epsilon_N)$. In this case put $\tilde{f}=f_n$ and finish the construction. Observe that $$\begin{gathered}
|\tilde{f}|+\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N) \leq k^0+ n^\star\cdot
\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)+\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N) \leq \\
k^0 +
\left(\frac{k_0}{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}-2\right) \cdot \bar{{{\operatorname{\mathbf {s}}}}}(k_0,N) +\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)
\leq k^0+k_0 - \bar{{{\operatorname{\mathbf {s}}}}}(k_0,N) < m_N.\end{gathered}$$
[Case 2]{}. $ \exists h \in {\operatorname{\mathsf {F}}}^{I_N}_{f_n, \bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} \
{\overline{{m}_{({C_N})^{h}}}} < {\overline{{m}_{({C_N})^{f}}}} \cdot(1-\epsilon_N)$. Using \[lem1\] we can assume that $|h|=|f_n|+\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)$.
Consider the partition of $({C_N})^{f_n}$ given by $h$, i.e. $$(C_N)^{f_n}=({C_N})^{h} \cup {\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}({C_N})^{f_n} \setminus
({C_N})^{h}{\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}.$$ Note that by considering the worst case we get $$\begin{gathered}
\frac{|({C_N})^{h}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|} \geq \\
\frac{(1-\varepsilon_N)^{m^0_{f_n}}\varepsilon_N^{m^1_{f_n}} \cdot
\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} - \delta_N
}{(1-\varepsilon_N)^{m^0_{f_n}}\varepsilon_N^{m^1_{f_n}}+\delta_N -
\left((1-\varepsilon_N)^{m^0_{f_n}}\varepsilon_N^{m^1_{f_n}}\cdot
\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} -
\delta_N\right)} \geq \\
\frac{\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} -
\dfrac{\delta_N}{(1-\varepsilon_N)^{m^0_{f_n}}
\varepsilon_N^{m^1_{f_n}}}
}{1-\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}+\dfrac{2\delta_N}{(1-\varepsilon_N)^{m^0_{f_n}}
\varepsilon_N^{m^1_{f_n}}}}.\end{gathered}$$ Moreover, since $\delta_N \leq \dfrac{1}{2} \varepsilon_N^{m_N}$, we have $$\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} \geq
2\cdot\frac{\delta_N}{(1-\varepsilon_N)^{m^0_{f_n}}\cdot
\varepsilon_N^{m^1_{f_n}}},$$ and thus $$\frac{\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} -
\dfrac{\delta_N}{(1-\varepsilon_N)^{m^0_{f_n}}
\varepsilon_N^{m^1_{f_n}}}
}{1-\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}+\dfrac{2\delta_N}{(1-\varepsilon_N)^{m^0_{f_n}}
\varepsilon_N^{m^1_{f_n}}}} \geq \dfrac{1}{2} \varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}.$$
It follows that $$\begin{gathered}
\frac{1}{{\overline{{m}_{({C_N})^{f}}}}}\cdot {\overline{{m}_{({C_N})^{f_n} \setminus ({C_N})^{h}}}}
\geq \left(1+\frac{n}{2} \epsilon_N
\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\right)\cdot\frac{|({C_N})^{f_n}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|} -\\
(1-\epsilon_N)\cdot\frac{|({C_N})^{h}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|}=\end{gathered}$$ $$\begin{gathered}
\frac{|({C_N})^{f_n}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|}+ \frac{n}{2} \epsilon_N \varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}
\cdot \frac{|({C_N})^{f_n}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|}- \frac{|({C_N})^{h}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|}+ \\
\epsilon_N \frac{|({C_N})^{h}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|}=\end{gathered}$$ $$\begin{gathered}
1+\frac{n}{2} \epsilon_N \varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} \cdot
\frac{|({C_N})^{f_n}|}{\left|({C_N})^{f_n} \setminus
({C_N})^{h}\right|} + \epsilon_N
\frac{|({C_N})^{h}|}{\left|({C_N})^{f_n}
\setminus ({C_N})^{h}\right|} \geq\\
1+ \frac{n}{2}\epsilon_N\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} +
\epsilon_N \cdot \frac{\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} -
\dfrac{\delta_N}{(1-\varepsilon_N)^{m^0_{f_n}}
\varepsilon_N^{m^1_{f_n}}}
}{1-\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}+\dfrac{2\delta_N}{(1-\varepsilon_N)^{m^0_{f_n}}
\varepsilon_N^{m^1_{f_n}}}}
\geq \\
1+ \frac{n}{2}\epsilon_N\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}+
\frac{1}{2}\epsilon_N\varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)} \geq
1+\frac{n+1}{2} \epsilon_N \varepsilon_N^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}.\end{gathered}$$
Let $\{h_1, \dots, h_{2^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}}\}$ be the list of all functions in ${\operatorname{\mathsf {F}}}^{I_N}$ such that ${{\operatorname{\mathsf {dom}}}}(h_i)={{\operatorname{\mathsf {dom}}}}(h) \setminus {{\operatorname{\mathsf {dom}}}}(f_n)$. Without loss of generality we can assume $f_n \cup h_1=h$. The sets $({C_N})^{f_n \cup h_2}, \dots, ({C_N})^{f_n \cup h_{2^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}}}$ define a partition of the set $({C_N})^{f_n} \setminus ({C_N})^{h}$. Since $${\overline{{m}_{({C_N})^{f_n} \setminus ({C_N})^{h}}}}
\geq {\overline{{m}_{({C_N})^{f}}}}\cdot\left(1+\frac{n+1}{2}\epsilon_N \varepsilon_N
^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\right)$$ it follows that there exists $2 \leq \ell \leq 2^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$ such that $${\overline{{m}_{({C_N})^{f_n \cup h_{\ell}}}}} \geq
{\overline{{m}_{({C_N})^{f}}}}\cdot\left(1+\frac{n+1}{2}\epsilon_N \varepsilon_N
^{\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\right).$$ Let $f_{n+1}=f_n \cup h_{\ell}$. This completes the induction. ${\hspace{0.1in} \square \vspace{0.1in}}$
[Proof of \[comb\]]{}. Suppose that $\overline{\overline{{m}_{({C_N})^{f}}}} =a_0\geq 2\epsilon_N$. Without loss of generality we can assume that $\alpha_{m_{({C_N})^f}}=1$, that is $\overline{\overline{{m}_{({C_N})^f}}}={\overline{{m}_{({C_N})^f}}}$. This is because if we succeed in proving the theorem for the distribution $\alpha_{m_{({C_N})^f}} \cdot {m}_{({C_N})^f}$ then it must be true for ${m}_{({C_N})^f}$ as well.
Apply \[small\] to get $f'
\in {\operatorname{\mathsf {F}}}^{I_N}_{f,k_0-\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$ such that $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{f',\bar{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\
{\overline{{m}_{({C_N})^{g}}}} \geq a_0(1-\epsilon_N).$$ Let $u_N$ be the smallest integer greater than $\log_2(8/\epsilon_N^2)$ and define by induction sequences $\{f_i,a_i,b_i: i \leq u_N\}$ such that
1. $b_0=1$ and $f_0=f'$,
2. $a_i,b_i \in {{\mathbb R}}$ for $i \leq u_N$,
3. $|b_i-a_i| \leq 2^{-i}$ for $i \leq u_N$,
4. $f_{i+1} \in
{\operatorname{\mathsf {F}}}^{I_N}_{f_i,\bar{{{\operatorname{\mathbf {s}}}}}^{(i+1)}(k_0,N)-\bar{{{\operatorname{\mathbf {s}}}}}^{(i+2)}(k_0,N)}$ for $i<u_N$,
5. $ \forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{f_i,\bar{{{\operatorname{\mathbf {s}}}}}^{(i+1)}(k_0,N)}\
a_i(1-\epsilon_N) \leq {\overline{{m}_{({C_N})^{g}}}} \leq b_i(1+\epsilon_N).$
Suppose that $a_i,b_i$ and $f_i$ are defined and let $c = {\overline{m_{(C_N)^{f_i}}}}$. Observe that $c \geq
a_0\cdot(1-\epsilon_N) > \epsilon_N$.
If $|c-a_i| \leq 2^{-i-1}$ then let $a_{i+1}=a_i$ and $b_{i+1}=c$. Apply \[small\] to get $f_{i+1}
\in {\operatorname{\mathsf {F}}}^{I_N}_{f_i,\bar{{{\operatorname{\mathbf {s}}}}}^{(i+1)}(k_0,N)-\bar{{{\operatorname{\mathbf {s}}}}}^{(i+2)}(k_0,N)}$ such that $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{f_{i+1},\bar{{{\operatorname{\mathbf {s}}}}}^{(i+2)}(k_0,N)}\
{\overline{{m}_{({C_N})^{g}}}} \leq b_{i+1}(1+\epsilon_N).$$ Otherwise let $a_{i+1}=c$ and $b_{i+1}=b_i$ and let $f_{i+1}
\in {\operatorname{\mathsf {F}}}^{I_N}_{h,\bar{{{\operatorname{\mathbf {s}}}}}^{(i+1)}(k_0,N)-\bar{{{\operatorname{\mathbf {s}}}}}^{(i+2)}(k_0,N)}$ be such that $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{f_{i+1},\bar{{{\operatorname{\mathbf {s}}}}}^{(i+2)}(k_0,N)}\
{\overline{{m}_{({C_N})^{g}}}} \geq a_{i+1}(1-\epsilon_N).$$ Put $f^\star = f_{u_N}$. Note that by the choice of $u_N$, $|b_{u_N} -a_{u_N}| \leq
\epsilon_N^2/8$. In addition, $\bar{{{\operatorname{\mathbf {s}}}}}^{(u_N+1)}(k_0,N)> \bar{{{\operatorname{\mathbf {s}}}}}^{(2u_N)}(k_0,N)=\tilde{{{\operatorname{\mathbf {s}}}}}(k_0,N)$. Since ${\overline{m_{(C_N)^{f^\star}}}}$ is equal to either $a_{u_N}$ or $b_{u_N}$, and $a_{u_N} \geq \varepsilon_N$, a simple computation shows that for every $g \in {\operatorname{\mathsf {F}}}^{I_N}_{f^\star,\tilde{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$, $${\overline{m_{(C_N)^{f^\star}}}}\cdot(1-2\epsilon_N) \leq a_{u_N}(1-\epsilon_N) \leq
{\overline{{m}_{({C_N})^{g}}}} \leq b_{u_N}(1+\epsilon_N) \leq
{\overline{m_{(C_N)^{f^\star}}}}\cdot(1+2\epsilon_N),$$ and $${\overline{{m}_{({C_N})^{g}}}} \geq a_{u_N}(1-\epsilon_N) \geq
a_{0}(1-2\epsilon_N)={\overline{m_{(C_N)^{f}}}}\cdot(1-2\epsilon_N).~{\hspace{0.1in} \square \vspace{0.1in}}$$
Before we start proving \[crucialgeneral\] we need to prove several facts concerning distributions. The following notation will be used in the sequel.
1. $v_k = \left|2^{I_0 \cup \dots
\cup I_{k-1}}\right|$ for $k \in \omega $.
2. If ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^{N}$ and $k<N$ then let $w_k({\operatorname{\mathbf {F}}})=
\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}k}\right|$.
Suppose that ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^{N+1}$ and $m$ is a distribution on $2^{I_0 \cup \dots
\cup I_{N}}$.
1. Let $m^+_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}$ be the distribution on $(C_N)^{{\operatorname{\mathbf {F}}}(N)}$ given by $$m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^+(s)=\sum \left\{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(t): s \subseteq t
\in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}
\right\} \text{ for } s \in (C_N)^{{\operatorname{\mathbf {F}}}(N)} .$$
2. For $N_0\leq N$ and $t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N_0}$, let $m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t$ be a distribution on $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}$ defined as $$m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t(s)=m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(t^\frown s) \text{ for } s \in
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}.$$
3. Let $m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^-$ be the distribution on $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}$ defined as $$m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^-(t)={\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t}} \text{ for } t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}.$$
\[new0\] Suppose that $N_0 \leq N$, ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^{N+1}$ and ${\operatorname{\mathbf {G}}}\in
{\operatorname{\mathsf {F}}}^{N_0,N+1}_{{\operatorname{\mathbf {F}}},h}$ for some $h \in \omega^\omega $. Then $$\left(m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}_{N_0}}=m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^t.$$
[[Proof]{}. ]{}Fix $t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N_0}=({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N_0}$ and observe that for $s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}$, $$\begin{gathered}
\left(m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}}(s)=
\frac{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}|}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}_{N_0}|}
\cdot m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(t^\frown s)=\\
\frac{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}|}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}_{N_0}|} \cdot
\frac{v_{N+1}}{w_{N+1}({\operatorname{\mathbf {F}}})} m(t^\frown s)=\frac{v_{N+1}}{w_{N+1}({\operatorname{\mathbf {G}}})}
\cdot m(t^\frown s)=
m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^t(s).~{\hspace{0.1in} \square \vspace{0.1in}}\end{gathered}$$
\[new0a\] Suppose that ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^{N+1}$ and ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N,N+1}_{{\operatorname{\mathbf {F}}}, h}$ for some $h \in \omega^\omega $. Then $$(m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^+)_{(C_N)^{{\operatorname{\mathbf {G}}}(N)}}=m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^+.$$
[[Proof]{}. ]{}Similar to the proof of \[new0\]. ${\hspace{0.1in} \square \vspace{0.1in}}$
\[new1\] Suppose that $N_0 \leq N$, ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^{N+1}$ and $t \in
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N_0}$. Then $$\overline{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t}} \geq w_{N_0}({\operatorname{\mathbf {F}}})\cdot {\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t}}.$$
[[Proof]{}. ]{}Note that $$w_{N_0}({\operatorname{\mathbf {F}}}) \cdot m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(t^\frown s) \leq
\frac{w_{N_0}({\operatorname{\mathbf {F}}})}{w_{N+1}({\operatorname{\mathbf {F}}})} = \frac{1}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}_{N_0}|} .~{\hspace{0.1in} \square \vspace{0.1in}}$$
The next two lemmas will be crucial in the recursive computations of distributions.
\[new2\] Suppose that $N_0 \leq N$, ${\operatorname{\mathbf {F}}},{\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N+1}$, ${\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}[N_0,N]={\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}[N_0,N] $ and $t \in
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N_0} \cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N_0}$. Then $$\alpha_{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t}\cdot m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t =
\alpha_{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^t }\cdot m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^t.$$ In particular, if ${\operatorname{\mathbf {F}}}^\star \in {\operatorname{\mathsf {F}}}^{N_0,N+1}_{{\operatorname{\mathbf {F}}},h}$ for some $h\in
\omega^\omega $ then $$\frac{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N_0{}^\frown
{\operatorname{\mathbf {F}}}^\star {{\mathord{\restriction}}}[N_0,N] }}}}}{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}}=\frac{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N_0{}^\frown {\operatorname{\mathbf {F}}}^\star {{\mathord{\restriction}}}[N_0,N]}}}}}{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}}}.$$
[[Proof]{}. ]{}Note that under the assumptions the distributions $m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}^t$ and $m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^t$ have the same domain and the fraction $\dfrac{m(t^\frown s)}{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(s)}$ has the constant value for both ${\operatorname{\mathbf {F}}}$ and ${\operatorname{\mathbf {G}}}$. ${\hspace{0.1in} \square \vspace{0.1in}}$
\[new3\] Suppose that ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^{N+1}$ and ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N+1}_{{\operatorname{\mathbf {F}}},h}$ for some $h \in \omega^\omega$. Then $${\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}}=\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}(N)}}(t)\cdot
\frac{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {G}}}(N)}}}}}{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}}.$$
[[Proof]{}. ]{}For $t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}$, $$\frac{m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {F}}}(N)}}(t)}{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}}=
\frac{\displaystyle
\sum_{s' \in (C_N)^{{\operatorname{\mathbf {F}}}(N)}} \dfrac{v_{N+1}}{w_{N+1}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}(N){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}} \cdot m(t^\frown s')
}
{\displaystyle\sum_{s' \in
(C_N)^{{\operatorname{\mathbf {F}}}(N)}} \dfrac{v_{N+1}}{{w_{N+1}({\operatorname{\mathbf {F}}})}} \cdot m(t^\frown
s')
}
=
\frac{w_N({\operatorname{\mathbf {F}}})}{w_N({\operatorname{\mathbf {G}}})} .$$ Therefore $$\begin{gathered}
\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}(N)}}(t)\cdot
\frac{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {G}}}(N)}}}}}{{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}} =
\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}} \frac{w_N({\operatorname{\mathbf {F}}})}{w_N({\operatorname{\mathbf {G}}})}\cdot
{\overline{m^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {G}}}(N)}}}} =\\
\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}} \frac{w_N({\operatorname{\mathbf {F}}})}{w_N({\operatorname{\mathbf {G}}})}\cdot
\sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}} \frac{v_{N+1}}{w_{N+1}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {G}}}(N){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}}\cdot m(s) = \\
\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}} \sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}
\frac{v_{N+1}}{w_{N+1}({\operatorname{\mathbf {G}}})}\cdot m(s)=
\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}} \sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}
m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(s)=
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}}.{\hspace{0.1in} \square \vspace{0.1in}}\end{gathered}$$
We will need one more definition:
Suppose that $m$ is a distribution on $X$ and $U \subseteq X$. Let $ {m_{[U]}}$ be the distribution on $X$ defined as $${m_{[U]}}(x)=\left\{
\begin{array}{ll}
m(x) & \text{if }x \in U\\
0 & \text{otherwise}
\end{array}\right. \text{ for } x \in X.$$
Now we are ready to prove theorem \[crucialgeneral\]. For technical reasons we will need a somewhat stronger result stated below.
\[crucialgeneralmore\] Suppose that $N_0 < N$ are natural numbers, $h^0,h_0 \in
\prod_{i<N} m_i$ satisfy $h_0(i)+h^0(i) \leq m_i$ for $i <N$, ${\operatorname{\mathbf {F}}}\in
{\operatorname{\mathsf {F}}}^N_{\emptyset, h^0}$ and $m$ is a distribution on $2^{I_0 \cup \dots \cup I_{N-1}}$ such that $${\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}} \geq \frac{2 \sum_{i=N_0}^N
\epsilon_i}{\prod_{i=N_0}^N (1-8\epsilon_i)}.$$ There exist ${\operatorname{\mathbf {F}}}^\star \in
{\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}, h_0-{{\operatorname{\mathbf {s}}}}(h_0)}$ and $U^\star \subseteq 2^{I_0 \cup \dots
\cup I_{N-1}}$ such that $${\overline{ \left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}}} \geq
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot \prod_{i=N_0}^{N-1} (1-8\epsilon_i) -
\sum_{i=N_0}^{N-2} \epsilon_i,$$ and for any ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}^\star, {{\operatorname{\mathbf {s}}}}(h_0)}$ and $t \in
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}M_0}$, $M_0 \in [N_0,N)$, $${\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^t}} \geq
{\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}^t}}\cdot\prod_{i=M_0}^{N-1}
(1-4\epsilon_i).$$
[[Proof]{}. ]{}First notice that \[crucialgeneral\] follows from \[crucialgeneralmore\]. If ${\operatorname{\mathbf {F}}}^\star$ and $U^\star$ are as required, then for all ${\operatorname{\mathbf {G}}}\in
{\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}^\star, {{\operatorname{\mathbf {s}}}}(h_0)}$, $$\begin{gathered}
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}} \geq {\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}} \geq
\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N_0}}
{\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}^t}} \geq \\
\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N_0}} \left({\overline{
\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}^t}}\cdot\prod_{i=N_0}^{N-1}
(1-4\epsilon_i)\right)\geq \\
\prod_{i=N_0}^{N-1}
(1-4\epsilon_i) \cdot \left(\sum_{t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star {{\mathord{\restriction}}}N_0}}
{\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}^t}}\right) =
\prod_{i=N_0}^{N-1}
(1-4\epsilon_i) \cdot {\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}}}\geq \\
\prod_{i=N_0}^{N-1}
(1-4\epsilon_i)\cdot
\left({\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot \prod_{i=N_0}^{N-1} (1-8\epsilon_i) -
\sum_{i=N_0}^{N-2} \epsilon_i \right) \geq \\
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot\prod_{i=N_0}^{N-1} (1-8\epsilon_i)^2 -
\sum_{i=N_0}^{N-2} \epsilon_i.\end{gathered}$$
We will proceed by induction on $N$. If $N=N_0$ then the theorem is trivially true. Thus, suppose that the result holds for some $N \geq N_0$ and consider $N+1$. Let ${\operatorname{\mathbf {F}}}\in {\operatorname{\mathsf {F}}}^{N_0,N+1}_{\emptyset, h^0}$ and let $m$ be a distribution on $2^{I_0 \cup \dots \cup I_{N}}$ such that $${\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\geq \frac{2\sum_{i=N_0}^N
\epsilon_i}{\prod_{i=N_0}^N(1-\epsilon_i)}.$$
Recall that by \[new0a\], $$\overline{\overline{m^+_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\geq {\overline{m^+_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}={\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\geq 2\epsilon_N,$$ and apply \[comb\] with $m=m^+_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}$, $k^0 = \left|{{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{\operatorname{\mathbf {F}}}(N){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}\right|$, $k_0=h_0(N)$ to get $\tilde{f}_0
\in {\operatorname{\mathsf {F}}}^{I_N}_{{\operatorname{\mathbf {F}}}(N),k_0-\tilde{{{\operatorname{\mathbf {s}}}}}(k_0,N)}$ such that $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}_0,\tilde{{{\operatorname{\mathbf {s}}}}}(k_0,N)}\
{\overline{{m^+}_{({{{\operatorname{\mathbf{ C}}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} \geq
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot (1-2\epsilon_N).$$
Let $\{s_i: 1 \leq i \leq w_N({\operatorname{\mathbf {F}}})\}$ be an enumeration of $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}$. By induction, build a sequence $\{\tilde{f}_{i}: i \leq w_N({\operatorname{\mathbf {F}}})\}$ such that
1. $\tilde{f}_i \subseteq \tilde{f}_{i+1}$,
2. $k_0 - |{{\operatorname{\mathsf {dom}}}}(\tilde{f}_i)| \geq \tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+1)}(k_0,N)$,
3. for every $i \geq 1$ one of the following conditions holds:
1. $ \forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}_i,\tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+1)}(k_0,N) }\
{\overline{{m^{s_i}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} < \dfrac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}$,
2. for all $g \in {\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}_{i}, \tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+1)}}$, $${\overline{{m^{s_i}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown\tilde{f}_{i}}}}} \cdot
(1- 2\epsilon_N) \leq {\overline{{m^{s_i}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} \leq
{\overline{{m^{s_i}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown\tilde{f}_{i}}}}} \cdot
(1+ 2\epsilon_N).$$
Suppose that $\tilde{f}_i$ is given. If $$\forall g \in {\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}_i,\tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+3)}(k_0,N) }\
{\overline{{m^{s_{i+1}}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} < \frac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}$$ then put $\tilde{f}_{i+1}=\tilde{f}_{i}$.
Otherwise, let $\tilde{f}_{i+1}' \in
{\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}_i,\tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+3)}(k_0,N) }$ be chosen so that $${\overline{{m^{s_{i+1}}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown\tilde{f}_{i+1}'}}}} \geq
\frac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}.$$ In particular, by \[new1\], $\overline{\overline{{m^{s_{i+1}}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown\tilde{f}_{i+1}'}}}}\geq 2\epsilon_N$. Let $\tilde{k}=k_0 - |{{\operatorname{\mathsf {dom}}}}{\tilde{f}_{i+1}'}|$. By \[comb\], there exist $\tilde{f}_{i+1} \in
{\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}_{i+1}',\tilde{k} - \tilde{{{\operatorname{\mathbf {s}}}}}(\tilde{k},N)
}$ such that for all $g \in {\operatorname{\mathsf {F}}}^{I_N}_{\tilde{f}_{i+1}, \tilde{{{\operatorname{\mathbf {s}}}}}(\tilde{k},N)}$, $${\overline{{m^{s_{i+1}}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown\tilde{f}_{i+1}}}}} \cdot
(1+ 2\epsilon_N) \geq {\overline{{m^{s_{i+1}}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} \geq
{\overline{{m^{s_{i+1}}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown\tilde{f}_{i+1}}}}} \cdot
(1- 2\epsilon_N).$$ Note that $\tilde{k}\geq k_0 -
|{{\operatorname{\mathsf {dom}}}}{\tilde{f}_{i}}|-\tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+3)}(k_0,N)$. Using the induction hypothesis we get that $\tilde{k} \geq \tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+1)}(k_0,N)-\tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+3)}(k_0,N)
\geq \tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+2)}(k_0,N)$. It follows that $\tilde{{{\operatorname{\mathbf {s}}}}}(\tilde{k},N) \geq \tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+3)}(k_0,N)$ and ${k}_0 - |{{\operatorname{\mathsf {dom}}}}(\tilde{f}_{i+1})| \geq
\tilde{{{\operatorname{\mathbf {s}}}}}^{(2i+3)}(k_0,N)$, which finishes the induction.
Let ${\operatorname{\mathbf {F}}}^\star(N) =\tilde{f}_{w_N({\operatorname{\mathbf {F}}})}$. Since $w_N({\operatorname{\mathbf {F}}}) \leq \left|2^{I_0
\cup \dots \cup I_{N-1}}\right|$ it follows that ${{\operatorname{\mathbf {s}}}}(k_0,N) \leq \tilde{{{\operatorname{\mathbf {s}}}}}^{(2w_N({\operatorname{\mathbf {F}}})+1)}(k_0,N)$. Thus ${\operatorname{\mathbf {F}}}^\star(N) \in
{\operatorname{\mathsf {F}}}^{I_N}_{{\operatorname{\mathbf {F}}}(N),h_0(N)-{{\operatorname{\mathbf {s}}}}(k_0,N)}$.
Observe that ${\overline{{m^{s}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
g}}}}=m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}(s)$ for every $s \in
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}$. In particular, ${\overline{{m^{s}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {F}}}^\star(N)}}}}=m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s)$.
By the construction, for every $g \in
{\operatorname{\mathsf {F}}}^{I_N}_{{\operatorname{\mathbf {F}}}^\star(N),{{\operatorname{\mathbf {s}}}}(k_0,N)}$ and $s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}$, $$m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {F}}}^\star(N)}}(s)\cdot \frac{1-2\epsilon_N}{1+2\epsilon_N}\leq
{\overline{{m^{s}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} \leq
m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s)\cdot
\frac{1+2\epsilon_N}{1-2\epsilon_N}$$ or otherwise $${\overline{{m^{s}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}}} \leq
\frac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})} \quad \text{ and }\quad
{\overline{{m^{s}}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}}\leq
\frac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}.$$
Moreover, by the choice of $\tilde{f}_0$, for every $g \in
{\operatorname{\mathsf {F}}}^{I_N}_{{\operatorname{\mathbf {F}}}^\star(N),{{\operatorname{\mathbf {s}}}}(k_0,N)}$, $${\overline{{m^+}_{({{{\operatorname{\mathbf{ C}}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} \geq
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot(1-2\epsilon_N).$$
Even though we do not have much control over the values of $m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s)$ we can show that many of them are larger than $\dfrac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}$. Let $$U=\left\{s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}: m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s) \geq
\frac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}\right\}.$$ Note that for every $g \in
{\operatorname{\mathsf {F}}}^{I_N}_{{\operatorname{\mathbf {F}}}^\star(N),{{\operatorname{\mathbf {s}}}}(k_0,N)}$, $$\begin{gathered}
(1-2\epsilon_N) \cdot {\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}} \leq
{\overline{{m^+}_{({{{\operatorname{\mathbf{ C}}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} =
{\overline{{m}_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} ={\overline{{m^-}_{({{{\operatorname{\mathbf{ C}}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}}} \leq\\
\sum_{s \in U}
m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown g}}(s) + \sum_{s \in
({{{\operatorname{\mathbf{ C}}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N} \setminus U} m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
g}}(s) \leq \\
\frac{1+2\epsilon_N}{1-2\epsilon_N}\cdot\sum_{s \in U} m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {F}}}^\star(N)}}(s) + w_N({\operatorname{\mathbf {F}}})\cdot \frac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}\cdot
\frac{1+2\epsilon_N}{1-2\epsilon_N} \leq \\
2\epsilon_{N}\cdot \frac{1+2\epsilon_N}{1-2\epsilon_N} + \frac{1+2\epsilon_N}{1-2\epsilon_N}\cdot\sum_{s \in U} m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s). \end{gathered}$$ It follows that $$\sum_{s \in U} m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s)
\geq \frac{(1-2\epsilon_N)^2}{1+2\epsilon_N}\cdot {\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}} -
2\epsilon_N \geq (1-8\epsilon_N)\cdot {\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}} -
2\epsilon_{N}.$$ Define distribution $m^\star$ on $2^{I_0\cup \dots \cup I_{N-1}}$ as $$m^\star(s) = \left\{
\begin{array}{ll}
m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s) & \text{if }
m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s) \geq
\dfrac{2\epsilon_N}{w_N({\operatorname{\mathbf {F}}})}\\
0 & \text{otherwise}
\end{array}\right. .$$
Clearly, $${\overline{m^\star_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}}}}=\sum_{s \in U} m^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s) \geq\\
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot(1-8\epsilon_N) - 2\epsilon_{N}\geq
\frac{2 \sum_{i=N_0}^{N-1}
\epsilon_i}{\prod_{i=N_0}^{N-1} (1-8\epsilon_i)}.$$ Apply the induction hypothesis to $m^\star$, ${\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N$ and $h_0{{\mathord{\restriction}}}N$ to obtain ${\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N$ and $V^\star$ as in \[crucialgeneralmore\]. Let $$U^\star = \left\{s \in 2^{I_0 \cup \dots \cup I_N}: s {{\mathord{\restriction}}}I_0 \cup
\dots \cup I_{N-1} \in V^\star
\cap U\right\}.$$ It remains to check that ${\operatorname{\mathbf {F}}}^\star$ and $U^\star$ have the required properties. $$\begin{gathered}
{\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}}} = \sum_{s\in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star
{{\mathord{\restriction}}}N}} \left({m_{[U^\star]}}\right)^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}} =\sum_{s\in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star
{{\mathord{\restriction}}}N}} \left({m_{[V^\star]}}^\star\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}}(s) =\\
{\overline{ \left({m_{[V^\star]}}^\star\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}}}} \geq
{\overline{m^\star_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}}}}\cdot \prod_{i=N_0}^{N-1} (1-8\epsilon_i) -
\sum_{i=N_0}^{N-2} \epsilon_i \geq\\
\left({\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot(1-8\epsilon_N) - 2\epsilon_{N}\right)
\cdot \prod_{i=M_0}^{N-1} (1-8\epsilon_i) - \sum_{i=M_0}^{N-2}
\epsilon_i \geq\\
{\overline{m_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}}}\cdot \prod_{i=M_0}^{N}
(1-8\epsilon_i) - \sum_{i=M_0}^{N-1}
\epsilon_i,\end{gathered}$$ which gives the first condition.
To verify the second condition suppose that ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N_0,N+1}_{{\operatorname{\mathbf {F}}}^\star, {{\operatorname{\mathbf {s}}}}(h_0)}$, $M_0
\in [N_0,N]$ and $t \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}M_0}$. By the inductive hypothesis we have that $${\overline{\left({m_{[V^\star]}}^\star\right)^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}}} \geq
{\overline{\left({m_{[U^\star]}}^\star\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}}^t}}\cdot\prod_{i=M_0}^{N-1} (1-4\epsilon_i).$$
By \[new2\] and \[new3\], $$\begin{gathered}
{\overline{\left({m_{[U^\star]}}\right)^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}}} = \sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
\left({m_{[U^\star]}}\right)^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s)\cdot
\frac{{\overline{\left({m_{[U^\star]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {G}}}(N)}}}}}{{\overline{\left({m_{[U^\star]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}}}}=\\
\sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
\left({m_{[U^\star]}}\right)^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s)\cdot
\frac{{\overline{\left({m_{[U^\star]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {G}}}(N)}}}}}{{\overline{\left({m_{[U^\star]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {F}}}^\star(N)}}}}}=\\
\sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
\left({m_{[U^\star]}}\right)^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(t)\cdot
\frac{{\overline{\left({m_{[U]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {G}}}(N)}}}}}{{\overline{\left({m_{[U]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {F}}}^\star(N)}}}}}.\end{gathered}$$ Now $$\begin{gathered}
\sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
\left({m_{[U^\star]}}\right)^-_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N{}^\frown {\operatorname{\mathbf {F}}}^\star(N)}}(s)\cdot
\frac{{\overline{\left({m_{[U]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {G}}}(N)}}}}}{{\overline{\left({m_{[U]}}\right)^s_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N{}^\frown
{\operatorname{\mathbf {F}}}^\star(N)}}}}} \geq \\
\sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
\left({m_{[U^\star]}}^\star\right)^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}(s)\cdot
\frac{1-2\epsilon_N}{1+2\epsilon_N} =\sum_{t \subseteq s \in ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}
\left({m_{[V^\star]}}^\star\right)^t_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}N}}(s)\cdot
\frac{1-2\epsilon_N}{1+2\epsilon_N} \geq \\
\frac{1-2\epsilon_N}{1+2\epsilon_N}\cdot
{\overline{\left({m_{[V^\star]}}^\star\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}}^t}}\cdot\prod_{i=M_0}^{N-1} (1-4\epsilon_i)
\geq {\overline{\left({m_{[V^\star]}}^\star\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}}^t}}\cdot\prod_{i=M_0}^{N} (1-4\epsilon_i) =\\
{\overline{\left({m_{[U^\star]}}\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}^t}}\cdot\prod_{i=M_0}^{N} (1-4\epsilon_i),\end{gathered}$$ which concludes the proof.
Measures and norms {#six}
==================
In this section we will examine the consequences of the combinatorial results proved earlier on measures on $2^\omega $.
For $U \subseteq 2^I$, $[U]=\{x \in
2^\omega : x {{\mathord{\restriction}}}I \in U\}$.
If $p \subseteq 2^{<\omega}$ is a tree, $s \in p$, and $N \in \omega $, then
1. $[p]$ denotes the set of branches of $p$,
2. $p_s = \{t \in p: t \subseteq s \text{ or } s \subseteq t\}$,
3. $p^N= p {{\mathord{\restriction}}}(I_0 \cup \dots \cup I_{N-1})$.
We will identify product with concatenation, i.e., $(s,t) $ with $s^\frown t$, and similarly for infinite products. Most of the time we will also identify $p$ with $[p]$.
Let $\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}$ be the measure on $({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}$ defined as the product of counting measures on the coordinates. In other words, if $s \in 2^{I_k}$ then $$\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}([s])=\left\{
\begin{array}{ll}
\left|(C_k)^{{{\mathbf F}}(k)}\right|^{-1} & \text{if } s \in (C_k)^{{{\mathbf F}}(k)}\\
0 & \text{otherwise}
\end{array}\right. .$$
Given a perfect set $p \in {\operatorname{\mathsf {Perf}}}$, $$\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(p)=\lim_{N \rightarrow \infty} \frac{\left|p^N \cap
({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}{{\mathord{\restriction}}}N}\right|}{\left|({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}{{\mathord{\restriction}}}N} \right|}.$$ Note that $\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(p)=\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}p\cap ({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}{\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$.
For a function $f \in \omega^\omega $ define $\log_{{{\operatorname{\mathbf {s}}}}}(f) \in \omega^\omega $ as $$\log_{{{\operatorname{\mathbf {s}}}}}(f)(N)=\max\left\{k: {{\operatorname{\mathbf {s}}}}^{(k\cdot l_N)}(f(N),N)> 0\right\}.$$
For $h_1, h_s \in \omega^\omega $ define $h_1 \simeq h_2$ if $\log_{{{\operatorname{\mathbf {s}}}}}(h_1)=\log_{{{\operatorname{\mathbf {s}}}}}(h_2)$. Clearly $\simeq $ is an equivalence relation.
Let ${\mathcal X} $ be the collection of functions $f \in
\omega^\omega $ such that
1. $\lim_{m \rightarrow \infty}\log_{{{\operatorname{\mathbf {s}}}}}(f)(m)=\infty,$
2. $f = \min\{g: f \simeq g\}$.
For $f \in \omega^\omega $ define functions $\bar{f},f^- \in {\mathcal
X} $ as follows: $\overline{f} = {\mathcal X} \cap \{g: f \simeq g\},$ and $$f^-(n)=\left\{
\begin{array}{ll}
\min\{k: \log_{{{\operatorname{\mathbf {s}}}}}(k,n)=\log_{{{\operatorname{\mathbf {s}}}}}(f(n),n)-1\}& \text{if
}\log_{{{\operatorname{\mathbf {s}}}}}(f(n),n)>0\\
0 & \text{otherwise}
\end{array}\right. .$$
If $f \in {\mathcal X}$ and $ n \in \omega $ let $i_f(n)=\max\{k: \log_{{{\operatorname{\mathbf {s}}}}}(f)(k) \leq n\}.$
[**Remarks**]{}. Note that ${\mathcal X} \neq \emptyset$. By P5, $\overline{h} \in {\mathcal
X} $, where $h(k)=m_k$ for $k\in \omega$. Also, $\lim_{n \rightarrow \infty} i_f(n)=\infty$ for $f \in
{\mathcal X} $. The purpose of the restriction put on the set ${\mathcal X} $ is to make the mapping $f \mapsto \log_{{{\operatorname{\mathbf {s}}}}}(f)$ one-to-one. In practice, we will only use the fact that if $\log_{{{\operatorname{\mathbf {s}}}}}(f)(n)=0$ then $f(n)=0$.
For a perfect set $p \subseteq 2^\omega $, ${{\mathbf F}}\in {\operatorname{\mathsf {F}}}^\omega$, $N\in \omega $ and $h \in {\mathcal X} $, define $${[\![p, {{\mathbf F}}, h]\!]}_N=\inf \left\{\mu_{({{\operatorname{\mathbf{ C}}}})^{\mathbf G}}(p) : {\mathbf G}
\in {\operatorname{\mathsf {F}}}^{N,\omega}_{{{\mathbf F}},h}\right\}.$$ We will write ${[\![p, {{\mathbf F}}, h]\!]}$ instead of ${[\![p, {{\mathbf F}}, h]\!]}_0$.
The following easy lemma lists some basic properties of these notions.
\[easy2\]
1. The sequence $\left\{\dfrac{\left|p^N \cap
({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}{{\mathord{\restriction}}}N}\right|}{\left|({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}{{\mathord{\restriction}}}N} \right|}: k
\in \omega\right\}$ is monotonically decreasing for every $p \in {\operatorname{\mathsf {Perf}}}$,
2. ${[\![p, {{\mathbf F}}_1,h_1]\!]}_N \geq {[\![p, {{\mathbf F}}_2, h_2]\!]}_N$ if ${{\mathbf F}}_1 \in
{\operatorname{\mathsf {F}}}^{N,\omega}_{{{\mathbf F}}_2,h_2-h_1}$,
3. if $p_1 \cap p_2 = \emptyset$ then ${[\![p_1\cup p_2, {{\mathbf F}}, h]\!]}_N \geq {[\![p_1, {{\mathbf F}}, h]\!]}_N +{[\![p_2, {{\mathbf F}}, h]\!]}_N$.
[[Proof]{}. ]{}(1) is obvious, and (2) follows from \[easy1\].
\(3) Take $ \varepsilon>0$ and let ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N,\omega}_{{{\mathbf F}},h}$ be such that $${[\![p_1\cup p_2, {{\mathbf F}},
h]\!]}_N+\varepsilon \geq \mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p_1 \cup p_2).$$ Now $$\begin{gathered}
{[\![p_1\cup p_2, {{\mathbf F}},
h]\!]}_N+\varepsilon \geq \mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p_1 \cup p_2)\geq
\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p_1 \cup p_2) \geq \\
\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p_1)+\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p_2) \geq
{[\![p_1, {{\mathbf F}}, h]\!]}_N
+{[\![p_2, {{\mathbf F}}, h]\!]}_N.\end{gathered}$$ Thus ${[\![p_1\cup p_2, {{\mathbf F}},
h]\!]}_N+\varepsilon \geq {[\![p_1, {{\mathbf F}}, h]\!]}_N
+{[\![p_2, {{\mathbf F}}, h]\!]}_N$ and the inequality follows. ${\hspace{0.1in} \square \vspace{0.1in}}$
The following two theorems are the key to the whole construction.
\[main\] Suppose that $\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(p)>0$, $h \in {\mathcal X} $ and $0<\varepsilon <1$. Then there exist $p^\star \subseteq p$, $h^\star \in {\mathcal X} $, $N_0 \in \omega $ and ${{\mathbf F}}^\star \in
{\operatorname{\mathsf {F}}}^{N_0, \omega}_{{\operatorname{\mathbf {F}}}, h-h^\star}$ such that $$\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}^\star}}(p^\star) \geq
(1-\varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(p), \quad
{[\![p^\star,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]} \geq
(1-2\varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(p)$$ and $$\forall N \ \forall s \in (p^\star)^{N} \
{[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N>0.$$ Moreover, we can require that $h^\star(N)=\overline{{{\operatorname{\mathbf {s}}}}(h)}(N)=h^-(N)$ for $N \geq N_0$.
[[Proof]{}. ]{}Find $N_0\in \omega $ such that
1. $\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(p)>\dfrac{2\sum_{i=N_0}^\infty
\epsilon_i}{\prod_{i=N_0}^\infty (1-8\epsilon_i)},$
2. $\prod_{i=N_0}^\infty (1-4\epsilon_i) < \varepsilon $,
3. $\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(p)\cdot
{\prod_{i=N_0}^\infty (1-8\epsilon_i)} - \sum_{i=N_0}^\infty
\epsilon_i \geq (1-\varepsilon)\cdot \mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(p),$
4. $h(N)>0$ for $N\geq N_0$.
For $N \in \omega $ let $m^N$ be the distribution on $2^{I_0
\cup \dots \cup I_{N-1}}$ defined as $$m^N(s) = \left\{
\begin{array}{ll}
2^{-|\bigcup_{i <
N} I_i|}& \text{if }s \in p^N\\
0& \text{otherwise}
\end{array} \right. .$$ Note that ${\overline{m^N}}$ is the counting measure of $p^N$.
Use \[crucialgeneralmore\] to find ${\operatorname{\mathbf {F}}}^\star_N \in {\operatorname{\mathsf {F}}}^{N_0,N}_{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N, h{{\mathord{\restriction}}}N - {{\operatorname{\mathbf {s}}}}(h{{\mathord{\restriction}}}N)}$ and $U^\star_N \subseteq 2^{I_0 \cup \dots \cup I_{N-1}}$ such that $$\begin{gathered}
{\overline{\left({m_{[U^\star_N]}}^N\right)_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star_N}}}} =
\frac{\left|p^N \cap U^\star_N \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star_N}\right|}{\left| ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star_N}\right|}
\geq \frac{\left|p^N \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}\right|}{\left| ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N}\right|}\cdot
{\prod_{i=N_0}^\infty (1-8\epsilon_i)} - \sum_{i=N_0}^\infty \epsilon_i,\end{gathered}$$ and for $M_0 \in
[N_0,N)$, $s \in p_s^N {{\mathord{\restriction}}}I_0\cup \dots \cup I_{M_0-1}$ and ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{M_0,N}_{{\operatorname{\mathbf {F}}}^\star_N, {{\operatorname{\mathbf {s}}}}(h{{\mathord{\restriction}}}N)}$, $$\frac{\left|p_s^N \cap U^\star_N \cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}} \right|}
{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}
\geq \frac{\left|p_s^N \cap U^\star_N \cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star} \right|}
{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}\right|}\cdot {\prod_{i=M_0}^\infty (1-4\epsilon_i)}.$$ By compactness, there exist ${\operatorname{\mathbf {F}}}^\star \in {\operatorname{\mathsf {F}}}^\omega $ and $U^\star
\subseteq 2^{<\omega}$ such that $$\forall N\ \exists M \geq N \ {\mathopen\ifcase2{}\oo\or
\big(\or\Big(\else\oo\fi}{\operatorname{\mathbf {F}}}^\star {{\mathord{\restriction}}}N =
{\operatorname{\mathbf {F}}}^\star_M {{\mathord{\restriction}}}N \ \&\ (U^\star)^N =
(U^\star_M)^N{\mathclose\ifcase2{}\oo\or
\big)\or\Big)\else\oo\fi}.$$
Put $p^\star = p \cap U^\star$ and note that, by \[crucialgeneralmore\], for every $N \geq N_0$ there exists $M \geq N$ such that $$\begin{gathered}
\frac{\left|(p^\star)^N \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N} \right|}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}|} =
\frac{\left|(p^M \cap U^\star_M )^N \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N} \right|}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}|} =
\frac{\left|(p^M \cap U^\star_M)^N\cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}_M^\star{{\mathord{\restriction}}}N} \right|}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}_M^\star{{\mathord{\restriction}}}N}|}\geq\\
\frac{\left|p^M \cap U^\star_M \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star_M} \right|}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star_M}|} \geq
\frac{\left|p^M \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}M}\right|}{\left| ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}M}\right|}\cdot
{\prod_{i=N_0}^\infty (1-8\epsilon_i)} - \sum_{i=N_0}^\infty
\epsilon_i \geq\\
\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(p)\cdot
{\prod_{i=N_0}^\infty (1-8\epsilon_i)} - \sum_{i=N_0}^\infty
\epsilon_i \geq (1-\varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(p).\end{gathered}$$ It follows that $$\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}(p^\star)=\lim_{N \rightarrow \infty}
\frac{\left|(p^\star)^N \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N} \right|}{|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N}|} \geq
(1-\varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(p).$$
Suppose that $s \in (p^\star)^{M_0}$ for some $M_0\geq N_0$. As above, for $N\geq M_0$ and ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{M_0,N}_{{\operatorname{\mathbf {F}}}^\star_N, {{\operatorname{\mathbf {s}}}}(h{{\mathord{\restriction}}}N)}$, the inequality $$\frac{\left|p_s^N \cap U^\star_N \cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}} \right|}
{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}
\geq \frac{\left|p_s^N \cap U^\star_N \cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star} \right|}
{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}\right|}\cdot {\prod_{i=M_0}^\infty (1-4\epsilon_i)},$$ translates to $$\begin{gathered}
\forall {\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{M_0, \omega}_{{\operatorname{\mathbf {F}}}^\star, {{\operatorname{\mathbf {s}}}}(h)} \
\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p^\star_s) \geq \mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}^\star}}(p^\star_s)
\cdot \prod_{i=M_0}^\infty (1-4\epsilon_i)\geq \\
\frac{1}{|(C_{M_0})^{{\operatorname{\mathbf {F}}}^\star(M_0)}|}
\cdot \prod_{i=M_0}^\infty (1-4\epsilon_i)>0.\end{gathered}$$ It follows that if $s \in (p^\star)^{M_0}$, $M_0 \geq N_0$ then for all $ {\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{M_0,
\omega}_{{\operatorname{\mathbf {F}}}^\star, {{\operatorname{\mathbf {s}}}}(h)} $, $$\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p^\star_s) \geq
(1-\varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}^\star}}(p^\star_s)>0.$$ Define $$h^\star(N)=\left\{
\begin{array}{ll}
\overline{{{\operatorname{\mathbf {s}}}}(h)}(N)& \text{if } N \geq N_0\\
0& \text{otherwise}
\end{array}\right. \text{ for } N \in \omega .$$ Suppose that $s \in (p^\star)^N$. If $N \geq N_0$ then the above estimates show that $${[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N\geq
(1-\varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}^\star}}(p^\star_s)>0.$$ If $N<N_0$ then by \[easy2\](3), $$\begin{gathered}
{[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N \geq \sum_{s \subseteq t \in
(p^\star)^{N_0}} {[\![p^\star_t,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N = \\
\frac{w_{N}({\operatorname{\mathbf {F}}}^\star)}{w_{N_0}({\operatorname{\mathbf {F}}}^\star)}\cdot\sum_{s \subseteq t \in
(p^\star)^{N_0}} {[\![p^\star_t,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_{N_0}
>0.\end{gathered}$$ Finally note that for ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{\omega}_{{\operatorname{\mathbf {F}}}^\star, h^\star}$, $$\begin{gathered}
\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p^\star) = \sum_{t \in
(p^\star)^{N_0}} \mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p^\star_s) \geq \sum_{t \in
(p^\star)^{N_0}} \mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}(p^\star_s) \cdot
(1-\varepsilon) =\\
(1-\varepsilon)\cdot \mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star}}(p^\star) \geq
(1-\varepsilon)^2\cdot \mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(p) \geq (1-2 \varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(p).\end{gathered}$$ It follows that $${[\![p^\star,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]} \geq (1-2 \varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}(p).~{\hspace{0.1in} \square \vspace{0.1in}}$$
\[main1\] Suppose that $M_0 \in \omega $, $ \varepsilon<1$ and $\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(A)=1$. Let $p \subseteq 2^\omega $ and $h \in {\mathcal X} $ be such that $$\forall N \ \forall s \in (p)^{N} \ {[\![p,{\operatorname{\mathbf {F}}},h]\!]}_N>0.$$
There exist $p^\star$, $h^\star \in {\mathcal X}$ and ${{\mathbf F}}^\star \in
{\operatorname{\mathsf {F}}}^{N_0, \omega}_{{\operatorname{\mathbf {F}}}, h-h^\star}$ such that
1. $p^\star \subseteq p \cap A$,
2. $h^\star {{\mathord{\restriction}}}M_0=h{{\mathord{\restriction}}}M_0$,
3. $\forall N\geq M_0 \ \log_{{{\operatorname{\mathbf {s}}}}}(h^\star)(N) = \log_{{{\operatorname{\mathbf {s}}}}}(h)(N)-1$,
4. $\forall s \in p^\star \ {[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N>0$,
5. $\forall s \in (p)^{M_0} \ {[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_{M_0} \geq
(1-4\varepsilon)\cdot {[\![p_s,{\operatorname{\mathbf {F}}},h]\!]}_{M_0}$.
[[Proof]{}. ]{} Let $\alpha = \min\left\{{[\![p_s,{\operatorname{\mathbf {F}}},h]\!]}_{M_0}: s \in (p)^{M_0}\right\}.$ Fix $ \varepsilon >0$ and for every $s \in (p)^{M_0}$ find $N_0^s\geq M_0$ as in \[main\]. Let $N_0\geq \max\left\{N_0^s: s \in (p)^{M_0}\right\}$ be such that $\log_{{{\operatorname{\mathbf {s}}}}}(h)(N_0)>0$.
Fix an enumeration $\{s_i: 0<i \leq \ell\}$ of $(p)^{M_0}$, and define sequences $\{{\operatorname{\mathbf {F}}}_i,h_i : i \leq \ell\}$ and $\{p^\star_i: 0<i \leq \ell\}$ such that
1. ${\operatorname{\mathbf {F}}}_0={\operatorname{\mathbf {F}}}$, $h_0=h$,
2. $h_i \in {\mathcal X} $ for $i \leq \ell$,
3. $p^\star_i \subseteq p_{s_i} \cap A$,
4. ${{\mathbf F}}_{i+1} \in {\operatorname{\mathsf {F}}}^{{N_0},\omega}_{{{\mathbf F}}_i,
h_{i}-{{\operatorname{\mathbf {s}}}}(h_i)}$,
5. $h_{i+1}(N)={{\operatorname{\mathbf {s}}}}(h_i)(N)$ for $N \geq N_0$, $i < \ell$,
6. $\forall i \leq \ell \ \forall N<N_0 \ h_i(N)=0$,
7. ${[\![p^\star_i,{\operatorname{\mathbf {F}}}_i,h_i]\!]}_{M_0} \geq
(1-4\varepsilon)\cdot\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}_i}}(p_{s_i})$,
8. $\forall N \ \forall s \in (p^\star_i)^{N} \ {[\![p^\star_i,{\operatorname{\mathbf {F}}}_i,h_i]\!]}_N>0.$
Suppose that ${\operatorname{\mathbf {F}}}_i^\star$, $ h_i^\star$ are given for some $i<\ell$. Find $q_{i+1} \subseteq p_{s_{i+1}} \cap A$ such that $\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}_i}}(q_{i+1}) \geq (1-\varepsilon) \mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}_i}}(p_{s_i}).$ Let $p_{i+1}$, ${\operatorname{\mathbf {F}}}_{i+1}$ and $ h_{i+1}$ be obtained by applying \[main\] to $q_{i+1},\
{\operatorname{\mathbf {F}}}_i$ and $h_i$. After $\ell$ steps we have constructed functions ${\operatorname{\mathbf {F}}}_\ell$, $h_\ell$ and a set $p^\star=\bigcup_{i \leq \ell} p_i$. Functions ${\operatorname{\mathbf {F}}}_\ell$ and $\overline{h_\ell}=h^-$ will define walues of ${\operatorname{\mathbf {F}}}^\star$ and $h^\star$ for $N \geq N_0$.
Define for $N \in \omega $, $$h^\star(N)=\left\{
\begin{array}{ll}
h(N)&\text{if } N<M_0\\
h^-(N)& \text{if } M_0\leq N\\
\end{array}\right.$$ and ${\operatorname{\mathbf {F}}}^\star(N)={\operatorname{\mathbf {F}}}_\ell(N)$ for $N\geq N_0$. It remains to define the values of ${\operatorname{\mathbf {F}}}^\star(N)$ for $N<N_0$.
Define ${\operatorname{\mathbf {F}}}^\star {{\mathord{\restriction}}}N_0$ by the following requirements:
1. ${\operatorname{\mathbf {F}}}^\star {{\mathord{\restriction}}}M_0={\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}M_0$,
2. ${\operatorname{\mathbf {F}}}^\star \in {\operatorname{\mathsf {F}}}^{M_0,\omega}_{{\operatorname{\mathbf {F}}}, h-h^-}$,
3. for $N<N_0$ and $s \in (p^\star)^N$, $$p^\star_s
\cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N_0}
\neq \emptyset \rightarrow \left(\forall {\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N,
N_0}_{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N_0, h^\star{{\mathord{\restriction}}}N_0} \
p^\star_s \cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\neq \emptyset\right).$$
More precisely, by induction on $N \in [M_0, N_0)$ define sequences $\left\{{\operatorname{\mathbf {F}}}^{N}_i:~i~\leq~v_{N}\right\}$ and $\left\{h^{N}_i: i
\leq v_N\right\}$ such that
1. $h_0^{M_0}=h{{\mathord{\restriction}}}N_0$, ${\operatorname{\mathbf {F}}}^{M_0}_0= {\operatorname{\mathbf {F}}}{{\mathord{\restriction}}}N_0$, ${\operatorname{\mathbf {F}}}^{N+1}_{0}={\operatorname{\mathbf {F}}}^N_{v_N}$ and $h^{N+1}_0=h^N_{v_N}$ for $N \geq M_0$,
2. $\forall N<N_0 \ \forall i \leq v_N \ h^{N}_i {{\mathord{\restriction}}}N=h^{N}_0 {{\mathord{\restriction}}}N$,
3. $h^{N}_{i+1}=h^{N}_0 {{\mathord{\restriction}}}N{}^\frown {{\operatorname{\mathbf {s}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}h_i{{\mathord{\restriction}}}[N,N_0){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi} $ for $ i \leq v_{N}$,
4. ${\operatorname{\mathbf {F}}}^{N}_{i+1} \in {\operatorname{\mathsf {F}}}^{N,N_0}_{{\operatorname{\mathbf {F}}}^{N}_{i},h_i^{N}-h_{i+1}^{N}}$,
5. if $s$ is the $i$’th element of $(p)^{N}$ then exactly one of the following two cases holds:
1. $ \forall {\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N,N_0}_{{\operatorname{\mathbf {F}}}^{N}_{i},h^{N}_{i}} \ ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}} \cap
(p_{s})^{N_0} \neq \emptyset$,
2. $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}^{N}_{i}} \cap
(p_{s})^{N_0} = \emptyset$.
The construction is straightforward. If case (5a) holds, then we define ${\operatorname{\mathbf {F}}}^N_{i+1}={\operatorname{\mathbf {F}}}^N_{i}$, otherwise there exists ${\operatorname{\mathbf {G}}}\in
{\operatorname{\mathsf {F}}}^{N,N_0}_{{\operatorname{\mathbf {F}}}^{N}_{i},h_{i}^{N}}$ such that $({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}} \cap
(p_{s})^{N_0} = \emptyset$, and we put ${\operatorname{\mathbf {F}}}^{N}_{i+1}={\operatorname{\mathbf {G}}}$.
Observe that for $N \geq M_0$, $h^\star(N)=h^-(N)={{\operatorname{\mathbf {s}}}}^{(l_N)}(h)(N)=h^{N+1}_{v_N}(N)$. Therefore we can carry out this construction provided that $\log_{{{\operatorname{\mathbf {s}}}}}(h)(N)>0$. However, by the choice of $ {\mathcal X} $, if $\log_{{{\operatorname{\mathbf {s}}}}}(h)(N)=0$ then $h(N)=0$ and the required condition is automatically met.
Finally let $${\operatorname{\mathbf {F}}}^\star(N)=\left\{
\begin{array}{ll}
{\operatorname{\mathbf {F}}}(N)&\text{if } N<M_0\\
{\operatorname{\mathbf {F}}}^{N}(N)& \text{if } M_0\leq N<N_0\\
{\operatorname{\mathbf {F}}}_\ell(N)&\text{if } N\geq N_0
\end{array}\right. .$$
We will show that $p^\star$, ${\operatorname{\mathbf {F}}}^\star$ and $h^\star$ have the required properties. Conditions (1)–(3) of \[main1\] are obvious.
To check (5) consider $s \in (p^\star)^{M_0}$. By the choice of $N_0$, $p^\star$ and ${\operatorname{\mathbf {F}}}_\ell$ we have $$\begin{gathered}
{[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_{M_0} \geq \min\left\{{\operatorname{\mathbf {G}}}\in
{\operatorname{\mathsf {F}}}^{M_0,N_0}_{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N_0, h^\star{{\mathord{\restriction}}}N_0}:
\frac{\left|(p^\star_s)^{N_0} \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}{\left({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}\right\}\cdot (1-4
\varepsilon ) = \\
\min\left\{{\operatorname{\mathbf {G}}}\in
{\operatorname{\mathsf {F}}}^{M_0,N_0}_{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N_0, h^\star{{\mathord{\restriction}}}N_0}:
\frac{\left|(p_s)^{N_0} \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}{\left({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}\right\}\cdot (1-4
\varepsilon ) \geq \\
{[\![p_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_{M_0}\cdot (1-4 \varepsilon ) \geq
{[\![p_s,{\operatorname{\mathbf {F}}},h]\!]}_{M_0}\cdot (1-4 \varepsilon ).\end{gathered}$$
To verify (4) we have to show that ${[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N>0$ for $s \in (p^\star)^N$. If $N \geq N_0$ it follows from the construction of ${\operatorname{\mathbf {F}}}_\ell$. If $N<N_0$ then $${[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N \geq (1-4 \varepsilon )\cdot
\min\left\{\frac{\left|(p^\star_s)^{N_0} \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}: {\operatorname{\mathbf {G}}}\in
{\operatorname{\mathsf {F}}}^{M_0,N_0}_{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N_0,
h^\star{{\mathord{\restriction}}}N_0}\right\}.$$ By the choice of ${\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N_0$, for all ${\operatorname{\mathbf {G}}}\in
{\operatorname{\mathsf {F}}}^{M_0,N_0}_{{\operatorname{\mathbf {F}}}^\star{{\mathord{\restriction}}}N_0,
h^\star{{\mathord{\restriction}}}N_0}$, $$\frac{\left|(p^\star_s)^{N_0} \cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|}{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}\right|} \neq 0.$$ It follows that ${[\![p^\star_s,{\operatorname{\mathbf {F}}}^\star,h^\star]\!]}_N>0$. ${\hspace{0.1in} \square \vspace{0.1in}}$
Definition of $ {\mathcal P} $ {#seven}
==============================
In this section we will define a partial order ${\mathcal P} $ having properties A0 – A2 from \[newtrick\]. This will conclude the proof of \[biggie\].
We start by defining a partial ordering $ {\mathcal Q} $ that will be used in the definition of $ {\mathcal P} $.
Let $ {\mathcal Q} $ be the following partial order:
$(p,{{\mathbf F}}, h) \in {\mathcal Q} $ if
1. $p \in {\operatorname{\mathsf {Perf}}}$, ${{\mathbf F}}\in {\operatorname{\mathsf {F}}}^\omega$, $h \in {\mathcal X} $,
2. $\left|{{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{{\mathbf F}}(k){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}\right|+h(k)\leq m_k$ for every $k$,
3. $p \subseteq ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}$,
4. $\forall s \in p^N \ {[\![p_s,{\operatorname{\mathbf {F}}},h]\!]}_N>0$.
For $(p^1,{{\mathbf F}}_1, h_1), (p^2,{{\mathbf F}}_2, h_2) \in {\mathcal Q}$ define $(p^1,{{\mathbf F}}_1, h_1) \geq (p^2,{{\mathbf F}}_2, h_2)$ if
1. $p^1 \subseteq p^2$,
2. ${{\mathbf F}}_1 \in {\operatorname{\mathsf {F}}}^\omega_{{{\mathbf F}}_2,h_2-h_1}$.
To see that $ {\mathcal Q} $ has the fusion property we define $\geq_n$:
For $ n>0$ define $(p^1,{{\mathbf F}}_1, h_1) \geq_n (p^2,{{\mathbf F}}_2, h_2)$ if
1. $(p^1,{{\mathbf F}}_1, h_1) \geq (p^2,{{\mathbf F}}_2, h_2)$,
2. $\forall s \in (p^2)^{n^\star}\
{[\![p^1_s, {{\mathbf F}}_1, h_1]\!]}_{n^\star} \geq (1-2^{-n-1})\cdot {[\![p^2_s, {{\mathbf F}}_2,
h_2]\!]}_{n^\star}$,
3. $h_1 {{\mathord{\restriction}}}n^\star = h_2 {{\mathord{\restriction}}}n^\star$,
4. ${{\mathbf F}}_1 {{\mathord{\restriction}}}n^\star = {{\mathbf F}}_2 {{\mathord{\restriction}}}n^\star$,
where $n^\star = i_{h_1}(n).$
Note that (2) implies that $(p^1)^{n^\star}=(p^2)^{n^\star}$.
\[qfp\] ${\mathcal Q} $ has the fusion property.
[[Proof]{}. ]{}Suppose that $\left\{(p^k,{{\mathbf F}}_k, h_k): k \in \omega \right\}$ is a sequence of conditions such that $(p^{k+1},{{\mathbf F}}_{k+1}, h_{k+1})\geq_{k+1}(p^k,{{\mathbf F}}_k, h_k)$ for each $k$. Let $n^\star(k)=i_{h_{k+1}}(k)$. Note that $\lim_{k \rightarrow \infty} n^\star(k)=\infty$. Define
1. $h=\bigcup_{k \in \omega } h_k {{\mathord{\restriction}}}n^\star(k)$,
2. ${{\mathbf F}}= \bigcup_{k \in \omega } {{\mathbf F}}_k {{\mathord{\restriction}}}n^\star(k)$,
3. $p= \bigcup_{k \in \omega } (p^k)^{n^\star(k)}.$
Observe that $h$, ${{\mathbf F}}$ and $p$ are well defined.
Suppose that $s \in p^{n^\star(k_0)}$, ${\operatorname{\mathbf {G}}}\in {\operatorname{\mathsf {F}}}^{N,\omega}_{{\operatorname{\mathbf {F}}},h}$ and $k
\geq k_0 $, and note that $$\frac{\left|(p_s)^{n^\star(k)}
\cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}n^\star(k)}\right|}{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}n^\star(k)}\right|}=
\dfrac{\left|(p_s^k)^{n^\star(k)}
\cap
({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}n^\star(k)}\right|}{\left|({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}{{\mathord{\restriction}}}n^\star(k)}\right|}
\geq {[\![p^k_s,{\operatorname{\mathbf {F}}}_k,h_k]\!]}.$$ Therefore $\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}}}(p_s) \geq \inf_{k}{[\![p^k_s,{\operatorname{\mathbf {F}}}_k,h_k]\!]}$. Hence, $$\begin{gathered}
{[\![p_s,{\operatorname{\mathbf {F}}},h]\!]}_{n^\star(k_0)} \geq {[\![p^{k_0}_s,{\operatorname{\mathbf {F}}}_{k_0},h_{k_0}]\!]}_{n^\star(k_0)}\cdot
\prod_{k> k_0}\left(1-\frac{1}{2^{k+1}}\right) \geq\\
\left(1-\frac{1}{2^{k_0+1}}\right)\cdot
{[\![p^{k_0}_s,{\operatorname{\mathbf {F}}}_{k_0},h_{k_0}]\!]}_{n^\star(k_0)}>0.\end{gathered}$$ The same computation shows that $(p,{\operatorname{\mathbf {F}}},h) \geq_k (p^k,{\operatorname{\mathbf {F}}}_k,h_k)$. ${\hspace{0.1in} \square \vspace{0.1in}}$
\[main2\] Suppose that $(p,{{\mathbf F}},h) \in {\mathcal Q}$.
If $q \subseteq p$ and $\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(q)>0$ then there exist $q^\star \subseteq q$, ${\operatorname{\mathbf {F}}}^\star$ and $ h^\star\in
{\mathcal X} $ such that $(q^\star, {\operatorname{\mathbf {F}}}^\star, h^\star) \in {\mathcal Q} $ and $(q^\star, {\operatorname{\mathbf {F}}}^\star, h^\star) \geq (p,{{\mathbf F}},h)$.
If $n \in \omega $ and $A \subseteq p$ is such that $\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}(A)=1$ then there exist $q^\star \subseteq p\cap
A$, ${\operatorname{\mathbf {F}}}^\star$ and $ h^\star\in
{\mathcal X} $ such that $(q^\star, {\operatorname{\mathbf {F}}}^\star, h^\star) \in {\mathcal Q} $ and $(q^\star, {\operatorname{\mathbf {F}}}^\star, h^\star) \geq_n (p,{{\mathbf F}},h)$.
[[Proof]{}. ]{}The first part follows from \[main\] and the second from \[main1\]. ${\hspace{0.1in} \square \vspace{0.1in}}$
The following theorem shows that $ {\mathcal Q} $ satisfies condition A2 defined in section \[out\].
\[qa2\] For every $(p, {\operatorname{\mathbf {F}}}, h) \in {\mathcal Q} $, $ n \in \omega$, $X \in
[2^\omega]^{\leq \boldsymbol\aleph_0} $, and ${{\mathbf t}}\in {\operatorname{\mathsf {Perf}}}$ such that $\mu({{\mathbf t}})>0$, $$\mu{\mathopen\ifcase2{}\oo\or
\big(\or\Big(\else\oo\fi}{\mathopen\ifcase1{}\oo\or
\big\{\or\Big\{\else\oo\fi}z\in 2^\omega : \exists (q, {\operatorname{\mathbf {G}}},
f) \geq_n (p, {\operatorname{\mathbf {F}}}, h)\
X \cup (q +
{{\mathbb Q}})
\subseteq {{\mathbf t}}+{{\mathbb Q}}+z{\mathclose\ifcase1{}\oo\or
\big\}\or\Big\}\else\oo\fi}{\mathclose\ifcase2{}\oo\or
\big)\or\Big)\else\oo\fi}=1.$$
[[Proof]{}. ]{} Suppose that $(p,{{\mathbf F}},h) \in {\mathcal Q} $ and ${{\mathbf t}}$ is a perfect set of positive measure.
We will need the following observation:
$$\mu\left(\left\{z\in 2^\omega : \mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}p \cap
({{\mathbf t}}+z){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}>0\right\}\right)>0.$$
[[Proof]{}. ]{}Consider the space $ p \times 2^\omega $ equipped with the product measure $(\mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}{{\mathord{\restriction}}}p)~\times~\mu$. Let $Z = \{(x,z)\in p\times 2^\omega : z \in {{\mathbf t}}+x\}$. Note that $\mu{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(Z)_x{\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}=\mu({{\mathbf t}}+x)=\mu({{\mathbf t}})>0$ for each $x$. By the Fubini theorem $$\left\{z: \mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(Z)^z{\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}>0\right\}$$ has positive measure. But $$(Z)^z = \{x \in p: z \in {{\mathbf t}}+x\}=\{x\in p: x \in {{\mathbf t}}+z\} = p \cap ({{\mathbf t}}+z).~{\hspace{0.1in} \square \vspace{0.1in}}$$
Let $X
\subseteq 2^\omega $ be a countable set. Put $Z_X = \{z \in 2^\omega : X \subseteq {{\mathbf t}}+{{\mathbb Q}}+z\}$. Note that $Z_X$ has measure one. Thus, without loss of generality, we can assume that $X= \emptyset$.
For each $s \in p$ let $$Z_s = \left\{z \in 2^\omega : \mu_{({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}p_s
\cap ({{\mathbf t}}+z){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}>0\right\}.$$ By the lemma, $\mu(Z_s)>0$ for each $s$. Let $Z = \bigcap_{s \in p} (Z_s+{{\mathbb Q}})$. This is the measure one set we are looking for.
Fix $z \in Z$ and $n \in \omega $. Note that $\mu_{({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {F}}}}}({{\mathbf t}}+{{\mathbb Q}}+z)=1$ and apply \[main2\]. ${\hspace{0.1in} \square \vspace{0.1in}}$
Let $ {\mathcal P} \subseteq {\mathcal Q} \times {\mathcal Q} $ be the collection of elements ${\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^1,{{\mathbf F}}_1, h), (p^2,{{\mathbf F}}_2, h){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$ such that
1. $ \forall k \ {{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{{\mathbf F}}_1(k){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}={{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{{\mathbf F}}_2(k){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$,
2. $ \forall k \ \forall s \in {{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{{\mathbf F}}_1(k){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi} \ {\mathopen\ifcase2{}\oo\or
\big(\or\Big(\else\oo\fi}{{\mathbf F}}_1(k)(s)=1
\text{ or } {{\mathbf F}}_2(k)(s)=1{\mathclose\ifcase2{}\oo\or
\big)\or\Big)\else\oo\fi}$.
For ${\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^1,{{\mathbf F}}_1, h_1), (q_1,{\operatorname{\mathbf {G}}}_1, h_1){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi},
{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^2,{{\mathbf F}}_2, h_2), (q_2,{\operatorname{\mathbf {G}}}_2, h_2){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi} \in {\mathcal P} $ and $n \in \omega $ define
${\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^1,{{\mathbf F}}_1, h_1), (q_1,{\operatorname{\mathbf {G}}}_1, h_1){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}\geq_n
{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^2,{{\mathbf F}}_2, h_2), (q_2,{\operatorname{\mathbf {G}}}_2, h_2){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$ if
$(p^1,{{\mathbf F}}_1, h_1) \geq_n (p^2,{{\mathbf F}}_2, h_2)$ and $(q_1,{\operatorname{\mathbf {G}}}_1, h_1) \geq_n (q_2,{\operatorname{\mathbf {G}}}_2, h_2)$.
Strictly speaking, the partial order used in the proof of \[newtrick\] was a subset of ${\operatorname{\mathsf {Perf}}}\times {\operatorname{\mathsf {Perf}}}$ while $ {\mathcal
P} $ defined above has more complicated structure. Nevertheless it is easy to see that it makes no difference in the proof of \[newtrick\] as conditions A1 and A2 refer only to the first coordinate of ${\mathcal P} $.
$ {\mathcal P} $ has the fusion property.
[[Proof]{}. ]{}Follows immediately from the definition of $ {\mathcal P} $ and \[qfp\]. ${\hspace{0.1in} \square \vspace{0.1in}}$
Next we show that ${\mathcal P} $ satisfies A1.
For every $ {{\mathbf p}}\in
{\mathcal P} $, $n \in \omega $ and $z \in 2^\omega $ there exists ${{\mathbf q}}\geq_n
{{\mathbf p}}$ such that $q_1 \subseteq H+z$ or $q_2 \subseteq H+z$.
[[Proof]{}. ]{}Suppose that ${\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^1,{{\mathbf F}}_1, h), (p^2,{{\mathbf F}}_2, h){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi} \in {\mathcal P} $ and $z \in 2^\omega $.
[Case 1]{}. There exist infinitely many $k$ such that $z {{\mathord{\restriction}}}I_k
\in {{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{{\mathbf F}}_1(k){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$.
It follows from the definition of $ {\mathcal P} $ that in this case there exists $i \in \{1,2\}$ and infinitely many $k$ such that ${{\mathbf F}}_i(k)(z {{\mathord{\restriction}}}I_k)=1$. In particular, since $p^i \subseteq
({{\operatorname{\mathbf{ C}}}})^{{{\mathbf F}}_i}$, for every $x \in p^i$, $$\exists^\infty k \ x {{\mathord{\restriction}}}I_k \not\in C_k+z {{\mathord{\restriction}}}I_k.$$ Thus, $p^i \subseteq H+z$.
[Case 2]{}. $z {{\mathord{\restriction}}}I_k \in {{\operatorname{\mathsf {dom}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}{{\mathbf F}}_1(k){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$ for finitely many $k$.
Let $n^\star =i_h(n)$. Define for $ k \in \omega $, and $i=1,2$ $${\operatorname{\mathbf {G}}}_i(k)=\left\{
\begin{array}{ll}
{{\mathbf F}}_i(k)& \text{if } k \leq n^\star\\
{{\mathbf F}}_i(k) \cup (z {{\mathord{\restriction}}}I_k, 0) & \text{if } k > n^\star
\end{array}\right. ,$$ $q_i = p^i \cap ({{\operatorname{\mathbf{ C}}}})^{{\operatorname{\mathbf {G}}}_i}$ and $$f(k)=\left\{
\begin{array}{ll}
h(k)& \text{if } k \leq n^\star\\
{{\operatorname{\mathbf {s}}}}{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}h(k),k{\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi} & \text{if } k > n^\star
\end{array}\right. .$$
Clearly ${\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(q_1,{\operatorname{\mathbf {G}}}_1, f), (q_2,{\operatorname{\mathbf {G}}}_2, f){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}\geq_n
{\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^1,{{\mathbf F}}_1, h), (p^2,{{\mathbf F}}_2, h){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$ and the same argument as in the first case shows that it has the required properties. ${\hspace{0.1in} \square \vspace{0.1in}}$
Next we show that $ {\mathcal P} $ satisfies A2.
For every ${{\mathbf p}}\in {\mathcal P} $, $ n \in \omega
$, $X \in
[2^\omega]^{\leq \boldsymbol\aleph_0} $, $i=1,2$ and ${{\mathbf t}}\in {\operatorname{\mathsf {Perf}}}$ such that $\mu({{\mathbf t}})>0$, $$\mu{\mathopen\ifcase2{}\oo\or
\big(\or\Big(\else\oo\fi}{\mathopen\ifcase1{}\oo\or
\big\{\or\Big\{\else\oo\fi}z\in 2^\omega : \exists {{\mathbf q}}\geq_n {{\mathbf p}}\
X \cup (q_i +
{{\mathbb Q}})
\subseteq {{\mathbf t}}+{{\mathbb Q}}+z{\mathclose\ifcase1{}\oo\or
\big\}\or\Big\}\else\oo\fi}{\mathclose\ifcase2{}\oo\or
\big)\or\Big)\else\oo\fi}=1.$$
[[Proof]{}. ]{} Suppose that ${\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(p^1, {{\mathbf F}}_1, h), (p^2, {{\mathbf F}}_2, h){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi} \in {\mathcal P}
$, $n \in \omega $, $X
\subseteq 2^\omega $ is a countable set, and ${{\mathbf t}}$ is a perfect set of positive measure. Without loss of generality we can assume that $i=1$. Consider the set $$Z={\mathopen\ifcase1{}\oo\or
\big\{\or\Big\{\else\oo\fi}z\in 2^\omega : \exists (q, {\operatorname{\mathbf {G}}},
f) \geq_n (p^1, {\operatorname{\mathbf {F}}}_1, h)\
X \cup (q +
{{\mathbb Q}})
\subseteq {{\mathbf t}}+{{\mathbb Q}}+z{\mathclose\ifcase1{}\oo\or
\big\}\or\Big\}\else\oo\fi}.$$ By \[qa2\], $\mu(Z)=1$. Fix $z \in Z$ and let $(p',{{\mathbf F}}'_1,h') \geq_n (p^1, {{\mathbf F}}_1,h)$ be such that $p' + {{\mathbb Q}}\subseteq {{\mathbf t}}+{{\mathbb Q}}+z$. Now define ${{\mathbf F}}_2'$ by putting ${{\mathbf F}}_2'(s)=1$ for every $s \in {{\operatorname{\mathsf {dom}}}}({{\mathbf F}}_1') \setminus {{\operatorname{\mathsf {dom}}}}({{\mathbf F}}_2)$. Clearly, ${\mathopen\ifcase1{}\oo\or
\big(\or\Big(\else\oo\fi}(q, {{\mathbf F}}_1', h'), (p^2, {{\mathbf F}}_2', h'){\mathclose\ifcase1{}\oo\or
\big)\or\Big)\else\oo\fi}$ is the condition we are looking for. ${\hspace{0.1in} \square \vspace{0.1in}}$
[**Acknowledgements**]{} We are grateful to Andrzej Ros[ł]{}anowski for devoting many hours to the much needed proofreading of this paper. Thanks to his perseverance, hopefully the process of reading this paper does not resemble the process of writing it.
[10]{}
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Fred Galvin and Arnold W. Miller. -sets and other singular sets of real numbers. , 17(2):145–155, 1984.
Fred Galvin, Jan Mycielski, and Robert Solovay. Strong measure zero sets. , pages A–280, 1973.
Richard Laver. On the consistency of [B]{}orel’s conjecture. , 137(3-4):151–169, 1976.
Janusz Pawlikowski. All [S]{}ierpiński sets are strongly meager. , 35:281–285, 1996.
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[^1]: First author partially supported by NSF grant DMS 95-05375 and Alexander von Humboldt Foundation
[^2]: Second author partially supported by Basic Research Fund, Israel Academy of Sciences, publication 607
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce a convolutional neural network for inferring a compact disentangled graphical description of objects from 2D images that can be used for volumetric reconstruction. The network comprises an encoder and a twin-tailed decoder. The encoder generates a disentangled *graphics code*. The first decoder generates a volume, and the second decoder reconstructs the input image using a novel training regime that allows the *graphics code* to learn a separate representation of the 3D object and a description of its lighting and pose conditions. We demonstrate this method by generating volumes and disentangled graphical descriptions from images and videos of faces and chairs.'
author:
- 'Edward Grant, Pushmeet Kohli, Marcel van Gerven'
bibliography:
- 'egbib.bib'
title: Deep Disentangled Representations for Volumetric Reconstruction
---
16SubNumber
Introduction
============
Images depicting natural objects are 2D representations of an underlying 3D structure from a specific viewpoint in specific lighting conditions.
This work demonstrates a method for recovering the underlying 3D geometry of an object depicted in a single 2D image or video. To accomplish this we first encode the image as a separate description of the shape and transformation properties of the input such as lighting and pose. The shape description is used to generate a volumetric representation that is interpretable by modern rendering software.
State of the art computer vision models perform recognition by learning hierarchical layers of feature detectors across overlapping sub-regions of the input space. Invariance to small transformations to the input is created by sub-sampling the image at various stages in the hierarchy.
In contrast, computer graphics models represent visual entities in a canonical form that is disentangled with respect to various realistic transformations in 3D, such as pose, scale and lighting conditions. 2D images can be rendered from the graphics code with the desired transformation properties.
A long standing hypothesis in computer vision is that vision is better accomplished by inferring such a disentangled graphical representation from 2D images. This process is known as ‘de-rendering’ and the field is known as ‘vision as inverse graphics’ [@yuille2006vision].
One obstacle to realising this aim is that the de-rendering problem is ill-posed. The same 2D image can be rendered from a variety of 3D objects. This uncertainty means that there is normally no analytical solution to de-rendering. There are however, solutions that are more or less likely, given an object class or the class of all natural objects.
Recent work in the field of vision as inverse graphics has produced a number of convolutional neural network models that accomplish de-rendering [@kulkarni2015deep; @tatarchenko2015single; @yang2015weakly]. Typically these models follow an encoding / decoding architecture. The encoder predicts a compact 3D graphical representation of the input. A control signal is applied corresponding with a known transformation to the input and a decoder renders the transformed image. We use a similar architecture. However, rather than rendering an image from the graphics code, we generate a full volumetric representation.
Unlike the disentangled graphics code generated by existing models, which is only renderable using a custom trained decoder, the volumetric representation generated by our model is easily converted to a polygon mesh or other professional quality 3D graphical format. This allows the object to be rendered at any scale and with other rendering techniques available in modern rendering software.
Related work
============
Several models have been developed that generate an disentangled representation given a 2D input, and output a new image subject to a transformation.
Kulkarni *et al*. proposed the Deep Convolutional Inverse Graphics Network (DC-IGN) trained using Stochastic Gradient Variational Bayes [@kulkarni2015deep]. This model encodes a factored latent representation of the input that is disentangled with respect to changes in azimuth, elevation and light source. A decoder renders the graphics code subject to the desired transformation as a 2D image. Training is performed with batches in which only a single transformation or the shape of the object are different. The activations of the graphics code layer chosen to represent the static parameters are clamped as the mean of the activations for that batch on the forward pass. On the backward pass the gradients for the corresponding nodes are set to their difference from this mean. The method is demonstrated by generating chairs and face images transformed with respect to azimuth, elevation and light source.
Tatarchenko *et al*. proposed a similar model that is trained in a fully supervised manner [@tatarchenko2015single]. The encoder takes a 2D image as input and generates a graphics code representing a canonical 3D object form. A signal is added to the code corresponding with a known transformation in 3D and the decoder renders a new image corresponding with that transformation. This method is also demonstrated by generating rotated images of cars and chairs.
Yang *et al*. demonstrated an encoder / decoder model similar to the above but utilize a recurrent structure to account for long-term dependencies in a sequence of transformations, allowing for realistic re-rendering of real face images from different azimuth angles [@yang2015weakly].
Spatial Transformer Networks (STN) allow for the spatial manipulation of images and data within a convolutional neural network [@jaderberg2015spatial]. The STN first generates a transformation matrix given an input, creates a grid of sampling points based on the transformation and outputs samples from the grid. The module is trained using back-propagation and transforms the input with an input dependent affine transformation. Since the output sample can be of arbitrary size, these modules have been used as an efficient down-sampling method in classification networks. STNs transform existing data by sampling but they are not generative, so cannot make predictions about occluded data, which is necessary when predicting 3D structure.
Girdhar *et al*. and Rezende *et al*. present methods for volumetric reconstructing from 2D images but do not generate disentangled representations [@girdhar2016learning; @rezende2016unsupervised].
The contribution of this work is an encoding / decoding model that generates a compact graphics code from 2D images and videos that is disentangled with respect to shape and the transformation parameters of the input, and that can also be used for volumetric reconstruction. To our knowledge this is the first work that generates a disentanlged graphical representation that can be used to reconstruct volumes from 2D images. In addition, we show that Spatial Transformer Networks can be used to replace max-pooling in the encoder as an efficient sampling method. We demonstrate this approach by generating a compact disentangled graphical representation from single 2D images and videos of faces and chairs in a variety of viewpoint and lighting conditions. This code is used to generate volumetric representations which are rendered from a variety of viewpoints to show their 3D structure.
Model
=====
Architecture
------------
As shown in Figure \[fig:network\], the network has one encoder, a *graphics code* layer and two decoders. The *graphics code* layer is separated into a *shape code* and a *transformation code*. The encoder takes as input an 80 $\times$ 80 pixel color image and generates the *graphics code* following a series of convolutions, point-wise randomized rectified linear units (RReLU) [@xu2015empirical], down-sampling Spatial Transformer Networks and max pooling. Batch normalization layers are used after each convolutional layer to speed up training and avoid problems with exploding and vanishing gradients [@ioffe2015batch].
![**Network architecture:** The network consists of an encoder (A), a volume decoder (B) and an image decoder (C). The encoder takes as input a 2D image and generates a 3D *graphics code* through a series of spatial convolutions, down-sampling Spatial Transformer Networks and max pooling layers. This code is split into a *shape code* and a *transformation code*. The volume decoder takes the *shape code* as input and generates a prediction of the volumetric contents of the input. The image decoder takes the *shape code* and the *transformation code* as input and reconstructs the input image.[]{data-label="fig:network"}](2D23D2tail){width="\textwidth"}
The two decoders are connected to the *graphics code* by switches so that the message from the *graphics code* is passed to either one of the decoders. The first decoder is the volume decoder. The volume decoder takes the *shape code* as input and generates an $80 \times 80 \times 80$ voxel volumetric prediction of the encoded shape. This is accomplished by a series of volumetric convolutions, point-wise RReLU and volumetric up-sampling. A parametric rectified linear unit (PReLU) [@he2015delving] is substituted for the RReLU in the output layer. This is done to avoid the saturation problems with rectified linear units early in training but allows for learning an activation threshold later in training, corresponding with the positive-valued output targets.
The second decoder reconstructs the input image with the correct pose and lighting, showing that pose and lighting parameters of the input are contained in the *graphics code*. The image decoder takes as input both the *shape code* and the *transformation code*, and generates a reconstruction of the original input image. This is accomplished by a series of spatial convolutions, point-wise RReLU, spatial up-sampling and point-wise PReLU in the final layer. During training, the backward pass from the image decoder to the *shape code* is blocked (see Figure \[fig:training\]). This encourages the *shape code* to only represent shape, as it only receives an error signal from the volume decoder.
![**Network training:** In the forward pass the *shape code* (Z1) and the *transformation code* (Z2) receive a signal from the encoder (E). The volume decoder (D1) receives input only from the *shape code*. The image decoder (D2) receives input from the *shape code* and the *transformation code*. On the backward pass the signal from the image decoder to the *shape code* is suppressed to force it to only represent shape.[]{data-label="fig:training"}](forback){width="40.00000%"}
The volume decoder only requires knowledge about the shape of the input since it generates binary volumes that are invariant to pose and lighting. However, the image decoder must generate a reconstruction of the original image which is not invariant to shape, pose or lighting. Both decoders have access to the *shape code* but only the image decoder has access to the *transformation code*. This encourages the network to learn a *graphics code* that is disentangled with respect to shape and transformations.
The network can be trained differently depending on whether pose and lighting conditions need to be encoded. If the only objective is to generate volumes from the input then the image decoder can be switched off during training. In this case the *graphics code* will learn to be invariant to viewpoint and lighting. If the volume decoder and image decoder are both used during training the *graphics code* learns a disentangled representation of shape and transformations.
Spatial transformer networks
----------------------------
Spatial Transformer Networks (STNs) perform input dependent geometric transformations on images or sets of feature maps [@jaderberg2015spatial]. There are two STNs in our model (see Figure \[fig:network\]).
Each STN comprises a localisation network, a grid generator and sampling grid. The localisation network takes the activations of the previous layer as input and regresses the parameters of an affine transformation matrix. The grid generator generates a sampling grid of ($x,y$) coordinates corresponding with the desired height and width of the output. The sampling grid is obtained by multiplying the generated grid with the transformation matrix. In our model this takes the form: $$\begin{aligned}
\begin{pmatrix}
x_{i}^{s} \\
y_{i}^{s}
\end{pmatrix}
= \mathcal{T}_{\theta} (G_{i})=
\begin{bmatrix}
\theta_{11} & \theta_{12} & \theta_{13} \\
\theta_{21}& \theta_{22} & \theta_{23}
\end{bmatrix}
\begin{pmatrix}
x_{i}^{t} \\
y_{i}^{t} \\
1
\end{pmatrix}\end{aligned}$$
Where ($x_{i}^{t},y_{i}^{t}$) are the generated grid coordinates and ($x_{i}^{s},y_{i}^{s}$) define the sample points. The transformation matrix $\mathcal{T}_{\theta}$ allows for cropping, scale, translation, scale, rotation and skew. Cropping and scale, in particular allow the STN to focus on the most important region in a feature map.
STNs have been shown to improve performance in convolutional network classifiers by modelling attention and transforming feature maps. Our model uses STNs in a generative setting to perform efficient down-sampling and assist the network in learning invariance to pose and lighting.
The first STN in our model is positioned after the first convolutional layer. It uses a convolutional neural network to regress the transformation coefficients. This localisation network consists of four $5\times5$ convolutional layers, each followed by batch normalization and the first three also followed by $2\times2$ max pooling.
The second STN in our model is positioned after the second convolutional layer and regresses the transformation parameters with a convolutional network consisting of two $5\times5$ an one $6\times6$ convolutional layers each followed by batch normalization and the last two also by $2\times2$ max pooling.
Data
----
The model was trained using $16,000$ image-volume pairs generated from the Basel Face Model [@paysan20093d]. Images of size $80\times80$ were rendered in RGB from five different azimuth angles and three ambient lighting settings. Volumes of size $80\times80\times80$ were created by discretizing the triangular mesh generated by the Basel Face Model.
Experimental Results
====================
Training
--------
We evaluated the model’s volume prediction capacity by training it on $16,000$ image-volume pairs. Each example pair was shown to the network only once to discourage memorization of the training data.
Training was performed using the Torch framework on a single NVIDIA Tesla K80 GPU. Batches of size 10 were given as input to the encoder and forward propagated through the network. The mean-squared error of the predicted and target volumes was calculated and back-propagated using the Adam learning algorithm [@kingma2014adam]. The initial learning rate was set to $0.001$.
Volume Predictions from Images of Faces
---------------------------------------
In this experiment we used the network to generate volumes from a single 2D images. The network was presented with unseen face images as input and generated 3D volume predictions. The image decoder was not used in this experiment.
The predicted volumes were binarized with a threshold of $0.01$. A triangular mesh was generated from the coordinates of active voxels using Delaunay triangulation. The patch was smoothed and the resulting image rendered using OpenGL and Matlab’s [trimesh]{} function.
Figure \[fig:ABC\](a) shows the input image, network predictions, ground truth, nearest neighour in the input space and the ground truth of the nearest neighour. The nearest neighbour was determined by searching the training images for the image with the smallest pixel-wise distance to the input. The generated volumes are visibly different depending on the shape of the input.
Figure \[fig:ABC\](b) shows the network output for the same input presented from different viewpoints. The images in the first row are the inputs to the network and the second row contains the volumes generated from each input. These are shown from the same viewpoint for comparison. The generated volumes are visually very similar, showing that the network generated volumes that are invariant to the pose of the input.
Figure \[fig:ABC\](c) shows the network output for the same face presented in different lighting conditions. The first row images are the inputs and the second row are the generated volumes also shown from the same viewpoint for comparison. These volumes are also visually very similar to each other showing that the network output appears invariant to lighting conditions in the input.
[cc]{}
[1.1]{} ![**Generated volumes:** Qualitative results showing the volume predicting capacity of the network on unseen data. (a) First column: network inputs. Columns 2-4 (white): network predictions shown from three viewpoints. Columns 5-7 (black): ground truth from the same viewpoints. Column 8: nearest neighbour image. Columns 9-11 (blue): nearest neighbour image ground truth. (b) Each column is an input/output pair. The inputs are in the first row. Each input is the same face viewed from a different position. The generated volumes in the second row are shown from the same viewpoint for comparison. (c) Each column is an input/output pair. The inputs are in the first row. Each input is the same face in different lighting conditions.[]{data-label="fig:ABC"}](predictions3 "fig:"){width="\textwidth"}
\
[0.75]{} ![**Generated volumes:** Qualitative results showing the volume predicting capacity of the network on unseen data. (a) First column: network inputs. Columns 2-4 (white): network predictions shown from three viewpoints. Columns 5-7 (black): ground truth from the same viewpoints. Column 8: nearest neighbour image. Columns 9-11 (blue): nearest neighbour image ground truth. (b) Each column is an input/output pair. The inputs are in the first row. Each input is the same face viewed from a different position. The generated volumes in the second row are shown from the same viewpoint for comparison. (c) Each column is an input/output pair. The inputs are in the first row. Each input is the same face in different lighting conditions.[]{data-label="fig:ABC"}](pose2 "fig:"){width="100.00000%"}
[0.52]{} ![**Generated volumes:** Qualitative results showing the volume predicting capacity of the network on unseen data. (a) First column: network inputs. Columns 2-4 (white): network predictions shown from three viewpoints. Columns 5-7 (black): ground truth from the same viewpoints. Column 8: nearest neighbour image. Columns 9-11 (blue): nearest neighbour image ground truth. (b) Each column is an input/output pair. The inputs are in the first row. Each input is the same face viewed from a different position. The generated volumes in the second row are shown from the same viewpoint for comparison. (c) Each column is an input/output pair. The inputs are in the first row. Each input is the same face in different lighting conditions.[]{data-label="fig:ABC"}](light2 "fig:"){width="\textwidth"}
Nearest Neighbour Comparison
----------------------------
The network’s quantitative performance was benchmarked using a nearest neighbour test. A test set of 200 image / volume pairs was generated using the Basel Face Model (ground truth). The nearest neighbour to each test image in the training set was identified by searching for the training set image with the smallest pixel-wise Euclidean distance to the test set image (nearest neighbour). The network generated a volume for each test set input (prediction).
Nearest neighbour error was determined by measuring the mean voxel-wise Euclidean distance between the ground truth and nearest neighbour volumes. Prediction error was determined by measuring the mean voxel-wise Euclidean distance between the ground truth volumes and the predicted volumes.
A paired-samples t-test was conducted to compare error score in predicted and nearest neighbour volumes. There was a significant difference in the error score for predictions ($M=0.0096$, $SD=0.0013$) and nearest neighbours ($M=0.017$, $SD=0.0038$) conditions; $t(199)=-21.5945$,$ p=4.7022e-54$.
These results show that network is better at predicting volumes than using the nearest neighbour.
Internal Representations
------------------------
In this experiment we tested the ability of the encoder to generate a *graphics code* that can be used to generate a volume that is invariant to pose and lighting. Since the volume encoder doesn’t need pose and lighting information we didn’t use the image decoder in this experiment.
To test the invariance of the encoder with respect to pose, lighting and shape we re-trained the model without using batch normalization. Three sets of 100 image batches were prepared where two of these parameters were clamped and the target parameter was different. This makes it possible to measure the variance of activations for changes in pose, lighting and shape. The set-wise mean of the mean variance of activations in each batch was compared for all layers in the network.
Figure \[fig:invariance\](a) shows that the network’s heightened sensitivity to shape relative to pose and lighting begins in the second convolutional layer. There is a sharp increase in sensitivity to shape in the *graphics code*, which is much more sensitive to shape than pose or lighting, and more sensitive to pose than lighting. This relative invariance to pose and lighting is retained in the volume decoder.
Figure \[fig:invariance\](b) shows a visual representation of the activations for the same face with different poses. The effect of the first STN can be seen in the second convolutional layer activations which are visibly warped. The difference in the warp depending on the pose of the face suggests that the STNs may be helping to create invariance to pose later in the network. The example input images have a light source which is directed from the left of the camera. The second convolutional layer activations show a dark area on the right side of each face which is less evident in the first convolutional layer, suggesting that shadowing is an important feature for predicting the 3D shape of the face.
[cc]{}
[0.6]{} ![**Invariance to pose and lighting:** (a) The relative mean standard deviation (SD) of activations in each network layer is compared for changes in shape, pose and lighting. Image is the input image, E1-E3 are the convolutional encoder layers, Z is the *graphics code*, D1-D3 are the convolutional decoder layers and Volume is the generated volume. In the input, changes to pose account for the highest SD. By the second convolutional layer the network is more sensitive to changes in shape than pose or lighting. The *graphics code* is much more sensitive to shape than pose or lighting. (b) The first row is five images of the same face from different viewpoints. Rows 2-4 show sampled encoder activations for the input image at the top of each column. The last row shows sampled *graphics code* activations reshaped into a square.[]{data-label="fig:invariance"}](layerSDc "fig:"){width="\textwidth"}
[c]{}
[0.5]{} ![**Invariance to pose and lighting:** (a) The relative mean standard deviation (SD) of activations in each network layer is compared for changes in shape, pose and lighting. Image is the input image, E1-E3 are the convolutional encoder layers, Z is the *graphics code*, D1-D3 are the convolutional decoder layers and Volume is the generated volume. In the input, changes to pose account for the highest SD. By the second convolutional layer the network is more sensitive to changes in shape than pose or lighting. The *graphics code* is much more sensitive to shape than pose or lighting. (b) The first row is five images of the same face from different viewpoints. Rows 2-4 show sampled encoder activations for the input image at the top of each column. The last row shows sampled *graphics code* activations reshaped into a square.[]{data-label="fig:invariance"}](features2 "fig:"){width="\textwidth"}
\
Disentangled Representations
----------------------------
In this experiment we tested the network’s ability to generate a compact 3D description of the input that is disentangled with respect to the shape of the object and transformations such as pose and lighting.
In order to generate this description we used the same network as in the volume generation experiment but with an additional fully connected RReLU layer of size $3,000$ in the encoder to compensate for the increased difficulty of the task.
During training, images were given as input to the encoder which generated an activity vector of $200$ scalar values. These were divided in the *shape code* comprising $185$ values and the *transformation code* comprising $15$ values. The network was trained on $16,000$ image / volumes pairs with batches of size $10$.
The switches connecting the encoder to the decoders were adjusted after every three training batches to allow the volume decoder and the image decoder to see the same number of examples. The volume decoder only received the *shape code*, whereas the image decoder received both the *shape code* and the *transformation code*.
To test if the *shape code* and the *transformation code* learned the desired invariance we measured the mean standard deviation of activations for batches where only one of shape, pose or lighting conditions were changed. The same batches as in the invariance experiment were used.
Figure \[fig:disentangled\](a) shows the relative mean standard deviation of activations of each layer in the encoder, *graphics code* and image decoder. The bifurcation at point Z on the plot shows that the two codes learned to respond differently to the same input. The *shape code* learned to be more sensitive to changes in shape than pose or lighting, and the *transformation code* learned to be more sensitive to changes in pose and lighting than shape.
To make sure the image decoder used the shape code to reconstruct the input we compared the output of the image decoder with input only from the *shape code*, the *transformation code* and both together. Figure \[fig:disentangled\](b) shows the output of the volume decoder and image decoder on a number of unseen images. The first column shows the input to the network. The second column shows the output of the image decoder with input only from the *shape code*. The third column shows the same for the output of the *transformation code*. The fourth column shows the combined output of the *shape code* and the *transformation code*. The fifth column shows the output of the volume decoder.
[cc]{}
[0.6]{} ![**Disentangled representations:** (a) The relative mean standard deviation (SD) of activations in the encoder, *shape code*, *transformation code* and image decoder is compared for changes in shape, pose and lighting. The *shape code* is most sensitive to changes in shape. The *transformation code* is most sensitive to changes in pose and lighting. Error bars show standard deviation. (b) The output of the volume decoder and image decoder on a number of unseen images. The first column is the input image. The second column is the image decoded from the *shape code* only. The third column is the image decoded from the *transformation code* only. The fourth column is the image decoded from the *shape code* and the *transformation code*. The fifth column is the output of the volume decoder shown from the same viewpoint for comparison.[]{data-label="fig:disentangled"}](layerSD2Zc "fig:"){width="\textwidth"}
[c]{}
[0.5]{} ![**Disentangled representations:** (a) The relative mean standard deviation (SD) of activations in the encoder, *shape code*, *transformation code* and image decoder is compared for changes in shape, pose and lighting. The *shape code* is most sensitive to changes in shape. The *transformation code* is most sensitive to changes in pose and lighting. Error bars show standard deviation. (b) The output of the volume decoder and image decoder on a number of unseen images. The first column is the input image. The second column is the image decoded from the *shape code* only. The third column is the image decoded from the *transformation code* only. The fourth column is the image decoded from the *shape code* and the *transformation code*. The fifth column is the output of the volume decoder shown from the same viewpoint for comparison.[]{data-label="fig:disentangled"}](codes "fig:"){width="\textwidth"}
\
Face Recognition in Novel Pose and Lighting Conditions
------------------------------------------------------
To measure the invariance and representational quality of the *shape code* we tested it on a face recognition task.
The point-wise Euclidean distance between the *shape code* generated by an image was measured for a batch of $150$ random images including one image that was the same face with a different pose (target). The random images were ordered from the smallest to greatest distance and the rank of the target was recorded. This was repeated $100$ times and an identical experiment was performed for pose. The mean rank for the same face with a different pose was $11.08$. The mean rank of the same face with different lighting was $1.02$. This demonstrates that the *shape code* can be used as a pose and lighting invariant face classifier.
To test if the *shape code* was more invariant to pose and lighting than the full *graphics code* we repeated this experiment using the full *graphics code*. The mean rank for the same face with a different pose was $26.86$. The mean rank of the same face with different lighting was $1.14$. This shows that the *shape code* was relatively more invariant to pose and lighting than the full *graphics code*.
Volume Predictions from Videos of Faces
---------------------------------------
To test if video input improved the quality of the generated volumes we adapted the encoder to take video as input and compared to a single image baseline. $10,000$ video / volume pairs of faces were created. Each video consisted of five RGB frames of a face rotating from left facing profile to right facing profile in equidistant degrees of rotation. The same network architecture was used as in experiment 4.5. For the video model the first layer was adapted to take the whole video as input. For the single image baseline model, single images from each video were used as input.
To test the performance difference between video and single image inputs a test set of 500 video / volume pairs was generated. Error was measured using the mean voxel-wise distance between ground truth and volumes generated by the network. For the video network the entire video was used as input. For the single image baseline each frame of the video was given separately as input to the network and the generated volume with the lowest error was used as the benchmark.
A paired-samples t-test was conducted to compare error score in volumes generated from volumes and single images. There was a significant difference in the error score for video based volume predictions ($M=0.0073$, $SD=0.0009$) and single image based predictions ($M=0.0089$, $SD=0.0014$) conditions; $t(199)=-13.7522$, $1.0947e-30$.
These results show that video input results in superior volume reconstruction performance compared with single images.
Volume Predictions from Images of Chairs
----------------------------------------
In this experiment we tested the capacity of the network to generate volume predictions from objects with more variable geometry. $5000$ Volume / image pairs of chairs were created from the ModelNet dataset [@wu20153d]. The images were $80 \times 80$ RGB images and the volumes were $30 \times 30 \times 30$ binary volumes. The predicted volumes were binarized with a threshold of $0.2$. Both decoders were used in this experiment. The *shape code* consisted of $599$ activations and the *transformation code* consisted of one activation. The *shape code* was used to reconstruct the volumes. Both the *shape code* and *transformation code* were used to reconstruct the input.
Figure \[fig:AAAA\] demonstrates the network’s capacity to generate volumetric predictions of chairs from novel images.
[cc]{}
[1.2]{} ![**Generated chair volumes:** Qualitative results showing the volume predicting capacity of the network on unseen data. First column: network inputs. Columns 2-4 (Yellow): network predictions shown from three viewpoints. Columns 5-7 (black): ground truth from the same viewpoints. Column 8: nearest neighbour image in the training set. Columns 9-11 (blue): nearest neighbour image ground truth.[]{data-label="fig:AAAA"}](chairs "fig:"){width="\textwidth"}
\
Interpolating the Graphics Code
-------------------------------
In order to qualitatively demonstrate that the *graphics code* in experiment $4.8$ was disentangled with respect to shape and pose, we swapped the *shape code* and *transformation code* of a number of images and generated new images from the interpolated code using the image decoder. Figure \[fig:AAAAA\] shows the output of the image decoder using the interpolated code. The shape of the chairs in the generated images is most similar to the shape of the chairs in the images used to generate the *shape code*. The pose of each chair is most similar to the pose of the chairs in the images used to generate the *transformation code*. This demonstrates that the *graphics code* is disentangled with respect to shape and pose.
[cc]{}
[1.2]{} ![**Interpolated code:** Qualitative results combining the and *transformation code* from different images. First row: images used to generate the *shape code*. Second row: images used to generate the *transformation code*. Last row: Image decoder output.[]{data-label="fig:AAAAA"}](interp "fig:"){width="\textwidth"}
\
Discussion
==========
We have shown that a convolutional neural network can learn to generate a compact graphical representation that is disentangled with respect to shape, and transformations such as lighting and pose. This representation can be used to generate a full volumetric prediction of the contents of the input image.
By comparing the activations of batches corresponding with a specific transformation or the shape of the image, we showed that the network can learn to represent a *shape code* that is relatively invariant to pose and lighting conditions. By adding an additional decoder to the network that reconstructs the input image, the network can learn to represent a *transformation code* that represents the pose and lighting conditions of the input.
Extending the approach to real world scenes requires consideration of the viewpoint of the generated volume. Although the volume is invariant in the sense that it contains all the information necessary to render the generated object from any viewpoint, a canonical viewpoint was used for all volumes so that they were generated from a frontal perspective. Natural scenes do not always have a canonical viewpoint for reference. One possible solution is to generate a volume from the same viewpoint as the input. Experiments show that this approach is promising but further work is needed.
In order to learn, the network requires image-volume pairs. This limits the type of data that can be used as volumetric datasets of sufficient size, or models that generate them are limited in number. A promising avenue for future work is incorporating a professional quality renderer into the decoder structure. This theoretically allows for 3D graphical representations to be learned, provided that the rendering process is approximately differentiable.\
**Acknowledgements:** Thanks to Thomas Vetter for access to the Basel Face Model.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Rock-scissors-paper game, as the simplest model of intransitive relation between competing agents, is a frequently quoted model to explain the stable diversity of competitors in the race of surviving. When increasing the number of competitors we may face a novel situation because beside the mentioned unidirectional predator-prey-like dominance a balanced or peer relation can emerge between some competitors. By utilizing this possibility in the present work we generalize a four-state predator-prey type model where we establish two groups of species labeled by even and odd numbers. In particular, we introduce different invasion probabilities between and within these groups, which results in a tunable intensity of bidirectional invasion among peer species. Our study reveals an exceptional richness of pattern formations where five quantitatively different phases are observed by varying solely the strength of the mentioned inner invasion. The related transition points can be identified with the help of appropriate order parameters based on the spatial autocorrelation decay, on the fraction of empty sites, and on the variance of the species density. Furthermore, the application of diverse, alliance-specific inner invasion rates for different groups may result in the extinction of the pair of species where this inner invasion is moderate. These observations highlight that beyond the well-known and intensively studied cyclic dominance there is an additional source of complexity of pattern formation that has not been explored earlier.'
author:
- 'D. Bazeia'
- 'B.F. de Oliveira'
- 'A. Szolnoki'
title: |
Invasion controlled pattern formation\
in a generalized multi-species predator-prey system
---
INTRODUCTION
============
To explain the diversity among competing species or states is a fundamental problem not only in biology, or ecology but also in social sciences [@chesson_ares00; @hauert_s02; @aguiar_n09]. One of the possible mechanisms that explains the stable coexistence of unequal species is the presence of intransitive relation or in other words cyclic dominance between competitors [@laird_e08; @traulsen_jtb12]. In game theory this relation can be well described by the so-called rock-scissors-paper game [@szolnoki_jrsif14]. Paper is cut by scissors, scissors are crushed by rock, and finally rock is wrapped by paper. In this way the circle ends and establishes the above described relation. In the absence of a superior competitor all the mentioned members can survive and hence diversity is preserved [@bazeia_epl18].
Interestingly, this relation is not a merely abstract model, but can be directly detected in several real-life systems [@kirkup_n04; @kelsic_n15], including microbes [@paquin_n83; @kerr_n02], social amoebas [@shibasaki_prsb18], or even plant communities [@lankau_s07; @cameron_jecol09]. Significant scientific efforts have been made in the last decade which clarified the possible consequences of different variations of the basic model [@szabo_pr07; @wang_wx_pre11; @szczesny_pre14; @szolnoki_njp15; @frey_pa10; @szolnoki_pre16; @valyi_15]. In spatially structured populations the topology of interaction graph is proved to be a decisive factor which determines whether an oscillatory state emerges or not [@masuda_prsb07; @szabo_jpa04; @masuda_jtb08]. Furthermore, the mobility of competing species is identified as an important factor to maintain diversity [@reichenbach_n07; @bazeia_epl17; @mobilia_g16; @armano_srep17; @avelino_pre18], but some research groups also underline the nontrivial role of mutations [@mobilia_jtb10; @park_c18; @park_c18c; @nagatani_jtb19]. Additionally, a recent work, obtained from off-lattice simulations, revealed the critical role of density on the original problem of maintaining diversity [@avelino_epl18]. It is worth noting that cyclic dominance can also emerge in systems where the values of payoff matrix, which characterizes the basic relation of different microscopic states or strategies, do not necessarily predict such interaction. Instead, this relation could be the result of a collective behavior due to the limited interactions with neighbors in a spatial system where effective multi-point interactions emerge [@szolnoki_pre10b; @dobramysl_jpa18; @szolnoki_njp14; @gao_l_srep15b; @roman_jtb16; @szolnoki_epl15].
Naturally, the number of competing species are not necessarily limited to three, but can be extended to four, five [@roman_jsm12; @lutz_jtb13; @vukov_pre13; @avelino_pla14; @rulquin_pre14; @intoy_jsm13] or even more species [@szabo_jpa05; @avelino_pre14; @szabo_pre08b; @brown_pre17; @avelino_pla17; @esmaeili_pre18]. This makes the food-web more complex where the relation between two members is not restricted to a unidirectional predator-prey type, but also a balanced, or bidirectional relation can also emerge. This chance allows new kind of solutions, including alliances or associations, to emerge [@szabo_jpa05; @szabo_pre08]. Beside the topological complexity of food-web an additional freedom is the heterogeneity of invasion rates between species. In some cases the latter fact alone is capable to change the final state significantly [@perc_pre07b; @masuda_jtb08; @he_q_pre10; @szolnoki_srep16b; @cazaubiel_jtb17; @liu_a_epl17].
In this work we follow this research avenue and generalize a previously introduced four-species model where every species has two preys in a cyclic manner [@avelino_pre12b]. As a result, some relations between species become unbiased or balanced because these peer species mutually invade each other. This fact allows us to distinguish the strengths of unidirectional and bidirectional invasions and establish a tunable parameter that characterizes the inner relations of peer species. Our key observation is the stationary pattern of the resulting evolutionary process can be varied intensively by tuning the inner invasion rate of peer species exclusively. The resulting phases can be distinguished quantitatively with the help of appropriate order parameters. These observations emphasize that not only the complex topology of a food-web, but also the varying invasion rates between related species can be the source of diverse patterns of the stationary states.
THE MODEL
=========
In the following we generalize a previously introduced cyclically dominated May-Leonard-type model [@frey_pa10] of four species [@avelino_pre12b]. Initially, empty sites, labeled by 0, and all competing species, labeled by $i=1 \dots 4$, are distributed uniformly on a $L \times L$ square grid where periodic boundary conditions are applied. At each time step a randomly chosen active individual interacts with one of the four nearest neighbor passive sites by executing the following elementary steps.
If the passive site is empty then the active individual reproduces by filling the empty site with probability $\mu$. When a motion step is applied then the active and passive individuals switch their positions with probability $m$. The last elementary step is the so-called predation when the active predator kills the passive prey and generates an empty site in the lattice.
Importantly, as an extension of the earlier introduced basic model [@avelino_pre12b], we distinguish different predation probabilities between species depending on whether their labels are odd or even. In particular, as Fig. \[def\] illustrates, an active $i$ player predates a passive $i+1$ species and generates an empty site with probability $p_1$. However, the predation between species $i$ and species $i+2$ happens with probability $p_2$. (Naturally labels are always considered cyclically to keep $i=1 \dots 4$ interval.) In this way we can distinguish predation strength between predator-prey pairs where invasion is unidirectional and between peer species where bidirectional invasions can happen. The members of latter pairs, like species 1 and 3, or species 2 and 4, are equally strong because they can mutually invade each other and keep a balanced relation, as it is stressed by dashed arrows in Fig. \[def\]. Interestingly, such a peer pair can form a defensive alliance against an external predator species that would dominate one of the members of the mentioned pair otherwise. Just to give an example, the invasion of species 2 toward species 3 can be avoided if species 1 is present and protects peer member species 3.
![Invasions between competing species. Solid arrows indicate the unidirectional invasions between primary predator-prey species which happen with probability $p_1$, while dashed arrows indicate bidirectional invasions between peer species that happen with probability $p_2 \leq p_1$.[]{data-label="def"}](fig1)
Summing up our model definition, the simulation algorithm can be given as follows. At each time step an active site and a neighboring passive site are chosen randomly. After we decide whether a mobility, a reproduction, or a predation elementary step is executed. Their relative weights are: $m=0.5$, $\mu=0.25$ and $p=0.25$. If the mobility step is chosen, then the active and passive sites exchange their positions. Note that the passive site can be any individual or an empty space. If the reproduction step is chosen, then the active species can duplicate itself only if the passive site was empty. In case of predation step we first consider the labels of the active $i$ species and the passive neighbor. If the label of the passive species is $i+1$ then the latter will disappear with probability $p_1$. Alternatively, if the label of passive species is $i+2$ then it will die out with probability $p_2$. Evidently, if the passive site is occupied by a predator species of the active species, or passive and active sites are occupied by identical species, or the active site is empty then nothing happens.
In our generalized model the key parameter is the value of $p_2$, which controls the inner, or bidirectional invasion between peer species. Notably, the gradual variation of $p_2$ allows us to bridge two previously studied independent models [@avelino_pre12b]. More precisely, in the $p_2=0$ limit we get back the so-called $I_4$ model where partnerships of peer species, such as $\{1+3\}$ or $\{2+4\}$, emerge and occupy different spatial regions. In the other extreme limit, when $p_2=p_1=1$ the model becomes equivalent to the so-called $II_4$ model where peer domains diminish and homogeneous spirals with four-arms characterize the stationary state [@avelino_pre12b]. As noted, in our present work we apply a relatively high mobility rate ($m=0.5$) comparing to the basic model of Ref.[@avelino_pre12b]. In this way emerging spirals of invasion fronts are not suppressed by low mobility rate, as it was observed earlier.
A full Monte Carlo step or in other words a full generation involves $N= L \times L$ interactions or elementary steps described above. We should stress that a sufficiently high system size is necessary, otherwise we can easily obtain misleading results. To illustrate this we present the stationary pattern of an $5000 \times 5000$ system in Fig. \[zoom\] which was obtained at $p_2=0.005$. Here species are colored in agreement with the color-code used in model definition of Fig. \[def\]. The snapshot of Fig. \[zoom\] depicts large homogeneous domains whose linear size can easily exceed an $L=300$ lattice site (one of these spots is framed by a square of latter size). This example illustrates nicely that during the simulations we faced serious finite-size problems [@lutz_g17], but luckily in the $p_2>0.01$ region $L=2000$ linear system size was generally enough to gain data, which are free from finite-size problems.
![Stationary state of an $5000 \times 5000$ system after 10000 generations obtained at $p_2=0.005$. The square in the centre of the pattern shows a $300 \times 300$ homogeneous area that is occupied exclusively by species 3. This example illustrates that the typical system size used by ordinary numerical works would result in misleading conclusions in our present model.[]{data-label="zoom"}](fig2){width="6.3cm"}
RESULTS
=======
We first present our main observations how the characteristic patterns change by varying only the $p_2$ value between 0 and 1, while $p_1=1$ is kept fixed. To obtain a general overview about the emerging patterns we provide in [@scan] an animation showing the typical spatiotemporal patterns in dependence of $p_2$. Based on this we can identify five characteristic regions as a function of invasion strength. The typical patterns of these phases and the separated state of $p_2=0$ case are plotted in Fig. \[snapshots\].
![Representative patterns of different phases in dependence of $p_2$ invasion rate. The values are $p_2=0$ (a), 0.02 (b), 0.06 (c), 0.12 (d), 0.25 (e), and 1 (f). Snapshots of stationary states were taken after 10000 generations for a $500 \times 500$ system.[]{data-label="snapshots"}](fig3)
The qualitative description of different phases can be given as follows. If $p_2$ is large enough, shown in panel (f) of Fig. \[snapshots\], then we can observe clear four-color rotating spirals that characterizes typical four-state systems where species dominate cyclically each other similarly to the extended Lotka-Volterra type dynamics [@szabo_pre04; @peltomaki_pre08; @hua_epl13]. When we start decreasing the value of $p_2$ the four-color vortices are replaced by three-color vortices, as illustrated in panel (e) of Fig. \[snapshots\].
By decreasing the value of $p_2$ further we enter to a phase where domains composed by peer species first emerge. This phenomenon is shown in panel (d). Since the relation of peer species is balanced therefore the borders which separate them are not as sharp as domain walls previously observed for unidirectional invasion. This effect becomes more pronounced for smaller $p_2$ values as shown in panels (a)-(c). In parallel the three-color vortices disappear. Such vortices are always the source of propagating waves, hence in the absence of them one would expect increased characteristic length of domains. On the other hand, however, the effective mix of peer species (between 1 and 3 or between 2 and 4) is still intensive which prevents typical length from growing. Both effects are weakened if we decrease $p_2$ even further, shown in panel (c), which results in smooth interfaces separating domains of different peer species. Simultaneously, homogeneous spots within such a two-species domain become also larger. This state is illustrated in panel (b) of Fig. \[snapshots\] signaling an enlarged typical length. Consequently, the densities of species fluctuate strongly in time which may involve serious finite size effects. For example, when the system size is comparable to the typical length of domains then the actual portions of species could be significantly different at a specific time. Such a situation is illustrated in panel (d) of Fig. \[snapshots\] where the temporary portions of blue and green are seemingly higher than the portions of red and yellow colors. But we can also observe reversed effect on panel (b) where the majority of sites are occupied by the $\{1+3\}$ alliance. Evidently, this contradicts to the basic symmetry of our model, shown in Fig. \[def\], that can only be restored if the system size is large enough.
![Spatial autocorrelation functions, obtained at a $20000 \times 20000$ system size, for different values of $p_2$ when the system is in the stationary state. The dashed line, drawn at $C(r=\ell) = 0.25$, indicates the threshold value of autocorrelation which is used to define the characteristic length scale.[]{data-label="autocorr"}](fig4)
As we already stressed, this enhanced characteristic length was illustrated in Fig. \[zoom\]. It is worth stressing that this low-$p_2$ state is significantly different from the limit case of $I_4$ model that is shown in panel (a) of Fig. \[snapshots\]. In the latter case, the lack of mutual invasion between peer species results in a perfect mixture of these species, which makes the typical length fall again.
To allow readers to collect general impressions about the dynamics of pattern formation for different characteristic $p_2$ values, we provide an animation where time evolutions are shown simultaneously in [@multi].
Inspired by the qualitative picture depicted above we made quantitative measurements for a more accurate description. First, we measure the typical length which characterizes the stationary states of different phases. For this goal we calculate the spatial autocorrelation function at different $p_2$ values in the long time limit when system evolved onto a stationary state. More precisely, we measure the function
$$C(r) = \displaystyle \sum_{|\vec{r}|=x+y}\dfrac{C(\vec{r})}{{\rm
min} (2N-(x+y+1), x+y+1)}\ ,
\label{eq2}$$
where $x$ and $y$ are the coordinates of a species in the position $\vec{r}$ on the lattice, while $C(\vec{r})$ is defined as $$C(\vec{r}) = \dfrac{1}{C(0)} \int_{\mathcal{S}}
\varphi(\vec{r})
\varphi(\vec{r}+\vec{r^{'}}) d^2\vec{r^{'}}\ .
\label{eq1}$$ Here $\varphi(\vec{r}) = \phi(\vec{r}) - \langle
\phi\rangle$ and $\phi(\vec{r})$ represents the species in the position $\vec{r}$ on the lattice in the stationary state. Naturally, $\vec{r^{'}}$ spans the whole lattice, hence $\mathcal{S}$ denotes the domain of integral. Also, in agreement with general notation, $\langle \phi(t) \rangle$ represents the spatial mean value of $\phi$ when the system relaxed into the stationary state. According to the model definition, we use 0 for the empty sites, and 1, 2, 3, 4 for species red, blue, yellow, and green, respectively, as also indicated in Fig. \[def\].
The above defined function is plotted for some representative $p_2$ values in Fig. \[autocorr\]. To estimate the typical length we determine the critical $r$ value for all cases where the value of $C (r)$ function decays below the 0.25 threshold value. For comparison this value is also plotted by a horizontal dashed line in Fig. \[autocorr\]. As these plots illustrate, the characteristic length derived from the autocorrelation function behaves in a largely non-monotonous way in dependence of the invasion rate $p_2$.
![The characteristic length, $\ell$, in dependence of control parameter $p_2$. The specific $p_2$ values where the length changes the sign of its growth tendency indicate the transition points separating different phases whose typical patterns are illustrated in Fig. \[snapshots\]. These values are marked by dashed vertical lines. To gain reliable results we used $20000 \times 20000$ system size. The error bars are comparable to the size of symbols.[]{data-label="length"}](fig5)
This behavior becomes more transparent in Fig. \[length\] where the above defined characteristic length is plotted for different $p_2$ values. We note that only the $p_2<0.5$ interval is shown here because there is no observable difference between stationary states above $p_2=0.5$. In general the characteristic length decays by increasing $p_2$ value, but this curve depicts several local minimum and local maximum, which are signaled by vertical dashed lines on the plot. The related $p_2$ values are marked on the top of the figure. Importantly, these critical values mark the transition points which separate the different phases we described earlier.
Next we also measure other parameters to confirm the importance of critical $p_2$ values we detected regarding to the characteristic length. First, we present the mean value of empty sites, $\overline{\rho}_0$, which was already proved to be an insightful quantity to characterize stationary states in previous studies [@avelino_pre12b; @avelino_pre14]. The results for our present model are summarized in Fig. \[empty\]. Again, for better visibility we only show the relevant $p_2<0.5$ region here. Similarly to the characteristic length parameter, the portion of empty sites also shows a non-monotonous dependence as $p_2$ is varied. Notably, the position of the local maximum at $p_2=0.21$ and the position of the local minimum at $p_2=0.28$ are in good agreement with the critical values we found in connection to the characteristic length parameter. On the other hand, the other two critical $p_2$ values, which are also marked by vertical dotted lines in Fig. \[empty\], remain hidden through the lens of $\rho_0$ parameter. The lack of observable breaking points in $\overline{\rho}_0 (p_2)$ function at small $p_2$ values suggests that when inner bidirectional invasions of peer species are too weak then the resulting concentration of empty sites becomes too small to sign the transition points reliably.
![The mean value of empty space, $\overline{\rho}_0$, as a function of $p_2$. The critical $p_2$ values where this quantity starts decaying (at $p_2=0.21$) or growing (at $p_2=0.28$) are in good agreement with the values obtained from the tendency change of characteristic length $\ell$ shown in Fig. \[length\]. $\overline{\rho}_0$, however, is an insensitive parameter to sign the transition points observed at small $p_2$ values. For comparison they are still marked by dotted lines in this plot.[]{data-label="empty"}](fig6)
As we already argued the typical length and the resulting stationary pattern may change significantly by varying the invasion rate between peer species. This effect can be captured indirectly by measuring the standard deviation of $\rho_i(t)$ ($i \in [1,\dots,4] $) functions in the stationary state. When the typical length becomes comparable to the applied system size then the expected symmetry of four species may be broken temporarily which leads to high fluctuation in the time dependence of these functions.
To reveal this effect we monitored the time dependence of all $i=1 \dots 4$ species in the stationary state and calculated their standard deviations. The results for different $p_2$ values are plotted in Fig. \[fluctuations\]. Due to the fundamental symmetry of our model here we present only the average of standard deviations for all species, because this quantity behaves similarly for all four $i$ values. This curve basically confirms our expectation, namely, the positions of local minimum and local maximum values are in good agreement with those obtained for other quantities.
It is worth noting that the enhanced fluctuation in the intermediate $0.04<p_2<0.21$ region is the direct consequence of how partnerships work between peer species. More precisely, as we already noted, species 1 and 3 can form a sort of alliance against species 2 and 4. If species 2 invades species 3 then a neighboring species 1 can strike back. Similarly, the invasion of species 4 against species 1 can be weakened by a neighboring species 3. If $p_2$ is small then this alliance cannot function well and the invasion fronts become smooth due to clear ranks between neighboring species. However, if $p_2$ is high enough then we get back the previously classified $II_4$ model [@avelino_pre12b] where homogeneous domains form four-arm spirals. Between these two extremes the partnership between peer species are functioning partly, which results in highly irregular invasion fronts and enhanced fluctuation of species. This effect can be detected clearly in Fig. \[fluctuations\].
![The average of standard deviations of $\rho_i(t)$ $(i=1\dots 4)$ functions in dependence of $p_2$. Similarly to previous plots the positions of previously detected transition points are marked by vertical lines. The standard deviation is calculated from $10000$ generations in the stationary state where the curve is the average of $600$ independent runs on a $2000 \times 2000$ grid size.[]{data-label="fluctuations"}](fig7)
From the fact how fluctuation depends on $p_2$ and from the representative patterns of different phases shown in Fig. \[snapshots\] we may conclude that partnerships of peer species play a decisive role on the emergence of first two phases at small $p_2$ values. More precisely, here the expected spirals, generated by the cyclic dominance between even and odd labeled species, disappear and they are replaced by the direct competition of alliances composed by peer species. Here the yellow-red species of $\{1+3\}$ and the blue-green species of $\{2+4\}$ are equal in strength because of the symmetry of the food-web shown in Fig. \[def\]. This symmetry, however, can be easily broken if we apply unequal inner invasion strengths for different alliances. A conceptually similar effect has already been observed for three-member alliances in multi-species systems [@szabo_pre08b; @szolnoki_epl15]. More precisely, if two cyclic dominating alliance compete then the one in which the inner invasion is faster can prevail and crowed out the alternative alliance where the inner invasion is slower [@perc_pre07b].
To confirm the possible conceptual similarity with our present model we generalize our model further and introduce alliance-specific inner bidirectional invasion rates in the rest of this work. In particular, we introduce $p_3 \ne p_2$ invasion rates between peer species 2 and 4 as it is shown in the inset of top panel of Fig. \[general\]. Technically, we keep $p_3$ constant while the value of $p_2$ is varied gradually.
As expected, the alliance of $\{1+3\}$ species cannot survive if $p_2$ is too small comparing to $p_3$ because they are dominated by the $\{2+4\}$ alliance where inner invasion, hence the resulting mix of species, is more intensive. The probability of the extinction for different fixed $p_3$ values is plotted in the top panel of Fig. \[general\]. Here an individual simulation was aborted after 5000 steps if no extinction occurred. The plotted values are the average of 1000 independent runs at fixed system size. We stress that the extinction of $\{1+3\}$ species is not a finite-size effect in the present case, as may happen even for the symmetric $p_2=p_3$ model if the system size is too small. Instead, in the present non-symmetric case it is a straightforward consequence of the dominance of $\{2+4\}$ alliance. Naturally, the expected extinction time may depend on the system size, but the extinction probability function converges to a limit case as we increase the system size gradually. This phenomenon is illustrated in the bottom panel of Fig. \[general\], where we plotted the extinction probabilities for different $L$ values at fixed $p_3=0.25$ value. This plot demonstrates that the usage of $L=500$ linear size can predict the large system size limit qualitatively well.
As Figure \[general\] suggests the critical $p_2$ value where the original four-species system becomes a two-species system is decreasing as we decrease $p_3$. In the limit case it tends to $p_2 \approx 0.07$ which is the transition point between the second and third phases in the symmetric model. This behavior indirectly supports our previous conjecture that the patterns characterize the low $p_2$ value regime is principally determined by the competition of alliances composed by peer species.
![Top panel shows the extinction probability of the $\{1+3\}$ alliance as a function of $p_2$ for different fixed values of $p_3$. In this case $p_3$ is the inner bidirectional invasion strength between species 2 and 4, while probability $p_2$ still represents the similar rate between species 1 and 3. The applied invasion rates are summarized in the inset of top panel. These results were obtained from $1000$ simulation runs for each data point in a $500 \times 500$ grid size where every simulation run is aborted after $5000$ steps if no extinction occurs. Bottom panel depicts the extinction probabilities for different system sizes at fixed $p_3=0.25$. The applied linear sizes are shown in the legend.[]{data-label="general"}](fig8a "fig:") ![Top panel shows the extinction probability of the $\{1+3\}$ alliance as a function of $p_2$ for different fixed values of $p_3$. In this case $p_3$ is the inner bidirectional invasion strength between species 2 and 4, while probability $p_2$ still represents the similar rate between species 1 and 3. The applied invasion rates are summarized in the inset of top panel. These results were obtained from $1000$ simulation runs for each data point in a $500 \times 500$ grid size where every simulation run is aborted after $5000$ steps if no extinction occurs. Bottom panel depicts the extinction probabilities for different system sizes at fixed $p_3=0.25$. The applied linear sizes are shown in the legend.[]{data-label="general"}](fig8b "fig:")
DISCUSSION AND CONCLUSIONS
==========================
To maintain biological and ecological diversity is a fundamental challenge for mankind and this cannot be solved without gaining deeper insights about the basic mechanisms which drive permanent evolution. The problem is hard because interactions among competitors can easily result in a complicated food-web with subtle topology. For instance, closed loops in such food-webs can provide a higher level of complexity that cannot be observed in a system where the food-web is characterized by a tree-like graph. In particular, in the presence of loops new kind of solutions, like cyclic time development of competing species may emerge. But beyond topological obstacles an additional difficulty can also emerge when the intensities of interaction are significantly different among competing species or agents [@szabo_pre08].
In this work we followed the latter research path by generalizing a previously established model of four interacting species with intransitive relations [@avelino_pre12b]. Our main motivation was to distinguish the bidirectional inner invasion rate between peer species and the unidirectional invasions characterize primary predator-prey partners. In this way the resulting mixing between peer species, who form a protective alliance against external species, can be tuned via a single parameter.
According to our key observation the strength of inner invasion within an alliance of peer species can play a decisive role on the resulting stationary state and several quantitatively different characteristic patterns can be detected as the related control parameter is varied. In these phases the microscopic mechanisms which are responsible for the emerging pattern can be different. Rather counter-intuitively the primary unidirectional predator-prey type invasions become dominant when the mutual invasions within peer species are intensive, while the competition of alliances acts as the leading pattern formation process when this inner bidirectional invasion is moderate.
In dependence of the mentioned $p_2$ control parameter we have observed five distinct phases where the emerging spatiotemporal patterns are different. The related transition points which separate these phases can be detected accurately by introducing appropriate order parameters. The characteristic length, which is calculated from the spatial autocorrelation function, is proved to be the most sensitive parameter which signals all emerging transition points. The measuring of standard deviation of time-dependent density functions of competing species is also proved to be an effective quantity to detect these transition points. The breaking points of latter parameter agree with those predicted by the $p_2$ dependence of characteristic length. For sake of completeness we have also measured the mean value of empty sites, which was reported as a useful parameter to quantify stationary states in earlier studies [@avelino_pre14; @avelino_pre12b]. The $p_2$ dependence of this quantity signals some of these transition points at the same positions as they were marked by the previously mentioned quantities. This parameter, however, becomes ineffective to sign transition points when its average value is too small due to the moderate inner invasion between peer species.
We have generalized our model further by introducing alliance-specific inner invasion strengths, hence the resulting effective mixture between species 1 and 3 become different from the inner mixture of species 2 and 4. In this way we can break the fundamental symmetry between competing alliances and demonstrate that it has a decisive role on the final outcome if the strengths of inner invasion rates are different enough. Indirectly, the latter observation also supports our argument that in the low $p_2$ value region the leading mechanism which determines the pattern formation is the competition of alliances formed by peer species.
From these observations we can conclude that the diverse invasion strengths between predator-prey partners may play an important role on the final state similarly to the pure topology of food-web. Therefore the classification of stable solutions based solely on the geometry of interactions is not satisfactory and more careful investigations are necessary when we try to predict the final stable solutions of a multi-species interacting system.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This research was supported by the Brazilian agencies CAPES, CNPq, Fundação Araucária, INCT-FCx, Paraiba State Research Foundation (Grant 0015/2019) and by the Hungarian National Research Fund (Grant K-120785).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we propose a new statistical inference method for massive data sets, which is very simple and efficient by combining divide-and-conquer method and empirical likelihood. Compared with two popular methods (the bag of little bootstrap and the subsampled double bootstrap), we make full use of data sets, and reduce the computation burden. Extensive numerical studies and real data analysis demonstrate the effectiveness and flexibility of our proposed method. Furthermore, the asymptotic property of our method is derived.'
author:
- ' Xuejun MA [^1] Shaochen WANG [^2] Wang ZHOU [^3]'
title: ' **Statistical inference in massive datasets by empirical likelihood** '
---
> [*Keywords*]{}: Bootstrap; divide-and-conquer; hypothesis test; empirical likelihood.
> [*MSC2010 subject classifications*]{}: Primary 62G10; secondary 62G05.
Introduction
============
With the rapid development of science and technologies, massive data can be collected at a large speed, especially in internet and financial fields. It is generally recognized that two major challenges in large-scale learning are estimation and inference due to large amount of computation.
For statistical inference on massive data sets, [@Kleiner2014] proposed the bag of little bootstrap (BLB) to assess the quality of estimators. However, they used only a small number of random subsets, and partial observations from each subset. This implies less efficiency in application. So, [@Sengupta2016] developed the subsampled double bootstrap (SDB) method which not noly saves cost computation, but also takes more information of full data than BLB. Compared with the traditional bootstrap (TB), BLB and SDB save the computation cost. However, BLB and SDB have some disadvantages. Similar to traditional bootstrap, they still sample from full dataset, and repeat the whole process many times. The computational cost is still expensive. On the other hand, they do not use the full data since about 63% of data points are contained in each resample.
In addition, [@Wang2018] proposed subsampling method to make inference for Logistic regression. Subsampling method was first proposed by [@Ma2015] for linear regression. Generally speaking, it is a two-step subsampling algorithm. The first step is to get the weight of each data point. In the second step, the weighted estimator is obtained by combining resample subset with subsampling weights. In order to get the optimal subsampling strategy, [@Wang2018] suggested two methods, minimum mean squared error (mMSE) and minimum variance-covariance (mVC). These methods make use of partial data, and rely on the weighted subsampling estimation. Although their efficiency of estimation is high, but their inference does not works well since the subsampling method aims at estimator in nature. Furthermore, one has to estimate the variance-covariance matrix.
In this paper, we propose combining divide-and-conquer (DAC) and empirical likelihood (EL). As we know, DAC is a very effective estimation method for massive data. Firstly, it split entire datasets into $K$ subsets, and each subset is analyzed separately. Secondly, we combine all subset results via average. [@Chen2014] called it “split-and-conquer", and applied it to the generalized linear model with sparse structure. [@Shi2018]) studied the M-estimators with cubic rate of convergence by DAC, and proved that its convergence rate is faster than the original M-estimator. We also refer to [@Zhang2013]. On the other hand, EL ([@Owen1988; @Owen1990; @Owen2001]) is a powerful nonparametric method to make inference on parameters of population without assuming the form of the underlying distribution, such as mean, quantiles and regression parameters. We will take advantage of DAC and EL. Compared with BLB and SDB, we not only take full data information, but also save the cost computation. Our method is very simple and efficient. It has two steps. In the first step, we split the sample into random subsets and the estimate of each subset is obtained. In the second step, the estimates are regarded as one sample from a population so that one can apply EL to this simplified sample.
The rest of this article is organized as follows. In Section \[sec2\], we explain our method in details, and establish its theoretical property. In Section \[sec3\], we assess the finite sample performance of proposed method via Monte Carlo simulations. A real data set is analyzed in Section \[sec4\]. All technical proofs of main results are postponed to Appendix.
Methodology {#sec2}
===========
Let $\mathcal{X}_{n}=\{X_{1}, \dots, X_{n}\}$ be a sample consisting of independent and identically distributed observations form some unknown $q$ dimensional distribution $F$. The parameter of interest is $\theta=\theta(F)\in {\mathbb{R}}^{p}$. Its estimator is ${\widehat}{\theta}_{n}={\widehat}{\theta}(\mathcal{X}_{n})$, which could be maximum likelihood estimator, M-estimator, sample correlation coefficient, U-statistics and many others. In this paper, we mainly focus on the inference of $\theta$. Here is our method.
We first divide the full data set into $K$ blocks randomly, say $\mathcal{X}_{1n_{1}},\dots, \mathcal{X}_{Kn_{K}}$, and then compute $\{ {\widehat}{\theta}_{1n_{1}}={\widehat}{\theta}(\mathcal{X}_{1n_{1}}), \dots {\widehat}{\theta}_{Kn_{K}}={\widehat}{\theta}(\mathcal{X}_{Kn_{K}})\}$. For simplicity, we assume $n_{j}=m$ for all $1\leq j\leq K$. The DAC estimator is defined by $$\widetilde{\theta}_{n}=\frac{1}{K}\sum_{j=1}^{K} {\widehat}{\theta}_{jm}.$$
Now, we discuss the asymptotic properties of $\widetilde{\theta}_n$. We assume that $p$ and $q$ are fixed and $K, m \to \infty$. Besides, we need the following assumptions.
\[assumption1\] $$\sqrt{m}({\widehat}{\theta}_{km}- \theta) = \frac{1}{\sqrt{m}}\sum_{i=1}^{m}\eta_{ki}+R_{k m},\quad k=1, \dots, K,$$ where $\eta_{ki}=(\eta_{ki1},\cdots,\eta_{kip})^\top$ and $R_{km}=(R_{km1},\cdots,R_{kmp})^\top$. Here $\eta_{k1}, \dots, \eta_{km}$ are independent and identically distributed vectors with zero mean, non-singular covariance matrix $\Sigma$ and ${\mathbb{E}}\|\eta_{k1}\|^4<\infty$. $R_{km}$ are the remainder terms, which satisfy $R_{km}=o_{p}(1)$.
\[assumption2\]
1. $R_n:=\frac{1}{\sqrt{K}}\sum_{k=1}^{K} R_{km} =o_{p}(1)$.
2. $\max_{1\leq k\leq K} \| R_{km}\|=o_{p}(m^{-\alpha})$ for some $\alpha>0$.
3. $K=O(m^{4\alpha})$.
Assumption \[assumption1\] is a commonly used condition. This is the Bahadur representation of ${\widehat}{\theta}_{n}$, which has very rich literatures. For example, [@He1996] studied the Bahadur representations for a general class of M-estimators. [@Arcones1996] explored the Bahadur representation of $L_{p}$ regression estimators. Assumption \[assumption2\] is about the rate convergence of the remainder term in the Bahadur representation, i.e., It implies that $$\sqrt{n}(\widetilde{\theta}_{n}- \theta) = \frac{1}{\sqrt{n}}\sum_{k=1}^K\sum_{i=1}^{n}\eta_{ki}+R_{n}.$$ This is a very mild condition.
\[theorem1\] Under Assumptions \[assumption1\]–\[assumption2\], we have $$\sqrt{n}\Big( \widetilde{\theta}_n -\theta \Big)\stackrel{d}{\longrightarrow}N(0,\Sigma),$$ as $m, K\to \infty$, where $\stackrel{d}{\longrightarrow}$ denotes convergence in distribution.
Theorem \[theorem1\] implies that if the usual estimator based on the whole sample has the asymptotic normal distribution, the DAC estimator $ \widetilde{\theta}_n$ has the same asymptotic distribution. However, the covariance matrix $\Sigma$ is usually unknown. One has to estimate it first when applying Theorem \[theorem1\] to make further statistical inference. Sometimes its estimator is hardly obtained. So we propose to use EL as follows.
Since the blocks are disjoint, ${\widehat}{\theta}_{1m}, \dots, {\widehat}{\theta}_{Km}$ are independent. We can regard them as one sample and apply EL to make inference on $\theta$. For notational convenience, let $Y_{km}={\sqrt{m}}{\widehat}{\theta}_{k m}$ and $\mu=\sqrt{m}\theta$. Hence, the empirical likelihood ratio for $\mu$ is given by $$\label{eq20}
\mathcal{R}(\mu)=\max\left\{ \prod_{k=1}^{K}K\omega_{k} ~\Big|~~\sum_{k=1}^{K}\omega_{k}Y_{k m}= \mu, \omega_{k}\geq 0,\quad \sum_{k=1}^{K}\omega_{k}=1 \right\}.$$ By the Lagrange multipliers method, we can find the maximum point $$\omega_{k}= \frac{1}{K} \frac{1}{1 + \lambda^\top(Y_{km}- \mu)},$$ where $\lambda=\lambda(\mu)$ satisfies the equation given by $$\label{eq24}
0=\frac{1}{K}\sum_{k=1}^{K} \frac{Y_{km}- \mu}{1 + \lambda^\top(Y_{km}- \mu)}.$$ As in [@Owen1990], we can get the follow Wilks’ theorem.
\[theorem2\] Under Assumptions \[assumption1\]–\[assumption2\], we have $$-2 \log \mathcal{R}(\mu) \stackrel{d}{\longrightarrow} \chi^{2}_{p}$$ as $K, m\to \infty$.
The accuracy of each block estimator increases as $m$ increases. The power of EL increases as $K$ becomes greater. So there is a trade-off between $K$ and $m$. But we are studying massive data, $K$ and $m$ are large enough to guarantee the accuracy of each step’s inference. In simulations, we set $n=10^5$, $K=\{50, 100, 150\}$. The numerical results show that our proposed method is not sensitive to $K$.
Compared with the BLB and SDB, our method provides a specific asymptotic distribution to make inference on $\theta$. It is unnecessary to apply bootstrap to specify critical values. This reduces the computation burden a lot.
Now, we discuss the computational times of our proposed method, BLB and SDB. Let $t(m)$ be the computational time to estimate ${\widehat}{\theta}_{m}$ based on a sample of size $m$. $c(K)$ denotes the cost time of EL based on $K$ blocks. Table \[table1\] presents the comparison. In Table \[table1\], the column “Estimation time" means the corresponding time measured in second when one runs Case 1 of Example \[example2\] in Section \[sec3\]. As for the other notation, $b$ is the subset size, $S$ is the number of subsets, $R$ is the number of sampled subsets. The detailed setting is shown in Section \[sec3\]. We run R language with version 3.5.2 in the desktop computer with Intel(R) Core(TM)CPU i7-4770 3.40GHz processor and 16.0GB RAM. Here we select $b$ of BLB and SDB to be a litle big so that most information of data can be used. From Table \[table1\], one can see that our method reduces the computation burden a lot.
Method Cost time Estimation time (seconds)
------------ ------------------------ ------------- ---------------------------
BLB $R\times S\times t(b)$ $b=n^{0.6}$ 26.528
$b=n^{0.8}$ 209.810
SDB $ S\times t(b)$ $b=n^{0.6}$ 6.810
$b=n^{0.8}$ 38.363
Our method $K\times t(m) + c(K)$ $K=50$ 1.031
$K=100$ 1.158
$K=150$ 1.285
: The computational time for different methods.
\[table1\]
Simulations {#sec3}
===========
In this section, we investigate the finite sample performance of our proposed method. We also compare it with several existing alternatives in the literature. Example \[example1\] is designed for linear model. Example \[example2\] is for Logistic regression. Based on the suggestion in [@Shi2018], the numbers of subsets for steps 1 and 2 are 2000 and $10^4$ respectively in mMSE and mVC. As in [@Kleiner2014] and [@Sengupta2016], we set subset size $b=n^{\gamma}$ with $\gamma=0.6$ and $0.8$. The numbers of subsets in BLB and SDB are 20 and 500 respectively. The number of sampled subset is 100 in BLB. Furthermore, we set the replications of TB to be 100, $K=\{50,100, 150\}$ and $n=10^5$. We report empirical sizes and powers for different distributions. Each experiment is repeated 500 times at the nominal level $\alpha=0.05$.
\[example1\] We consider the linear model: $Y=X^\top\beta + \varepsilon$. Here $\beta$ is a $7\times 1$ vector with all coordinates 0.2 and $X$ comes from the 7-dimensional multivariate normal distribution $N(0, \Sigma)$, where $\Sigma=(\rho_{ij})$ and $\rho_{ij}=0.2^{|i- j|}$. $\varepsilon$ comes from three distributions:
- The normal distribution, $N(0,1)$.
- $t$ distribution, $t$(10).
- Mixed normal distribution, $0.5 N(1, 1) + 0.5 N(-1, 1)$.
Table \[table2\] shows the empirical sizes when we are testing $H_0: \beta_j=0.2$. Table \[table3\] summaries the lengths of confidence intervals by different methods. We can obtain the following conclusions.
1. Regardless of distribution of $\varepsilon$, the empirical size of our proposed method outperformes BLB, SDB, and is slightly better than TB at many cases. Our method is not sensitive to the selection of $K$ since their results are similar.
2. The empirical sizes of BLB and SDB are close to zero. The possible reason is that the lengths of their confidence intervals are very long, especially when $\gamma=0.6$. Compared with BLB and SDB, our method is similar to TB. We also note that in Table \[table3\], the lengths in one row are almost the same. This is due to the fact that all $\beta_j$ are set to be equal.
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 0.044 0.046 0.046 0.060 0.062 0.056 0.064
K=100 0.060 0.036 0.046 0.054 0.048 0.054 0.074
K=150 0.042 0.034 0.038 0.044 0.044 0.056 0.068
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.002
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.068 0.042 0.048 0.074 0.060 0.080 0.076
2 K=50 0.044 0.080 0.038 0.060 0.060 0.060 0.052
K=100 0.032 0.066 0.032 0.054 0.066 0.066 0.046
K=150 0.030 0.054 0.032 0.050 0.064 0.058 0.058
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.002 0.002 0.000 0.000 0.000 0.004
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.044 0.078 0.040 0.064 0.072 0.082 0.080
3 K=50 0.066 0.060 0.042 0.054 0.062 0.052 0.060
K=100 0.054 0.060 0.042 0.062 0.054 0.046 0.046
K=150 0.060 0.056 0.048 0.064 0.054 0.050 0.048
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.002 0.002 0.000 0.002 0.000 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.070 0.074 0.050 0.082 0.062 0.058 0.066
: Empirical sizes comparison for Example \[example1\].
\[table2\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 0.013 0.013 0.013 0.013 0.013 0.013 0.013
K=100 0.013 0.013 0.013 0.013 0.013 0.013 0.013
K=150 0.013 0.013 0.013 0.013 0.013 0.013 0.013
BLB($n^{0.6}$) 0.106 0.107 0.111 0.110 0.109 0.109 0.106
BLB($n^{0.8}$) 0.034 0.035 0.034 0.035 0.035 0.034 0.034
SDB($n^{0.6}$) 0.127 0.129 0.129 0.129 0.129 0.129 0.127
SDB($n^{0.8}$) 0.042 0.042 0.043 0.042 0.042 0.042 0.042
TB 0.012 0.012 0.012 0.012 0.012 0.012 0.012
2 K=50 0.014 0.014 0.014 0.014 0.014 0.014 0.014
K=100 0.014 0.015 0.015 0.015 0.015 0.014 0.014
K=150 0.014 0.015 0.014 0.015 0.014 0.015 0.014
BLB($n^{0.6}$) 0.120 0.122 0.122 0.121 0.122 0.123 0.118
BLB($n^{0.8}$) 0.037 0.039 0.038 0.038 0.038 0.038 0.037
SDB($n^{0.6}$) 0.142 0.144 0.144 0.144 0.144 0.145 0.141
SDB($n^{0.8}$) 0.046 0.047 0.047 0.047 0.048 0.048 0.046
TB 0.014 0.014 0.014 0.014 0.014 0.014 0.014
3 K=50 0.018 0.018 0.018 0.018 0.018 0.018 0.018
K=100 0.018 0.018 0.018 0.018 0.018 0.018 0.018
K=150 0.018 0.018 0.018 0.018 0.018 0.018 0.018
BLB($n^{0.6}$) 0.150 0.155 0.155 0.154 0.155 0.153 0.150
BLB($n^{0.8}$) 0.047 0.048 0.049 0.049 0.049 0.048 0.047
SDB($n^{0.6}$) 0.179 0.182 0.183 0.182 0.182 0.182 0.178
SDB($n^{0.8}$) 0.059 0.060 0.060 0.060 0.060 0.060 0.059
TB 0.017 0.018 0.017 0.018 0.017 0.017 0.017
: Lengths of confidence interval comparison for Example \[example1\].
\[table3\]
\[example2\] In this example, we consider a $p$-dimensional multiple Logistic regression model. Given covariates $Z_{i}\in {\mathbb{R}}^{p}$, $${\mathbb{P}}(Y_{i}=1|Z_{i})=\frac{\exp(Z_{i}^\top \beta)}{1+ \exp(Z_{i}^\top \beta)}, \quad i=1,\dots, n,$$ where $Y_{i}\in\{0,1\}$ is the response and $\beta$ is a $p$-dimensional unknown parameter. The interesting problem is to test the hypothesis: $\beta_{j}=\beta_{j0}$ for some $1\leq j\leq p$, or $\beta=\beta_{0}$.
We let $\beta$ be a $7\times 1$ vector with all coordinates equal to 0.2. $Z_i$ comes from seven distributions which were used in [@Shi2018].
- $N(0, \Sigma)$, $\Sigma=(\rho_{ij})$ with $\rho_{ij}=0.5^{I(i\neq j)}$, where $I(\cdot)$ is the indicator function.
- $N(1.5, \Sigma)$.
- $0.5 N(1, \Sigma) + 0.5 N(-1, \Sigma)$.
- The multivariate $t$ distribution $t_{3}(0, \Sigma)/10$, with degrees of freedom 3.
- The multivariate exponential distribution whose components are independent and each has an exponential distribution with a rate parameter of 2.
- $0.5 N(-2.14, \Sigma) + 0.5 N(-2.9, \Sigma)$.
Here, Cases 2 and 5 produce imbalanced data. Case 6 produces rare events data.
Tables \[table4\]-\[table7\] show the empirical sizes and powers. When we consider powers of test, the null hypothesis is that the parameter $\beta_j$ is zero. Tables \[table8\] and \[table9\] summarize the lengths of confidence intervals. We draw the following conclusions.
1. Regardless of imbalanced data or the rare events data, the empirical sizes of our proposed method are close to the nominal level, which implies our method performs well. Moreover, the empirical power is very close to 1. The differences among three values of K is not significantly.
2. As $\gamma$ increases, the performance of BLB and SDB becomes better. However, they are worse than our method. TB slightly inflated rejection probabilities under the null hypothesis. From Tables \[table8\] and \[table9\], the length of confidence intervals decreases as $\gamma$ increases. When $\gamma=0.6$, it is ten times as long as TB, which results in the lower empirical size.
3. In terms of empirical powers, mVC and mMSE outperform BLB and SDB. Compared with mVC and mMSE, our method is better, especially in the case imbalanced data and rare events data in terms of empirical sizes and powers.
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 0.054 0.064 0.052 0.052 0.038 0.040 0.048
K=100 0.056 0.062 0.044 0.050 0.050 0.038 0.056
K=150 0.070 0.060 0.062 0.050 0.052 0.050 0.048
mVC 0.056 0.076 0.058 0.044 0.038 0.086 0.072
mMSE 0.068 0.046 0.068 0.062 0.074 0.078 0.044
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.002 0.000
BLB($n^{0.8}$) 0.004 0.002 0.000 0.004 0.000 0.000 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.066 0.068 0.058 0.056 0.042 0.042 0.072
2 K=50 0.044 0.050 0.054 0.058 0.066 0.052 0.060
K=100 0.050 0.046 0.050 0.060 0.054 0.032 0.052
K=150 0.054 0.052 0.060 0.070 0.054 0.040 0.052
mVC 0.076 0.098 0.092 0.086 0.094 0.078 0.070
mMSE 0.076 0.086 0.086 0.074 0.082 0.098 0.060
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.002 0.000 0.000 0.004 0.002 0.000 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.062 0.054 0.064 0.076 0.074 0.050 0.070
3 K=50 0.060 0.054 0.058 0.040 0.040 0.040 0.058
K=100 0.060 0.048 0.048 0.036 0.032 0.046 0.058
K=150 0.074 0.044 0.050 0.054 0.040 0.046 0.056
mVC 0.054 0.064 0.084 0.070 0.058 0.066 0.058
mMSE 0.066 0.078 0.048 0.066 0.084 0.072 0.084
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.002 0.000 0.000 0.002
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.080 0.054 0.076 0.060 0.054 0.050 0.066
: Empirical sizes comparison for Cases 1-3 in Example \[example2\].
\[table4\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
4 K=50 0.058 0.058 0.058 0.048 0.062 0.068 0.062
K=100 0.056 0.062 0.058 0.054 0.052 0.054 0.044
K=150 0.046 0.062 0.058 0.054 0.058 0.054 0.060
mVC 0.078 0.072 0.070 0.080 0.054 0.070 0.066
mMSE 0.070 0.080 0.058 0.068 0.060 0.066 0.068
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.002 0.002 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.070 0.082 0.062 0.074 0.064 0.060 0.084
5 K=50 0.066 0.062 0.046 0.074 0.046 0.058 0.048
K=100 0.056 0.058 0.054 0.070 0.048 0.066 0.050
K=150 0.060 0.060 0.066 0.084 0.044 0.068 0.052
mVC 0.060 0.082 0.066 0.090 0.070 0.068 0.066
mMSE 0.074 0.074 0.048 0.068 0.052 0.070 0.064
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.002 0.000 0.000 0.002
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.062 0.070 0.074 0.070 0.060 0.074 0.064
6 K=50 0.064 0.070 0.048 0.050 0.070 0.050 0.066
K=100 0.060 0.078 0.048 0.068 0.058 0.042 0.052
K=150 0.062 0.074 0.056 0.062 0.056 0.044 0.060
mVC 0.126 0.160 0.132 0.154 0.134 0.152 0.124
mMSE 0.140 0.152 0.146 0.146 0.154 0.142 0.152
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.002 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.078 0.082 0.068 0.076 0.076 0.054 0.070
: Empirical sizes comparison for Cases 4-6 in Example \[example2\].
\[table5\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=100 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mVC 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mMSE 1.000 1.000 1.000 1.000 1.000 1.000 1.000
BLB($n^{0.6}$) 0.884 0.894 0.848 0.896 0.864 0.872 0.880
BLB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SDB($n^{0.6}$) 0.910 0.900 0.910 0.908 0.876 0.900 0.884
SDB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TB 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2 K=50 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=100 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mVC 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mMSE 1.000 1.000 1.000 1.000 1.000 1.000 1.000
BLB($n^{0.6}$) 0.464 0.472 0.428 0.458 0.480 0.478 0.488
BLB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SDB($n^{0.6}$) 0.006 0.010 0.006 0.008 0.010 0.004 0.008
SDB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TB 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3 K=50 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=100 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mVC 0.944 0.960 0.944 0.966 0.950 0.960 0.968
mMSE 0.976 0.976 0.978 0.976 0.952 0.986 0.962
BLB($n^{0.6}$) 0.030 0.070 0.042 0.066 0.042 0.044 0.032
BLB($n^{0.8}$) 0.994 0.988 0.998 0.998 0.994 0.996 0.994
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.998 1.000 1.000 1.000 0.998 1.000 0.998
TB 1.000 1.000 1.000 1.000 1.000 1.000 1.000
: Empirical powers comparison for Cases 1-3 in Example \[example2\].
\[table6\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
4 K=50 0.976 0.982 0.972 0.962 0.972 0.962 0.964
K=100 0.972 0.970 0.964 0.956 0.978 0.960 0.950
K=150 0.974 0.976 0.956 0.956 0.962 0.956 0.966
mVC 0.354 0.388 0.340 0.364 0.366 0.356 0.372
mMSE 0.400 0.352 0.384 0.416 0.368 0.356 0.376
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.002 0.000 0.002
BLB($n^{0.8}$) 0.206 0.230 0.224 0.230 0.210 0.226 0.230
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.014 0.018 0.006 0.016 0.010 0.012 0.006
TB 0.982 0.980 0.984 0.968 0.986 0.972 0.970
5 K=50 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=100 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mVC 1.000 0.998 1.000 1.000 0.998 1.000 1.000
mMSE 1.000 1.000 1.000 1.000 1.000 1.000 1.000
BLB($n^{0.6}$) 0.394 0.418 0.406 0.442 0.424 0.466 0.394
BLB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SDB($n^{0.6}$) 0.002 0.000 0.012 0.000 0.010 0.004 0.004
SDB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TB 1.000 1.000 1.000 1.000 1.000 1.000 1.000
6 K=50 0.996 0.996 0.998 0.984 0.994 0.996 0.994
K=100 0.994 0.998 0.998 0.990 0.988 0.994 0.996
K=150 0.994 1.000 0.998 0.986 0.992 0.998 0.990
mVC 0.918 0.922 0.944 0.930 0.958 0.922 0.958
mMSE 0.942 0.962 0.954 0.944 0.950 0.954 0.958
BLB($n^{0.6}$) 0.000 0.000 0.006 0.000 0.002 0.000 0.000
BLB($n^{0.8}$) 0.312 0.310 0.358 0.370 0.346 0.324 0.354
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.020 0.030 0.040 0.022 0.042 0.040 0.030
TB 0.994 0.998 0.996 0.992 0.994 0.996 0.992
: Empirical powers comparison for Cases 4-6 in Example \[example2\].
\[table7\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 0.037 0.037 0.037 0.037 0.037 0.037 0.037
K=100 0.037 0.037 0.037 0.037 0.037 0.037 0.037
K=150 0.038 0.038 0.037 0.037 0.037 0.037 0.037
mVC 0.048 0.048 0.048 0.048 0.048 0.048 0.048
mMSE 0.047 0.047 0.047 0.047 0.047 0.047 0.047
BLB($n^{0.6}$) 0.313 0.313 0.319 0.311 0.311 0.316 0.313
BLB($n^{0.8}$) 0.097 0.096 0.098 0.096 0.097 0.097 0.098
SDB($n^{0.6}$) 0.370 0.370 0.370 0.370 0.369 0.370 0.369
SDB($n^{0.8}$) 0.120 0.120 0.120 0.121 0.121 0.121 0.120
TB 0.035 0.035 0.035 0.035 0.035 0.035 0.035
2 K=50 0.050 0.049 0.049 0.049 0.049 0.050 0.049
K=100 0.050 0.050 0.050 0.050 0.050 0.050 0.050
K=150 0.051 0.051 0.051 0.051 0.051 0.051 0.051
mVC 0.050 0.050 0.050 0.050 0.050 0.050 0.050
mMSE 0.047 0.047 0.047 0.047 0.047 0.047 0.047
BLB($n^{0.6}$) 0.416 0.424 0.423 0.422 0.419 0.424 0.423
BLB($n^{0.8}$) 0.131 0.133 0.130 0.132 0.132 0.133 0.132
SDB($n^{0.6}$) 0.503 0.502 0.500 0.500 0.499 0.500 0.501
SDB($n^{0.8}$) 0.162 0.162 0.162 0.162 0.162 0.162 0.162
TB 0.047 0.047 0.047 0.047 0.047 0.047 0.047
3 K=50 0.082 0.082 0.082 0.082 0.082 0.082 0.082
K=100 0.083 0.083 0.083 0.083 0.083 0.083 0.083
K=150 0.084 0.084 0.084 0.083 0.083 0.084 0.083
mVC 0.103 0.103 0.103 0.103 0.103 0.103 0.103
mMSE 0.096 0.095 0.096 0.096 0.096 0.096 0.095
BLB($n^{0.6}$) 0.707 0.702 0.697 0.696 0.686 0.701 0.690
BLB($n^{0.8}$) 0.218 0.219 0.217 0.217 0.219 0.214 0.216
SDB($n^{0.6}$) 0.827 0.826 0.827 0.824 0.825 0.824 0.825
SDB($n^{0.8}$) 0.270 0.268 0.268 0.269 0.268 0.268 0.269
TB 0.078 0.078 0.078 0.078 0.078 0.078 0.078
: Lengths of confidence interval for Cases 1-3 in Example \[example2\]
\[table8\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
4 K=50 0.207 0.207 0.208 0.207 0.206 0.206 0.206
K=100 0.213 0.213 0.213 0.214 0.213 0.213 0.212
K=150 0.219 0.218 0.218 0.218 0.218 0.218 0.218
mVC 0.250 0.249 0.250 0.250 0.250 0.250 0.250
mMSE 0.240 0.240 0.241 0.240 0.241 0.241 0.240
BLB($n^{0.6}$) 1.793 1.790 1.810 1.785 1.800 1.772 1.792
BLB($n^{0.8}$) 0.531 0.541 0.539 0.534 0.531 0.540 0.538
SDB($n^{0.6}$) 2.129 2.118 2.119 2.128 2.128 2.128 2.131
SDB($n^{0.8}$) 0.662 0.663 0.661 0.662 0.663 0.663 0.661
TB 0.191 0.191 0.192 0.190 0.192 0.190 0.190
5 K=50 0.052 0.052 0.052 0.052 0.052 0.052 0.052
K=100 0.052 0.052 0.052 0.052 0.052 0.052 0.052
K=150 0.053 0.053 0.053 0.053 0.053 0.053 0.053
mVC 0.065 0.065 0.065 0.065 0.065 0.065 0.065
mMSE 0.064 0.064 0.064 0.064 0.064 0.064 0.064
BLB($n^{0.6}$) 0.441 0.435 0.441 0.440 0.442 0.439 0.430
BLB($n^{0.8}$) 0.138 0.136 0.138 0.139 0.138 0.136 0.138
SDB($n^{0.6}$) 0.523 0.523 0.522 0.521 0.523 0.522 0.523
SDB($n^{0.8}$) 0.170 0.169 0.169 0.170 0.169 0.169 0.169
TB 0.049 0.049 0.049 0.049 0.049 0.049 0.049
6 K=50 0.178 0.178 0.178 0.178 0.178 0.178 0.179
K=100 0.181 0.181 0.182 0.182 0.182 0.182 0.182
K=150 0.185 0.186 0.185 0.185 0.186 0.185 0.185
mVC 0.098 0.098 0.098 0.098 0.098 0.098 0.098
mMSE 0.093 0.093 0.093 0.093 0.093 0.093 0.093
BLB($n^{0.6}$) 1.521 1.512 1.538 1.516 1.522 1.530 1.526
BLB($n^{0.8}$) 0.466 0.466 0.472 0.459 0.463 0.467 0.468
SDB($n^{0.6}$) 1.806 1.815 1.803 1.813 1.816 1.806 1.811
SDB($n^{0.8}$) 0.579 0.578 0.576 0.578 0.579 0.577 0.579
TB 0.168 0.167 0.168 0.169 0.167 0.168 0.167
: Lengths of confidence interval for Cases 4-6 in Example \[example2\]
\[table9\]
A real data {#sec4}
===========
In this section, we apply the proposed method to a census income data set, which aims to determine whether a person makes \$50K or more a year. The data can be obtained from <https://archive.ics.uci.edu/ml/datasets/census+income>, with 48,842 observations in total. As in [@Wang2018], the response variable is whether a person’s income exceeds \$50K a year. The explanatory variables are as follows:
- $X_{1}$: age
- $X_{2}$: final weight (Fnlwgt)
- $X_{3}$: highest level of education in numerical form (Education-num)
- $X_{4}$: capital loss (Capital-loss);
- $X_{5}$: hours worked per week (Hours-per-week).
There are 11,687 individuals (23.929%) in the data whose income exceeds \$50K a year. In order to eliminate the effect of scale, we have scaled and centered each explanatory variable so that they have mean 0 and variance 1. To evaluate the performance of the above methods, we replicate each method 500 times since these methods split sample randomly. We report the average estimate and the average proportion of rejecting the null hypothesis that the regression coefficient is zero by all methods.
Table \[table10\] shows the result. The traditional Logistic regression (TLR) indicates that all coefficients are significant, not equal to 0 under the nominal level 5%. Our method is consistent to the traditional Logistic regression. Compared with $K=150$ and $K=50$, $K=100$ is better since each block sample contains enough data points. For $\beta_{3}$, the average proportion of rejecting the null hypothesis by mVC, mMSE, BLB and SDB are much lower than 1 while ours are 1. It implies that our proposed method works in cases where others don’t work.
[cccc ccc]{}\
Method & $\beta_{1}$ & $\beta_{2}$ & $\beta_{3}$ & $\beta_{4}$ & $\beta_{5}$ & $\beta_{6}$\
\
TLR & -1.514 & 0.630 & 0.063 & 0.877 & 0.226 & 0.521\
\
K=50 & -1.525 & 0.637 & 0.063 & 0.885 & 0.229 & 0.529\
K=100 & -1.537 & 0.644 & 0.063 & 0.896 & 0.231 & 0.538\
K=150 & -1.549 & 0.651 & 0.062 & 0.905 & 0.234 & 0.547\
mVC & -1.510 & 0.627 & 0.066 & 0.876 & 0.225 & 0.527\
mMSE & -1.514 & 0.634 & 0.059 & 0.876 & 0.229 & 0.518\
\
TLR & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000\
\
K=50 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000\
K=100 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000\
K=150 & 1.000 & 1.000 & 1.000 & 1.000 & 0.990 & 1.000\
mVC & 1.000 & 1.000 & 0.860 & 1.000 & 1.000 & 1.000\
mMSE & 1.000 & 1.000 & 0.770 & 1.000 & 1.000 & 1.000\
BLB($n^{0.6}$) & 1.000 & 1.000 & 0.000 & 1.000 & 0.910 & 1.000\
BLB($n^{0.8}$) & 1.000 & 1.000 & 0.750 & 1.000 & 1.000 & 1.000\
SDB($n^{0.6}$) & 1.000 & 1.000 & 0.000 & 1.000 & 1.000 & 1.000\
SDB($n^{0.8}$) & 1.000 & 1.000 & 0.010 & 1.000 & 1.000 & 1.000\
\[table10\]
Appendix {#appendix .unnumbered}
========
[**Proof of Theorem \[theorem1\]**]{}.
From Assumption \[assumption1\], we can get $$\sqrt{m} ({\widehat}{\theta}_{km}-\theta )=\frac{1}{\sqrt{m}}\sum_{i=1}^{m}\eta_{ki}+R_{km}.$$ Hence, $$\begin{aligned}
\label{eqA1}
\sqrt{n}(\widetilde{\theta} - \theta ) & = \sqrt{n} \Big{(}\frac{1}{K}\sum_{k=1}^{K} {\widehat}{\theta}_{km} -\theta \Big{)} \notag \\
&= \frac{\sqrt{n}}{K}\sum_{k=1}^{K} ({\widehat}{\theta}_{km} -\theta ) \notag \\
&= \frac{1}{\sqrt{K}}\sum_{k=1}^{K} \sqrt{m} ({\widehat}{\theta}_{km} -\theta ) \notag \\
&=\frac{1}{\sqrt{K}}\sum_{k=1}^{K} W_{km}+ \frac{1}{\sqrt{K}}\sum_{k=1}^{K} R_{km},\end{aligned}$$ where $W_{km}=\frac{1}{\sqrt{m}}\sum_{i=1}^{m}\eta_{k,i}$. From Assumption \[assumption2\], we get the last term in (\[eqA1\]) is $o_{p}(1)$.
Now, we prove that $\frac{1}{\sqrt{K}}\sum_{k=1}^{K} W_{km} $ has the asymptotic normality distribution. Let $V_{km}=c^\top W_{km}$, then ${\mathbb{E}}(V_{km})=0, Var(V_{km})=c^\top \Sigma c=\sigma^2$. By the Cramér-Wold theorem, we only need to prove $$\frac{1}{\sqrt{K}}\sum_{k=1}^{K} V_{km} \stackrel{d}{\longrightarrow}N(0,\sigma^2)$$ for each fixed $c\in\mathbb{R}^p\setminus\{0\}$.
Since $V_{km}$ is a normalized sum of $K$ independent and identically distributed random variables, it follows from Linderberg’s CLT that $${\mathbb{E}}e^{\imath tV_{mk}(u)}=e^{-t^2\sigma^2/2}+o(t^2),$$ for any real $t\in {\mathbb{R}}$. Here $\imath=\sqrt{-1}$.
Hence, $$\begin{aligned}
&{\mathbb{E}}\exp\left\{ \imath t\frac{1}{\sqrt{K}}\sum_{k=1}^K V_{km}\right\}\\
&=\left({\mathbb{E}}e^{\imath tK^{-1/2} V_{km}}\right)^K\\
&=\Big(e^{-(tK^{-1/2})^2\sigma^2/2}+o(tK^{-1/2})^2\Big)^K\to e^{-t^2\sigma^2/2},\end{aligned}$$ as $K\to\infty$. The proof of Theorem \[theorem1\] is completed.
For proving Theorem \[theorem2\], we need the following two lemmas.
\[lem-1\] Let $Z_{K}=\max_{1\leq k\leq K} \|Y_{km}- \mu \|$. Under the conditions of Theorem \[theorem2\], we have $$Z_{K}=o_p(K^{1/2})$$ as $K, m\to \infty$.
Note that $$Y_{km} - \mu =\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}+R_{km}.$$ Since $\eta_{ki}$’s are independent and identically distributed random vectors with mean zero and finite fourth moment, $$\begin{aligned}
{\mathbb{P}}\Big(\max_{1\leq k\leq K} \Big\|\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}\Big\|>\epsilon \sqrt{K}\Big)
&\leq \sum_{k=1}^K {\mathbb{P}}\Big( \Big\|\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}\Big\|>\epsilon \sqrt{K} \Big) \\
&\leq K ( \epsilon \sqrt{K})^{-4} {\mathbb{E}}\Big\|\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}\Big\|^4\\
&=O(K^{-1})\end{aligned}$$ as $K, m\to \infty$. On the other hand, from the Assumption \[assumption2\], we get $${\mathbb{P}}\Big(\max_{1\leq i\leq K} \|R_{km}\|>\epsilon K^{1/2}\Big)\to 0$$ as $K, m\to \infty$. So we can complete the proof.
\[lem-2\] Let $$S_K =\frac{1}{K}\sum_{k=1}^{K}(Y_{km}-\mu)(Y_{km}-\mu)^\top.$$ Under the conditions of Theorem \[theorem2\], we have $S_K \stackrel{p}{\longrightarrow} \Sigma $ as $K,m\to \infty$.
Note that $$\begin{aligned}
& (Y_{km}-\mu)(Y_{km}-\mu)^\top \\
&=\Big(\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}+R_{km}\Big)\Big(\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}+R_{km}\Big)^\top\\
&=\frac{1}{m}\Big(\sum_{i=1}^m\eta_{ki}\Big)\Big(\sum_{i=1}^m\eta_{ki}\Big)^\top
+2\frac{1}{\sqrt{m}}\Big(\sum_{i=1}^m\eta_{ki}\Big)R_{km}^\top + R_{km}R_{km}^\top.\end{aligned}$$ Now we consider the convergence of the $(j,l)$ element of $S_K$ for $1\leq j,l\leq p$. For any $\epsilon>0$, $$\begin{aligned}
& {\mathbb{P}}\Big(K^{-1}\Big|\frac{1}{\sqrt{m}}\sum_{k=1}^K (\sum_{i=1}^m\eta_{kij})R_{kml}\Big| > \epsilon\Big)\\
&\leq {\mathbb{P}}\Big(\max_{1\leq k\leq K}\Big| \Big(\sum_{i=1}^m\eta_{kij}\Big)R_{kml}\Big| > \sqrt{m}\epsilon\Big) \\
&\leq \sum_{k=1}^K {\mathbb{P}}\Big(| \sum_{i=1}^m\eta_{kij}| >C m^{1/2+\alpha} \Big) + {\mathbb{P}}\Big(\max_{1\leq k\leq K}|R_{kml}| >C^{-1}m^{-\alpha} \epsilon \Big),\end{aligned}$$ where $C$ is a constant which will go to infinity finally. Since $\eta_{kij}$, $k,i=1,2,\cdots$ are independent and identically distributed random variables with mean zero and finite fourth moment, $${\mathbb{P}}\Big(| \sum_{i=1}^m\eta_{kij}| > Cm^{1/2+\alpha}\Big)\leq C^{-4}m^{-2-4\alpha} {\mathbb{E}}| \sum_{i=1}^m\eta_{kij}|^4 =C^{-4}O(m^{-4\alpha}).$$ It follows from Assumption \[assumption2\] that $${\mathbb{P}}\Big(K^{-1}|\sum_{k=1}^K (\sum_{i=1}^m\eta_{kij})R_{kml}| > \epsilon\Big) \to 0,$$ if we let $K,m\to \infty$ as a first step, then let $C\to \infty$ as a second step. Similarly, $${\mathbb{P}}\Big(K^{-1}|\sum_{k=1}^K R_{kmj}R_{kml}|>\epsilon\Big)\to 0$$ as $K,m\to \infty$. It remains to consider $$\label{eqA2}
(Km)^{-1} \sum_{k=1}^K\Big(\sum_{i=1}^m \eta_{kij}\Big)\Big(\sum_{i=1}^m \eta_{kil}\Big)=(Km)^{-1} \sum_{k=1}^K\sum_{i=1}^m \eta_{kij}\eta_{kil}+(Km)^{-1} \sum_{k=1}^K\sum_{1\leq i_1\not=i_2\leq m}\eta_{ki_1j}\eta_{ki_2l}.$$ The second sum on the right hand side of equality in converges to zero in probability by Markov’s inequality as $K,m\to \infty$. The first sum on the right hand side of equality in converges to the $(j,l)$ element of $\Sigma$ in probability by law of large numbers. Combining all above completes the proof.
[**Proof Theorem \[theorem2\]**]{}. (\[eq24\]) can be re-expression as $$\label{eqA3}
f(\lambda)=\frac{1}{K}\sum_{k=1}^{K} \frac{Y_{km}- \mu}{1 + \lambda^\top(Y_{km}- \mu)}=0.$$ Let $\lambda=\| \lambda\|\theta$, where $\theta\in \Theta$ is a unit vector, and $\Theta$ denotes the set of unit vector in $\mathbb{R}^p$. In the following, we show $$\|\lambda\|= O_{p}(K^{-1/2}).$$
Let $$U_{km}= \lambda^\top(Y_{km}- \mu).$$ Using the representation $1/(1+ U_{km})=1 - U_{m,k}/ (1+U_{km})$, and $\theta^\top f(\lambda)=0$, we have $$\label{eqA4}
\theta^\top (\bar{Y}_{Km}- \mu) =\| \lambda \| \theta^\top \tilde{S} \theta,$$ where $$\tilde{S}=\frac{1}{K}\sum_{k=1}^{K}\frac{(Y_{km}-\mu)(Y_{km}-\mu)^\top}{1+ U_{km}}$$ and $$\bar{Y}_{Km} = \frac{1}{K} \sum_{k=1}^{K}Y_{km}.$$ Since $0<\omega_{k}<1$, we have $1+U_{m, k} >0$, hence $$\begin{aligned}
\|\lambda\|\theta^\top S_K \theta & \leq \|\lambda\|\theta^\top \tilde{S} \theta (1+ \max_{1\leq k\leq K} U_{km})\\
&\leq \|\lambda\|\theta^\top \tilde{S} \theta (1+ \|\lambda\| Z_{K})\\
&= \theta^\top (\bar{Y}_{Km}- \mu)(1+ \|\lambda\| Z_{K}).\end{aligned}$$ The last equality follows by (\[eqA4\]). Hence, $$\|\lambda\|[ \theta^\top S_K\theta - \theta^\top(\bar{Y}_{Km}-\mu) Z_{K}] \leq \theta^\top (\bar{Y}_{Km}-\mu).$$ By the central limit theorem, $\bar{Y}_{Km}-\mu=O_{p}(K^{-1/2})$. Lemma \[lem-1\] shows $Z_{K}=o_{p}(K^{1/2})$. By Lemma \[lem-2\], the smallest eigenvalue of $S$ always has a positive lower bound in probability. Combing these three facts, it gives $$\|\lambda\|[ \theta^\top S_K\theta + O_{p}(K^{-1/2}) o_{p}(K^{1/2}) ] = O_{p}(K^{-1/2}).$$ So, we have $$\|\lambda\|= O_{p}(K^{-1/2}).$$ Furthermore, $$\label{eqA5}
\max_{1\leq k\leq K}|U_{km}|= O_{p}(K^{-1/2})o_{p}(K^{-1/2})=o_{p}(1).$$ Expanding (\[eqA3\]) gives $$\begin{aligned}
\label{eqA6}
0&=\frac{1}{K}\sum_{k=1}^{n} (Y_{km}- \mu) \Big{\{} 1 - U_{km} + \frac{U_{km}^{2}}{1 + U_{km}} \Big{\}} \notag \\
&=(\bar{Y}_{km}- \mu) - S_K \lambda + \frac{1}{K}\sum_{k=1}^{K} \frac{(Y_{km}- \mu)U_{km}^{2}}{1 + U_{km}}.\end{aligned}$$ The final term in (\[eqA6\]) above has a norm bounded by $$\frac{1}{K}\sum_{k=1}^{K}\|Y_{km}- \mu \| ^{3} \|\lambda\| ^2 |1+Y_{km}|^{-1} = o_{p}(K^{1/2})O_{p}(K^{-1}) O_{p}(1)=o_{p}(K^{-1/2}).$$ So, $$\lambda = S_K^{-1}(\bar{Y}_{km}-\mu) + \beta,$$ with $\beta =o_{p}(K^{-1/2})$. By (\[eqA5\]), we may expand $$\log\Big{(} 1+ U_{m,k} \Big{)}= U_{m, k} - \frac{1}{2}U_{m,k}^{2} + \eta_{k}$$ where for some finite $B >0, 1\leq k\leq K$, $${\mathbb{P}}(| \eta_{k}|\leq B|U_{km}|^3 )\to 1$$ as $K\to \infty$ and $m\to \infty$.
We can verify the follow the identities after some algebra $$\begin{aligned}
-2 \log \mathcal{R}(\mu) &=2\sum_{k=1}^{K}\log \Big{(} 1+ U_{km} \Big{)} \\
&= 2\sum_{k=1}^{K} \Big{(} U_{km} - \frac{1}{2}U_{km}^{2} + \eta_{k} \Big{)} \\
&=2K\lambda^\top(\bar{Y}_{Km} -\mu) -K\lambda^\top S_K\lambda + 2 \sum_{k=1}^{K} \eta_{i}\\
&= K(\bar{Y}_{Km} -\mu)^\top S_K^{-1} (\bar{Y}_{Km} -\mu) -K\beta^\top S_K^{-1}\beta + 2 \sum_{k=1}^{K} \eta_{k}.\end{aligned}$$ By Theorem \[theorem1\] and Lemma \[lem-2\] $$K(\bar{Y}_{km}-\mu)^\top S_K^{-1} (\bar{Y}_{km}-\mu) \stackrel{d}{\longrightarrow} \chi^{2}_{p}.$$ The second and third terms are $o_{p}(1)$ since $$K\beta^\top S_K^{-1}\beta=Ko_{p}(K^{-1/2})O_{p}(1)o_{p}(K^{-1/2})=o_{p}(1),$$ $$\Big{|} \sum_{k=1}^{K} \eta_{k} \Big{|}\leq B\|\lambda\|^3 \sum_{k=1}^{K} \| Y_{km}-\mu\|^{3} =O_{p}(K^{-3/2})o_{p}(K^{3/2})=o_{p}(1).$$ Combing above, we can finish the proof.
[11]{}
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[^1]: School of Mathematical Sciences, Soochow University, 215006, Suzhou, China, stamax360@outlook.com
[^2]: School of Mathematics, South China University of Technology, Guangzhou, 510640, P.R. China. mascwang@scut.edu.cn
[^3]: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, 117546, Singapore. stazw@nus.edu.sg
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the [*toroidal expanse*]{} of an embedded graph $G$, that is, the size of the largest toroidal grid contained in $G$ as a minor. In the course of this work we introduce a new embedding density parameter, the [*stretch*]{} of an embedded graph $G$, and use it to bound the toroidal expanse from above and from below within a constant factor depending only on the genus and the maximum degree. We also show that these parameters are tightly related to the planar [*crossing number*]{} of $G$. As a consequence of our bounds, we derive an efficient constant factor approximation algorithm for the toroidal expanse and for the crossing number of a surface-embedded graph with bounded maximum degree.'
author:
- 'Markus Chimani[^1]'
- 'Petr Hliněný [^2]'
- 'Gelasio Salazar[^3]'
title: 'Toroidal Grid Minors and Stretch in Embedded Graphs[^4] '
---
[**Keywords:**]{} Graph embeddings, compact surfaces, face-width, edge-width, toroidal grid, crossing number, stretch
[**AMS 2010 Subject Classification:**]{} 05C10, 05C62, 05C83, 05C85, 57M15, 68R10
Introduction {#sec:intro}
============
In their development of the Graph Minors theory towards the proof of Wagner’s Conjecture [@RoSeGMXX], Robertson and Seymour made extensive use of surface embeddings of graphs. Robertson and Seymour introduced parameters that measure the density of an embedding, and established results that are not only central to the Graph Minors theory, but are also of independent interest. We recall that the [*face-width*]{} $\fw(G)$ of a graph $G$ embedded in a surface $\Sigma$ is the smallest $r$ such that $\Sigma$ contains a noncontractible closed curve (a [*loop*]{}) that intersects $G$ in $r$ points.
\[thm:fw-minor\] For any graph $H$ embedded on a surface $\Sigma$, there exists a constant $c:=c(H)$ such that every graph $G$ that embeds in $\Sigma$ with face-width at least $c$ contains $H$ as a minor.
This theorem, and other related results, spurred great interest in understanding which structures are forced by imposing density conditions on graph embeddings. For instance, Thomassen [@Th94] and Yu [@Yu97] proved the existence of spanning trees with bounded degree for graphs embedded with large enough face-width. In the same paper, Yu showed that under strong enough connectivity conditions, $G$ is Hamiltonian if $G$ is a triangulation.
Large enough density, in the form of edge-width, also guarantees several nice coloring properties. We recall that the [*edge-width*]{} $\ew(G)$ of an embedded graph $G$ is the length of a shortest noncontractible cycle in $G$. Fisk and Mohar [@FM94] proved that there is a universal constant $c$ such that every graph $G$ embedded in a surface of Euler genus $g >0$ with edge-width at least $c\log{g}$ is $6$-colorable. Thomassen [@Th93] proved that larger (namely $2^{14g+6}$) edge-width guarantees $5$-colorability. More recently, DeVos, Kawarabayashi, and Mohar [@DKM08] proved that large enough edge-width actually guarantees $5$-choosability.
In a direction closer to our current interest, Fiedler et al. [@FHRR95] proved that if $G$ is embedded with face-width $r$, then it has $\floor{r/2}$ pairwise disjoint contractible cycles, all bounding discs containing a particular face. Brunet, Mohar, and Richter [@BMR96] showed that such a $G$ contains at least $\floor{(r-1)/2}$ pairwise disjoint, pairwise homotopic, non-separating (in $\Sigma$) cycles, and at least $\floor{(r-1)/8} -1$ pairwise disjoint, pairwise homotopic, separating, noncontractible cycles. We remark that throughout this paper, “homotopic” refers to “freely homotopic” (that is, not to “fixed point homotopic”).
For the particular case in which the host surface is the torus, Schrijver [@Sc93] unveiled a beautiful connection with the geometry of numbers and proved that $G$ has at least $\floor{3r/4}$ pairwise disjoint noncontractible cycles, and proved that the factor $3/4$ is best possible.
The [*toroidal $p \times q\>$-grid*]{} is the Cartesian product $C_p\Box C_q$ of the cycles of sizes $p$ and $q$. See Figure \[fig:torgrid\]. Using results and techniques from [@Sc93], de Graaf and Schrijver [@dS94] showed the following:
\[thm:deGS\] Let $G$ be a graph embedded in the torus with face-width $\fw(G)=r\ge 5$. Then $G$ contains the toroidal $\floor{2r/3} \times \floor{2r/3}\>$-grid as a minor.
De Graaf and Schrijver also proved that $\floor{2r/3}$ is best possible, by exhibiting (for each $r\ge 3$) a graph that embeds in the torus with face-width $r$ and that does not contain a toroidal -grid as a minor. As they observe, their result shows that $c=\ceil{3m/2}$ is the smallest value that applies in (Robertson-Seymour’s) Theorem \[thm:fw-minor\] for the case of $H=C_m\Box C_m$.
Our focus: toroidal expanse, stretch, and crossing number
---------------------------------------------------------
Along the lines of the aforementioned de Graaf-Schrijver result, our aim is to investigate the largest size (meaning the number of vertices) of a toroidal grid minor contained in a graph $G$ embedded in an arbitrary orientable surface of genus greater than zero. We do not restrict ourselves to square proportions of the grid and define this parameter as follows.
\[def:texpanse\] The [*toroidal expanse*]{} of a graph $G$, denoted by $\Tex(G)$, is the largest value of $p\cdot q$ over all integers $p,q\geq3$ such that $G$ contains a toroidal $p \times q\>$-grid as a minor. If $G$ does not contain $C_3\Box C_3$ as a minor, then let $\Tex(G)=0$.
Our interest is both in the structural and the algorithmic aspects of the toroidal expanse.
The “bound of nontriviality” $p,q\geq3$ required by Definition \[def:texpanse\] is natural in the view of toroidal embeddability —the degenerate cases $C_2\Box C_q$ are planar, while $C_p\Box C_q$ has orientable genus one for all $p,q\geq3$. It is not difficult to combine results from [@BMR96] and [@dS94] to show that for each positive integer $g>0$ there is a constant $c:=c(g)$ with the following property: if $G$ embeds in the orientable surface $\Sigma_g$ of genus $g$ with face-width $r$, then $G$ contains a toroidal $c\cdot r \times c\cdot r\>$-grid as a minor; that is, $\Tex(G) = \Omega(r^2)$.
On the other hand, it is very easy to come up with a sequence of graphs $G$ embedded in a fixed surface with face-width $r$ and arbitrarily large $\Tex(G)/ r^2$: it is achieved by a natural toroidal embedding of $C_r \Box C_{q}$ for arbitrarily large $q$. This inadequacy of face-width to estimate the toroidal expanse of an embedded graph is to be expected, due to the one-dimensional character of this parameter.
To this end, we introduce a new density parameter of embedded graphs that captures the truly two-dimensional character of our problem; the [*stretch of an embedded graph*]{} in Definition \[def:stretch\]. Using this tool, we unveil our main result—a tight two-way relationship between the toroidal expanse of a graph $G$ in an orientable surface and its [*crossing number*]{} $\crg(G)$ in the plane. We furthermore provide an approximation algorithm for both these numbers under an assumption of a sufficiently dense embedding. A simplified summary of the main results follows:
\[thm:main-overview\] Let $\Sigma$ be an orientable surface of fixed genus $g>0$, and let $\Delta$ be an integer. There exist constants $r_0,c_0,c_1,c_2>0$, depending only on $g$ and $\Delta$, such that the following holds: If $G$ is a graph of maximum degree $\Delta$ embedded in $\Sigma$ with face-width at least $r_0$, then
- $c_0\cdot\crg(G) \leq \Tex(G) \leq c_1\cdot\crg(G)$, and
- there is a polynomial time algorithm that outputs a drawing of $G$ in the plane with at most $c_2\cdot\crg(G)$ crossings.
The density assumption that $\fw(G)\geq r_0$ is unavoidable for (a). Indeed, consider a very large planar grid plus an edge. Such a graph clearly admits a toroidal embedding with face-width $1$. By suitably placing the additional edge, such a graph would have arbitrarily large crossing number, and yet no $C_3\Box C_3$ minor. However, one could weaken this restriction a bit by considering “nonseparating” face-width instead, as we are going to do in the proof. Furthermore, we shall show later (Section \[sec:nodensity\]) how to remove the density assumption $\fw(G)\geq r_0$ completely for the algorithm (b), using additional results of [@MR2874100].
Regarding the constants $r_0,c_0,c_1,c_2$ we note that, in our proofs,
- $r_0$ is exponential in $g$ (of order $2^g$),
- $c_1$ is independent of $g,\Delta$, and
- $c_2$ and $1/c_0$ are quadratic in $\Delta$ and exponential in $g$ (of order $8^g$).
The rest of this paper is structured as follows. In Section \[sec:prelims\] we present some basic terminology and results on graph drawings and embeddings, and introduce the key concept of stretch of an embedded graph. In Section \[sec:mainresults\] we give a commentated walkthrough on the lemmas and theorems leading to the proof of Theorem \[thm:main-overview\]. The exact value of the constants $r_0,c_0,c_1,c_2$ is given there as well. Some of the presented statements seem to be of independent interest, and their (often long and technical) proofs are deferred to the later sections of the paper. Final Section \[sec:concluding\] then outlines some possible extensions of the main theorem and directions for future research.
Preliminaries {#sec:prelims}
=============
We follow standard terminology of topological graph theory, see Mohar and Thomassen [@MT01] and Stillwell [@St93]. We deal with undirected multigraphs by default; so when speaking about a [*graph*]{}, we allow multiple edges and loops. The vertex set of a graph $G$ is denoted by $V(G)$, the edge set by $E(G)$, the number of vertices of $G$ (the [*size*]{}) by $|G|$, and the maximum degree by $\Delta(G)$.
In this section we lay out several concepts and basic results relevant to this work, and introduce the key new concept of stretch of an embedded graph.
Graph drawings and embeddings in surfaces {#sub:gdes}
-----------------------------------------
We recall that in a [*drawing*]{} of a graph $G$ in a surface $\Sigma$, vertices are mapped to points and edges are mapped to simple curves (arcs) such that the endpoints of an arc are the vertices of the corresponding edge; no arc contains a point that represents a non-incident vertex. For simplicity, we often make no distinction between the topological objects of a drawing (points and arcs) and their corresponding graph theoretical objects (vertices and edges). A [*crossing*]{} in a drawing is an intersection point of two edges (or a self-intersection of one edge) in a point other than a common endvertex. An [*embedding*]{} of a graph in a surface is a drawing with no edge crossings.
If we regard an embedded graph $G$ as a subset of its host surface $\Sigma$, then the connected components of $\Sigma \setminus G$ are the [*faces*]{} of the embedding. We recall that the vertices of the [ *topological dual*]{} $G^*$ of $G$ are the faces of $G$, and its edges are the edge-adjacent pairs of faces of $G$. There is a natural one-to-one correspondence between the edges of $G$ and the edges of $G^*$, and so, for an arbitrary $F\subseteq E(G)$, we denote by $F^*$ the corresponding subset of edges of $E(G^*)$. We often use lower case Greek letters (such as $\alpha, \beta, \gamma$) to denote dual cycles. The rationale behind this practice is the convenience to regard a dual cycle as a simple closed curve, often paying no attention to its graph-theoretical properties.
Let $G$ be a graph embedded in a surface $\Sigma$ of genus $g > 0$, and let $C$ be a two-sided surface-nonseparating cycle of $G$. We denote by $G\cutt C$ the graph obtained by [*cutting $G$ through $C$*]{} as follows. Let $F$ denote the set of edges not in $C$ that are incident with a vertex in $C$. Orient $C$ arbitrarily, so that $F$ gets naturally partitioned into the set $L$ of edges to the left of $C$ and the set $R$ of edges to the right of $C$. Now contract (topologically) the whole curve representing $C$ to a point-vertex $v$, to obtain a pinched surface, and then naturally split $v$ into two vertices, one incident with the edges in $L$ and another incident with the edges in $R$. The resulting graph $G\cutt C$ is thus embedded on a surface $\Sigma'$ such that $\Sigma$ results from $\Sigma'$ by adding one handle. Clearly $E(G\cutt C)=E(G)\sem E(C)$, and so for every subgraph $F\subseteq G\cutt C$ there is a unique naturally corresponding subgraph $\hat F\subseteq G$ (on the same edge set), which we call the [*lift of $F$ into $G$*]{}.
The “cutting through” operation is a form of a standard surface surgery in topological graph theory, and we shall be using it in the dual form too, as follows. Let $G$ be a graph embedded in a surface $\Sigma$ and $\gamma\subseteq G^*$ a dual cycle such that $\gamma$ is two-sided and $\Sigma$-nonseparating. Now cut the surface along $\gamma$, discarding the set $E'$ of edges of $G$ that are severed in the process. This yields an embedding of $G-E$ in a surface with two holes. Then paste two discs, one along the boundary of each hole, to get back to a compact surface. We denote the resulting embedding by $G\cutt\gamma$, and say that this is obtained by [ *cutting $G$ along $\gamma$*]{}. Note that we may equivalently define $G\cutt\gamma$ as the embedded $(G^*\cutt\gamma)^*$. Note also that $V(G\cutt\gamma)=V(G)$, and that the previous definition of a [*lift*]{} applies also to this case.
Graph crossing number
---------------------
We further look at drawings of graphs (in the plane) that allow edge crossings. To resolve ambiguity, we only consider drawings where no three edges intersect in a common point other than a vertex. The [*crossing number*]{} $\crg(G)$ of a graph $G$ is then the minimum number of edge crossings in a drawing of $G$ in the plane.
For the general lower bounds we shall derive on the crossing number of graphs we use the following results on the crossing number of toroidal grids (see [@BeR; @JS01; @KR; @RBe]).
\[thm:crossing-CpCq\] For all nonnegative integers $p$ and $q$, $\crg(C_p\Box C_q)\geq
\frac12(p-2)q$. Moreover, $\crg(C_p\Box C_q) = (p-2)q$ for $p=3,4,5$.
We note that this result already yields the easy part of Theorem \[thm:main-overview\](a):
\[cor:crossing-texp\] Let $G$ be a graph embedded on a surface. Then $\crg(G)\geq\frac1{12}\Tex(G)$.
Let $q \ge p \ge 3$ be integers that witness $\Tex(G)$ (that is, $G$ contains $C_p\Box C_q$ as a minor, and $\Tex(G)=pq$). It is known [@GS01] that if $G$ contains $H$ as a minor, and $\Delta(H)=4$, then $\crg(G)\geq\frac14\crg(H)$. We apply this bound with $H=C_p\Box C_q$. By Theorem \[thm:crossing-CpCq\], we then have for $p\in\{3,4,5\}$ that $\crg(G)\geq\frac14(p-2)q\geq\frac1{12}pq$, and for $p\geq6$ we obtain $\crg(G)\geq\frac14\cdot\frac12(p-2)q\geq\frac1{12}pq$.
Curves on surfaces and embedded cycles
--------------------------------------
For the rest of the paper, we shall exclusively focus on orientable surfaces, and for each $g\ge 0$ we let $\Sigma_g$ denote the orientable surface of genus $g$. Note that in an embedded graph, paths are simple curves and cycles are simple closed curves in the surface, and hence it makes good sense to speak about their homotopy. In particular, there are no one-sided cycles embedded in $\Sigma_g$.
If $B$ is a path or a cycle of a graph, then the [*length*]{} $\len(B)$ of $B$ is its number of edges. We recall that the [*edge-width*]{} $\ew(G)$ of an embedded graph $G$ is the length of a shortest noncontractible cycle in $G$. The [*nonseparating edge-width*]{} $\ewn(G)$ is the length of a shortest nonseparating (and hence also noncontractible) cycle in $G$. It is easy to see that the face-width $\fw(G)$ of $G$ equals one half of the edge-width of the vertex-face incidence graph of $G$. It is also an easy exercise to show that $\ew(G^*)\geq\fw(G)\geq\frac{\ew(G^*)}{\lfloor\Delta(G)/2\rfloor}$. In this paper, we are primarily interested in graphs of bounded degree. We can thus regard $\ew(G^*)$ as a suitable (easier to deal with) replacement for $\fw(G)$.
For a cycle (or an arbitrary subgraph) $C$ in a graph $G$, we call a path $P\subset G$ a [*$C$-ear*]{} if the ends $r,s$ of $P$ belong to $C$, but the rest of $P$ is disjoint from $C$. We allow $r=s$, i.e., a $C$-ear can also be a cycle. A $C$-ear $P$ is a (with respect to an orientable embedding of $G$) if the two edges of $P$ incident with the ends $r,s$ are embedded on opposite sides of $C$. The following simple technical claim is useful.
\[lem:kl2\] If $C$ is a nonseparating cycle in an embedded graph $G$ of length $\len(C)=\ewn(G)$, then all $C$-switching ears in $G$ have length at least $\frac12\ewn(G)$.
Seeking a contradiction, we suppose that there is a $C$-switching ear $D$ of length $<\frac12\ewn(G)$. The ends of $D$ on $C$ determine two subpaths $D_1,D_2\subseteq C$ (with the same ends as $D$), labeled so that $||D_1|| \le ||D_2||$. Then $D\cup D_1$ (respectively, $D\cup D_2$) is a nonseparating cycle, as witnessed by $D_2$ (respectively, $D_1$). Since $\len(D_1)\leq\frac12\len(C)$, then $$\len(D\cup D_1)\leq\len(D)+\frac12\len(C)
<\biggl(\frac12+\frac12\biggr)\len(C)=\ewn(G)\,,$$ a contradiction.
Even though surface surgery can drastically decrease (and also increase, of course) the edge-width of an embedded graph in general, we now prove that this is not the case if we cut through a short cycle (in Lemma \[lem:cutdew\] we shall establish a surprisingly powerful extension of this simple claim).
\[lem:dew2\] Let $G$ be a graph embedded in the orientable surface $\Sigma_g$ of genus $g\geq2$, and let $C$ be a nonseparating cycle in $G$ of length $\len(C)=\ewn(G)$. Then $\ewn(G\cutt C)\geq\frac12\ewn(G)$.
Let $c_1,c_2$ be the two vertices of $G\cutt C$ that result from cutting through $C$, i.e., $\{c_1,c_2\}=V(G\cutt C)\sem V(G)$. Let $D\subseteq G\cutt C$ be a nonseparating cycle of length $\ewn(G\cutt C)$. If $D$ avoids both $c_1,c_2$, then its lift $\hat D$ in $G$ is a nonseparating cycle again, and so $\ewn(G)\leq\len(D)=\ewn(G\cutt
C)$. If $D$ hits both $c_1,c_2$ and $P\subseteq D$ is (any) one of the two subpaths with the ends $c_1,c_2$, then the lift $\hat P$ is a $C$-switching ear in $G$. Thus, by Lemma \[lem:kl2\], $$\ewn(G\cutt C)=\len(D)\geq\len(\hat P)\geq\frac12\ewn(G)\,.$$
In the remaining case $D$, up to symmetry, hits $c_1$ and avoids $c_2$. Then its lift $\hat D$ is a $C$-ear in $G$. If $\hat D$ itself is a cycle, then we are done as above. Otherwise, $\hat D\cup C\subseteq G$ is the union of three nontrivial internally disjoint paths with common ends, forming exactly three cycles $A_1,A_2,A_3\subseteq\hat D\cup C$. Since $D$ is nonseparating in $G\cutt C$, each of $A_1,A_2,A_3$ is nonseparating in $G$, and hence $\len(A_i)\geq\ewn(G)$ for $i=1,2,3$. Since every edge of $\hat D\cup C$ is in exactly two of $A_1,A_2,A_3$, we have $\len(A_1)+\len(A_2)+\len(A_3)=2\len(C)+2\len(\hat D)
=2\ewn(G)+2\len(\hat D)$ and $\len(A_1)+\len(A_2)+\len(A_3)\geq3\ewn(G)$, from which we get $$\ewn(G\cutt C)=\len(D)=\len(\hat D)\geq\frac12\ewn(G)\,.\tag*{\qedhere}$$
Many arguments in our paper exploit the mutual position of two graph cycles in a surface. In topology, the [*geometric intersection number*]{}[^5] $i(\alpha,\beta)$ of two (simple) closed curves $\alpha,\beta$ in a surface is defined as $\min\{ \alpha'\cap \beta'\}$, where the minimum is taken over all pairs $(\alpha',\beta')$ such that $\alpha'$ (respectively, $\beta'$) is homotopic to $\alpha$ (respectively, $\beta$). For our purposes, however, we prefer the following slightly adjusted discrete view of this concept. Let $A\not=B$ be cycles of a graph embedded in a surface $\Sigma$. Let $P\subseteq A\cap B$ be a connected component of the graph intersection $A\cap B$ (a path or a single vertex), and let $f_A,f_A'\in E(A)$ (respectively, $f_B,f_B'\in E(B)$) be the edges immediately preceding and succeeding $P$ in $A$ (respectively, $B$). See Figure \[fig:defineleap\]. Then $P$ is called a [*leap of A,B*]{} if there is a sufficiently small open neighborhood $\Omega$ of $P$ in $\Sigma$ such that the mentioned edges meet the boundary of $\Omega$ in this cyclic order; $f_A,f_B,f_A',f_B'$ (i.e., $A$ and $B$ [*meet transversely in $P$*]{}). Note that $A\cap B$ may contain other components besides $P$ that are not leaps.
\[def:leaping\] Two cycles $A,B$ of an embedded graph are in a [*$k$-leap position*]{} (or simply [*$k$-leaping*]{}), if their intersection $A\cap B$ has exactly $k$ connected components that are leaps of $A,B$. If $k$ is odd, then we say that $A,B$ are in an [*odd-leap position*]{}.
We now observe some basic properties of the $k$-leap concept:
- If $A,B$ are in an odd-leap position, then necessarily each of $A,B$ is noncontractible and nonseparating.
- It is not always true that $A,B$ in a $k$-leap position have geometric intersection number exactly $k$, but the parity of the two numbers is preserved. Particularly, $A,B$ are in an odd-leap position if and only if their geometric intersection number is odd. (We will not directly use this fact herein, though.)
- We will later prove (Lemma \[lem:3pp\]) that the set of embedded cycles that are odd-leaping a given cycle $A$ satisfies the useful [*$3$-path condition*]{} (cf. [@MT01 Section 4.3]).
Stretch of an embedded graph
----------------------------
In the quest for another embedding density parameter suitable for capturing the two-dimensional character of the toroidal expanse and crossing number problems, we put forward the following concept improving upon the original “orthogonal width” of [@HS07].
\[def:stretch\] Let $G$ be a graph embedded in an orientable surface $\Sigma$. The [*stretch $\stretch(G)$ of $G$*]{} is the minimum value of $\len(A)\cdot\len(B)$ over all pairs of cycles $A,B\subseteq G$ that are in a one-leap position in $\Sigma$.
As we noted above, if $A,B$ are in an odd-leap position, then both $A$ and $B$ are noncontractible and nonseparating. Thus it follows that $\stretch(G)\geq\ewn(G)^2$. We postulate that stretch is a natural two-dimensional analogue of edge-width, a well-known and often used embedding density parameter. Actually, one may argue that the dual edge-width is a more suitable parameter to measure the density of an embedding, and so we shall mostly deal with [*dual stretch*]{}—the stretch of the topological dual $G^*$—later in this paper (starting since Lemma \[lem:cr-stretch-torus\] and Section \[sec:mainresults\]). Analogously to face-width, we can also define the [*face stretch*]{} of $G$ as one quarter of the stretch of the vertex-face incidence graph of $G$, and this is to be discussed later in Section \[sec:facestretch\].
We note in passing that although our paper does not use nor provide an algorithm to compute the stretch of an embedding, this can be done efficiently on any surface [@CCH13].
We now prove several basic facts about the stretch of an embedded graph. We start with an easy observation.
\[lem:thstr\] If $C$ is a nonseparating cycle in an embedded graph $G$, and $P$ is a $C$-switching ear in $G$, then $\stretch(G)\leq\len(C)\cdot\big(\len(P)+\frac12\len(C)\big)
\leq2\len(C)\cdot\len(P)$.
The ends of $P$ partition $C$ into two paths $C_1,C_2\subseteq C$, which we label so that $||C_1|| \le ||C_2||$. (In a degenerate case, $C_1$ can be a single vertex). Thus $\len(C_1)\leq\frac12\len(C)$. Since $C$ and $P\cup C_1$ are in a one-leap position, we have $\stretch(G)\leq\len(C)\cdot(\len(P)+\len(C_1))\leq\len(C)\cdot2\len(P)$.
A tight relation of stretch to the topic of our paper is illustrated in the following claims.
\[lem:cr-stretch-torus\] If $G$ is a graph embedded in the torus, then $\crg(G)\leq\stretch(G^*)$.
Let $\alpha,\beta\subseteq G^*$ be a pair of dual cycles witnessing $\stretch(G^*)$, and let $K:=E(\alpha)^*$, $L:=E(\beta)^*\sem K$, and $M:=E(\alpha\cap \beta)^*$. Note that $K,L$, and $M$ are edge sets in $G$. Then, by cutting $G$ along $\alpha$, we obtain a plane (cylindrical) embedding $G_0$ of $G-K$. It is easily possible to draw the edges of $K$ into $G_0$ in one parallel “bunch” along the fragment of $\beta$ such that they cross only with edges of $L$ and $M\subseteq K$ (indeed, crossings between edges of $K$ are necessary when $M\not=\emptyset$), thus getting a drawing of $G$ in the plane. See Figure \[fig:addedges\]. The total number of crossings in this particular drawing, and thus the crossing number of $G$, is at most $|K|\cdot|L|+|K|\cdot|M|=
|K|\cdot(|L|+|M|)=\len(\alpha)\cdot\len(\beta)=\stretch(G^*)$.
\[cor:texp-stretch-torus\] If $G$ is a graph embedded in the torus, then $\Tex(G)\leq12\stretch(G^*)$.
This follows immediately using Corollary \[cor:crossing-texp\].
We finish this section by proving an analogue of Lemma \[lem:dew2\] for the stretch of an embedded graph, showing that this parameter cannot decrease too much if we cut the embedding through a short cycle. This will be important to us since cutting through handles of embedded graphs will be our main inductive tool in the proofs of lower bounds on $\crg(G)$ and $\Tex(G)$.
\[lem:str4\] Let $G$ be a graph embedded in the orientable surface $\Sigma_g$ of genus $g\geq2$, and let $C$ be a nonseparating cycle in $G$ of length $\len(C)=\ewn(G)$. Then $\stretch(G\cutt C)\geq\frac14\stretch(G)$.
Let $c_1,c_2$ be the two vertices of $G\cutt C$ that result from cutting through $C$, i.e., $\{c_1,c_2\}=V(G\cutt C)\sem V(G)$. Suppose that $\stretch(G\cutt C)=ab$ is attained by a pair of one-leaping cycles $A,B$ in $G\cutt C$, with $a=\len(A)$ and $b=\len(B)$. Our goal is to show that $\stretch(G)\leq4ab$. Using Lemma \[lem:dew2\] and the fact that both $A,B$ are nonseparating, we get $$\label{eq:abrho}
a,b\geq\ewn(G\cutt C)\geq\frac12\ewn(G)=\frac12\len(C)
.$$
Suppose first that both $c_1,c_2\in V(A\cup B)$. Then there exists a path $P\subseteq A\cup B$ connecting $c_1$ to $c_2$ such that $\len(P)\leq\frac12(a+b)$. Clearly, its lift $\hat P$ is a $C$-switching ear in $G$, and so by Lemma \[lem:thstr\] and (\[eq:abrho\]), $$\begin{aligned}
\stretch(G)\;&\leq\; \len(C)\cdot\big(\len(\hat P)+\frac12\len(C)\big)
\leq\len(C)\cdot\frac12(a+b+\len(C))
\\
&\leq\; \frac12(2ba+2ab+4ab)=4ab=4\,\stretch(G\cutt C)
.\end{aligned}$$
Finally suppose that, up to symmetry, $c_2\not\in V(A\cup B)$ but possibly $c_1\in V(A\cup B)$. Consider the lift $\hat A$ in $G$ (which is a $C$-ear in the case $c_1\in V(A)$). We define $\bar A$ to be $\hat A$ if $\hat A$ is a cycle, and otherwise $\bar A=\hat A\cup C_0$ where $C_0\subseteq C$ is a shortest subpath with the same ends in $C$ as $\hat A$. We define $\bar B$ analogously. With the help of a simple case-analysis, it is straightforward to verify that the cycles $\bar A,\bar B$ form a one-leaping pair in $G$, and so again using Lemma \[lem:thstr\] we obtain $$\begin{aligned}
\stretch(G)\;&\leq\; \len(\bar A)\cdot\len(\bar B)\leq (a+\frac12\len(C))\cdot(b+\frac12\len(C))
\\ &\leq\; (a+a)\cdot(b+b)=4ab=4\,\stretch(G\cutt C).\tag*{\qedhere}\end{aligned}$$
Breakdown of the proof of Theorem \[thm:main-overview\] {#sec:mainresults}
=======================================================
In this section we shall state the results leading to the proof of Theorem \[thm:main-overview\], which is given in Section \[sub:mainproof\]. The proofs of (most of) these statements are long and technical, and so they are deferred to the later sections of the paper.
To reach our main goal, i.e., to provide a proof for Theorem \[thm:main-overview\], we aim to:
- extend the upper estimate of Lemma \[lem:cr-stretch-torus\] to surfaces of higher genus than the torus; and
- provide asymptotically matching lower bounds on $\crg(G)$ and $\Tex(G)$ in terms of the dual stretch of $G$.
While the upper bounds are given (cf. Lemma \[lem:cr-stretch-torus\]) for the crossing number, the lower bounds here will be investigated for the toroidal expanse. At first glance, goal (I) would appear to be much easier than (II), but it is not really so straightforward due to some complications in expressing the upper bound (cf. Theorem \[thm:upper-cr\] below). Such difficulties are to be expected: for instance, a graph embedded in the double torus could have a huge toroidal grid living in one of the handles, and yet very small dual stretch due to a very small dual edge width in the other handle.
Since we will frequently deal with dual graphs in our arguments, we introduce several conventions in order to help comprehension. When we add an adjective [*dual*]{} to a graph term, we mean this term in the topological dual of the (currently considered) graph. We will denote the faces of an embedded graph $G$ using lowercase letters, treating them as vertices of its dual $G^*$. As we already mentioned in Section \[sub:gdes\], we use lowercase Greek letters to refer to subgraphs (cycles or paths) of $G^*$, and when there is no danger of confusion, we do not formally distinguish between a graph and its embedding. In particular, if $\alpha\subseteq G^*$ is a dual cycle, then $\alpha$ also refers to the loop on the surface determined by the embedding $G$. Finally, we will denote by $\ewnd(G):=\ewn(G^*)$ the nonseparating edge-width of the dual $G^*$ of $G$, and by $\stretchd(G):=\stretch(G^*)$ the dual stretch of $G$.
Estimating the toroidal expanse {#sub:toridalexp}
-------------------------------
We first give some basic lower bound estimates for the toroidal expanse, aimed at goal (II) above. These estimates ultimately rely on the following basic result, which appears to be of independent interest. Loosely speaking, it states that if a graph has two collections of cycles that mimic the topological properties of the cycles that build up a $p\times q$-toroidal grid, then the graph does contain such a grid as a minor. We say that a pair $(C,D)$ of curves in the torus is a [*basis*]{} (for the fundamental group) if there are no integers $m,n$ such that $C^m$ is homotopic to $D^n$.
\[thm:two-cycle-families\] Let $G$ be a graph embedded in the torus. Suppose that $G$ contains a collection $\{C_1,\dots,C_p\}$ of $p\geq3$ pairwise disjoint, pairwise homotopic cycles, and a collection $\{D_1,\dots,D_q\}$ of $q\geq3$ pairwise disjoint, pairwise homotopic cycles. Further suppose that the pair $(C_1,D_1)$ is a basis. Then $G$ contains a $p\times q$-toroidal grid as a minor.
The proof of this statement is in Section \[sec:grids\].
Now recall that in the torus $\ewn(G)=\ew(G)$, and so $\fw(G)\geq\frac{\ewnd(G)}{\lfloor\Delta(G)/2\rfloor}$. Hence, for instance, one can formulate Theorem \[thm:deGS\] in terms of nonseparating dual edge-width. Along these lines we shall derive the following as a consequence of Theorem \[thm:two-cycle-families\] (the proof is also in Section \[sec:grids\]):
\[thm:agrid-torus\] Let $G$ be a graph embedded in the torus and $k:=\ewnd(G)$. Let $\ell$ be the largest integer such that, in the dual graph $G^*$, there exists a dual cycle $\alpha$ of length $k$ and the shortest $\alpha$-switching dual ear has length $\ell$ (recall from Lemma \[lem:kl2\] that $\ell\geq k/2$). If $k\geq5\lfloor\Delta(G)/2\rfloor$, then $G$ contains as a minor the toroidal grid of size $$\left\lceil\frac \ell{\lfloor\Delta(G)/2\rfloor}\right\rceil
\>\times\>
\left\lfloor\frac23\left\lceil
\frac k{\lfloor\Delta(G)/2\rfloor}
\right\rceil\right\rfloor
\,.\smallskip$$
Hence the toroidal expanse of $G$ is at least $\big\lceil\frac \ell{\lfloor\Delta(G)/2\rfloor}\big\rceil
\cdot\big\lfloor\frac23\lceil\frac k{\lfloor\Delta(G)/2\rfloor}\rceil\big\rfloor$. On the other hand, since $\fw(G)\ge\frac
k{\lfloor\Delta(G)/2\rfloor}$, by Theorem \[thm:deGS\] it follows that the toroidal expanse of $G$ is at least $\big\lfloor\frac23\big\lceil\frac
k{\lfloor\Delta(G)/2\rfloor}\big\rceil\big\rfloor^2$. Therefore our estimate becomes useful roughly whenever $\ell>\frac23k$. Now by Lemma \[lem:thstr\] (applied to $G^*$), we have $\stretch^*(G) \le
k\cdot(\ell + k/2)$, and so $\ell>\frac23k$ whenever $\stretch^*(G)
> \frac{7}{6}k^2$.
Moreover, Theorem \[thm:agrid-torus\] can be reformulated in terms of $\stretch^*(G)$ (instead of “$\ell\cdot k$”). This reformulation is important for the general estimate on the toroidal expanse of $G$:
\[cor:agrid-torus\] Let $G$ be a graph embedded in the torus with $\ewnd(G)\geq5\lfloor\Delta(G)/2\rfloor$. Then $$\Tex(G)\>\geq\> \frac{2}{7}\,
\big\lfloor{\Delta(G)}/2\big\rfloor^{-2} \cdot\stretchd(G)
\>\geq\> \frac87\Delta(G)^{-2} \cdot\stretchd(G)
\,.$$ Furthermore, for any $\varepsilon>0$ there is a $k_0:=k_0(\Delta,\varepsilon)$ such that if $\ewnd(G)> k_0$, then $\Tex(G)\geq(\frac8{21}-\varepsilon)\cdot
\lfloor{\Delta(G)}/2\rfloor^{-2}\cdot\stretchd(G)$.
For the proof of this statement, we again refer to Section \[sec:grids\].
Stepping up to orientable surfaces of genus $g>1$, we use Lemmas \[lem:dew2\] and \[lem:str4\] and Corollary \[cor:agrid-torus\] iteratively ($g-1$ times), cutting through shortest nonseparating dual cycles. This easily leads by induction to the following lower estimate:
\[cor:agrid-all\] Let $G$ be a graph embedded in the orientable surface $\Sigma_g$, $g\geq1$, such that $\ewnd(G)\geq5\cdot2^{g-1}\lfloor\Delta(G)/2\rfloor$. Then $$\Tex(G)\>\geq\> \frac{2}{7}\,4^{1-g}
\big\lfloor{\Delta(G)}/2\big\rfloor^{-2} \cdot\stretchd(G)
\>\geq\> \frac17\,2^{5-2g}\Delta(G)^{-2} \cdot\stretchd(G)
\,.\tag*{\qed}$$
This bound is, unfortunately, not strong enough to give the desired conclusion for $g\geq2$, but it is nevertheless useful in the course of deriving a stronger estimate later on (cf. Lemma \[lem:kl-to-stretch\]).
Algorithmic upper estimate for higher surfaces
----------------------------------------------
We now tackle task (I): to give an algorithmically efficient upper bound on the crossing number of a graph embedded in $\Sigma_g$.
Peter Brass conjectured the existence of a constant $c$ such that the crossing number of a toroidal graph on $n$ vertices is at most $c\Delta n$. This conjecture was proved by Pach and Tóth [@pachtoth]. Moreover, Pach and Tóth showed that for every orientable surface $\Sigma$ there is a constant $c_\Sigma$ such that the crossing number of an $n$-vertex graph embeddable on $\Sigma$ is at most $c_\Sigma \Delta n$; this result was extended to any surface by B[ö]{}r[ö]{}czky, Pach, and Tóth [@BPT06]. The constant $c_\Sigma$ proved in these papers is exponential in the genus of $\Sigma$. This was later refined by Djidjev and Vrt’o [@DV12], who decreased the bound to $\ca O(g\Delta n)$, and proved that this is tight within a constant factor.
At the heart of these results lies the technique of (perhaps recursively) cutting along a suitable [*planarizing*]{} subgraph (most naturally, a set of short cycles), and then redrawing the missing edges without introducing too many crossings. Our techniques and aims are of a similar spirit, although our cutting process is more delicate, due to our need to (eventually) find a matching lower bound for the number of crossings in the resulting drawing. Our cutting paradigm is formalized in the following definition.
\[def:good-planarizing\] Let $G$ be a graph embedded in the orientable surface $\Sigma_g$. A sequence $(G_1,C_1),(G_2,C_2), \dots, (G_g,C_g)$ is called a [*good planarizing sequence for $G$*]{} if the following holds for $i=1,\dots,g$, letting $G_0=G$:
- $G_i$ is a graph embedded in $\Sigma_{g-i}$,
- $C_i$ is a nonseparating cycle in $G_{i-1}$ of length $\ewn(G_{i-1})$, and
- $G_i$ results by cutting the embedding $G_{i-1}$ through $C_i$.
We implicitly associate such a planarizing sequence with the values $\{k_i,\ell_i\}_{i=1,\ldots,g}$, where $k_i=\len(C_i)$ and $\ell_i$ is the length of a shortest $C_i$-switching ear in $G_{i-1}$, for $i=1,\dots,g$.
In order to upper bound the crossing number of an embedded graph, we make use of good planarizing sequences in the dual graph, as stated in the following result.
\[thm:upper-cr\] Let $G$ be a graph embedded in $\Sigma_g$. Let $(G_1^*,\gamma_1), \dots, (G_g^*,\gamma_g)$ be a good planarizing sequence for the topological dual $G^*$, with associated lengths $k_1,\ell_1,\dots,k_g,\ell_g$. Then $$\label{eq:uppercr}
\crg(G)\>\leq\>
3\cdot\left(2^{g+1}-2-g\right)\cdot\max
\{k_i\ell_i\}_{i=1,2,\ldots,g}
\,.$$ There is an algorithm that produces a drawing of $G$ in the plane with at most crossings in time $\ca O(n\log n)$ for fixed $g$.
The proof of this theorem is given in Section \[sec:drawing-upper\].
Bridging the approximation gap
------------------------------
Let us briefly revise where we stand in our way towards proving Theorem \[thm:main-overview\]. The right hand side of part (a) already follows from Corollary \[cor:crossing-texp\], and so to finish this part we need an estimate of the form $\Tex(G)=\Omega(\crg(G))$. We currently have a lower bound for $\Tex(G)$ in terms of $\stretch^*(G)$ (Corollary \[cor:agrid-all\]) and an upper bound for $\crg(G)$ in terms of $\max \{k_i\ell_i\}$. It may thus appear that our next task is to bridge the gap by proving that $\stretchd(G) = \Omega(\max
\{k_i\ell_i\})$. As it happens, no such statement is true in general, and so we need to find a way around this difficulty.
The following is a key technical claim that allows us to bridge the aforementioned gap.
\[lem:kl-to-stretch\] Let $H$ be a graph embedded in the orientable surface $\Sigma_g$. Let $k:=\ewnd(H)$, and let $\ell$ be the largest integer such that there is a cycle $\gamma$ of length $k$ in $H^*$ whose shortest $\gamma$-switching ear has length $\ell$. Assume $k\geq2^g$. Then there exists an integer $g'$, $0< g'\leq g$, and a subgraph $H'$ of $H$ embedded in $\Sigma_{g'}$ such that $$\ewnd(H')\geq 2^{g'-g}k
\qquad\mbox{and}\qquad
\stretchd(H')\geq 2^{2g'-2g}\cdot k\ell
\,.$$
In a nutshell, the main idea behind the proof of this statement is to cut along handles that (may) cause small stretch, until we arrive to the desired toroidal $\Omega(k\times\ell)$ grid.
The arguments required to prove Lemma \[lem:kl-to-stretch\] span two sections. In Section \[sec:more\] we establish several basic results on the stretch of an embedded graph. As we believe this new parameter may be of independent interest, it makes sense to gather these results in a standalone section for possible further reference. The proof of Lemma \[lem:kl-to-stretch\] is then presented in Section \[sec:finding\].
The importance of Lemma \[lem:kl-to-stretch\] is its crucial role in establishing the following result, the final step in bridging the approximation gap.
\[cor:kl-to-stretch\] Let $G$ be a graph embedded in $\Sigma_g$. Let $(G_1^*,\gamma_1), \dots, (G_g^*,\gamma_g)$ be a good planarizing sequence of $G^*$, with associated lengths $k_1,\ell_1,\dots,k_g,\ell_g$. Suppose that $\ewnd(G)\geq5\cdot2^{g-1}\lfloor\Delta(G)/2\rfloor$. Then $$\Tex(G) \>\geq\>
\frac{1}{7}\,2^{3-2g}
\big\lfloor{\Delta(G)}/2\big\rfloor^{-2}
\cdot \max \{k_i\ell_i\}_{i=1,2,\dots,g}
\,.$$ Consequently, $$\crg(G) \>\geq\>
\frac{1}{21}\,2^{1-2g}
\big\lfloor{\Delta(G)}/2\big\rfloor^{-2}
\cdot \max \{k_i\ell_i\}_{i=1,2,\dots,g}
\,.$$
Let $j$ be the smallest integer such that $k_j\ell_j=\max \{k_i\ell_i\}_{
i=1,2,\dots,g}$, and let $H:=G_{j-1}$ (in case $j=1$, recall that we set $G_0:=G$). Thus $H$ is embedded in a surface of genus $g_1=g-j+1$. An iterative application of Lemma \[lem:dew2\] yields that $\ewnd(H)/\lfloor\Delta(H)/2\rfloor\geq
5\cdot2^{g-1}\cdot2^{g_1-g} =5\cdot2^{g_1-1}$.
We now apply Lemma \[lem:kl-to-stretch\] to $H$. Thus the resulting graph $H'$ of genus $g'\geq1$ satisfies $\ewnd(H')/\lfloor\Delta(H')/2\rfloor\geq5\cdot2^{g'-1}$ and $\stretchd(H')\geq 2^{2g'-2g_1}\cdot k_j\ell_j\geq 2^{2g'-2g}\cdot
k_j\ell_j$. Note that, even though $H^*=G^*_{j-1}$ may not be a subgraph of $G^*$, we have that $H$ (and thus $H'$) is a subgraph of $G$, and so $\Tex(G) \geq \Tex(H')$. Using Corollary \[cor:agrid-all\] we finally get $$\begin{aligned}
\Tex(G)\geq& \,\Tex(H') \>\geq\> \frac{2}{7}\,4^{1-g'}
\big\lfloor{\Delta(H')}/2\big\rfloor^{-2} \cdot\stretchd(H')
\\
\geq&\> \frac{1}{7}\,2^{3-2g'}
\big\lfloor{\Delta(G)}/2\big\rfloor^{-2}
\cdot2^{2g'-2g}k_j\ell_j
\>=\>\frac{1}{7}\,2^{3-2g}
\big\lfloor{\Delta(G)}/2\big\rfloor^{-2} \cdot k_j\ell_j
\,.\end{aligned}$$ The second inequality then follows immediately by Corollary \[cor:crossing-texp\].
Proof of Theorem \[thm:main-overview\] {#sub:mainproof}
--------------------------------------
Having deferred the long and technical proofs of the previous subsections for the later sections of the paper, all the ingredients are now in place to prove Theorem \[thm:main-overview\].
The right hand side inequality in (a) follows from Corollary \[cor:crossing-texp\] (with $c_1=12$), and the left hand side follows at once by combining Theorem \[thm:upper-cr\] and Corollary \[cor:kl-to-stretch\]. Finally we note that part (b) follows from Theorem \[thm:upper-cr\] and (the crossing number inequality in) Corollary \[cor:kl-to-stretch\].
Finding grids in the torus {#sec:grids}
==========================
In this section we prove Theorems \[thm:two-cycle-families\] and \[thm:agrid-torus\] and Corollary \[cor:agrid-torus\].
Let $\alpha, \beta$ be oriented simple closed curves such that $(\alpha,\beta)$ is a basis, and such that $\alpha$ and $\beta$ intersect (cross) each other exactly once. Using a standard surface homeomorphism argument (cf. [@St93]), we may assume without loss of generality that each $C_i$ has the same homotopy type as $\alpha$ (we assign an orientation to the cycles $C_i$ to ensure this). Thus it follows that the cycles $D_j$ may be oriented in such a way that there exist integers $r\ge 0,s\ge 1$ such that the homotopy type of each $D_j$ is $\alpha^r\beta^s$.
We assume without loss of generality that $p\geq q\geq3$. We let $C_+:=C_1\cup C_2\cup\dots\cup
C_p$ and $D_+:=D_1\cup D_2\cup\dots\cup D_q$. We shall assume that among all possible choices of the collections $\{C_1,\dots,C_p\}$ and $\{D_1,\ldots,D_q\}$ that satisfy the conditions in the theorem (for the given values of $p$ and $q$), our collection $\cc:=\{C_1,\ldots,C_p\}$ minimizes $|E(C_+)\sem E(D_+)|$.
The indices of the $C_i$-cycles (respectively, the $D_j$-cycles) are read modulo $p$ (respectively, modulo $q$). We may assume that the cycles $C_1,C_2,\dots,C_p$ appear in this cyclic order around the torus; that is, for each $i=1,2,\ldots,p$, one of the cylinders bounded by $C_i$ and $C_{i+1}$ does not intersect any other curve in $\cc$. Moreover, we may choose this labeling so that $\beta$ intersects $C_1,C_2,\ldots,C_p$ in this cyclic order.
At first glance it may appear that it is easy to get the desired grid as a minor of $C_+\cup D_+$, since every $D_j$ has to intersect each $C_i$ in some vertex of $G$ (this follows since each pair $(C_i,D_j)$ is a basis). There are, however, two possible complications. First, two cycles $C_i, D_j$ could have many “zigzag” intersections, with $D_j$ intersecting $C_i$, then $C_{i+1}$, then $C_i$ again, etc. Second, $D_j$ may “wind” many times in the direction orthogonal to $C_i$. These are the problems to overcome in the upcoming proof.
We start by showing that, even though we may intersect some $C_i$ several times when traversing some $D_j$, it follows from the choice of $\cc$ that, after $D_j$ intersects $C_i$, it must hit either $C_{i-1}$ or $C_{i+1}$ before coming back to $C_i$.
\[clm\_ear\] No $C_+$-ear contained in $D_+$ has both ends on the same cycle $C_i$.
Suppose that there is a $C_+$-ear $P\subset D_+$ with both ends on the same $C_i$. Modify $C_i$ by following $P$ in the appropriate section, and let $C_i'$ be the resulting cycle. The families $\{C_1,\ldots,C_{i-1},C_i',C_{i+1},\ldots ,C_p\}$ and $\{D_1,\ldots,D_q\}$ satisfy the conditions in the theorem. The fact that $|E(C_1\cup \cdots \cup C_{i-1} \cup C_{i}' \cup C_{i+1} \cdots \cup C_p)
\setminus E(D_+)| < |E(C_+ \setminus D_+)|$ contradicts the choice of $\{C_1,\ldots,C_p\}$.
For any cycle $C$, a [*quasicycle*]{} is a graph-homomorphic image of $C$ without degree-$1$ vertices, implicitly retaining its cyclic ordering of vertices.
Let $D_j'$ be a quasicycle in $G$ homotopic to $D_1$, with its same orientation. We say that $D_j'$ is [*$C_+$-ear good*]{} if (cf. Claim \[clm\_ear\]) no $C_+$-ear contained in $D_j'$ has both ends on the same $C_i$. The [*rank*]{} $s_j$ of $D_j'$ is the number of connected components of $C_+\cap D_j'$. By traversing $D_j'$ once and registering each time it intersects a curve in $\cc$, starting with (some intersection with) $C_1$, we obtain an [*intersection sequence*]{} $a_j(i)$, $i=1,\dots,s_j$, where each $a_j(i)$ is in $\{1,\dots,p\}$. Since we chose the starting point of the traversal of $D_j'$ so that the first curve of $\cc$ it intersects is $C_1$, it follows that $a_j(1)=1$. We read the indices of this subsequence modulo $s_j$. We denote by $Q_{j,t}$, $t=1,2,\dots,s_j$, the path of $D_j'$ (possibly a single vertex) forming the corresponding intersection with the cycle $C_{a_j(t)}$, and by $T_{j,t}$ the path of $D_j'$ between $Q_{j,t}$ and $Q_{j,t+1}$. If $D_j'$ is $C_+$-ear good then $a_j(t+1)\not=a_j(t)$, and hence in this case $|a_j(t+1)-a_j(t)|\in\{1,p-1\}$ for $t=1,2,\dots,s_j$.
A collection of $C_+$-ear good quasicycles $D_1',D_2',\dots,D_q'$ in $G$ is [*quasigood*]{} if it satisfies the property that whenever $D_n'$ intersects $D_m'$ in a path $P$ (counting also the case of a self-intersection with $m=n$), the following hold up to symmetry between $n$ and $m$: (i) $P\subseteq Q_{n,x}$ for an appropriate index $x$ of the intersection sequence of $D_n'$ for which $a_n(x-1)=a_n(x+1)$ and $a_n(x)-a_n(x-1)\in\{1,1-p\}$; and (ii) the path $T_{n,x-1}\cup Q_{n,x}\cup T_{n,x}$ of $D_n'$ stays locally on one side of the (embedded) quasicycle $D_m'$. Informally, this means that if $D_n'$ intersects $D_m'$ in $P$, then $D_n'$ makes a $C_{a_n(x-1)}$-ear with $P$ “touching” $D_m'$ from the left side. For further reference we say that $D_n'$ is locally on the [*left side*]{} of the intersection $P$.
Since $D_j$ is clearly a $C_+$-ear good quasicycle for each $j=1,2,\ldots,q$, it follows that $D_1,D_2,\ldots,D_q$ is a quasigood collection. Now among all choices of a quasigood collection $D_1',D_2',\dots,D_q'$ in $G$, we select one minimizing the sum of the ranks of its quasicycles. For each $D_j'$, as above we let $s_j$ denote its rank.
\[clm\_quasi\] For all $1\leq j\leq q$ the intersection sequence of $D_j'$ satisfies $a_j(t-1)\not=a_j(t+1)$ for any $1<t\leq s_j$. Consequently, $D_1',D_2',\dots,D_q'$ is a collection of pairwise disjoint cycles in $G$.
The conclusion that $D_1',D_2',\ldots,D_q'$ is a collection of pairwise disjoint cycles directly follows from the first statement in the claim, since it is a quasigood collection. We hence focus on the first statement in the following.
The main idea in the proof is quite simple: if $a_j(t-1)=a_j(t+1)$, then we could modify $D_j'$ rerouting it through $C_{a_j(t-1)}$ instead of $T_{j,t-1}\cup Q_{j,t}\cup T_{j,t}$, thus decreasing $s_j$ (and hence the total sum of the ranks) by $2$, and consequently contradicting the choice of $\dd:=\{D_1',D_2',\ldots,D_q'\}$. We now formalize this idea.
Let $\cylin_i$ denote the cylinder bounded by $C_i$ and $C_{i+1}$. Note that if for some $j,t$ we have $a_j(t-1)=a_j(t+1)$ and $a_j(t)-a_j(t-1)\in\{-1,p-1\}$, then necessarily for some $t'$ we must have $a_j(t'-1)=a_j(t'+1)$ and $a_j(t')-a_j(t'-1)\in\{1,1-p\}$. So, seeking a contradiction, we may suppose that there exist $j,t$ such that $a_j(t-1)=a_j(t+1)=i$ and $a_j(t)=i+1$. Then the path $P=T_{j,t-1}\cup Q_{j,t}\cup T_{j,t}$ is drawn in $\cylin_i$ with both ends on $C_i$ and “touching” (i.e., not intersecting transversally) $C_{i+1}$. We denote by $R_0\subset\cylin_i$ the open region bounded by $P$ and $C_i$, and by $P'$ the section of the boundary of $R_0$ not belonging to $D_j'$.
Assuming that $R_0$ is minimal over all choices of $j$ for which $a_j(t-1)=a_j(t+1)$, we show that no $D_m'$, $m\in\{1,\dots,q\}$, intersects $R_0$. Indeed, if some $D_m'$ intersected $R_0$, then $D_m'$ could not enter $R_0$ across $P$ by the “stay on one side” property of a quasigood collection. Hence $D_m'$ should enter and leave $R_0$ across $P'\subseteq C_i$, but not touch $Q_{j,t}\subseteq C_{i+1}$, by the minimality of $R_0$. But then, $D_m'$ would make a $C_+$-ear with both ends on $C_i$, contradicting the assumption that $D_m'$ is $C_+$-ear good.
Now we form $D_j^o$ as the symmetric difference of $D_j'$ with the boundary of $R_0$ (so that $D_j^o$ follows $P'$). To argue that $D_1',\dots,D_j^o,\dots,D_q'$ is a quasigood collection again, it suffices to verify all possible new intersections of $D_j^o$ along $P'$. Suppose there is an $D_n'$ such that its intersection $Q_{n,x}$ with $C_i$ contains some internal vertex of $P'$. Since $D_n'$ is disjoint from (the open region) $R_0$, it will “stay on one side” of $D_j^o$. If $Q_{n,x}$ intersects $D_j'$, then $D_n'$ must be locally on the left side of this intersection, and so it is also on the left side of the intersection with $D_j^o$. If, on the other hand, $Q_{n,x}$ is disjoint from $D_j'$, then the adjacent paths $T_{n,x-1}$ and $T_{n,x}$ have to connect to $C_{i-1}$ by Claim \[clm\_ear\], and so we have $a_n(x)=i$ and $a_n(x-1)=a_n(x+1)=i-1$ as required by the definition for $D_n'$ on the left side. Let $\dd^o$ be the collection derived from $\dd$ by substituting $D_j'$ with $D_j^o$. In every case, $\dd^o$ is quasigood as well, but the sum of the ranks of its elements is strictly smaller (by $2$) than it is for $\dd$. This contradicts the choice of $\dd$.
\[clm\_wonce\] There is a collection of $q$ pairwise disjoint, pairwise homotopic noncontractible cycles in $G$, each of which has a connected nonempty intersection with each cycle in $\cc$.
It follows from Claim \[clm\_quasi\] that the intersection sequence of each $D_j'$ is a $t$-fold repetition of the subsequence $\langle 1,2,\ldots,p\rangle$, for some nonnegative integer $t$. If $t=1$, we are obviously done, so assume $t \ge 2$. Our task is to “shortcut” each $D_j'$ such that it “winds only once” in the direction orthogonal to $\alpha$ (more formally, to modify each $D_j'$ so that its homotopy type is $\alpha^r \beta$ for some integer $r$).
Note that, for all $i=1,\dots,p$, every $C_i$-ear contained in any $D_j'$ is $C_i$-switching by Claim \[clm\_quasi\]. Each such ear naturally inherits an orientation from $D_j'$, so that after leaving $C_i$ it intersects $C_{i+1},
C_{i+2}, \ldots, C_{i-1}$ in this order, and then intersects $C_i$ again. Let $T_1 \subset D_1'$ be any $C_1$-ear, and let $x_1,y_1$ be their start and end points, respectively. Then let $W_1\subset C_1$ be (any) one of the two paths contained in $C_1$ with endpoints $x_1, y_1$. It is clear that the cycle $D_1''=T_1\cup W_1$ is a simple closed curve that has a connected nonempty intersection with each $C_i$. Our final task is to find, for each $j=2,\ldots,q$, a homotopic, similarly constructed cycle $D_j''$, so that the cycles $D_1'', D_2'',\ldots,D_j''$ are pairwise disjoint.
Since $D_1''$ is not homotopic to $D_1'$, every $D_j'$ has to intersect $D_1''$ in $W_1$; this intersection is a path $P_j$ (possibly a single vertex). Since the curves $D_j'$ are pairwise disjoint, it follows that the paths $P_j$ are also pairwise disjoint. For $j=2,\ldots,q$, let $x_j$ be the point in $P_j$ closest to $x_1$, and let $T_j'$ be the unique $C_1$-ear starting at $x_j$. Now let $T_j$ be the unique $C_j$-ear starting on a vertex in $T_j'$, and let $W_j\subset C_j$ be the path joining the ends of $T_j$ that is disjoint from $T_1$. Finally, set $D_j''=T_j\cup
W_j$, for $j=2,\ldots,q$. It is straightforward to check that the curves $D_1'', D_2'', \ldots, D_q''$ satisfy the required properties.
To conclude the proof of Theorem \[thm:two-cycle-families\], we let $\{D_1'', D_2'', \ldots, D_q''\}$ be the collection guaranteed by this last claim. For each $i=1,2,\ldots,p$ and $j=1,2,\ldots,q$, we contract the path $C_i\cap D_j''$ to a single vertex (unless it already is a single vertex). Since the curves $D_1'',D_2'',\ldots,D_q''$ are pairwise disjoint and pairwise homotopic, it directly follows that the resulting graph is isomorphic to a subdivision of the $p\times q$-toroidal grid.
First we show the following.
\[cl:ellcycles\] $G$ has a set of at least $\frac \ell{\lfloor\Delta/2\rfloor}$ pairwise disjoint cycles, all homotopic to $\alpha$.
Let $F$ be the set of those edges of $G$ intersected by $\alpha$. Let $\alpha_1, \alpha_2$ be loops very close to and homotopic to $\alpha$, one to each side of $\alpha$, so that the cylinder bounded by $\alpha_1$ and $\alpha_2$ that contains $\alpha$ intersects $G$ only in the edges of $F$. Now we cut the torus by removing the (open) cylinder bounded by $\alpha_1$ and $\alpha_2$, thus leaving an embedded graph $H:=G-F$ on a cylinder $\cylin$ with boundary curves (“rims”) $\alpha_1$ and $\alpha_2$. Let $\delta$ be a curve on $\cylin$ connecting a point of $\alpha_1$ to a point of $\alpha_2$, such that $\delta$ has the fewest possible points in common with the embedding $H$. We note that we may clearly assume that the $p$ points in which $\delta$ intersects $H$ are vertices.
We claim that $p\geq \frac \ell{\lfloor\Delta/2\rfloor}$. Indeed, if $p< \frac \ell{\lfloor\Delta/2\rfloor}$, then the union of all faces incident with the $p$ vertices intersected by $\delta$ would contain a dual path $\beta$ of length at most $p\cdot \lfloor\Delta/2\rfloor <
\frac \ell{\lfloor\Delta/2\rfloor}\cdot
\lfloor\Delta/2\rfloor=\ell$. Such $\beta$ would be an $\alpha$-switching dual ear in $G^*$ of length less than $\ell$, a contradiction.
We now cut open the cylinder $\cylin$ along $\delta$, duplicating each vertex intersected by $\delta$. As a result we obtain a graph $H'$ embedded in the rectangle with sides $\alpha_1,\delta_1,\alpha_2,\delta_2$ in this cyclic order, so that $\delta_1$ (respectively, $\delta_2$) contains $p$ vertices $w_i^1, i=1,2,\ldots,p$ (respectively, $w_i^2, i=1,2,\ldots,p$).
We note that there is no vertex cut of size at most $p-1$ in $H'$ separating $\{w_1^1,\dots,w_p^1\}$ from $\{w_1^2,\ldots,w_p^2\}$, for such a vertex cut would imply the existence of a curve $\varepsilon$ from $\alpha_1$ to $\alpha_2$ on $\cylin$ intersecting $H$ in fewer than $p$ points, contradicting our choice of $\delta$. Thus applying Menger’s Theorem we obtain $p$ pairwise disjoint paths from $\{w_1^1,\dots,w_p^1\}$ to $\{w_1^2,\ldots,w_p^2\}$ in $H'$. Moreover, it follows by planarity of $H'$ that each of these paths connects $w_i^1$ to the corresponding $w_i^2$ for $i=1,\dots,p$. By identifying back $w_i^1$ and $w_i^2$ for $i=1,\dots,p$, we get a collection of $p$ pairwise disjoint cycles in $H$, each of them homotopic to $\alpha$.
We have thus proved the existence of a collection $\cc$ of $\ell/\dee$ pairwise disjoint, pairwise homotopic noncontractible cycles. By Theorem \[thm:deGS\], since $\fw(G) \ge \ewnd(G)/\dee$, it follows that $G$ also contains two collections $\dd,\ee$ of cycles such that: (i) the cycles in $\dd$ are noncontractible, pairwise disjoint, and pairwise homotopic; (ii) the cycles in $\ee$ are noncontractible, pairwise disjoint, and pairwise homotopic; (iii) for any $D\in\dd$ and $E\in\ee$, the pair $(D,E)$ is a basis; and (iv) each of $|\dd|$ and $|\ee|$ is at least $\big\lfloor\frac23\lceil\frac k{\lfloor\Delta(G)/2\rfloor}\rceil\big\rfloor$. Let $C\in\cc$, $D\in\dd$, and $E\in\ee$. From properties (i)–(iii) it follows that either $(C,D)$ or $(C,E)$ is a basis. Therefore the result follows from Theorem \[thm:two-cycle-families\].
Let $k:=\ewnd(G)$, and let $\ell$ and $\alpha$ be as in Theorem \[thm:agrid-torus\]. By Lemma \[lem:thstr\], . Let $r=\big\lceil\frac k{\lfloor\Delta(G)/2\rfloor}\big\rceil$. Since $r\geq5$, it follows that $\lfloor2r/3\rfloor\geq\frac67(2r/3)=\frac47r$ (with equality at $r=7$). Letting $s=\big\lceil\frac\ell{\lfloor\Delta(G)/2\rfloor}\big\rceil$ we then get, by Theorem \[thm:agrid-torus\], $$\Tex(G) \geq s\cdot\left\lfloor\frac23r\right\rfloor\geq
\frac47rs \geq \frac47k\ell \cdot\big\lfloor{\Delta(G)}/2\big\rfloor^{-2}
\geq \frac27\stretchd(G) \cdot\big\lfloor{\Delta(G)}/2\big\rfloor^{-2}
\,.$$ In order to get the better asymptotic estimate $\Tex(G)\geq(\frac8{21}-\varepsilon)\cdot
\lfloor{\Delta(G)}/2\rfloor^{-2}\cdot\stretchd(G)$, we directly apply Theorem \[thm:deGS\] in the case $s\leq2r/3$; otherwise, we use the stronger bound $\stretchd(G)\leq
k\ell+k\cdot k/2\leq k(\ell+3\ell/4)=\frac74k\ell$.
Drawing embedded graphs into the plane {#sec:drawing-upper}
======================================
In this section, we prove Theorem \[thm:upper-cr\]. That is, we provide an efficient algorithm that, given a graph $G$ embedded in some orientable surface, yields a drawing of $G$ (with a controlled number of crossings) in the plane. Although our algorithm takes an embedded graph as its input, we might as well take the non-embedded graph as input without any loss of efficiency; indeed, Mohar [@Mo99] showed that, for any fixed genus $g$, there is a linear time algorithm that takes as input any graph $G$ embeddable in $\Sigma_g$ and outputs an embedding of $G$ in $\Sigma_g$.
We start with an informal outline of the proof.
We proceed in $g$ steps, working at the $i$-th step with the pair $(G_i^*,\gamma_i)$. For convenience, let $G_0=G$, and define $F_i=E(G_{i-1})\sem E(G_{i})=E(\gamma_i)$. The idea at the $i$-th step is to cut from $G_{i-1}$ the edges intersected by $\gamma_i$ (that is, the set $F_i$). We could then to draw these edges into the embedded graph $G_i$ along the route determined by a $\gamma_i$-switching ear of length $\ell_i$ in $G_{i-1}$. This would result in at most $k_i(\ell_i+k_i)$ new crossings in $G_i$ (similarly as in Figure \[fig:addedges\]). For technical reasons, we consider routing the edges of each $F_i$ in one bunch (i.e., along the same route), even though routing every edge separately could perhaps save a small number of crossings.
In reality, the situation is not as simple as in the previous sketch. The main complication comes from the fact that subsequent cutting (step $j>i$) could “destroy” the chosen route for $F_i$ (or at least part of it). Then it would be necessary to perform a further re-routing for the edges of $F_i$ in step $j$. This could essentially happen in each subsequent step until the end of the process (when obtaining planar $G_g$).
We handle this complication in two ways: Proof-wise, we track a possible insertion route (and its necessary modifications) for $F_i$ through the full cutting process. In particular, we show that the final insertion route is never longer than $\ell_i+\ell_{i+1}+\dots+\ell_g$, for each index $i$. Another detail one has to take care of, is to ensure that such a detour for $F_i$ would not produce significantly more additional crossings than $k_j\ell_j$, over all $j=i+1,\ldots g$; this holds as long as $k_j$ is never much smaller than $k_i$ (cf. Lemma \[lem:dew2\]).
Algorithmically, we will reinsert all edges $\bigcup_{i=1}^g F_i$ only at the very end, into $G_g$. The previously tracked routes are then upper bounds for the so-achieved solution.
In the proof, we briefly use the concept of an *angle* of a pair of edges in an embedded graph. For this, we recall that the [ *rotation*]{} of a vertex $v$ in an embedded graph is the (say, counterclockwise, by convention) cyclic order in which the edges incident with $v$ leave this vertex. Suppose now that the rotation of a degree-$d$ vertex is $e_0,e_1,\ldots, e_{d-1}$, and let $(e_i,e_j)$ be an ordered pair. Then the [*angle*]{} of $(e_i,e_j)$ is the set of edges $\{e_i,
e_{i+1}, \ldots, e_{j-1}, e_j\}$ (with indices read modulo $d$).
As outlined in the sketch above, we proceed in $g$ steps. At the $i$-th step, for $i=1,2,\ldots,g$, we take the embedded graph $G_{i-1}$ and cut the surface open along $\gamma_i$, thus severing the edges in the set $F_i:=E(G_{i-1})\setminus E(G_i)=E(\gamma_i)$. This decreases the genus by one, and creates two holes, which we repair by pasting a closed disc on each hole. Thus we get the graph $G_i$ embedded in a compact surface with no holes.
\[cl:sumell\] Let $i=1,\ldots,g$, and let $f$ be an edge in $F_i$. Then, $f$ can be drawn into the plane graph $G_g$ with at most $\sum_{j=i}^g\ell_j$ crossings.
Let $i\in\{1,\ldots,g\}$ be fixed. In the graph $G_i$, we let $a,b$ denote the two new faces created by cutting $G_{i-1}$ along $\gamma_i$ (thus each of these faces contains one of the pasted closed discs). Let $f$ be an edge in $F_i$, with endpoints $f_a$ (incident with $a$ in $G_i$) and $f_b$ (incident with $b$ in $G_i$).
For each $j=i,i+1,\ldots,g$, we associate two faces $a_j(f), b_j(f)$ of $G_j$ with $f$. Loosely speaking, these faces are the natural heirs in $G_j$ of the faces $a$ and $b$, if we stand in $G_j$ on the vertices $f_a$ and $f_b$ (we note that $a,b$ are faces in $G_i$, but by the further cutting process, they may not be faces in $G_j$ for some $j > i$). The faces $a_j(f), b_j(f)$ are recursively defined as follows. First, let $a_i(f) = a$ and $b_i(f) = b$. Now suppose $a_{j-1}(f), b_{j-1}(f)$ have been defined for some $j$, $i< j \le g$. We then let $a_j(f)$ be the unique face $h$ of $G_j$ that satisfies the following: if $(e,e')$ is the pair of edges of $h$ incident with $f_a$, ordered so that the angle of $(e,e')$ in $G_{j}$ consists only of $e$ and $e'$, then the angle of $(e,e')$ in $G_{j-1}$ includes the edges of the face $a_{j-1}(f)$ that are incident with $f_a$. The face $b_j(f)$ is defined analogously.
The vertex $f_a$ (respectively, $f_b$) is incident to the face $a_g(f)$ (respectively, $b_g(f)$) in the plane embedding $G_g$. To finish the proof, it suffices to show that the dual distance between $a_g(f)$ and $b_g(f)$ in $G_g$ is at most $\sum_{j=i}^g\ell_j$. We prove this via induction over $j=i,i+1,\dots,g$, i.e., we show that the dual distance between $a_j(f)$ and $b_j(f)$ in $G_j$ is at most $\ell_i+\ell_{i+1}+\dots+\ell_j$.
This holds (with equality) for $j=i$ by the definition of $\ell_i$. For $j>i$, take a shortest dual path $\pi$ in $G_{j-1}$ connecting $a_{j-1}(f)$ to $b_{j-1}(f)$. Unless $\pi$ intersects $\gamma_j$, its length also bounds the dual distance in $G_j$. Assuming $\pi\cap\gamma_j\not=\emptyset$ in $G_{j-1}$, we can replace (in $G_j$) the section of $\pi$ between the first and the last intersection with $\gamma_j$ by a $\gamma_j$-switching ear of length $\ell_j$. It follows that the dual distance between $a_j(f)$ and $b_j(f)$ is at most $
\len(\pi)+\ell_j\leq \ell_i+\dots+\ell_{j-1}+\ell_j$, as claimed.
Now recall that $|F_i|=k_i$, for $i=1,\ldots,g$. From Claim \[cl:sumell\] it follows that the edges in $F_i$ can be added to the plane embedding $G_g$ by introducing at most $k_i \cdot \sum_{j=i}^g\ell_j$ crossings with the edges of $G_g$. This measure disregards any additionally crossings arising between edges of $F_i$. We add to $G_g$ the edges of $F_{g}$, then the edges of $F_{g-1}$, and so on. As we add the edges of $F_i$, in the worst case scenario each edge we add crosses every edge already or currently inserted; thus the total cost of adding the edges of $F_i$ is at most $k_i \cdot \sum_{j=i}^g\ell_j + k_i \cdot \sum_{j=i}^g k_j$. Overall, the edges $F_1 \cup F_2 \cup \cdots \cup F_g$ can be added to the plane embedding by introducing at most $\sum_{i=1}^g \left(k_i\cdot\sum_{j=i}^g (k_j+\ell_j)\right)$ crossings. Using that $2\ell_i\geq k_i$ (cf. Lemma \[lem:kl2\]), this process yields a drawing of $G$ in the plane with at most $$\begin{aligned}
\sum_{i=1}^g \left(k_i\cdot\sum_{j=i}^g (k_j+\ell_j)\right)\;&\leq\;
\sum_{i=1}^g \left(k_i\cdot\sum_{j=i}^g 3\ell_j\right)
\\
&=\; 3\sum_{j=1}^g \left(\ell_j\cdot\sum_{i=1}^j k_i\right)\end{aligned}$$ crossings. The inductive application of Lemma \[lem:dew2\] yields $k_i\leq2^{j-i}k_j$ for all $1\leq i<j\leq g$. Therefore $$\begin{aligned}
\nonumber
3\sum_{j=1}^g \left(\ell_j\cdot\sum_{i=1}^j k_i\right)\;&\leq\;
3\sum_{j=1}^g \ell_jk_j(2^{j-1}+\dots+2^1+2^0)
\\\nonumber
&=\; 3\sum_{j=1}^g k_j\ell_j(2^{j}-1)
\\\nonumber
&\leq\; 3\max_{1\le i \le g} \{k_i\ell_i\}\cdot(2^1+2^2+\dots+2^g-g)
\\\label{eq:3M}
&=\; 3\cdot(2^{g+1}-2-g)\cdot \max_{1\le i \le g} \{k_i\ell_i\}.\end{aligned}$$
We have thus shown how to produce a drawing of $G$ with at most $3\cdot(2^{g+1}-2-g)\cdot \max_{1\le i \le g} \{k_i\ell_i\}$ crossings. It remains to show how such a drawing can be computed efficiently from an embedding of $G$ in $\Sigma_g$. The algorithm runs two phases:
1. A good planarizing sequence $(G_1^*,\gamma_1), \dots, (G_g^*,\gamma_g)$ for $G^*$ is computed using $g$ calls to the $\ca O(n\log n)$ algorithm of Kutz [@Ku06], which finds a cycle witnessing nonseparating edge-width in orientable surfaces. During the computation, we represent $G^*$ by its rotation scheme which allows fast implementation of the cutting operation as well.
2. In the planar graph $G_g$, optimal insertion routes are found for all the missing edges $F=E(G)\sem E(G_g)$ using linear-time breadth-first search in $G_g^*$. A key observation is that we are looking for these insertion routes only between predefined pairs of faces $a_g(f)$ and $b_g(f)$ for each $f\in F$. Since each of $\{a_g(f) \ : f\in F_i\ \}$ and $\{b_g(f) \ : f\in F_i\ \}$ has at most $2^{g-i}$ elements for each $i=1,2,\ldots,g$, it follows that we need to perform at most $2^{g-1}+\dots+2^1+2^0<2^g$ searches in total (independently of $|F|$), a process that takes an overall linear time for fixed $g$. From the practical point of view, it may be worthwhile to mention that $|G_g|$ also serves as a natural upper bound for the considered faces.
In view of this, the overall runtime of the algorithm is $\ca O(n\log
n)$ for each fixed $g$.
More properties of stretch {#sec:more}
==========================
In this section, we establish several basic properties on the stretch of an embedded graph. Even though we could have alternatively included these in the next section, as we only require them in the proof of Lemma \[lem:kl-to-stretch\], we prefer to present them in a separate section, for an easier further reference of the basic properties of this new parameter which may be of independent interest.
We recall that a graph property $\ca P$ satisfies the [*$3$-path condition*]{} (cf. [@MT01 Section 4.3]) if the following holds: Let $T$ be a [*theta graph*]{} (a union of three internally disjoint paths with common endpoints) such that two of the three cycles of $T$ do not possess $\ca P$; then neither does the third cycle. In the proof of the following lemma we make use of halfedges. A [*halfedge*]{} is a pair $\langle e,v\rangle$ (“$e$ at $v$”), where $e$ is an edge and $v$ is one of the two ends of $e$.
\[lem:3pp\] Let $G$ be embedded on an orientable surface, and let $C$ be a cycle of $G$. The set of cycles of $G$ satisfies the $3$-path condition for the property of odd-leaping $C$. Furthermore, not all three cycles in any theta subgraph of $G$ can be odd-leaping $C$.
Let a theta graph $T\subseteq G$ be formed by three paths $T=T_1\cup T_2\cup T_3$ connecting the vertices $s,t$ in $G$. We consider a connected component $M$ of $C\cap T$. If $M=\emptyset$ or $M=C$, then the $3$-path condition trivially holds. Otherwise, $M$ is a path with ends $m_1,m_2$ in $G$. We denote by $f_1,f_2$ the edges in $E(C)\sem E(M)$ incident with $m_1,m_2$, respectively, and by $M^+$ the union of $M$ and the halfedges $\langle f_1,m_1\rangle$ and $\langle f_2,m_2\rangle$. We show that the number $q$ of leaps of $M^+$ summed over all three cycles in $T$ is always even.
If $m_i\not\in\{s,t\}$ for $i\in\{1,2\}$, then contracting the edge of $M$ incident to $m_i$ clearly does not change the number $q$. Iteratively applying this argument, we can assume that finally either (i) $m_1=m_2$ (and possibly $m_1\in\{s,t\}$), or (ii) $m_1=s$, $m_2=t$, and $M=T_1$. In case (i), $M^+$ leaps either none or two of the cycles of $T$ in the single vertex $m_1$, and so $q\in\{0,2\}$. Thus we assume for the rest of the proof that (ii) holds.
For $i=1,2,3$, let $e_i$ (respectively, $e_i'$) be the edge of $T_i$ incident with $s$ (respectively, $t$). By relabeling $e_1, e_2, e_3$ if needed, we may assume that the rotation around $s$ is one of the cyclic permutations $(e_1,f_1,e_2,e_3)$ or $(e_1,e_2,f_1,e_3)$. The rotation around $t$ could be any of the six cyclic permutations of $e_1',e_2',e_3',f_2$. This yields a total of twelve possibilities to explore. A routine analysis shows that in every case we get $q\in\{0,2\}$, except for the case in which the rotation around $s$ is $(e_1,e_2,f_1,e_3)$ and the rotation around $t$ is $(e_1',e_2',f_2,e_3')$; in this case, $M^+$ leaps twice the cycle $T_2\cup T_3$, and $q=4$.
Altogether, the number of leaps of $C$ summed over all three cycles in $T$ is even. Hence the number of cycles of $T$ which are odd-leaping with $C$ is also even, and the $3$-path condition follows.
The next claim shows that stretch (Definition \[def:stretch\]) could have been equivalently defined as an [*odd-stretch*]{}, using pairs of odd-leaping cycles instead of one-leaping cycles.
\[lem:odd-stretch\] Let $G$ be a graph embedded in an orientable surface. If $C,D$ is an odd-leaping pair of cycles in $G$, then $\stretch(G)\leq\len(C)\cdot\len(D)$.
We choose an odd-leaping pair $C,D$ that minimizes $\len(C)\cdot\len(D)$. Up to symmetry, $\len(C)\leq\len(D)$. Since $C\cap D\not=\emptyset$, there is a set $\ca D=\{D_1,\dots,D_k\}$ of pairwise edge-disjoint $C$-ears in $D$, such that $E(D_1)\cup\dots\cup E(D_k)=E(D)\sem E(C)$. By a simple parity argument, there exists a $C$-switching ear in $\ca D$. Hence if $|\ca D|=1$, then $C,D$ are one-leaping, and the lemma immediately follows.
If more than one $C$-ear in $\ca D$ is switching, then we pick, say, $D_1$ as the shorter of these. By the choice of $D$ we have $\len(D_1)\leq\frac12\len(D)$, and so by Lemma \[lem:thstr\] we have $$\stretch(G)\leq\len(C)\cdot\left(\len(D_1)+\frac12\len(C)\right)
\leq\len(C)\cdot\left(\frac12\len(D)+\frac12\len(D)\right) =
\len(C)\cdot\len(D)
\,,$$ as required.
In the remaining case, we have that $|\ca D|>1$ and exactly one $C$-ear in $\ca D$ (say $D_1$) is switching. We pick any $D_j\in\ca D$, $j>1$, let $u,v$ be the ends of $D_j$ on $C$, and compare the distance $d$ between $u$ and $v$ on $C$ with $\len(D_j)$. If $d>\len(D_j)$, then both cycles of $C\cup D_j$ containing $D_j$ are shorter than $\len(C)$, and one of them is odd-leaping with $D$ by Lemma \[lem:3pp\]. This contradicts the choice of $C$ (for the pair $C,D$, that is). Hence $\len(D_j)\geq d$, and summing these inequalities over all $j=1,\dots,k$ we get $\len(D_1)\leq\len(D)-s$, where $s$ is the distance between the ends of $D_1$ on $C$. Similarly as in Lemma \[lem:thstr\], we then get $$\stretch(G)\leq\len(C)\cdot(\len(D_1)+s)
\leq\len(C)\cdot\left(\len(D)-s+s\right)
= \len(C)\cdot\len(D).\tag*{\qedhere}$$
\[lem:cutdew\] Let $H$ be a graph embedded in an orientable surface of genus $g\geq2$, and let $A,B\subseteq H$ be a one-leaping pair of cycles witnessing the stretch of $H$, such that $\len(A)\leq\len(B)$. Then $\ewn(H\cutt A)\geq\frac12\ewn(H)$.
Let $C$ be a nonseparating cycle in $H\cutt A$ of length $\ewn(H\cutt A)$. If its lift $\hat C$ is a cycle again, then (since $\hat C$ is nonseparating in $H$) $\ewn(H)\leq\len(\hat C)=\ewn(H\cutt A)$, and we are done. Thus we may assume that $\hat C$ contains an $A$-ear $P\subseteq\hat C$ such that $A\cup P$ is a theta graph. Let $A_1,A_2\subseteq A$ be the subpaths into which the ends of $P$ divide $A$. By Lemma \[lem:3pp\], exactly two of the three cycles of $A\cup P$ are odd-leaping with $B$. One of these cycles is $A$; let the other one, without loss of generality, be $A_1\cup P$. Then $\len(A_1\cup P)\geq\len(A)$ using Lemma \[lem:odd-stretch\], and so $\len(P)\geq\len(A_2)$. Furthermore, $A_2\cup P$ is nonseparating in $H$, and we conclude that $$\ewn(H)\leq\len(A_2\cup P)\leq
2\len(P)\leq2\len(\hat C)=2\ewn(H\cutt A)
\,.\tag*{\qedhere}$$
At this point, an attentive reader may wonder why we do not use the cutting paradigm as in Lemma \[lem:cutdew\] in a good planarizing sequence for Theorem \[thm:upper-cr\] (Section \[sec:drawing-upper\]). Indeed, it would seem that the same proof as in Section \[sec:drawing-upper\] works in this new setting, and the added benefit would be an immediately matching lower bound in the form provided by Corollary \[cor:agrid-all\]. The caveat is that the proof of Theorem \[thm:upper-cr\] strongly uses the fact that subsequent cuts in a planarizing sequence do not involve [*much fewer*]{} edges (recall “$k_i\leq2^{j-i}k_j$ for all $1\leq i<j\leq g$” from the proof). If one cuts along the shortest cycle of a pair that witnesses the dual stretch, then the number of cut edges may jump up or down arbitrarily. Thus an attempted proof along the lines of the proof we gave in Section \[sec:drawing-upper\] would (inevitably?) fail at this point.
Finding a subgraph of large stretch {#sec:finding}
===================================
In this section we prove Lemma \[lem:kl-to-stretch\]. Therefore, we need to generalize the concepts of switching and leaping. Given an embedded graph $H$ and an embedded subgraph $D\subset G$, we want to talk about $D$-switching ears, and walks that are $k$-leaping $D$, also in cases when $D$ is a not necessarily a cycle. The essential property of a cycle used in these definitions is that it has two clearly defined sides. We generalize this feature (to subgraphs that are not necessarily cycles) to a property we call *polarity*.
Polarity
--------
Let $H$ be a graph cellularly embedded in a surface $\Sigma$, and let $D$ be a (not necessarily connected) subgraph of $H$. The [*$H$-induced*]{} embedding $\tilde D$ of the graph $D$ is determined by the system of $H$-rotations around vertices of $D$ restricted to $E(D)$. Intuitively, $\tilde D$ is obtained from the usual subembedding of $D$ in $\Sigma$ via replacing all non-cellular faces with discs. Notice that $\tilde D$ has a separate surface for each connected component of $D$. If $\tilde D$ can be face-bicolored, then we say that $D$ is [*bipolar in $H$*]{}, and we associate one chosen facial bicoloring of $\tilde D$ with $D$ (notice that this bicoloring is not unique when $D$ is not connected). We will refer to the facial colors of $\tilde D$ (white and black) as the [*$D$-polarities*]{} in $H$ (positive and negative, respectively).
More formally, for $v\in V(D)$ and $e\not\in E(D)$, the halfedge $\langle e,v\rangle$ has a [*positive*]{} ([*negative*]{}) $D$-polarity if the position of $e$ in the $H$-rotation around $v$ is between consecutive edges of a white (black) $\tilde D$-face. Clearly, a cycle in any orientable embedding is always bipolar. Also, if $D$ is bipolar, then it is Eulerian.
A $D$-ear $P$ is [*$D$-polarity switching*]{} if the halfedges of $P$ incident with the ends of $P$ are of distinct $D$-polarities. If $D$ is a cycle, then being “$D$-polarity switching” is equivalent to being “$D$-switching”. We now consider a (possibly closed) walk $W\subseteq H$. A proper subwalk $M$ of $W$ is called a [*polarity leap (of $W$ and $D$)*]{} if
- $M\subseteq D\cap W$ and neither the edge $f_0$ preceding $M$ in $W$ nor the edge $f_1$ succeeding $M$ in $W$ belong to $D$ (in particular, $M$ is neither a prefix nor a suffix of $W$), and
- the halfedges of $f_0,f_1$ incident with $M$ are of distinct $D$-polarities.
We say that $W$ is [*odd-leaping*]{} bipolar $D$ if the number of all proper subwalks of $W$ which are polarity leaps is odd; otherwise $W$ is [*even-leaping*]{} $D$. Notice that being “one-leaping” (Definition \[def:leaping\]) implies “odd-leaping” in this new sense.
The workhorse
-------------
Informally speaking, the intuition behind our proof of Lemma \[lem:kl-to-stretch\] is to suitably cut down the embedding $G$ to a smaller surface (destroying handles causing small stretch; remember our aim is to find a subgraph with large stretch), while approximately preserving $\gamma$ and its switching distance.
The main tool behind the proof of Lemma \[lem:kl-to-stretch\] is the following lemma. To make sense of this statement, and to grasp how this easily leads to the proof of Lemma \[lem:kl-to-stretch\], we refer the reader to the informal discussion provided immediately after the statement.
\[lem:cutstep\] Let $H$ be a graph embedded in an orientable surface. Suppose that:
- there is a bipolar dual subgraph $\delta$ in $H^*$;
- there exists a closed walk in $H^*$ that is odd-leaping $\delta$; and
- the shortest $\delta$-polarity switching ear in $H^*$ has length $h$.
Let $\alpha,\beta$ be a one-leaping pair (any one) of dual cycles in $H^*$ such that $\len(\alpha)\leq\len(\beta)$ and $\stretchd(H)=\len(\alpha)\cdot\len(\beta)$. Then, unless (d) $\len(\beta)\geq h$, the following hold:
- there is a bipolar dual subgraph $\delta_1$ (“induced” by $\delta$) in $(H\cutt \alpha)^*$;
- there exists a closed walk in $(H\cutt \alpha)^*$ that is odd-leaping $\delta_1$; and
- the shortest $\delta_1$-polarity switching ear in $(H\cutt \alpha)^*$ has length $h_1\geq h-\frac12\len(\alpha)$.
Conditions (a) and (a’) address the “preservation of $\gamma$” requisite from Lemma \[lem:kl-to-stretch\], and (c),(c’) address the necessarily long “switching distance”. Conditions (b) and (b’) have a purely technical purpose. Notice, for instance, that if (b) is true, then the embedding $H$ is not planar (and so the stretch of $H$ is well defined). Indeed, a closed walk odd-leaping a bipolar plane $\delta$ cannot exist since such a $\delta$ would equal its $H^*$-induced embedding $\tilde\delta$, which means that $\delta$ is face-bicolored, too; a simple parity argument then gives a contradiction. For a similar parity reason, (b) implies that a $\delta$-polarity switching ear in $H^*$ (implicitly required in (c)) must exist. Moreover, as we proceed in the cutting process, the non-planarity implied by (b’) guarantees that we will eventually arrive at the desired exceptional conclusion (d) $\len(\beta)\geq h$, which is the ultimately desired outcome for Lemma \[lem:cutstep\].
Recall the definition of cutting an embedding $H$ along a dual cycle $\alpha$. The dual graph $H^*\cutt\alpha=(H\cutt\alpha)^*$ is obtained from $H^*$ by successive contractions of all the dual edges in $E(\alpha)$ into one dual vertex $a$, and then “splitting” $a$ into two $a_1,a_2$ (giving the two $\alpha$-cut faces of $H\cutt\alpha$). This “stepwise contraction” perspective of cutting turns out to be very useful in our proof.
[*Proof of (a’)*]{}. Let $\varepsilon$ denote the subgraph of $H_1^*$ induced by the edges in $E(\delta)\sem E(\alpha)$. If $\alpha=\delta$, then clearly (d) $\len(\beta)\geq h$, and so we may assume that $\varepsilon$ is nonempty. We show that we can choose $\delta_1=\varepsilon$, under the assumption that $\alpha$ contains a $\delta$-polarity switching ear (the validity of this assumption follows since, if no such switching ear existed, then by (c) it would follow that $\len(\beta)\geq\len(\alpha)\geq h$, thus implying (d)).
The following is immediate from the definition of bipolarity:
\[factA\] If $f\in E(H^*)$ is not a loop-edge and not a $\delta$-polarity switching ear, then the dual graph $H^*\contract f$ (obtained by [*contraction*]{} of $f$) is embedded in the same surface as $H^*$, and the dual subgraph $\delta'$ induced by $E(\delta)\sem\{f\}$ in $H^*\contract f\>$ is bipolar again, where the $\delta'$-polarities are naturally inherited from the $\delta$-polarities.
Since we assume that $\alpha$ contains no $\delta$-polarity switching ear, we can iteratively apply Fact \[factA\] to all edges of $\alpha$ except some (the last one) $f_1\in E(\alpha)\sem E(\beta)$. In this way we get an “intermediate” embedding $H_1^*=H^*\contract\big(E(\alpha)\!\sem\!\{f_1\}\big)$ such that $f_1$ is a nonseparating dual loop-edge in $H_1^*$, and bipolar $\varepsilon_1\subseteq H_1^*$ is naturally derived from $\delta$. Let $a$ be the face of $H_1$ that is the double end of $f_1$, and let the $H_1^*$-rotation of edges around $a$ be $e_1,\dots,e_i,f_1,e_1',\dots,e_j',f_1$. The last step in the construction of $H_1^*$ (and of $\varepsilon$) is to remove $f_1$ and split $a$ into $a_1,a_2$ such that the $H_1^*$-rotation around $a_1$ (respectively, $a_2$) is $e_1,\dots,e_i$ (respectively, $e_1',\dots,e_j'$).
Clearly, $\varepsilon_1=\varepsilon$ stays bipolar in $H_1^*$ if $a\not\in V(\varepsilon_1)$, and so we assume $a\in V(\varepsilon_1)$. Let $\tilde\varepsilon$ denote the $H_1^*$-induced embedding of $\varepsilon$. Let $e_a$ and $e_b$ be the first and last element of the list $e_1,\ldots,e_i$, respectively, that are also edges of $\varepsilon$. Note that both ends of $f_1$ in the $H_1^*$-rotation around $a$ are between $e_b$ and $e_a$. Then, $e_b,e_a$ appear consecutively on a unique face $x$ of $\tilde\varepsilon$. Analogously, we find a face $x'$ at $a_2$ in $\tilde\varepsilon$. Loosely speaking, $x,x'$ are the dual $\tilde\varepsilon$-faces “inheriting” the two $H_1^*$-faces incident with $f_1$. If $f_1\not\in E(\varepsilon_1)$, then both halfedges of $f_1$ are of the same $\varepsilon_1$-polarity (by our assumption on $\alpha$), say positive. Hence both $\tilde\varepsilon$-faces $x$ and $x'$ will get (consistently) positive polarity, and so $\varepsilon$ is bipolar in $H_1^*$. If, on the other hand, $f_1\in E(\varepsilon_1)$, then one of the two faces incident with $f_1$ in the $H_1^*$-induced embedding $\tilde\varepsilon_1$ of $\varepsilon_1$ is positive, say the one containing edge(s) from $e_1,\dots,e_i$, and the other one is negative. Then the $\tilde\varepsilon$-face $x$ will be (consistently) positive and $x'$ negative. Thus also in this case $\varepsilon=\delta_1$ is bipolar in $H_1^*$.
[*Proof of (b’)*]{}. As in (a’), we may assume that $\alpha$ contains no $\delta$-polarity switching ear. We can make a similar assumption with $\beta$: if there is a $\delta$-polarity switching ear contained in $\beta$, then $\len(\beta)\geq h$ (that is, (d) holds). The following counterpart of Fact \[factA\], formulated for any closed dual walk $\psi$ in $H^*$, is easily derived from our definition of a leap.
\[factB\] Suppose $f\in E(H^*)$ is not a loop-edge and not a $\delta$-polarity switching ear, and denote by $\delta'$, $\psi'$ the dual subgraphs induced by $E(\delta)\sem\{f\}$ and $E(\psi)\sem\{f\}$ in $H^*\contract f$ (i.e., after contraction of $f$). Then the number of leaps of $\delta'$ and $\psi'$ in $H^*\contract f$ is the same as the number of leaps of $\delta$ and $\psi$ in $H^*$, with an exception when $f\in E(\psi)\sem E(\delta)$ and both ends of $f$ are incident with leaps of $\delta$ and $\psi$ in $H^*$ (in which case the two leaps vanish in $H^*\contract f$).
We now proceed in the same way as in (a’), and use the same notation $H_1^*,$ $f_1,a,$ $\varepsilon_1$, etc. Let $\omega$ be a dual closed walk in $H^*$ odd-leaping $\delta$, and $\omega_1$, $\beta_1$ denote the dual closed walks in $H_1^*$ induced by $E(\omega)\cap E(H_1^*)$ and $E(\beta)\cap E(H_1^*)$. By an iterative application of Fact \[factB\] to all edges in $E(\alpha)\sem\{f_1\}$, we get that the parity of leaping between $\delta$ and $\omega$ (respectively, $\delta$ and $\beta$) in $H^*$ is the same as that between $\varepsilon_1$ and $\omega_1$ (respectively, $\varepsilon_1$ and $\beta_1$) in $H_1^*$. Hence $\omega_1$ is odd-leaping $\varepsilon_1$, and $\beta_1$ is even-leaping $\varepsilon_1$, since $\beta$ contains no $\delta$-polarity switching ear in $H^*$ and so $\beta$ is not odd-leaping $\delta$.
We note that $a\in V(\beta_1)$ since $\alpha$ intersects $\beta$, and recall $f_1\not\in E(\beta)$. If $f_1\in E(\omega)$, then we moreover remove $f_1$ from $\omega_1$; this does not change the parity of leaping between $\varepsilon_1$ and $\omega_1$. We say that the dual walk $\omega_1$ [*passes through*]{} $a$ in $H_1^*$ if one edge of $\omega_1$ is from $e_1,\dots,e_i$ and the next edge of $\omega_1$ is among $e_1',\dots,e_j'$, or vice versa. Every time $\omega_1$ passes through $a$, we replace this pass by one iteration of the cycle $\beta_1$. The resulting closed dual walk $\omega_2$ in $H_1^*$ (which does not pass through $a$) is again odd-leaping $\varepsilon_1$, since $\beta_1$ is even-leaping $\varepsilon_1$. Then, the subgraph $\omega_0$ induced by $E(\omega_2)$ in the graph $H_1^*$ is a closed dual walk odd-leaping $\varepsilon=\delta_1$.
[*Proof of (c’)*]{}. Let $\sigma$ be a $\delta_1$-polarity switching ear in $H_1^*$ of length $h_1$. If $V(\sigma)$ contains both $\alpha$-cut faces $a_1,a_2$, then the lift $\hat\nu$ of a subpath $\nu\subseteq\sigma$ between $a_1$ and $a_2$ is a $\delta$-polarity switching ear, and hence $h\leq\len(\hat\nu)\leq h_1$, thus implying (c’). Otherwise, the lift $\hat\sigma$ in $H^*$ is an $(\alpha\cup\delta)$-ear which means that, for some subpath $\pi\subseteq\alpha$ of length at most $\frac12\len(\alpha)$ (possibly empty), $\hat\sigma\cup\pi$ is a $\delta$-ear. Since $\sigma$ is $\delta_1$-polarity switching in $H_1^*$, and the $\delta_1$-polarities are inherited from those of $\delta$ in $H^*$ by (a’) and Fact \[factA\], we conclude that $\hat\sigma\cup\pi$ is a $\delta$-polarity switching ear. Therefore, $h\leq\len(\hat\sigma\cup\pi)\leq h_1+\frac12\len(\alpha)$ as claimed.
Proof of Lemma \[lem:kl-to-stretch\]
------------------------------------
We proceed by induction, using Lemma \[lem:cutstep\]. Notice that all the conditions (a),(b),(c) of Lemma \[lem:cutstep\] are satisfied by the graph $H$, its bipolar dual cycle $\delta:=\gamma$, and by $h:=\ell$. Let $H_0=H$, $\gamma_0=\gamma$, and $\ell_0=\ell$. Until we reach the condition (d) $\len(\beta)\geq h$, we repeatedly apply Lemma \[lem:cutstep\] for $i=1,2,\dots$ to $H:=H_{i-1}$ and $\delta:=\gamma_{i-1},\, h:=\ell_{i-1}$, obtaining $H_{i}:=H\cutt\alpha$ and $\gamma_{i}:=\delta_1,\, \ell_i:=h_1$. After the maximum possible number $i$ of iterations in which (d) does not hold:
- the graph $H_i$ has genus $g-i$, and it is $i\leq g-1$ since (b’) implies nonplanarity of $H_i$;
- the nonseparating dual edge-width is $\ewnd(H_i)\geq2^{-i}\cdot\ewnd(H)>1$ (this follows by iterating Lemma \[lem:cutdew\] $i$ times); and
- the shortest $\gamma_i$-polarity switching ear in $H_i^*$ has length at least $\ell_i\geq 2^{-i}\cdot\ell$, since one can iterate $h_1\geq h-\frac12\len(\alpha)\geq h-\frac12\len(\beta)\geq \frac12h$ at each of the $i$ steps.
Hence (as no further iteration is possible), we have gotten an $i\leq g-1$ such that (cf. Lemma \[lem:cutstep\]) there exists a pair of odd-leaping dual cycles $\alpha_i,\beta_i$ in $H_i^*$ such that $\stretchd(H_i)=\len(\alpha_i)\cdot\len(\beta_i)$, and (d) $\len(\beta_i)\geq \ell_i$ holds. Consequently, $$\stretchd(H_i)=\len(\alpha_i)\cdot\len(\beta_i)\geq
\ewnd(H_i)\cdot \ell_i\geq 2^{-i}\ewnd(H)\cdot2^{-i}\ell
= 2^{-2i}\cdot k\ell
\,.$$ By setting $H'=H_i$ for $g'=g-i$, Lemma \[lem:kl-to-stretch\] follows.
Concluding remarks {#sec:concluding}
==================
There are several natural questions that arise.
#### Extension to nonorientable surfaces.
One can wonder whether our results, namely about approximating planar crossing number of an embedded graph, can also be extended to nonorientable surfaces of higher genus. Indeed, the upper-bound result of [@BPT06] holds for any surface, and there is an algorithm to approximate the crossing number for graphs embeddable in the projective plane [@GHLS08]. We currently do not see any reason why such an extension would be impossible.
However, the individual steps become much more difficult to analyze and tie together, since the “cheapest” cut through the embedding can cut (a) a handle along a two-sided loop, (b) an antihandle along a two-sided loop, or (c) a crosscap along a one-sided loop. Hence it then does not suffice to consider toroidal grids as the sole base case (and a usable definition of “nonorientable stretch” should reflect this), but the lower bound may also arise from a projective or Klein-bottle grid minor. Already for the latter, there are currently no non-trivial results known. We thus leave this direction for future investigation.
#### Dependency of the constants in Theorem \[thm:main-overview\] on $\Delta$ and $g$.
Taking a toroidal grid with sufficiently multiplied parallel edges (possibly subdividing them to obtain a simple graph) easily shows that a relation between the toroidal expanse and the crossing number must involve a factor of $\Delta^2$. Regarding an efficient approximation algorithm for the crossing number, general dependency on the maximum degree seems unavoidable as well, as is suggested by comparison with related algorithmic results. However, considering the so-called minor crossing number (see Section \[sec:minorcr\] below), one can avoid this dependency at least in a special case.
The exponential dependency of the constants and the approximation ratio on $g$, on the other hand, is very interesting. It pops up independently in multiple places within the proofs, and these occurrences seem unavoidable on a local scale, when considering each inductive step independently. However, it seems very hard to construct any example where such an exponential jump or decrease can actually be observed. It might be that a different approach with a global view can reduce the dependency in Theorem \[thm:main-overview\] to some $\mathit{poly}(g)$ factor, cf. also [@DV12].
Toroidal grids and minor crossing number {#sec:facestretch}
----------------------------------------
\[sec:minorcr\] The *minor crossing number* $\mcr(G)$ [@BFM06] is the smallest crossing number over all graphs $H$ that have $G$ as their minor. Hence it is, by definition and in contrast to the traditional crossing number, a well-behaved minor-monotone parameter. In general, however, minor crossing number is not any easier to compute [@Hl06] than ordinary crossing number. We note the following intuitive observation related to our topic: if $G$ is embedded in $\Sigma$ with face-width $r$, then $G$ is a surface minor of a graph $H$ (in particular, $H$ is embedded in $\Sigma$ as well) such that $\ewn(H)=r$. Indeed, consider a loop $\lambda$ in $\Sigma$ attaining $\fw(G)$ and split every vertex intersected by $\lambda$ into an edge “perpendicular” to $\lambda$. This results in desired $H$ (for formal details, see the proof of Lemma \[lem:cr-fstretch-torus\]).
For an embedded graph $G$, let $G_f$ denote the vertex-face incidence (bipartite) graph of $G$. It is well-known that $\fw(G)=\frac12\ew(G_f)$. We can analogously define the [*face stretch*]{} of an embedded graph $G$ as $\stretchf(G)=\frac14\stretch(G_f)$, and claim:
\[lem:cr-fstretch-torus\] Let $G$ be a graph embedded in an orientable surface $\Sigma$. Then there is a graph $H$ also embedded in $\Sigma$, such that $G$ is a minor of $H$ and $$\stretchd(H)\leq \stretchf(G)+\sqrt{\stretchf(G)} .$$
Let $A,B$ be one-leaping cycles of $G_f$ witnessing $\stretchf(G)$. When viewing $A$ and $B$ as simple loops $\alpha$ and $\beta$, respectively, on the surface $\Sigma$, they intersect the embedding of $G$ only in $a=\len(A)/2$ and $b=\len(B)/2$ vertex points. Consider a vertex $v$ of $G$ intersected by $\alpha$. We replace $v$ in the embedding with two new vertices $v_l,v_r$, where $v_l$ is incident with those edges of $v$ on the left-hand side of $\alpha$ and $v_r$ with the edges of $v$ on the right-hand side of $\alpha$. We join $v_l$ to $v_r$ with a new edge; it is “perpendicular” to $\alpha$ in the embedding in $\Sigma$ (Figure \[fig:fstr-split\]). Let $H_0$ be the new graph having $G$ as its minor. If $v$ belongs also to $\beta$, and there is an edge (or two) of $E(B)\sem E(A)$ in $G_f$ incident to $v$, then we position the corresponding one (or two) of $v_l,v_r$ right on this section of $\beta$ close to original $v$. So, $\beta$ intersects the embedded graph $H_0$ only in vertex points, as well. We apply the same construction to the vertices of $H_0$ intersected by $\beta$, resulting in the desired embedded graph $H$ having $G$ as its minor.
In $H$, the loop $\alpha$ now intersects exactly $a$ edges (and no vertex), while the loop $\beta$ intersects $b$ or $b+1$ edges. The latter case happens when $\alpha,\beta$ intersect each other in exactly one vertex point $v$ of $G$, and hence both $v_l,v_r$ belong to $\beta$ in $H'$. (Generally, this odd case is unavoidable in the situation illustrated in Figure \[fig:fstr-split\].) Therefore, up to symmetry between $\alpha,\beta$, $H$ witnesses that $\stretchd(H)\leq\min\{a(b+1),b(a+1)\}= ab+\min(a,b)\leq ab+\sqrt{ab}$, where $\stretchf(G)=ab$.
From Lemma \[lem:cr-stretch-torus\] we then immediately obtain:
\[cor:cr-fstretch-torus\] If $G$ is a graph embedded in the torus, then $\mcr(G)\leq\stretchf(G)+\sqrt{\stretchf(G)}$. Assuming $\fw(G)\geq5$, we have $\mcr(G)\leq\frac65\stretchf(G)$.
The next logical step is to translate the findings from Section \[sub:toridalexp\] to the face stretch notion. In the special case of the torus, this translation in fact makes some things simpler. Consider a graph embedded in the torus $\Sigma_1$. Let $\alpha$ be a loop in $\Sigma_1$ intersecting $G$ only in vertex points. When cutting along $\alpha$ we obtain a cylindrical surface $\Gamma$ with two borders, corresponding to the former left and right-hand sides of $\alpha$. We naturally obtain the graph $G'$ embedded on $\Gamma$ from $G$ by duplicating the vertices $v$ cut by $\alpha$ along the two borders. As in the previous proof, each copy of $v$ in $G'$ retains the edges formerly incident to $v$ on the respective side of $\alpha$ on $\Sigma_1$. We say that $G'$ embedded in $\Gamma$ is obtained by [*cutting $G$ along $\alpha$*]{}.
\[thm:facegrid-torus\] Let $G$ be a graph embedded in the torus $\Sigma_1$ with $k:=\fw(G)$. Let $\alpha$ be a loop in $\Sigma_1$ witnessing the face-width of $G$, and let $G'$ be a graph embedded in the cylinder $\Gamma$, obtained by cutting $G$ along $\alpha$. Among all pairs of points $x,y$ on the opposite bounderies of $\Gamma$, let $\ell$ be the least number of points in which a simple arc from $x$ to $y$ in $\Gamma$ intersects $G'$, not counting $x,y$ themselves. If $k\geq5$, then $G$ contains a toroidal $\floor{2k/3} \times \ell\>$-grid as a minor.
Analogously to Claim \[cl:ellcycles\] we prove that $G$ has a set of at least $\ell$ pairwise disjoint cycles, all homotopic to $\alpha$ in $\Sigma_1$. Then we finish as in the proof of Theorem \[thm:agrid-torus\], using Theorems \[thm:deGS\] and \[thm:two-cycle-families\].
\[lem:stretch-kl-torus\] Let $G$, $k\geq5$, and $\ell$ be as in Theorem \[thm:facegrid-torus\]. Then $\stretchf(G)\leq3k\ell$.
The proof is analogous to that of Lemma \[lem:thstr\], but slightly more complicated. Let $\gamma'$ be the curve in $\Gamma$ defining $\ell$ as above, and let $\gamma$ denote the corresponding curve back in $G$ in $\Sigma_1$. We can consider $\alpha$ and $\gamma$ as a cycle and a path, respectively, in the vertex-face incidence graph $G_f$. Let $\alpha\cap\gamma=\{a,b\}$ (where possibly $a=b$), and let $\alpha'$ denote the component of $\alpha\sem\{a,b\}$ having not more intersecting points with the drawing $G$ than the other component. Then $\alpha'\cup\gamma$ is a noncontractible loop intersecting $G$ in $\ell'\leq\ell+k/2+1$ points, as a simple case analysis shows (observe that, indeed, $\ell'$ may be larger than $\ell+k/2$ when some of $a,b$ are vertices of $G$). In particular, $\ell'\geq k\geq5$ and so $k/2\leq\ell+1$ and $\ell\geq2$. Therefore, $\alpha$ and $\alpha'\cup\gamma$ define a pair of one-leaping cycles in $G_f$ witnessing $\stretchf(G)\leq k\ell'\leq3k\ell$.
We may now conclude, in the toroidal case:
\[thm:main-minorcr\] Let $G$ be a graph embedded in the torus. If $\fw(G)\geq5$, then
- $\frac{10}{63}\cdot\mcr(G) \leq \Tex(G) \leq 12\cdot\mcr(G)$, and
- there is a polynomial time algorithm that computes a graph $H$ having $G$ as its minor and outputs a drawing of $H$ in the plane with at most $76\cdot\mcr(G)$ crossings.
Let $G$, $k\geq5$, and $\ell$ be as in Theorem \[thm:facegrid-torus\]. Combining Corollary \[cor:cr-fstretch-torus\] with Lemma \[lem:stretch-kl-torus\] we get $\mcr(G)\leq\frac{18}5k\ell$. Then, Theorem \[thm:facegrid-torus\] gives $\Tex(G)\geq\floor{2k/3}\cdot\ell\geq\frac47k\ell$ and the left-hand side of (a) follows. For the right-hand side, we simply use the fact that $\Tex(G)$ is minor monotone and apply Corollary \[cor:crossing-texp\] to the graph witnessing $\mcr(G)$.
For (b) we compute the graph $H$ from Lemma \[lem:cr-fstretch-torus\] and apply the algorithm of Theorem \[thm:main-overview\]. The resulting drawing of $H$ has at most $\frac{18}5k\ell$ crossings by the previous, and $\mcr(G)\geq\frac1{12}\cdot\frac47k\ell=\frac1{21}k\ell$. Hence the number of crossings in $H$ is at most $21\cdot\frac{18}5\mcr(G)\leq76\mcr(G)$.
Obviously, the approximation constants in Theorem \[thm:main-minorcr\] are very rough and can likely be improved a lot. However, the important point is that these constants are independent of the maximum degree. It is interesting to ask whether Theorem \[thm:main-minorcr\] can be extended to all orientable surfaces analogously to Theorem \[thm:main-overview\]. Although this seems quite plausible, there are complications similar to those seen already in the proofs of Lemmas \[lem:cr-fstretch-torus\] and \[lem:stretch-kl-torus\]. Consequently, the nice technical properties of stretch presented in Section \[sec:more\] cannot be straighforwardly extended to face stretch, and the whole question is left for future research.
Removing the density requirement {#sec:nodensity}
--------------------------------
Our algorithmic technique in Section \[sec:drawing-upper\] starts with a graph on a higher surface, and brings the graph to the plane without introducing too many crossings. As mentioned before, focusing only on surface-operations will inevitably require a certain lower bound on the density of the original embedding. However, we can naturally combine this algorithm with some other algorithmic results on inserting a *small* number of edges into a planar graph, to obtain a polynomial algorithm with essentially the same approximation ratio but without the density requirement. This combination of algorithms can be sketched as follows:
1. As long as the embedding density requirement of Theorem \[thm:main-overview\] is violated, we cut the surface along the violating loops. Let $K\subseteq E(G)$ be the set of edges affected by this; we know that $|K|$ is small, bounded by a function of $g$ and $\Delta$. Let $G_K:=G-K$.
2. By Theorem \[thm:upper-cr\], applied to $G_K$, we obtain a suitable set $F\subseteq E(G_K)$ such that $G_{KF}:=G_K-F$ is plane. ($F$ is the union of the edge sets corresponding to dual cycles in the considered dual planarizing sequence of $G_K$.)
3. We would like to apply independently [@MR2874100] to insert the edges of $K$ back to $G_{KF}$ with not many crossings, and Theorem \[thm:upper-cr\] to insert $F$ back to $G_{KF}$. The number of possible mutual crossing $|F|\cdot|K|$ is neglectable, but the real trouble is that [@MR2874100] is allowed to change the planar embedding of $G_{KF}$ and hence the insertion routes assumed by Theorem \[thm:upper-cr\] may no longer exist. Fortunately, the number of the insertion routes for $F$ is bounded in the genus (unlike $|F|$), and so the algorithm from [@MR2874100] can be adapted to respect these routes without a big impact on its approximation ratio.
Unfortunately, turning this simple sketch into a formal proof would not be short, due to the necessity to bring up many fine algorithmic details from [@MR2874100]. That is why we consider another option, allowing short self-contained proof at the expense of giving a weaker approximation guarantee. We use the following simplified formulation of the main result of [@MR2874100]. For a graph $H$ and a set of edges $K$ with ends in $V(H)$, but $K\cap
E(H)=\emptyset$, let $H+K$ denote the graph obtained by adding the edges $K$ into $H$.
\[thm:apxmei\] Let $H$ be a connected planar graph with maximum degree $\Delta$, $K$ an edge set with ends in $V(H)$ but $K\cap E(G)=\emptyset$, and $k=|K|$. There is a polynomial-time algorithm that finds a drawing of $H+K$ in the plane with at most $d\cdot\crg(H+K)$ crossings, where $d$ is a constant depending only on $\Delta$ and $k$. In this drawing, subgraph $H$ is drawn planarly, i.e., all crossings involve at least one edge of $K$.
An algorithmic strengthening of our Theorem \[thm:main-overview\] now reads:
\[thm:main-nodense\] Let $\Sigma$ be an orientable surface of fixed genus $g>0$, and let $\Delta$ be an integer constant. Assume $G$ is a graph of maximum degree $\Delta$ embedded in $\Sigma$. There is a polynomial time algorithm that outputs a drawing of $G$ in the plane with at most $c_3\cdot\crg(G)$ crossings, where $c_3$ is a constant depending on $g$ and $\Delta$.
Let $r_0,c_2$ be the constants from Theorem \[thm:main-overview\], depending on $g$ and $\Delta$. Recall that $r_0$ is nondecreasing in $g$, and so we may just fix it for the rest of the proof. If $\ewnd(G)<r_0\deee$, let $\gamma$ be the witnessing dual cycle of $G$. We cut $G$ along $\gamma$, and repeat this operation until we arrive at an embedded graph $G_K\subseteq G$ of genus $g_K<g$ such that $\ewnd(G_K)\geq r_0\deee$ (and hence $\fw(G_1)\geq r_0$). Let $K=E(G)\sem E(G_K)$ be the affected edges, where $|K|\leq g\,r_0\deee$ is bounded by a constant.
If $g_K=0$, then we simply finish by applying Theorem \[thm:apxmei\]. Otherwise, we apply the algorithm of Theorem \[thm:upper-cr\] to $G_K$, which results in a planar graph $G_{KF}\subseteq G_K$ and the edge set $F=E(G_K)\sem E(G_{KF})$, such that $F$ can be drawn into $G_{KF}$ using at most $c_2\cdot\crg(G_K)$ crossings by Theorem \[thm:main-overview\]. In this resulting drawing of $G_K$ we replace each crossing by a new subdividing vertex. This gives a planarly embedded graph $G_K'$ that contains a planarly embedded subdivision $G_{KF}'$ of $G_{KF}$. Let $F_2=E(G_K')\sem E(G_{KF}')$. Since we clearly may assume that every edge of $F$ required at least one crossing in $G_{KF}$, we have $|F_2|\leq2c_2\cdot\crg(G_K)$. Now we apply Theorem \[thm:apxmei\] to $H=G_{KF}'$ and $K$ (from the previous paragraph). This gives a drawing $G_F$ of $G_{KF}'+K$ with at most $d\cdot\crg(G_{KF}+K)$ crossings in the plane. The final task is to put back the edges of $F_2$ into $G_F$; note, however, that the planar subdrawing of $G_{KF}'$ within $G_F$ is generally different from the original embedding of $G_{KF}'$.
For the latter task use the following technical claim:
\[cl:edgeback\] Suppose $H$ is a connected graph embedded in the plane, and $e,f\not\in E(H)$ are two edges joining vertices of $H$ such that $H+f$ is a planar graph. If $e$ can be drawn in $H$ with $\ell$ crossings, then there is a planar embedding of $H+f$ in which $e$ can be drawn with at most $\ell+2\cdot\lfloor\Delta(H)/2\rfloor$ crossings.
Although [@HS06] does not explicitly handle the algorithmic aspect of Claim \[cl:edgeback\], it is easily seen there that the claimed drawing of $H+f+e$ can be found in polynomial time from the assumed drawing of $H+e$ (for the algorithm of [@MR2874100], for example, this is a simple special case).
Let $F_2=\{f_1,f_2,\dots,f_a\}$. By induction on $i=1,2,\dots,a$, we apply Claim \[cl:edgeback\] to $f:=f_i$ and $H:=G_{KF}'+f_1+\dots+f_{i-1}$, and simultaneously to each $e$ from $K$. As the final result we obtain a planar embedding of $G_{KF}'+F_2=G_K'$. Into this $G_K'$, we can draw $K$ with at most $|K|\cdot2\deee\cdot|F_2|+|K|^2/2$ additional crossings (compared to the number of crossings achieved by Theorem \[thm:apxmei\] to draw $K$ into $G_K$). By turning the vertices of $V(G_K')\sem V(G_K)$ back into edge crossings of $G_K$ this leads to a drawing of $G_K+K=G$ with at most $$\begin{aligned}
c_2\cdot\crg(G_K) +
d\cdot\crg(G_{KF}+K) &+
|K|\cdot2\deee |F_2|
+|K|^2/2 \\
\leq c_2\cdot\crg(G_K) +
d\cdot\crg(G_{KF}+K) &+
g\,r_0\Delta^2c_2\cdot\crg(G_K)
+(g\,r_0\Delta)^2/8\\
\leq (c_2 +
d &+
g\,r_0\Delta^2c_2)\cdot\crg(G)
+(g\,r_0\Delta)^2/8\end{aligned}$$ crossings where all the remaining terms are constants depending only on $g$ and $\Delta$.
[99]{}
[^1]: Faculty of Mathematics/Computer Science, Osnabrück University. Osnabrück, Germany.
[^2]: Faculty of Informatics, Masaryk University. Brno, Czech Republic. Supported by the Czech Science Foundation, Eurocores project GraDR GIG/11/E023, and project 14-03501S (since 2014).
[^3]: Instituto de Fisica, Universidad Autonoma de San Luis Potosi. San Luis Potosi, Mexico. Supported by CONACYT Grant 106432.
[^4]: This draws upon and extends partial results presented at ISAAC 2007 [@HS07] and SODA 2010 [@HC10].
[^5]: Note that this quantity is also called the “crossing number” of the curves, and a pair of curves may be said to be “$k$-crossing”. Such a terminology would, however, conflict with the graph crossing number, and we have to avoid it. Following [@HC10], we thus use the term “$k$-leaping”, instead.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we propose a new primal-dual algorithm for minimizing $f(\vx)+g(\vx)+h(\vA\vx)$, where $f$, $g$, and $h$ are proper lower semi-continuous convex functions, $f$ is differentiable with a Lipschitz continuous gradient, and $\vA$ is a bounded linear operator. The proposed algorithm has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. E.g., it reduces to the Chambolle-Pock algorithm when $f=0$ and the proximal alternating predictor-corrector when $g=0$. For the general convex case, we prove the convergence of this new algorithm in terms of the distance to a fixed point by showing that the iteration is a nonexpansive operator. In addition, we prove the $O(1/k)$ ergodic convergence rate in the primal-dual gap. With additional assumptions, we derive the linear convergence rate in terms of the distance to the fixed point. Comparing to other primal-dual algorithms for solving the same problem, this algorithm extends the range of acceptable parameters to ensure its convergence and has a smaller per-iteration cost. The numerical experiments show the efficiency of this algorithm.'
author:
- Ming Yan
bibliography:
- 'PM3O.bib'
date: 'Received: date / Accepted: date'
title: 'A new primal-dual algorithm for minimizing the sum of three functions with a linear operator[^1] '
---
Introduction
============
This paper focuses on minimizing the sum of three proper lower semi-continuous convex functions in the form of $$\begin{aligned}
\label{for:main_problem}
\vx^* \in \argmin_{\vx\in\cX} f(\vx) + g(\vx) + h\square l(\vA\vx),\end{aligned}$$ where $\cX$ and $\cS$ are two real Hilbert spaces, $h\square l:\cS\mapsto (-\infty,+\infty]$ is the infimal convolution defined as $h\square l(\vs)=\inf_{\vt}h(\vt)+l(\vs-\vt)$, $\vA:\cX\mapsto \cS$ is a bounded linear operator. $f:\cX\mapsto \vR$ and the conjugate function[^2] $l^*:\cS\mapsto\vR$ are differentiable with Lipschitz continuous gradients, and both $g$ and $h$ are proximal, that is, the proximal mappings of $g$ and $h$ defined as $$\begin{aligned}
\prox_{\lambda g}(\widetilde \vx)=(\vI+\lambda\partial g)^{-1}(\widetilde\vx):=\argmin_\vx~ \lambda g(\vx)+{1\over 2}\|\vx-\widetilde\vx\|^2\end{aligned}$$ have analytical solutions or can be computed efficiently. When $l(\vs)=\iota_{\{\vzero\}}(\vs)$ is the indicator function that returns zero when $\vs=\vzero$ and $+\infty$ otherwise, the infimal convolution $h\square l=h$, and the problem becomes $$\begin{aligned}
\vx^* \in \argmin_{\vx\in\cX} f(\vx) + g(\vx) + h(\vA\vx).\end{aligned}$$ A wide range of problems in image and signal processing, statistics and machine learning can be formulated into this form. Here, we give some examples.
[**Elastic net regularization [@zou_regularization_2005]:**]{} The elastic net combines the $\ell_1$ and $\ell_2$ penalties to overcome the limitations of both penalties. The optimization problem is $$\begin{aligned}
\textstyle \vx^*\in\argmin\limits_{\vx\in\vR^p}~ \mu_2\|\vx\|_2^2 + \mu_1\|\vx\|_1 + l(\vA\vx,\vb),\end{aligned}$$ where $\vA\in \vR^{n\times p}$, $\vb\in\vR^n$, and $l$ is the loss function, which may be nondifferentiable. The $\ell_2$ regularization term $\mu_2\|\vx\|_2^2$ is differentiable and has a Lipschitz continuous gradient.
[**Fused lasso [@tibshirani2005sparsity]:**]{} The fused lasso was proposed for group variable selection. Except the $\ell_1$ penalty, it includes a new penalty term for large changes with respect to the temporal or spatial structure such that the coefficients vary in a smooth fashion. The problem for fused lasso with the least squares loss is $$\begin{aligned}
\vx^*\in\argmin\limits_{\vx\in\vR^p} {1\over 2}\|\vA\vx-\vb\|_2^2 + \mu_1\|\vx\|_1 + \mu_2\|\vD\vx\|_1,\label{eqn:fusedlasso}\end{aligned}$$ where $\vA\in\vR^{n\times p}$, $\vb\in\vR^n$, and $$\begin{aligned}
\vD=\left[\begin{array}{ccccc}-1&1& & & \\ &-1&1& & \\ & & \dots & \dots & \\& & &-1&1\end{array}\right]\in\vR^{(p-1)\times p}.\end{aligned}$$
[**Image restoration with two regularizations:**]{} Many image processing problems have two or more regularizations. For instance, in computed tomography reconstruction, nonnegative constraint and total variation regularization are applied. The optimization problem can be formulated as $$\begin{aligned}
\vx^*\in\argmin\limits_{\vx\in\vR^n} {1\over 2}\|\vA\vx-\vb\|_2^2 + \iota_{C}(\vx) + \mu\|\vD\vx\|_1,\end{aligned}$$ where $\vx$ is the image to be reconstructed, $\vA\in \vR^{m\times n}$ is the linear forward projection operator that maps the image to the sinogram data, $\vb\in\vR^m$ is the measured sinogram data with noise, $\iota_{C}$ is the indicator function that returns zero if $\vx\in C$ (here, $C$ is the set of nonnegative vectors in $\vR^n$) and $+\infty$ otherwise, $\vD$ is a discrete gradient operator, and the last term is the (an)isotropic total variation regularization.
Before introducing algorithms for solving , we discuss special cases of with only two functions. When either $f$ or $g$ is missing, the problem reduces to the sum of two functions, and many splitting and proximal algorithms have been proposed and studied in the literature. Two famous groups of methods are Alternating Direction of Multiplier Method (ADMM) [@gabay1976dual; @glowinski1975approximation] and primal-dual algorithms [@PlayDual]. ADMM applied on a convex optimization problem was shown to be equivalent to Douglas-Rachford Splitting (DRS) applied on the dual problem by [@gabay1983chapter], and [@yan2014self] showed recently that it is also equivalent to DRS applied on the same primal problem. In fact, there are many different ways to reformulate a problem into a separable convex problem with linear constraints such that ADMM can be applied, and among these ways, some are equivalent. However, there will be always a subproblem involving $\vA$, and it may not be solved analytically depending on the properties of $\vA$ and the way ADMM is applied. On the other hand, primal-dual algorithms only need the operator $\vA$ and its adjoint operator $\vA^\top$[^3]. Thus, they have been applied in a lot of applications because the subproblems would be easy to solve if the proximal mappings for both $g$ and $h$ can be computed easily.
The primal-dual algorithms for two and three functions are reviewed by [@PlayDual] and specially for image processing problems by [@chambolle2016introduction]. When the differentiable function $f$ is missing, the primal-dual algorithm is Chambolle-Pock, (see e.g., [@pock2009algorithm; @esser2010general; @chambolle2011first]), while the primal-dual algorithm with $g$ missing (Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP$^2$O) or Proximal Alternating Predictor-Corrector (PAPC)) is proposed in [@loris2011generalization; @chen2013primal; @Drori2015209]. In order to solve problem with three functions, we can reformulate the problem and apply the primal-dual algorithms for two functions. E.g., we can let $\bar h([\vI;~\vA]\vx) = g(\vx)+h(\vA\vx)$ and apply PAPC or let $\bar g(\vx)=f(\vx)+g(\vx)$ and apply Chambolle-Pock. However, the first approach introduces more dual variables and may need more iterations to obtain the same accuracy. For the second approach, the proximal mapping of $\bar g$ may not be easy to compute, and the differentiability of $f$ is not utilized.
When all three terms are present, the problem was firstly considered in [@combettes2012primal] using a forward-backward-forward scheme, where two gradient evaluations and four linear operators are needed in one iteration. Later, [@condat2013primal] and [@vu2013splitting] proposed a primal-dual algorithm for , which we call Condat-Vu. It is a generalization of Chambolle-Pock by involving the differentiable function $f$ with a more restrictive range for acceptable parameters than Chambolle-Pock. As noted in [@PlayDual], there was no generalization of PAPC for three functions at that time. Later, a generalization of PAPC–a Primal-Dual Fixed-Point algorithm (PDFP)–is proposed by [@chen2016primal], in which two proximal mappings of $g$ are needed in each iteration. But, this algorithm has a larger range of acceptable parameters than Condat-Vu. Recently the Asymmetric Forward-Backward-Adjoint splitting (AFBA) is proposed by [@latafat2016asymmetric], and it includes Condat-Vu and PAPC as two special cases. This generalization has a conservative stepsize, which is the same as Condat-Vu, but only one proximal mapping of $g$ is needed in each iteration.
In this paper, we will give a new generalization of both Chambolle-Pock and PAPC. This new algorithm employs the same regions of acceptable parameters with PDFP and the same per-iteration complexity as Condat-Vu and AFBA. In addition, when $\vA=\vI$, it recovers the three-operator splitting scheme developed by [@davis2015three], which we call Davis-Yin. Note that Davis-Yin is a generalization of many existing two-operator splitting schemes such as forward-backward splitting [@PASSTY1979383], backward-forward splitting [@peng2016coordinate; @AMTL], Peaceman-Rachford splitting [@Douglas1956NumerSol; @PRS], and forward-Douglas-Rachford splitting [@FDRS].
The proposed algorithm has the following iteration:
$$\begin{aligned}
\vx^{k} & =\prox_{\gamma g} (\vz^k);\\
\vs^{k+1} & =\prox_{\delta h^*} \big((\vI-\gamma\delta \vA\vA^\top)\vs^k-{\delta}\nabla l^*(\vs^k) + {\delta}\vA\left(2\vx^k-\vz^k-{\gamma}\nabla f(\vx^k)\right)\big);\\
\vz^{k+1} & =\vx^k-{\gamma}\nabla f(\vx^k) -{\gamma}\vA^\top\vs^{k+1}.\end{aligned}$$
Here $\gamma$ and $\delta$ are the primal and dual stepsizes, respectively. The convergence of this algorithm requires conditions for both stepsizes, as shown in Theorem \[thm:main\]. Since this is a primal-dual algorithm for three functions ($h\square l$ is considered as one function) and it has connections to the three-operator splitting scheme in [@davis2015three], we name it as Primal-Dual Three-Operator splitting (PD3O). For simplicity of notation in the following sections, we may use $(\vz,\vs)$ and $(\vz^+,\vs^+)$ for the current and next iterations. Therefore, one iteration of PD3O from $(\vz,\vs)$ to $(\vz^+,\vs^+)$ is
\[for:PD3O\] $$\begin{aligned}
\vx & = \prox_{\gamma g} (\vz);\label{for:PD3O_iteration_a}\\
\vs^+ & =\prox_{\delta h^*} \big((\vI-\gamma\delta \vA\vA^\top)\vs-{\delta}\nabla l^*(\vs) + {\delta}\vA\left(2\vx-\vz-{\gamma}\nabla f(\vx)\right)\big); \label{for:PD3O_iteration_b}\\
\vz^+ & =\vx-{\gamma}\nabla f(\vx) -{\gamma}\vA^\top\vs^+.\label{for:PD3O_iteration_c}\end{aligned}$$
We define one PD3O iteration as an operator $\vT_{\textnormal{PD3O}}$ and $(\vz^+,\vs^+) = \vT_{\textnormal{PD3O}}(\vz,\vs)$.
The contributions of this paper can be summarized as follows:
- We proposed a new primal-dual algorithm for solving an optimization problem with three functions $f(\vx)+g(\vx)+h\square l(\vA\vx)$ that recovers Chambolle-Pock and PAPC for two functions with either $f$ or $g$ missing. Comparing to three existing primal-dual algorithms for solving the same problem: Condat-Vu ([@condat2013primal; @vu2013splitting]), AFBA ([@latafat2016asymmetric]), and PDFP ([@chen2016primal]), this new algorithm combines the advantages of all three methods: the low per-iteration complexity of Condat-Vu and AFBA and the large range of acceptable parameters for convergence of PDFP. The numerical experiments show the advantage of the proposed algorithm over these existing algorithms.
- We prove the convergence of the algorithm by showing that the iteration is an $\alpha$-averaged operator. This result is stronger than the result for PAPC in [@chen2013primal], where the iteration is shown to be nonexpansive only. Also, we show that Chambolle-Pock is equivalent to DRS under a different metric from the previous result that it is equivalent to a proximal point algorithm applied on the Karush-Kuhn-Tucker (KKT) conditions.
- We show the ergodic convergence rate for the primal-dual gap. The convergent sequence is different from that in [@Drori2015209] when it reduces to PAPC.
- This new algorithm also recovers Davis-Yin for minimizing the sum of three functions and thus many splitting schemes involving two operators such as forward-backward splitting, backward-forward splitting, Peaceman-Rachford splitting, and forward-Douglas-Rachford splitting.
- With additional assumptions on the functions, we show the linear convergence rate of PD3O.
The rest of the paper is organized as follows. We compare PD3O with several existing primal-dual algorithms and Davis-Yin in Section \[sec:compare\]. Then we show the convergence of PD3O for the general case and its linear convergence rate for special cases in Section \[sec:general\]. The numerical experiments in Section \[sec:numerical\] show the effectiveness of the proposed algorithm by comparing with other existing algorithms, and finally, Section \[sec:conclusion\] concludes the paper with future directions.
Connections to existing algorithms {#sec:compare}
==================================
In this section, we compare our proposed algorithm with several existing algorithms. In particular, we show that our proposed algorithm recovers Chambolle-Pock [@chambolle2011first], PAPC [@loris2011generalization; @chen2013primal; @Drori2015209], and Davis-Yin [@davis2015three]. In addition, we compare our algorithm with PDFP [@chen2016primal], Condat-Vu [@condat2013primal; @vu2013splitting], and AFBA [@latafat2016asymmetric].
Before showing the connections, we reformulate our algorithm by changing the update order of the variables and introducing $\bar\vx$ to replace $\vz$ (i.e., $\bar\vx = 2\vx-\vz-\gamma\nabla f(\vx)-\gamma \vA^\top\vs$). The reformulated algorithm is
\[for:PD3Ov2\] $$\begin{aligned}
\vs^+ & =\prox_{\delta h^*}\left(\vs-\delta \nabla l^*(\vs) + \delta\vA\bar\vx\right);\\
\vx^+ & =\prox_{\gamma g} (\vx -\gamma\nabla f(\vx)-{\gamma}\vA^\top\vs^+);\\
\bar\vx^+ & =2\vx^+-\vx +\gamma \nabla f(\vx)-\gamma \nabla f(\vx^+).\end{aligned}$$
Since the infimal convolution $h\square l$ is only considered by Vu in [@vu2013splitting]. For simplicity, we let $l=\iota_{\{\vzero\}}$, thus $l^*=0$, for the rest of this section.
Three special cases
-------------------
In this subsection, we show three special cases of our new algorithm: PAPC, Chambolle-Pock, and Davis-Yin.
[**PAPC**]{}: When $g=0$, i.e., the function $g$ is missing, we have $\vx=\vz$, and the iteration reduces to
\[for:PDFP2O\] $$\begin{aligned}
\vs^+ & =\prox_{\delta h^*} ((\vI-\gamma\delta \vA\vA^\top)\vs + \delta\vA\left(\vx-{\gamma}\nabla f(\vx)\right));\\
\vx^+ & =\vx-{\gamma}\nabla f(\vx) -{\gamma}\vA^\top\vs^+,\end{aligned}$$
which is PAPC in [@loris2011generalization; @chen2013primal; @Drori2015209]. The PAPC iteration is shown to be nonexpansive in [@chen2013primal], while we will show a stronger result that the PAPC iteration is $\alpha$-averaged with certain $\alpha\in(0,1)$ in Corollary \[cor:PDFP2O\]. In addition, we prove the convergence rate of the primal-dual gap using a different sequence from [@Drori2015209]. See Section \[sec:primal\_dual\].
[**Chambolle-Pock**]{}: Let $f=0$, i.e., the function $f$ is missing, then we have, from ,
$$\begin{aligned}
\vs^+ & =\prox_{\delta h^*} (\vs + \delta\vA\bar\vx);\\
\vx^+ & =\prox_{\gamma g} (\vx -{\gamma}\vA^\top\vs^+);\\
\bar\vx^+ & =2\vx^+-\vx,\end{aligned}$$
which is Chambolle-Pock in [@chambolle2011first]. We will show that Chambolle-Pock is equivalent to a Douglas-Rachford splitting under a metric in Corollary \[cor:CP\].
[**Davis-Yin**]{}: Let $\vA=\vI$ and $\gamma\delta=1$, then we have, from ,
\[for:3O\] $$\begin{aligned}
\vx &=\prox_{\gamma g} (\vz);\label{for:3O_iteration_a}\\
\vs^+ &= \prox_{{\delta}h^*}\left(\delta(2\vx-\vz-{\gamma}\nabla f(\vx))\right)\nonumber \\
&={\delta}(\vI-\prox_{{\gamma} h}) \left(2\vx-\vz-{\gamma}\nabla f(\vx)\right);\label{for:3O_iteration_b} \\
\vz^+ &=\vx-{\gamma}\nabla f(\vx) -{\gamma}\vs^+.\label{for:3O_iteration_c}\end{aligned}$$
Here we used the Moreau decomposition in . Combining together, we have $$\begin{aligned}
\vz^+ =\vz+ \prox_{{\gamma} h}\left(2\prox_{\gamma g} (\vz)-\vz-{\gamma}\nabla f(\prox_{\gamma g} (\vz))\right) -\prox_{\gamma g} (\vz),\end{aligned}$$ which is Davis-Yin in [@davis2015three].
Comparison with three primal-dual algorithms for three functions
----------------------------------------------------------------
In this subsection, we compare our algorithm with three primal-dual algorithms for solving the same problem .
[**PDFP**]{}: The PDFP algorithm [@chen2016primal] is developed as a generalization of PAPC. When $g=\iota_C$ ($C$ is a convex set in $\cX$), PDFP reduces to the Preconditioned Alternating Projection Algorithm (PAPA) proposed in [@krol2012preconditioned]. The PDFP iteration can be expressed as follows:
\[for:PDFP\] $$\begin{aligned}
\vs^+ &\textstyle =\prox_{\delta h^*} \left(\vs + {\delta}\vA\bar\vx\right); \\
\vx^+ &\textstyle =\prox_{\gamma g} (\vx-\gamma \nabla f(\vx)-\gamma\vA^\top\vs^+);\\
\bar\vx^+ &\textstyle =\prox_{\gamma g} (\vx^+-\gamma \nabla f(\vx^+)-\gamma\vA^\top\vs^+).\end{aligned}$$
Note that two proximal mappings of $g$ are needed in each iteration, while other algorithms only need one.
[**Condat-Vu**]{}: The Condat-Vu algorithm [@condat2013primal; @vu2013splitting] is a generalization of the Chambolle-Pock algorithm for problem . The iteration is
\[for:CV\] $$\begin{aligned}
\vs^+ & =\prox_{\delta h^*} (\vs + \delta\vA\bar\vx);\\
\vx^+ & =\prox_{\gamma g} (\vx -\gamma\nabla f(\vx)-{\gamma}\vA^\top\vs^+);\\
\bar\vx^+ & =2\vx^+-\vx.\end{aligned}$$
The difference between our algorithm and Condat-Vu is in the updating of $\bar\vx$. Because of the difference, our algorithm will be shown to have more freedom than Condat-Vu in choosing acceptable parameters. Note though [@vu2013splitting] considers a general form with infimal convolution, its condition for the parameters, when reduced to the case without the infimal convolution, is worse than that in [@condat2013primal]. The correct general condition has been given in [@lorenz2015inertial Theorem 5].
[**AFBA**]{}: AFBA is a very general operator splitting scheme, and one of its special cases [@latafat2016asymmetric Algorithm 5] can be used to solve the problem . The corresponding algorithm is
\[for:AFBA\] $$\begin{aligned}
\vs^+ & =\prox_{\delta h^*} (\vs + \delta\vA\bar\vx); \\
\vx^+ & =\bar\vx-\gamma\vA^\top(\vs^+-\vs);\\
\bar\vx^+ & =\prox_{\gamma g} (\vx^+-\gamma \nabla f(\vx^+)-\gamma\vA^\top\vs^+).\end{aligned}$$
The difference between AFBA and PDFP is in the update of $\vx^+$. Though AFBA needs only one proximal mapping of $g$ in each iteration, it has a more conservative range for the two parameters than PDFP and Condat-Vu.
The parameters for the four algorithms solving and their relations to the aforementioned primal-dual algorithms for the sum of two functions are given in Table \[tab:compare\]. Here $\|\vA\vA^\top\|:=\max\limits_{\|\vs\|=1}{\|\vA\vA^\top\vs\|}$.
$f\neq0,~g\neq0$ $f=0$ $g=0$
----------- ---------------------------------------------------------------------------------------------- ---------------- -------
PDFP $\gamma\delta\|\vA\vA^\top\|<1$; $\gamma/(2\beta)<1$ PAPC
Condat-Vu $\gamma\delta \|\vA\vA^\top\|+\gamma/(2\beta)\leq 1$ Chambolle-Pock
AFBA $\gamma\delta \|\vA\vA^\top\|/2+\sqrt{\gamma\delta \|\vA\vA^\top\|}/2+\gamma/(2\beta)\leq 1$ PAPC
PD3O $\gamma\delta\|\vA\vA^\top\|<1$; $\gamma/(2\beta)<1$ Chambolle-Pock PAPC
: The comparison of convergence conditions for PDFP, Condat-Vu, AFBA, and PD3O and their reduced primal-dual algorithms.[]{data-label="tab:compare"}
Comparing the iterations of PDFP, Condat-Vu, and PD3O in , , and , respectively, we notice that the difference is in the third step for updating $\bar\vx$. The third steps for these three algorithms are summarized below:
PDFP $\bar\vx^+ =\prox_{\gamma g} (\vx^+-\gamma \nabla f(\vx^+)-\gamma\vA^\top\vs^+)$
----------- -----------------------------------------------------------------------------------
Condat-Vu $\bar\vx^+ =2\vx^+-\vx$
PD3O $\bar\vx^+ = 2\vx^+-\vx +\gamma \nabla f(\vx)-\gamma \nabla f(\vx^+)$
Though there are two more terms ($\nabla f(\vx)$ and $\nabla f(\vx^+)$) in PD3O than Condat-Vu, $\nabla f(\vx)$ has been computed in the previous step, and $\nabla f(\vx^+)$ will be used in the next iteration. Thus, except that $\nabla f(\vx)$ has to be stored, there is no additional cost in PD3O comparing to Condat-Vu. However, for PDFP, the proximal operator $\prox_{\gamma g}$ is applied twice on different values in each iteration, and it will not be used in the next iteration. Therefore, the per-iteration cost is more than the other three algorithms. When the proximal mapping is simple, the additional cost can be relatively small compared to other operations in one iteration.
The proposed primal-dual algorithm {#sec:general}
==================================
Notation and preliminaries
--------------------------
Let $\vI$ be the identity operator defined on a real Hilbert space. For simplicity, we do not specify the space on which it is defined when it is clear from the context. Let $\vM={\gamma\over \delta}(\vI - \gamma\delta\vA\vA^\top)$ and $\langle \vs_1,\vs_2\rangle_\vM:=\langle \vs_1, \vM\vs_2\rangle$ for $\vs_1,~\vs_2\in\cS$. When $\gamma\delta$ is small enough such that $\vM$ is positive semidefinite, we define $\|\vs\|_\vM=\sqrt{\langle \vs,\vs\rangle_\vM}$ for any $\vs\in\cS$ and $\|(\vz,\vs)\|_{\vI,\vM}=\sqrt{\|\vz\|^2+\|\vs\|^2_\vM}$ for any $(\vz,\vs)\in \cX\times \cS$. Specially, when $\vM$ is positive definite, $\|\cdot\|_\vM$ and $\|(\cdot,\cdot)\|_{\vI,\vM}$ are two norms defined on $\cS$ and $\cX\times\cS$, respectively. From , we define $\vq_{h^*}(\vs^+)\in \partial h^*(\vs^+)$ and $\vq_g(\vx)\in\partial g(\vx)$ as: $$\begin{aligned}
\vq_{h^*}(\vs^+):=& \textstyle{\gamma}^{-1}\vM\vs-\nabla l^*(\vs)+\vA(2\vx-\vz-{\gamma}\nabla f(\vx))-{\delta}^{-1}\vs^+ \in \partial h^*(\vs^+),\\
\vq_g(\vx): =& \textstyle{\gamma}^{-1}(\vz-\vx)\in\partial g(\vx).\end{aligned}$$ An operator $\vT$ is nonexpansive if $\|\vT\vx_1 -\vT\vx_2\|\leq \|\vx_1-\vx_2\|$ for any $\vx_1$ and $\vx_2$. An operator $\vT$ is $\alpha$-averaged for $\alpha\in(0,1]$ if $\|\vT\vx_1 -\vT\vx_2\|^2\leq \|\vx_1-\vx_2\|^2-((1-\alpha)/\alpha)\|(\vI-\vT)\vx_1-(\vI-\vT)\vx_2\|^2$; firmly nonexpansive operators are $1/2$-averaged, and merely nonexpansive operators are $1$-averaged.
\[assum:1\] We assume that functions $f$, $g$, $h$, and $l$ are proper lower semi-continuous convex and there exists $\beta>0$ such that $$\begin{aligned}
\langle \vx_1-\vx_2,\nabla f(\vx_1)-\nabla f(\vx_2)\rangle\geq & \beta\|\nabla f(\vx_1)-\nabla f(\vx_2)\|^2, \label{eq:fcoco}\\
\langle \vs_1-\vs_2,\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle\geq &\beta\|\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\|^2_{\vM^{-1}}\label{eq:lcoco}\end{aligned}$$ are satisfied for any $\vx_1,~\vx_2\in \cX$ and $\vs_1,~\vs_2\in\cS$. We assume that $\vM$ is positive definite if $\nabla l^*$ is not a constant. When $\nabla l^*$ is a constant, the left hand side of equation is zero, and we say that the equation is satisfied for any positive $\beta$ and positive semidefinite $\vM$. Therefore, when $\nabla l^*$ is a constant, we only assume that $\vM$ is positive semidefinite.
Equation is satisfied if and only if $f$ has a $1/\beta$ Lipschitz continuous gradient when $f$ is proper lower semi-continuous convex (see [@bauschke2011convex Theorem 18.15]). We assume with $\vM$ for the simplicity of the results. The results are valid but complicated when $\vM$ is replaced by $\vI$ in . When $\nabla l^*$ is Lipschitz continuous, we can always find $\beta$ such that equation is satisfied if $\vM$ is positive definite. When $\det(\vM)=0$, we consider the case with $\nabla l^*$ being a constant only.
\[lemma4gap\] If Assumption \[assum:1\] is satisfied, we have $$\begin{aligned}
f(\vx_2)-f(\vx_1) \leq &\langle \nabla f(\vx_2),\vx_2-\vx_1\rangle -{\beta\over2} \|\nabla f(\vx_2)-\nabla f(\vx_1)\|^2, \label{lemma1a}\\
l^*(\vs_1)-l^*(\vs_2) \leq & \langle \nabla l^*(\vs_2),\vs_1-\vs_2\rangle+{(2\beta)^{-1}}\|\vs_1-\vs_2\|_\vM^2. \label{lemma1b}\end{aligned}$$
Given $\vx_2$, the function $f_{\vx_2}(\vx):=f(\vx)-\nabla f(\vx_2)^\top\vx$ is convex and has a Lipschitz continuous gradient with constant $1/\beta$. In addition, $\vx_2$ is a minimum of $f_{\vx_2}(\vx)$. Therefore, we have $$\begin{aligned}
f_{\vx_2}(\vx_2)\leq & \inf_{\vx\in\cX} f(\vx_1)-\nabla f(\vx_2)^\top\vx_1 +\langle \nabla f(\vx_1)-\nabla f(\vx_2),\vx-\vx_1\rangle + {1\over2\beta}\|\vx-\vx_1\|^2 \nonumber\\
= & f(\vx_1)-\nabla f(\vx_2)^\top\vx_1 -{\beta\over2}\|\nabla f(\vx_2)-\nabla f(\vx_1)\|^2,\end{aligned}$$ which gives $$\begin{aligned}
f(\vx_2)- f(\vx_1)\leq \langle\nabla f(\vx_2),\vx_2-\vx_1\rangle-{\beta\over2}\|\nabla f(\vx_2)-\nabla f(\vx_1)\|^2.\end{aligned}$$ When $\vM$ is positive definite, the inequality implies the Lipschitz continuity of $ \vM^{-1/2} \nabla l^*(\vM^{-1/2}\cdot)$ with constant $1/\beta$. Then the function $(2\beta)^{-1}\hat\vs^\top\hat\vs-l^*(\vM^{-1/2}\hat\vs)$ is convex. Therefore we have $$\begin{aligned}
{(2\beta)^{-1}}\hat\vs_1^\top\hat\vs_1-l^*(\vM^{-1/2}\hat\vs_1) \geq &{(2\beta)^{-1}}\hat\vs_2^\top\hat\vs_2-l^*(\vM^{-1/2}\hat\vs_2) \\
&+\left\langle {\beta^{-1}}\hat\vs_2-\vM^{-1/2}\nabla l^*(\vM^{-1/2}\hat\vs_2),\hat\vs_1-\hat\vs_2 \right\rangle\end{aligned}$$ Let $\vs_1=\vM^{-1/2}\hat\vs_1$ and $\vs_2=\vM^{-1/2}\hat\vs_2$, then we have $$\begin{aligned}
l^*(\vs_1) -l^*(\vs_2) \leq \langle\nabla l^*(\vs_2),\vs_1-\vs_2 \rangle+{(2\beta)^{-1}}\|\vs_1-\vs_2\|_\vM^2.\end{aligned}$$ When $\nabla l^*$ is a constant, is satisfied for any positive semidefinite $\vM$.
\[lemma:fundamental\] Consider the PD3O iteration with two inputs $(\vz_1,\vs_1)$ and $(\vz_2,\vs_2)$. Let $(\vz_1^+,\vs_1^+)=\PD(\vz_1,\vs_1)$ and $(\vz_2^+,\vs_2^+)=\PD(\vz_2,\vs_2)$. We also define $$\begin{aligned}
\vq_{h^*}(\vs_1^+):=& \textstyle {\gamma}^{-1}\vM\vs_1-\nabla l^*(\vs_1)+\vA(2\vx_1-\vz_1-{\gamma}\nabla f(\vx_1))-{\delta}^{-1}\vs_1^+ \in \partial h^*(\vs_1^+),\\
\vq_g(\vx_1):= & \textstyle {\gamma}^{-1}(\vz_1-\vx_1)\in\partial g(\vx_1),\\
\vq_{h^*}(\vs_2^+):=& \textstyle {\gamma}^{-1}\vM\vs_2-\nabla l^*(\vs_2)+\vA(2\vx_2-\vz_2-{\gamma}\nabla f(\vx_2))-{\delta}^{-1}\vs_2^+ \in \partial h^*(\vs_2^+),\\
\vq_g(\vx_2):= & \textstyle {\gamma}^{-1}(\vz_2-\vx_2)\in\partial g(\vx_2). \end{aligned}$$ Then, we have $$\begin{aligned}
& 2\gamma\langle \vs_1^+-\vs_2^+,\vq_{h^*}(\vs_1^+)-\vq_{h^*}(\vs_2^+)+\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle \nonumber\\
& +2\gamma\langle \vx_1-\vx_2,\vq_g(\vx_1)-\vq_g(\vx_2)+\nabla f(\vx_1)-\nabla f(\vx_2)\rangle\nonumber\\
= & \|(\vz_1,\vs_1)-(\vz_2,\vs_2)\|_{\vI,\vM}^2-\|(\vz_1^+,\vs_1^+)-(\vz_2^+,\vs_2^+)\|_{\vI,\vM}^2 -\|\vz_1-\vz_1^+-\vz_2+\vz_2^+\|^2 \nonumber\\
& -\|\vs_1-\vs_1^+-\vs_2+\vs_2^+\|_\vM^2 +2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vz_1-\vz_1^+-\vz_2+\vz_2^+\rangle. \label{eqn:fundamental_1}\end{aligned}$$
We consider the two terms on the left hand side of separately. For the first term, we have $$\begin{aligned}
& 2\gamma\langle \vs_1^+-\vs_2^+,\vq_{h^*}(\vs_1^+)-\vq_{h^*}(\vs_2^+) +\nabla l^*(\vs_1)-\nabla l^*(\vs_2) \rangle\nonumber\\
= &\textstyle 2\gamma\langle \vs_1^+-\vs_2^+,{\gamma}^{-1}\vM\vs_1-\nabla l^*(\vs_1)+\vA(\vz_1^+-\vz_1+\vx_1+\gamma\vA^\top\vs_1^+)-{\delta}^{-1}\vs_1^+\rangle\nonumber\\
&\textstyle -2\gamma\langle \vs_1^+-\vs_2^+,{\gamma}^{-1}\vM\vs_2-\nabla l^*(\vs_2)+\vA(\vz_2^+-\vz_2+\vx_2+\gamma\vA^\top\vs_2^+)-{\delta}^{-1}\vs_2^+\rangle\nonumber\\
& +2\gamma\langle \vs_1^+-\vs_2^+,\nabla l^*(\vs_1)-\nabla l^*(\vs_2) \rangle \nonumber\\
= & 2\langle\vs_1^+-\vs_2^+,\vs_1-\vs_1^+-\vs_2+\vs_2^+\rangle_\vM\nonumber\\
& + 2\gamma\langle \vA^\top(\vs_1^+-\vs_2^+),\vz_1^+-\vz_1+\vx_1-\vz_2^++\vz_2-\vx_2\rangle. \label{eq:equal_h}\end{aligned}$$ The updates of $\vz_1^+$ and $\vz_2^+$ in show $$\begin{aligned}
& 2\gamma\langle \vx_1-\vx_2,\vq_g(\vx_1)-\vq_g(\vx_2)+\nabla f(\vx_1)-\nabla f(\vx_2)\rangle\nonumber\\
= & 2\gamma\langle \vx_1-\vx_2,{\gamma}^{-1}(\vz_1-\vx_1)-{\gamma}^{-1}(\vz_2-\vx_2)+\nabla f(\vx_1)-\nabla f(\vx_2)\rangle\nonumber\\
= & 2\langle \vx_1-\vx_2,\vz_1-\vx_1+\gamma\nabla f(\vx_1)-\vz_2+\vx_2-\gamma\nabla f(\vx_2)\rangle\nonumber\\
= & 2\langle \vx_1-\vx_2,\vz_1-\vz_1^+-\gamma\vA^\top\vs_1^+ -\vz_2+\vz_2^++\gamma \vA^\top\vs_2^+\rangle. \label{eq:equal_fg}\end{aligned}$$ Combining both and , we have $$\begin{aligned}
& 2\gamma\langle \vs_1^+-\vs_2^+,\vq_{h^*}(\vs_1^+)-\vq_{h^*}(\vs_2^+)+\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle \nonumber\\
& +2\gamma\langle \vx_1-\vx_2,\vq_g(\vx_1)-\vq_g(\vx_2)+\nabla f(\vx_1)-\nabla f(\vx_2)\rangle \nonumber\\
= & 2\langle\vs_1^+-\vs_2^+,\vs_1-\vs_1^+-\vs_2+\vs_2^+\rangle_\vM + 2\gamma\langle \vA^\top(\vs_1^+-\vs_2^+),\vz_1^+-\vz_1-\vz_2^++\vz_2\rangle \nonumber\\
& + 2\gamma\langle \vA^\top(\vs_1^+-\vs_2^+),\vx_1-\vx_2\rangle - 2\langle \vx_1-\vx_2,\gamma\vA^\top\vs_1^+ -\gamma \vA^\top\vs_2^+\rangle \nonumber\\
& +2\langle \vx_1-\vx_2,\vz_1-\vz_1^+ -\vz_2+\vz_2^+\rangle \nonumber\\
= & 2\langle\vs_1^+-\vs_2^+,\vs_1-\vs_1^+-\vs_2+\vs_2^+\rangle_\vM \nonumber\\
& +2\langle \vx_1 -\gamma\vA^\top\vs_1^+-\vx_2+\gamma\vA^\top\vs_2^+,\vz_1-\vz_1^+ -\vz_2+\vz_2^+\rangle \nonumber\\
= & 2\langle\vs_1^+-\vs_2^+,\vs_1-\vs_1^+-\vs_2+\vs_2^+\rangle_\vM \nonumber\\
& +2\langle \vz_1^+ +\gamma\nabla f(\vx_1)-\vz_2^+-\gamma\nabla f(\vx_2),\vz_1-\vz_1^+ -\vz_2+\vz_2^+\rangle \label{eqn:gapfunction}\\
= & \|(\vz_1,\vs_1)-(\vz_2,\vs_2)\|_{\vI,\vM}^2-\|(\vz_1^+,\vs_1^+)-(\vz_2^+,\vs_2^+)\|_{\vI,\vM}^2 -\|\vz_1-\vz_1^+-\vz_2+\vz_2^+\|^2 \nonumber\\
& -\|\vs_1-\vs_1^+-\vs_2+\vs_2^+\|_{\vM}^2 +2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vz_1-\vz_1^+-\vz_2+\vz_2^+\rangle, \nonumber\end{aligned}$$ where the third equality comes from the updates of $\vz_1^+$ and $\vz_2^+$ in and the last equality holds because of $2\langle a,b\rangle =\|a+b\|^2-\|a\|^2-\|b\|^2$.
Note that Lemma \[lemma:fundamental\] only requires the symmetry of $\vM$.
Convergence analysis for the general convex case
------------------------------------------------
In this section, we show the convergence of the proposed algorithm in Theorem \[thm:main\]. We show firstly that the operator $\PD$ is a nonexpansive operator (Lemma \[lemma:nonexpansive\]) and then finding a fixed point $(\vz^*,\vs^*)$ of $\PD$ is equivalent to finding an optimal solution to (Lemma \[lemma:optimal\_solution\]).
\[lemma:nonexpansive\] Let $(\vz_1^+,\vs_1^+)=\PD(\vz_1,\vs_1)$ and $(\vz_2^+,\vs_2^+)=\PD(\vz_2,\vs_2)$. Under Assumption \[assum:1\],we have $$\begin{aligned}
& \|(\vz_1^+,\vs_1^+)-(\vz_2^+,\vs_2^+)\|_{\vI,\vM}^2 -\|(\vz_1,\vs_1)-(\vz_2,\vs_2)\|_{\vI,\vM}^2 \nonumber\\
\leq & -{2\beta-\gamma\over2\beta}\left(\|(\vz_1^+,\vs_1^+)-(\vz_1,\vs_1)-(\vz_2^+,\vs_2^+)+(\vz_2,\vs_2)\|_{\vI,\vM}^2\right).\label{eq:average}\end{aligned}$$ Furthermore, when $\vM$ is positive definite, the operator $\PD$ in is nonexpansive for $(\vz,\vs)$ if $\gamma\leq 2\beta$. More specifically, it is $\alpha$-averaged with $\alpha={2\beta\over 4\beta-\gamma}$.
Because of the convexity of $h^*$ and $g$, we have $$\begin{aligned}
0\leq & \langle \vs_1^+-\vs_2^+,\vq_{h^*}(\vs_1^+)-\vq_{h^*}(\vs_2^+)\rangle
+\langle \vx_1-\vx_2,\vq_g(\vx_1)-\vq_g(\vx_2)\rangle,\end{aligned}$$ where $\vq_{h^*}(\vs_1^+)$, $\vq_{h^*}(\vs_1^+)$, $\vq_g(\vx_1)$, and $\vq_g(\vx_2)$ are defined in Lemma \[lemma:fundamental\]. Then, the fundamental equality in Lemma \[lemma:fundamental\] gives $$\begin{aligned}
& \|(\vz_1^+,\vs_1^+)-(\vz_2^+,\vs_2^+)\|_{\vI,\vM}^2 - \|(\vz_1,\vs_1)-(\vz_2,\vs_2)\|_{\vI,\vM}^2 \nonumber\\
\leq & 2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vz_1-\vz_1^+-\vz_2+\vz_2^+\rangle -2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vx_1-\vx_2\rangle\nonumber\\
& -2\gamma\langle \vs_1^+-\vs_2^+,\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle \label{eq:nonexpansive_cp}\\
& \textstyle - \|\vz_1-\vz_1^+-\vz_2+\vz_2^+\|^2 - \|\vs_1-\vs_1^+-\vs_2+\vs_2^+\|_\vM^2.\nonumber\end{aligned}$$ Next we derive the upper bound of the cross terms in as follows: $$\begin{aligned}
& 2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vz_1-\vz_1^+-\vz_2+\vz_2^+\rangle -2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vx_1-\vx_2\rangle\nonumber\\
& -2\gamma\langle \vs_1^+-\vs_2^+,\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle \nonumber\\
= & 2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vz_1-\vz_1^+-\vz_2+\vz_2^+\rangle - 2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2), \vx_1-\vx_2\rangle \nonumber\\
& + 2\gamma\langle \vs_1-\vs_1^+-\vs_2+\vs_2^+,\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle -2\gamma\langle \vs_1-\vs_2,\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle \nonumber\\
\leq & 2\gamma\langle \nabla f(\vx_1)-\nabla f(\vx_2) , \vz_1-\vz_1^+-\vz_2+\vz_2^+\rangle - 2\gamma\beta \| \nabla f(\vx_1)-\nabla f(\vx_2)\|^2 \nonumber\\
& +2\gamma\langle \vs_1-\vs_1^+-\vs_2+\vs_2^+,\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\rangle -2\gamma\beta\|\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\|^2_{\vM^{-1}} \nonumber\\
\leq & \textstyle \epsilon \|\vz_1-\vz_1^+-\vz_2+\vz_2^+\|^2+\left({\gamma^2\over \epsilon}-2\gamma\beta\right)\| \nabla f(\vx_1)-\nabla f(\vx_2)\|^2 \nonumber\\
& \textstyle+\epsilon\|\vs_1-\vs_1^+-\vs_2+\vs_2^+\|_\vM^2+\left({\gamma^2\over \epsilon}-2\gamma\beta\right)\|\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\|^2_{\vM^{-1}}. \label{lemma2_a}\end{aligned}$$ The first inequality comes from the cocoerciveness of $\nabla f$ and $\nabla l^*$ in Assumption \[assum:1\], and the second inequality comes from the Cauchy-Schwarz inequality. Therefore, combing and gives $$\begin{aligned}
& \|(\vz_1^+,\vs_1^+)-(\vz_2^+,\vs_2^+)\|_{\vI,\vM}^2-\|(\vz_1,\vs_1)-(\vz_2,\vs_2)\|_{\vI,\vM}^2\nonumber\\
\leq & \textstyle - (1-\epsilon)\|(\vz_1^+,\vs_1^+)-(\vz_1,\vs_1)-(\vz_2^+,\vs_2^+)+(\vz_2,\vs_2)\|_{\vI,\vM}^2 \label{eq:nonexpansive_a}\\
& \textstyle +\left({\gamma^2\over \epsilon}-2\gamma\beta\right)(\| \nabla f(\vx_1)-\nabla f(\vx_2)\|^2+\|\nabla l^*(\vs_1)-\nabla l^*(\vs_2)\|_{\vM^{-1}}^2). \nonumber\end{aligned}$$ In addition, letting $\epsilon=\gamma/(2\beta)$, we have that $$\begin{aligned}
& \|(\vz_1^+,\vs_1^+)-(\vz_2^+,\vs_2^+)\|_{\vI,\vM}^2 -\|(\vz_1,\vs_1)-(\vz_2,\vs_2)\|_{\vI,\vM}^2 \nonumber\\
\leq & -{2\beta-\gamma\over2\beta}\left(\|(\vz_1^+,\vs_1^+)-(\vz_1,\vs_1)-(\vz_2^+,\vs_2^+)+(\vz_2,\vs_2)\|_{\vI,\vM}^2\right),\label{eq:nonexpansive_b}\end{aligned}$$ Furthermore, when $\vM$ is positive definite, $\PD$ is $\alpha-$averaged with $\alpha ={2\beta\over 4\beta-\gamma}$ under the norm $\|(\cdot,\cdot)\|_{\vI,\vM}$.
\[remark1\] For Davis-Yin (i.e., $\vA=\vI$ and $\gamma\delta=1$), we have $\vM=\vzero$ (therefore, $\nabla l^*$ is a constant) and becomes $$\begin{aligned}
\|\vz_1^+-\vz_2^+\|^2-\|\vz_1-\vz_2\|^2 \leq & - (1-\epsilon)\|\vz_1^+-\vz_1-\vz_2^++\vz_2\|^2 \\
& \textstyle +\left({\gamma^2\over \epsilon}-2\gamma\beta\right)\| \nabla f(\vx_1)-\nabla f(\vx_2)\|^2,\end{aligned}$$ which is similar to that of Remark 3.1 in [@davis2015three]. In fact, the nonexpansiveness of $\PD$ in Lemma \[lemma:nonexpansive\] can also be derived by modifying the result of the three-operator splitting under the new norm defined by $\|(\cdot,\cdot)\|_{\vI,\vM}$ (for positive definite $\vM$). The equivalent problem is $$\begin{aligned}
\left[\begin{array}{c} \vzero\\\vzero\end{array}\right]\in&\left[\begin{array}{cc}\nabla f&0\\0&\vM^{-1}\nabla l^*\end{array}\right]\left[\begin{array}{c} \vx\\\vs\end{array}\right]
+\left[\begin{array}{cc}\partial g&0\\0&0\end{array}\right]\left[\begin{array}{c} \vx\\\vs\end{array}\right] +\left[\begin{array}{cc}\vI&0\\0&\vM^{-1}\end{array}\right]\left[\begin{array}{cc} 0&\vA^\top\\-\vA&\partial h^*\end{array}\right]\left[\begin{array}{c} \vx\\\vs\end{array}\right].\end{aligned}$$ In this case, Assumption \[assum:1\] provides $$\begin{aligned}
&\left\langle\left[\begin{array}{cc}\nabla f&0\\0&\vM^{-1}\nabla l^*\end{array}\right]\left[\begin{array}{c} \vz_1\\\vs_1\end{array}\right]-
\left[\begin{array}{cc}\nabla f&0\\0&\vM^{-1}\nabla l^*\end{array}\right]\left[\begin{array}{c} \vz_2\\\vs_2\end{array}\right],\left[\begin{array}{c} \vz_1-\vz_2\\\vs_1-\vs_2\end{array}\right]
\right\rangle_{\vI,\vM}\\
\geq & \beta\left\|\left[\begin{array}{cc}\nabla f&0\\0&\vM^{-1}\nabla l^*\end{array}\right]\left[\begin{array}{c} \vz_1\\\vs_1\end{array}\right]-
\left[\begin{array}{cc}\nabla f&0\\0&\vM^{-1}\nabla l^*\end{array}\right]\left[\begin{array}{c} \vz_2\\\vs_2\end{array}\right]\right\|_{\vI,\vM}^2,\end{aligned}$$ i.e., $\left[\begin{array}{cc}\nabla f&0\\0&\vM^{-1}\nabla l^*\end{array}\right]$ is a $\beta$-cocoercive operator under the norm $\|(\cdot,\cdot)\|_{\vI,\vM}$. In addition, the two operators $\left[\begin{array}{cc}\partial g&0\\0&0\end{array}\right]$ and $\left[\begin{array}{cc}\vI&0\\0&\vM^{-1}\end{array}\right]\left[\begin{array}{cc} 0&\vA^\top\\-\vA&\partial h^*\end{array}\right]$ are maximal monotone under the norm $\|(\cdot,\cdot)\|_{\vI,\vM}$. Then we can modify the result of Davis-Yin and show the $\alpha$-averageness of $\PD$. However, the primal-dual gap convergence in Section \[sec:primal\_dual\] and linear convergence rate in Section \[sec:linear\] can not be obtained from [@davis2015three] because $\left[\begin{array}{cc}\nabla f&0\\0&0\end{array}\right]$ is not strongly monotone under the norm $\|(\cdot,\cdot)\|_{\vI,\vM}$. For the completeness, we provide the proof of the $\alpha$-averageness of $\PD$ here. In fact, in the proof can be stronger than the $\alpha$-averageness result when $\nabla l^*=\vzero$ or $\nabla f=\vzero$. For example, when $\nabla l^*=\vzero$, we have $$\begin{aligned}
& \|(\vz_1^+,\vs_1^+)-(\vz_2^+,\vs_2^+)\|_{\vI,\vM}^2-\|(\vz_1,\vs_1)-(\vz_2,\vs_2)\|_{\vI,\vM}^2\nonumber\\
\leq & \textstyle - {2\beta-\gamma\over2\beta} \|\vz_1-\vz_1^+-\vz_2+\vz_2^+\|^2 - \|\vs_1-\vs_1^+-\vs_2+\vs_2^+\|_\vM^2.\end{aligned}$$
\[cor:CP\] When $f=0$ and $l^*=0$, we have $$\begin{aligned}
& \|\vz_1^+-\vz_2^+\|^2 + \|\vs_1^+-\vs_2^+\|_\vM^2 - \|\vz_1-\vz_2\|^2 -\|\vs_1-\vs_2\|_\vM^2 \\
\leq & - \|\vz_1^+-\vz_1-\vz_2^++\vz_2\|^2 - \|\vs_1^+-\vs_1-\vs_2^++\vs_2\|_\vM^2,\end{aligned}$$ for any $\gamma$ and $\delta$ such that $\gamma\delta\|\vA\vA^\top\|\leq1$. It means that the Chambolle-Pock iteration is equivalent to a firmly nonexpansive operator under the norm $\|(\cdot,\cdot)\|_{\vI,\vM}$ when $\vM$ is positive definite.
Letting $f=0$ and $l^*=0$ in immediately gives the result.
The equivalence of the Chambolle-Pock iteration to a firmly nonexpansive operator under the norm defined by ${1\over \gamma}\|\vx\|^2-2\langle \vA\vx,\vs\rangle +{1\over\delta}\|\vs\|^2$ is shown in [@peng2016coordinate; @he2014convergence; @7025841] by reformulating Chambolle-Pock as a proximal point algorithm applied on the KKT conditions. Here, we show the firmly nonexpansiveness of the Chambolle-Pock iteration for $(\vz,\vs)$ under a different norm defined by $\|(\cdot,\cdot)\|_{\vI,\vM}$. In fact, the Chambolle-Pock iteration is equivalent to DRS applied on the KKT conditions based on the connection between PD3O and Davis-Yin in Remark \[remark1\] and that Davis-Yin reduces to DRS without the Lipschitz continous operator. The equivalence of the Chambolle-Pock iteration and DRS is also showed in [@o2017equivalence] using $(\vz,\sqrt{\vM}\vs)$ under the standard norm instead of $(\vz,\vs)$ under the norm $\|(\cdot,\cdot)\|_{\vI,\vM}$, therefore the convergence condition of Chambolle-Pock becomes $\gamma\delta\|\vA\vA^\top\|\leq 1$, if we do not consider the convergence of the dual variable $\vs$.
\[cor:PDFP2O\] When $g=0$ and $l^*=0$, $\PD$ reduces to the PAPC iteration, and it is $\alpha$-averaged with $\alpha={2\beta/(4\beta-\gamma)}$ when $\gamma<2\beta$ and $\gamma\delta\|\vA\vA^\top\|<1$.
In [@chen2013primal], the PAPC iteration is shown to be nonexpansive [only]{.nodecor} under a norm for $\cX\times \cS$ that is defined by $\sqrt{\|\vz\|^2+{\gamma\over \delta}\|\vs\|^2}$. Corollary \[cor:PDFP2O\] improves the result by showing that it is $\alpha$-averaged with $\alpha=2\beta/(4\beta-\gamma)$ under the norm $\|(\cdot,\cdot)\|_{\vI,\vM}$. This result appeared in [@davis2015convergence] previously.
\[lemma:optimal\_solution\]For any fixed point $(\vz^*,\vs^*)$ of $\PD$, $\prox_{\gamma g}(\vz^*)$ is an optimal solution to the optimization problem . For any optimal solution $\vx^*$ of the optimization problem satisfying $\vzero\in \partial g(\vx^*)+ \nabla f(\vx^*)+ \vA^\top\partial h\square l(\vA\vx^*)$, we can find a fixed point $(\vz^*,\vs^*)$ of $\PD$ such that $\vx^*=\prox_{\gamma g}(\vz^*)$.
If $(\vz^*,\vs^*)$ is a fixed point of $\PD$, let $\vx^*=\prox_{\gamma g}(\vz^*)$. Then we have $\vzero= \vz^*-\vx^*+\gamma\nabla f(\vx^*)+ \gamma\vA^\top\vs^*$ from , $\vz^*-\vx^*\in\gamma \partial g(\vx^*)$ from , and $\vA\vx^*\in\partial h^*(\vs^*)+\nabla l^*(\vs^*)$ from and . Therefore, $\vzero\in \gamma \partial g(\vx^*)+\gamma \nabla f(\vx^*)+\gamma \vA^\top\partial h\square l(\vA\vx^*)$, i.e., $\vx^*$ is an optimal solution for the convex problem .
If $\vx^*$ is an optimal solution for problem such that $\vzero\in \partial g(\vx^*)+ \nabla f(\vx^*)+ \vA^\top\partial h\square l(\vA\vx^*)$, there exist $\vq_g^*\in\partial g(\vx^*)$ and $\vq_{h}^*\in\partial h\square l(\vA\vx^*)$ such that $\vzero=\vq_g^* +\nabla f(\vx^*) + \vA^\top\vq_{h}^*$. Letting $\vz^*=\vx^*+\gamma \vq_g^*$ and $\vs^*=\vq_{h}^*$, we derive $\vx=\vx^*$ from , $\vs^+=\prox_{\delta h^*}(\vs^*-\delta\nabla l^*(\vs^*)+\delta\vA\vx^*)=\vs^*$ from , and $\vz^+=\vx^*-\gamma\nabla f(\vx^*)-\gamma \vA^\top\vs^*=\vx^*+\gamma \vq_g^*=\vz^*$ from . Thus $(\vz^*,\vs^*)$ is a fixed point of $\PD$.
\[thm:main\] Choose $\gamma$ and $\delta$ such that $\gamma<2\beta$ and Assumption \[assum:1\] is satisfied. Let $(\vz^*,\vs^*)$ be any fixed point of $\PD$, and $(\vz^k,\vs^k)_{k\geq0}$ is the sequence generated by PD3O.
1. The sequence $(\|(\vz^k,\vs^k)-(\vz^*,\vs^*)\|_{\vI,\vM})_{k\geq 0}$ is monotonically nonincreasing.
2. The sequence $(\|\PD(\vz^k,\vs^k)-(\vz^k,\vs^k)\|_{\vI,\vM})_{k\geq 0}$ is monotonically nonincreasing and converges to 0.
3. We have the following convergence rate $$\begin{aligned}
\|\PD(\vz^k,\vs^k)-(\vz^k,\vs^k)\|_{\vI,\vM}^2\leq {2\beta\over 2\beta-\gamma } {\|(\vz^0,\vs^0)-(\vz^*,\vs^*)\|_{\vI,\vM}^2\over k+1} \label{eqn:rate_fp}\end{aligned}$$ and $$\begin{aligned}
\|\PD(\vz^k,\vs^k)-(\vz^k,\vs^k)\|_{\vI,\vM}^2=o\left({1\over k+1}\right).\end{aligned}$$
4. $(\vz^k,\vs^k)$ weakly converges to a fixed point of $\PD$.
1\) Let $(\vz_1,\vs_1)=(\vz^k,\vs^k)$ and $(\vz_2,\vs_2)=(\vz^*,\vs^*)$ in , and we have $$\begin{aligned}
& \|(\vz^{k+1},\vs^{k+1})-(\vz^*,\vs^*)\|_{\vI,\vM}^2 -\|(\vz^k,\vs^k)-(\vz^*,\vs^*)\|_{\vI,\vM}^2 \nonumber\\
\leq & -{2\beta-\gamma\over2\beta}\|\PD(\vz^{k},\vs^{k})-(\vz^k,\vs^k)\|_{\vI,\vM}^2.\label{eq:average2}\end{aligned}$$ Thus, $\|(\vz^k,\vs^k)-(\vz^*,\vs^*)\|_{\vI,\vM}^2$ is monotonically decreasing as long as $\PD(\vz^{k},\vs^{k})-(\vz^k,\vs^k)\neq \vzero$ because $\gamma<2\beta$.
2\) Summing up from 0 to $\infty$, we have $$\sum_{k=0}^\infty\|\PD(\vz^k,\vs^k)-(\vz^k,\vs^k)\|_{\vI,\vM}^2\leq {2\beta\over 2\beta-\gamma}\|(\vz^0,\vs^0)-(\vz^*,\vs^*)\|_{\vI,\vM}^2$$ and thus the sequence $(\|\PD(\vz^k,\vs^k)-(\vz^k,\vs^k)\|_{\vI,\vM})_{k\geq 0}$ converges to 0. Furthermore, the sequence $(\|\PD(\vz^k,\vs^k)-(\vz^k,\vs^k)\|_{\vI,\vM})_{k\geq 0}$ is monotonically nonincreasing because $$\begin{aligned}
&\|\PD(\vz^{k+1},\vs^{k+1})-(\vz^{k+1},\vs^{k+1})\|_{\vI,\vM}= \|\PD(\vz^{k+1},\vs^{k+1})-\PD(\vz^{k},\vs^{k})\|_{\vI,\vM} \\
\leq & \|(\vz^{k+1},\vs^{k+1})-(\vz^{k},\vs^{k})\|_{\vI,\vM}=\|\PD(\vz^{k},\vs^{k})-(\vz^{k},\vs^{k})\|_{\vI,\vM}.\end{aligned}$$
3\) The $o(1/k)$ convergence rate follows from [@davis2014convergence Theorem 1]. $$\begin{aligned}
\|\PD(\vz^k,\vs^k)-(\vz^k,\vs^k)\|_{\vI,\vM}^2 \leq & {1\over k+1}\sum_{i=0}^k\|\PD(\vz^i,\vs^i)-(\vz^i,\vs^i)\|_{\vI,\vM}^2 \\
\leq & {2\beta\over 2\beta-\gamma } {\|(\vz^0,\vs^0)-(\vz^*,\vs^*)\|_{\vI,\vM}^2\over k+1}.\end{aligned}$$
4\) This follows from the Krasnosel’skii-Mann theorem [@bauschke2011convex Theorem 5.14].
The convergence of $\vz^k$ (or $\vx^k$) can be obtained under a weaker condition that $\vM$ is positive semidefinite, instead of positive definite if $\nabla l^*$ is a constant. In this case, the weak convergence of $(\vz^k,\sqrt{\vM}\vs^k)$ can be obtained under the standard norm. Furthermore, if we assume that $\cX$ is finite dimensional, we have the strong convergence of $\vz^k$ and thus $\vx^k$ because the proxmial operator in is nonexpansive.
Since the operator $\PD$ is $\alpha$-averaged with $\alpha={2\beta/(4\beta-\gamma)}$, the relaxed iteration $(\vz^{k+1},\vs^{k+1})=\theta_k\PD(\vz^k,\vs^k) + (1-\theta_k)(\vz^k,\vs^k)$ still converges if $\sum_{k=0}^\infty \theta_k (2-\gamma/(2\beta)-\theta_k) =\infty$.
$O(1/k)$-ergodic convergence for the general convex case {#sec:primal_dual}
--------------------------------------------------------
In this subsection, let $$\begin{aligned}
\cL(\vx,\vs)=f(\vx)+g(\vx)+\langle\vA\vx,\vs\rangle- h^*(\vs)-l^*(\vs).\end{aligned}$$ If $\cL$ has a saddle point, there exists $(\vx^*,\vs^*)$ such that $$\cL(\vx^*,\vs)\leq \cL(\vx^*,\vs^{*})\leq \cL(\vx,\vs^{*}),~\forall (\vx,\vs)\in\cX\times\cS.$$
Then, we consider the quantity $\cL(\vx^k,\vs)-\cL(\vx,\vs^{k+1})$ for all $(\vx,\vs)\in\cX\times \cS$. If $\cL(\vx^k,\vs)-\cL(\vx,\vs^{k+1})\leq 0$ for all $(\vx,\vs)$, then we have $$\cL(\vx^k,\vs)\leq \cL(\vx^k,\vs^{k+1})\leq \cL(\vx,\vs^{k+1}),$$ which means that $(\vx^k,\vs^{k+1})$ is a saddle point of $\cL$.
Let $\gamma\leq \beta$ and Assumption \[assum:1\] be satisfied. The sequence $(\vx^k,\vz^k,\vs^k)_{k\geq 0}$ is generated by PD3O. Define $$\begin{aligned}
\overline\vx^k = {1\over k+1}\sum_{i=0}^k\vx^i,\qquad \overline\vs^{k+1} = {1\over k+1}\sum_{i=0}^k\vs^{i+1}.\end{aligned}$$ Then we have $$\begin{aligned}
\cL(\bar\vx^k,\vs)-\cL(\vx,\bar\vs^{k+1})
\leq & {1\over 2(k+1)\gamma}\|(\vz,\vs)-(\vz^0,\vs^0)\|_{\vI,\vM}^2.\end{aligned}$$ for any $(\vx,\vs)\in\cX\times\cS$ and $\vz=\vx-\gamma\nabla f(\vx)-\gamma\vA^\top\vs$.
First, we provide an upper bound for $\cL(\vx^k,\vs)-\cL(\vx,\vs^{k+1})$, and to do this, we consider upper bounds for $\cL(\vx^k,\vs)-\cL(\vx^k,\vs^{k+1})$ and $\cL(\vx^k,\vs^{k+1})-\cL(\vx,\vs^{k+1})$.
For any $\vx\in\cX$, we have $$\begin{aligned}
& \cL(\vx^k,\vs^{k+1})-\cL(\vx,\vs^{k+1}) \nonumber\\
= & f(\vx^k)+g(\vx^k)+\langle \vA\vx^k,\vs^{k+1}\rangle-f(\vx)-g(\vx)-\langle\vA\vx,\vs^{k+1}\rangle \nonumber\\
\leq & \langle \nabla f(\vx^k),\vx^k-\vx\rangle -{(\beta/2)}\|\nabla f(\vx^k)-\nabla f(\vx)\|^2 \nonumber\\
& + \gamma^{-1}\langle \vx-\vx^k,\vx^k-\vz^k\rangle +\langle \vx^k-\vx,\vA^\top\vs^{k+1}\rangle \nonumber\\
= & \gamma^{-1}\langle \vz^{k+1}-\vz^{k},\vx-\vx^k\rangle-{(\beta/2)}\|\nabla f(\vx^k)-\nabla f(\vx)\|^2. \label{ergodic_a}\end{aligned}$$ The inequality comes from (with $\vx_1$ and $\vx_2$ being $\vx$ and $\vx^k$, respectively) and . The last equality holds because of .
On the other hand, combing and , we obtain $$\begin{aligned}
\vs^{k+1} &= \prox_{{\delta} h^*} ((\vI-\gamma\delta \vA\vA^\top)\vs^k-{\delta}\nabla l^*(\vs^k) + {\delta}\vA\left(2\vx^k-\vz^k-{\gamma}\nabla f(\vx^k)\right)) \\
&= \prox_{{\delta} h^*} (\vs^k+\gamma\delta \vA\vA^\top(\vs^{k+1}-\vs^k)-{\delta}\nabla l^*(\vs^k) + {\delta}\vA\left(\vx^k+\vz^{k+1}-\vz^k\right)).\end{aligned}$$ Therefore, for any $\vs\in\cS$, the following inequality holds. $$\begin{aligned}
& h^*(\vs^{k+1})-h^*(\vs) \nonumber\\
\leq & \langle \gamma^{-1}\vM(\vs^{k+1}-\vs^k)+\nabla l^*(\vs^k) - \vA\left(\vx^k+\vz^{k+1}-\vz^k\right),\vs-\vs^{k+1} \rangle \nonumber\\
= & \gamma^{-1}\langle \vs^{k+1}-\vs^k,\vs-\vs^{k+1} \rangle_\vM +\langle \nabla l^*(\vs^k),\vs-\vs^{k+1}\rangle \nonumber\\
& -\langle \vz^{k+1}-\vz^k,\vA^{\top}(\vs-\vs^{k+1}) \rangle -\langle \vA\vx^k,\vs-\vs^{k+1} \rangle. \label{inequal_h}\end{aligned}$$ Additionally, from , we have $$\begin{aligned}
l^*(\vs^{k+1}) \leq l^*(\vs^k) + \langle \nabla l^*(\vs^k),\vs^{k+1}-\vs^k\rangle + (2\beta)^{-1}\|\vs^{k+1}-\vs^k\|_\vM^2,~\label{dual_l1}\end{aligned}$$ and the convexity of $l^*$ gives $$\begin{aligned}
l^*(\vs) \geq l^*(\vs^k) + \langle \nabla l^*(\vs^k),\vs-\vs^k\rangle. ~\label{dual_l2}\end{aligned}$$ Combining , , and , we derive $$\begin{aligned}
& \cL(\vx^k,\vs)-\cL(\vx^k,\vs^{k+1}) \nonumber\\
= & \langle \vA\vx^k,\vs-\vs^{k+1}\rangle +h^*(\vs^{k+1})+l^*(\vs^{k+1})-h^*(\vs)-l^*(\vs)\nonumber\\
\leq & \gamma^{-1}\langle \vs-\vs^{k+1},\vs^{k+1}-\vs^k\rangle_\vM-\langle \vz^{k+1}-\vz^k,\vA^\top(\vs-\vs^{k+1})\rangle \nonumber\\
& +(2\beta)^{-1}\|\vs^{k+1}-\vs^k\|_\vM^2. \label{ergodic_b} \end{aligned}$$
Then, adding and together gives $$\begin{aligned}
& \cL(\vx^k,\vs)-\cL(\vx,\vs^{k+1}) \\
\leq & \gamma^{-1}\langle \vs-\vs^{k+1},\vs^{k+1}-\vs^k\rangle_\vM-\langle \vz^{k+1}-\vz^k,\vA^\top(\vs-\vs^{k+1})\rangle \\
& +\gamma^{-1}\langle \vz^{k+1}-\vz^{k},\vx-\vx^k\rangle-{(\beta/2)}\|\nabla f(\vx^k)-\nabla f(\vx)\|^2 \\
& +(2\beta)^{-1}\|\vs^{k+1}-\vs^k\|_\vM^2\\
= & \gamma^{-1}\langle \vs-\vs^{k+1},\vs^{k+1}-\vs^k\rangle_\vM +\gamma^{-1}\langle \vz^{k+1}-\vz^{k},\vz-\vz^{k+1}\rangle\\
& +\langle \vz^{k+1}-\vz^{k},\nabla f(\vx)-\nabla f(\vx^k)\rangle-{(\beta/2)}\|\nabla f(\vx^k)-\nabla f(\vx)\|^2\\
& +(2\beta)^{-1}\|\vs^{k+1}-\vs^k\|_\vM^2\\
\leq & (2\gamma)^{-1}(\|\vs-\vs^k\|_\vM^2-\|\vs-\vs^{k+1}\|_\vM^2-\|\vs^k-\vs^{k+1}\|_\vM^2) \\
& +(2\gamma)^{-1}(\|\vz-\vz^{k}\|^2-\|\vz-\vz^{k+1}\|^2-\|\vz^k-\vz^{k+1}\|^2) \\
& +(2\beta)^{-1}\|\vz^k-\vz^{k+1}\|^2+(2\beta)^{-1}\|\vs^{k+1}-\vs^k\|_\vM^2.\end{aligned}$$ That is $$\begin{aligned}
\cL(\vx^k,\vs)-\cL(\vx,\vs^{k+1}) \leq & (2\gamma)^{-1}\|(\vz^k,\vs^k)-(\vz,\vs)\|_{\vI,\vM}^2\nonumber\\
& -(2\gamma)^{-1}\|(\vz^{k+1},\vs^{k+1})-(\vz,\vs)\|_{\vI,\vM}^2 \label{eqn:gap_conv_a}\\
& -((2\gamma)^{-1}-(2\beta)^{-1})\|(\vz^k,\vs^k)-(\vz^{k+1},\vs^{k+1})\|_{\vI,\vM}^2. \nonumber\end{aligned}$$ When $\gamma\leq \beta$, we have $$\begin{aligned}
\cL(\vx^k,\vs)-\cL(\vx,\vs^{k+1})
\leq & (2\gamma)^{-1}\|(\vz,\vs)-(\vz^{k},\vs^k)\|_{\vI,\vM}^2 \nonumber\\
& -(2\gamma)^{-1}\|(\vz,\vs)-(\vz^{k+1},\vs^{k+1})\|_{\vI,\vM}^2. \label{conv:gap_one_step}\end{aligned}$$ Using the definition of $(\bar\vx^k,\bar\vs^{k+1})$ and the Jensen inequality, it follows that $$\begin{aligned}
\cL(\bar\vx^k,\vs)-\cL(\vx,\bar\vs^{k+1})
\leq & {1\over{k+1}}\sum_{i=0}^k \cL(\vx^i,\vs)-\cL(\vx,\vs^{i+1}) \nonumber\\
\leq & {1\over 2(k+1)\gamma}\|(\vz,\vs)-(\vz^0,\vs^0)\|_{\vI,\vM}^2.\end{aligned}$$ This proves the desired result.
The $O(1/k)$-ergodic convergence rate proved in [@Drori2015209] for PAPC is on a different sequence $\left({1\over k+1}\sum_{i=1}^{k+1}\vx^{i},{1\over k+1}\sum_{i=1}^{k+1}\vs^{i}=\bar\vs^{k+1}\right)$.
Linear convergence rate for special cases {#sec:linear}
-----------------------------------------
In this subsection, we provide some results on the linear convergence rate of PD3O with additional assumptions. For simplicity, let $(\vz^*,\vs^*)$ be a fixed point of $\PD$ and $\vx^*=\prox_{\gamma g}(\vz^*)$. In addition, we let $\langle \vs-\vs^*,\vq_{h^*}(\vs)-\vq_{h^*}^*\rangle\geq \tau_{h^*}\|\vs-\vs^*\|^2_\vM$ and $\langle \vs-\vs^*,\nabla l^*(\vs)-\nabla l^*(\vs^*)\rangle\geq \tau_{l^*}\|\vs-\vs^*\|^2_\vM$ for any $\vs\in\cS$, $\vq_{h^*}(\vs)\in\partial h^*(\vs)$, and $\vq_{h^*}^*\in\partial h^*(\vs^*)$. Then $\vM^{-1}\partial h^*$ and $\vM^{-1}\nabla l^*$ are restricted $\tau_{h^*}$-strongly monotone and restricted $\tau_{l^*}$-strongly monotone under the norm defined by $\|(\cdot,\cdot)\|_{\vI,\vM}$, respectively. Here we allow that $\tau_{h^*}=0$ and $\tau_{l^*}=0$ for merely monotone operators. Similarly, we let $\langle \vx-\vx^*,\vq_g(\vx)-\vq_g^*\rangle\geq \tau_g\|\vx-\vx^*\|^2$ and $\langle \vx-\vx^*,\nabla f(\vx)-\nabla f(\vx^*)\rangle \geq \tau_f\|\vx-\vx^*\|^2$ for any $\vx\in\cX$, $\vq_g(\vx)\in\partial g(\vx)$, and $\vq_g^*\in\partial g(\vx^*)$.
If $g$ has a $L_g$-Lipschitz continuous gradient, i.e., $$\|\nabla g(\vx)-\nabla g(\vy)\|\leq L_g\|\vx-\vy\|$$ and $(\vz^+,\vs^+)=\PD(\vz,\vs)$, then under Assumption \[assum:1\], we have $$\begin{aligned}
\textstyle \|\vz^+-\vz^*\|^2+\left(1+2\gamma\tau_{h^*}\right)\|\vs^+-\vs^*\|_\vM^2 \leq \rho\left(\|\vz-\vz^*\|^2+\left(1+2\gamma\tau_{h^*}\right)\|\vs-\vs^*\|_\vM^2 \right)\end{aligned}$$ where $$\begin{aligned}
\label{eqn:rho}
\textstyle \rho = \max\left( {{1-\left(2\gamma-{\gamma^2\over\beta}\right)\tau_{l^*}} \over 1 +2\gamma\tau_{h^*}}, 1- {\left(\left(2\gamma-{\gamma^2\over\beta}\right)\tau_f+2\gamma\tau_g\right)\over 1+\gamma L_g}\right).\end{aligned}$$ When, in addition, $\gamma<2\beta$, $\tau_{h^*}+\tau_{l^*}>0$, and $\tau_f+\tau_g>0$, we have that $\rho<1$ and the algorithm PD3O converges linearly.
Equation in Lemma \[lemma:fundamental\] with $(\vz_1,\vs_1)=(\vz,\vs)$ and $(\vz_2,\vs_2)=(\vz^*,\vs^*)$ gives
$$\begin{aligned}
&2\textstyle \gamma\langle \vs^+-\vs^*,\vq_{h^*}(\vs^+)-\vq_{h^*}^*+\nabla l^*(\vs)-\nabla l^*(\vs^*)\rangle \\
& + 2\gamma\langle \vx-\vx^*,\vq_g(\vx)-\vq_g^*+\nabla f(\vx)-\nabla f(\vx^*)\rangle \\
= &\textstyle \|(\vz,\vs)-(\vz^*,\vs^*)\|_{\vI,\vM}^2 - \|(\vz^+,\vs^+)-(\vz^*,\vs^*)\|_{\vI,\vM}^2 - \|(\vz,\vs)-(\vz^+,\vs^+)\|_{\vI,\vM}^2 \\
&\textstyle + 2\gamma\langle \vz-\vz^+,\nabla f(\vx)-\nabla f(\vx^*)\rangle.\end{aligned}$$
Rearranging it, we have $$\begin{aligned}
& \|(\vz^+,\vs^+)-(\vz^*,\vs^*)\|_{\vI,\vM}^2 \\
= & \|(\vz,\vs)-(\vz^*,\vs^*)\|_{\vI,\vM}^2 - \|(\vz,\vs)-(\vz^+,\vs^+)\|_{\vI,\vM}^2 + 2\gamma\langle \vz-\vz^+,\nabla f(\vx)-\nabla f(\vx^*)\rangle\\
& - 2\gamma\langle \vs^+-\vs^*,\vq_{h^*}(\vs^+)-\vq_{h^*}^*+\nabla l^*(\vs)-\nabla l^*(\vs^*)\rangle \\
& - 2\gamma\langle \vx-\vx^*,\vq_g(\vx)-\vq_g^*+\nabla f(\vx)-\nabla f(\vx^*)\rangle.\end{aligned}$$
The Cauchy-Schwarz inequality gives us $$\begin{aligned}
2\gamma\langle \vz-\vz^+,\nabla f(\vx)-\nabla f(\vx^*)\rangle \leq \|\vz-\vz^+\|^2+\gamma^2\|\nabla f(\vx)-\nabla f(\vx^*)\|^2,\end{aligned}$$ and the cocoercivity of $\nabla f$ shows $$\begin{aligned}
\textstyle -{\gamma^2\over \beta}\langle \vx-\vx^*,\nabla f(\vx)-\nabla f(\vx^*)\rangle\leq -\gamma^2\|\nabla f(\vx)-\nabla f(\vx^*)\|^2.\end{aligned}$$ Combining the previous two inequalities, we have $$\begin{aligned}
& -\|\vz-\vz^+\|^2 + 2\gamma\langle \vz-\vz^+,\nabla f(\vx)-\nabla f(\vx^*)\rangle\\
& - 2\gamma\langle \vx-\vx^*,\vq_g(\vx)-\vq_g^*+\nabla f(\vx)-\nabla f(\vx^*)\rangle\\
\leq & \textstyle - \left(2\gamma-{\gamma^2\over\beta}\right) \langle \vx-\vx^*,\nabla f(\vx)-\nabla f(\vx^*)\rangle - 2\gamma\langle \vx-\vx^*,\vq_g(\vx)-\vq_g^*\rangle \\
\leq & \textstyle - \left(\left(2\gamma-{\gamma^2\over\beta}\right)\tau_f+2\gamma \tau_g\right) \|\vx-\vx^*\|^2.\end{aligned}$$ Similarly, we derive $$\begin{aligned}
& - \|\vs-\vs^+\|_\vM^2 - 2\gamma\langle \vs^+-\vs^*,\vq_{h^*}(\vs^+)-\vq_{h^*}^*+\nabla l^*(\vs)-\nabla l^*(\vs^*)\rangle\\
= & \textstyle - \|\vs-\vs^+\|_\vM^2- 2\gamma\langle \vs^+-\vs,\nabla l^*(\vs)-\nabla l^*(\vs^*)\rangle \\
& -2\gamma\langle \vs-\vs^*,\nabla l^*(\vs)-\nabla l^*(\vs^*)\rangle - 2\gamma\langle \vs^+-\vs^*,\vq_{h^*}(\vs^+)-\vq_{h^*}^*\rangle \\
\leq & \textstyle - \left(2\gamma-{\gamma^2\over\beta}\right) \langle \vs-\vs^*,\nabla l^*(\vs)-\nabla f(\vs^*)\rangle - 2\gamma\langle \vs^+-\vs^*,\vq_{h^*}(\vs^+)-\vq_{h^*}^*\rangle \\
\leq & \textstyle - \left(2\gamma-{\gamma^2\over\beta}\right)\tau_{l^*} \|\vs-\vs^*\|_\vM^2-2\gamma\tau_{h^*}\|\vs^+-\vs^*\|^2_\vM.\end{aligned}$$
Therefore, we have $$\begin{aligned}
&\textstyle \|(\vz^+,\vs^+)-(\vz^*,\vs^*)\|_{\vI,\vM}^2 \\
\leq &\textstyle \|(\vz,\vs)-(\vz^*,\vs^*)\|_{\vI,\vM}^2 - \left(\left(2\gamma-{\gamma^2\over\beta}\right)\tau_f+2\gamma\tau_g\right)\|\vx-\vx^*\|^2 \\
&\textstyle - \left(2\gamma-{\gamma^2\over\beta}\right)\tau_{l^*}\|\vs-\vs^*\|_\vM^2-2\gamma\tau_{h^*} \|\vs^+-\vs^*\|_\vM^2.\end{aligned}$$ Finally $$\begin{aligned}
&\textstyle \|\vz^+-\vz^*\|^2+\left(1+2\gamma\tau_{h^*}\right)\|\vs^+-\vs^*\|_\vM^2 \\
\leq &\textstyle \|\vz-\vz^*\|^2+\left(1- \left(2\gamma-{\gamma^2\over\beta}\right)\tau_{l^*}\right)\|\vs-\vs^*\|_\vM^2 - \left(\left(2\gamma-{\gamma^2\over\beta}\right)\tau_f+2\gamma\tau_g\right)\|\vx-\vx^*\|^2 \\
\leq &\textstyle \|\vz-\vz^*\|^2+\left(1- \left(2\gamma-{\gamma^2\over\beta}\right)\tau_{l^*}\right)\|\vs-\vs^*\|_\vM^2 - {\left(\left(2\gamma-{\gamma^2\over\beta}\right)\tau_f+2\gamma\tau_g\right)\over 1+\gamma L_g}\|\vz-\vz^*\|^2\\
\leq &\textstyle \rho \left(\|\vz-\vz^*\|^2+(1+2\gamma\tau_{h^*})\|\vs-\vs^*\|_\vM^2\right),\end{aligned}$$ with $\rho$ defined in .
Numerical experiments {#sec:numerical}
=====================
In this section, we compare PD3O with PDFP, AFBA, and Condat-Vu in solving the fused lasso problem . Note that many other algorithms can be applied to solve this problem. For example, the proximal mapping of $\mu_2\|\vD \vx\|_1$ can be computed exactly and quickly [@condat2013direct], and Davis-Yin can be applied directly without using the primal-dual algorithms. Even for primal-dual algorithms, there are accelerated versions available [@chambolle2016ergodic]. However, it is not the focus of this paper to compare PD3O with all existing algorithms for solving the fused lasso problem. The numerical experiment in this section is used to validate and demonstrate the advantages of PD3O over other existing unaccelerated primal-dual algorithms: PDFP, AFBA, and Condat-Vu. More specifically, we would like to show the advantage of using a larger stepsize.
Here, only the number of iterations is recorded because the computational cost for each iteration is very close for all four algorithms. In this example, the proximal mapping of $\mu_1\|\vx\|_1$ is easy to compute, and the additional cost of one proximal mapping in PDFP can be ignored. The code for all the comparisons in this section is available at <http://github.com/mingyan08/PD3O>.
We use the same setting as [@chen2016primal]. Let $n=500$ and $p=10000$. $\vA$ is a random matrix whose elements follow the standard Gaussian distribution, and $\vb$ is obtained by adding independent and identically distributed Gaussian noise with variance 0.01 onto $\vA\vx$. For the parameters, we set $\mu_1=20$ and $\mu_2=200$.
![The comparison of four algorithms (PD3O, PDFP, Condat-Vu, and AFBA) on the fused lasso problem in terms of primal objective values and the distances of $\vx^k$ to the optimal solution $\vx^*$ with respect to iteration numbers. In the top figures, we fix $\lambda=1/8$ and let $\gamma=\beta,~1.5\beta,~1.99\beta$. In the bottom figures, we fix $\gamma=1.9\beta$ and let $\lambda = 1/80,~1/8,~1/4$. PD3O and PDFP perform better than Condat-Vu and AFBA because they have larger ranges for acceptable parameters and choosing large numbers for both parameters makes the algorithm converge fast. Note PD3O has a slightly lower per-iteration complexity than PDFP.[]{data-label="fig:flasso"}](fig/flasso_energy_1 "fig:"){width="49.00000%"}![The comparison of four algorithms (PD3O, PDFP, Condat-Vu, and AFBA) on the fused lasso problem in terms of primal objective values and the distances of $\vx^k$ to the optimal solution $\vx^*$ with respect to iteration numbers. In the top figures, we fix $\lambda=1/8$ and let $\gamma=\beta,~1.5\beta,~1.99\beta$. In the bottom figures, we fix $\gamma=1.9\beta$ and let $\lambda = 1/80,~1/8,~1/4$. PD3O and PDFP perform better than Condat-Vu and AFBA because they have larger ranges for acceptable parameters and choosing large numbers for both parameters makes the algorithm converge fast. Note PD3O has a slightly lower per-iteration complexity than PDFP.[]{data-label="fig:flasso"}](fig/flasso_LS_1 "fig:"){width="49.00000%"} ![The comparison of four algorithms (PD3O, PDFP, Condat-Vu, and AFBA) on the fused lasso problem in terms of primal objective values and the distances of $\vx^k$ to the optimal solution $\vx^*$ with respect to iteration numbers. In the top figures, we fix $\lambda=1/8$ and let $\gamma=\beta,~1.5\beta,~1.99\beta$. In the bottom figures, we fix $\gamma=1.9\beta$ and let $\lambda = 1/80,~1/8,~1/4$. PD3O and PDFP perform better than Condat-Vu and AFBA because they have larger ranges for acceptable parameters and choosing large numbers for both parameters makes the algorithm converge fast. Note PD3O has a slightly lower per-iteration complexity than PDFP.[]{data-label="fig:flasso"}](fig/flasso_energy_2 "fig:"){width="49.00000%"}![The comparison of four algorithms (PD3O, PDFP, Condat-Vu, and AFBA) on the fused lasso problem in terms of primal objective values and the distances of $\vx^k$ to the optimal solution $\vx^*$ with respect to iteration numbers. In the top figures, we fix $\lambda=1/8$ and let $\gamma=\beta,~1.5\beta,~1.99\beta$. In the bottom figures, we fix $\gamma=1.9\beta$ and let $\lambda = 1/80,~1/8,~1/4$. PD3O and PDFP perform better than Condat-Vu and AFBA because they have larger ranges for acceptable parameters and choosing large numbers for both parameters makes the algorithm converge fast. Note PD3O has a slightly lower per-iteration complexity than PDFP.[]{data-label="fig:flasso"}](fig/flasso_LS_2 "fig:"){width="49.00000%"}
We would like to compare the four algorithms with different parameters, and the result may guide us in choosing parameters for these algorithms in other applications. Here we let $\lambda=\gamma\delta$. Then recall that the parameters for Condat-Vu and AFBA have to satisfy $\lambda \left(2-2\cos ({p-1\over p}\pi)\right)+\gamma/(2\beta) \leq 1$ and $\lambda \left(1-\cos ({p-1\over p}\pi)\right)+\sqrt{\lambda (1-\cos ({p-1\over p}\pi))}/2+\gamma/(2\beta) \leq 1$, respectively, and those for PD3O and PDFP have to satisfy $\lambda \left(2-2\cos ({p-1\over p}\pi)\right) \leq 1$ and $\gamma< 2\beta$. Firstly, we fix $\lambda=1/8$ and let $\gamma=\beta,~1.5\beta,~1.99\beta$. For Condat-Vu and AFBA, we only show $\gamma=\beta$ because $\gamma=1.5\beta$ and $1.99\beta$ do not satisfy their convergence conditions. The primal objective values and the distances of $\vx^k$ to the optimal solution $\vx^*$ for these algorithms after each iteration are compared in Fig. \[fig:flasso\] (Top). The optimal objective value $f^*$ is obtained by running PD3O for 20,000 iterations. The results show that all four algorithms have very close performance when they converge ($\gamma=\beta$). Both PD3O and PDFP converge faster with a larger stepsize $\gamma$. In addition, the figure shows that the speed of all four algorithms almost linearly depends on the primal stepsize $\gamma$, i.e., increasing $\gamma$ by two reduces the number of iterations by half.
Then we fix $\gamma=1.9\beta$ and let $\lambda = 1/80,~1/8,~1/4$. The objective values and the distances to the optimal solution for these algorithms after each iteration are compared in Fig. \[fig:flasso\] (Bottom). Again, we can see that the performances for these three algorithms are very close when they converge ($\lambda=1/80$), and PD3O is slightly better than PDFP in terms of the number of iterations and the per-iteration complexity. However, when $\lambda$ changes from $1/8$ to $1/4$, there is no improvement in the convergence, and when the number of iterations is more than 2000, the performance of $\lambda=1/4$ is even worse than that of $\lambda=1/8$. This result also suggests that it is better to choose a slightly large $\lambda$ and the increase in $\lambda$ does not bring too much advantage if $\lambda$ is large enough (at least in this experiment). Both experiments demonstrate the effectiveness of having a large range of acceptable parameters.
Conclusion {#sec:conclusion}
==========
In this paper, we proposed a primal-dual three-operator splitting scheme PD3O for minimizing $f(\vx)+g(\vx)+h\square l(\vA\vx)$. It has primal-dual algorithms Chambolle-Pock and PAPC for minimizing the sum of two functions as special cases. Comparing to the three existing primal-dual algorithms PDFP, AFBA, and Condat-Vu for minimizing the sum of three functions, PD3O has the advantages from all these algorithms: a low per-iteration complexity and a large range of acceptable parameters ensuring the convergence. In addition, PD3O reduces to Davis-Yin when $\vA=\vI$ and special parameters are chosen. The numerical experiments show the effectiveness and efficiency of PD3O. We left the acceleration and other modifications of PD3O as future work.
Acknowledgments. {#acknowledgments. .unnumbered}
================
We would like to thank the anonymous reviewers for their helpful comments.
[^1]: The work was supported in part by the NSF grant DMS-1621798.
[^2]: The conjugate function $l^*$ of $l$ is defined as $l^*(\vs)=\sup_{\vt}\langle\vs,\vt\rangle-l(\vt)$. When $l(\vs)=\iota_{\{\vzero\}}(\vs)$, we have $l^*(\vs)=0$. The conjugate function of $h\square l$ is $(h\square l)^*=h^*+l^*$.
[^3]: The adjoint operator $\vA^\top$ is defined by $\langle \vs,\vA\vx\rangle_\cS = \langle \vA^\top\vs,\vx\rangle_\cX$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We explore the behavior of collective nuclear excitations under a multi-parameter deformation of the Hamiltonian. The Hamiltonian matrix elements have the form $P(|H_{ij}|)\propto 1/\sqrt{|H_{ij}|}\exp(-|H_{ij}|/V)$, with a parametric correlation of the type $\log \langle H(x)H(y)\rangle\propto
-|x-y|$. The studies are done in both the regular and chaotic regimes of the Hamiltonian. Model independent predictions for a wide variety of correlation functions and distributions which depend on wavefunctions and energies are found from parametric random matrix theory and are compared to the nuclear excitations. We find that our universal predictions are observed in the nuclear states. Being a multi-parameter theory, we consider general paths in parameter space and find that universality can be effected by the topology of the parameter space. Specifically, Berry’s phase can modify short distance correlations, breaking certain universal predictions.
---
23.5cm 1ex -40pt = .5ex
YCTP-N11-95\
June 1995
[Universal Predictions for Statistical\
Nuclear Correlations]{}
[Dimitri Kusnezov[^1] and David Mitchell]{}\
\
[*Center for Theoretical Physics, Sloan Physics Laboratory,*]{}\
[*Yale University, New Haven, CT 06520-8120 USA*]{}
1.5 cm
[**PACS numbers:**]{} 21.60.Fw, 24.60.Lz, 21.10.Re
Introduction
============
The statistics of nuclear excitations has been explored from the shell model to collective models, with studies ranging from the relation of observed quantum fluctuations to those in random matrix models, to the connection with chaos using classical limits of the Hamiltonian[@general]-[@shriner]. The agreement of various spectral properties with random matrix predictions has shown that certain simplifying assumptions can be made concerning fluctuations in nuclei. Once random matrix theory can be justified, certain results follow immediately. These studies of chaos in nuclei stem from attempts to extract a simplified behavior from the complexity of nuclear excitations. In this respect, random matrix theory has provided invaluable assistance in developing simple methods to compute complex behaviors. In the past, aside from the studies of constant random matrices and the relation to chaos, these models have been given a parameter dependence to model correlations in various nuclear systems, from heavy ion collisions[@weid], high spin physics[@aberg] to large amplitude collective motion[@aurel]. Recently it has been shown that Hamiltonians which have a parametric dependence can exhibit universal behavior[@SA], that is, there can exist model independent quantities in a given theory, providing the Hamiltonian has certain random matrix properties. In this article we study a wide class of observables and develop universal predictions. We further show that parametric deformations of nuclear Hamiltonians can be readily modeled by a simple translationally invariant parametric random matrix theory, even though the Hamiltonian does not apriori look like a random matrix. We further justify the use of parametric random matrix theory for collective nuclear excitations.
Collective Nuclear States
=========================
We have chosen to model collective nuclear excitations in the framework of the Interacting Boson Model (IBM)[@franco] for several reasons. One of our main objectives is to explore and categorize types of model independent predictions that exist in parametric quantum theories which exhibit classical chaos. The IBM is ideally suited for this since the classical limit has been extensively studied in recent years using coherent states [@Joe], and the complete chaotic behavior is now known for every value of the parameters[@Niall]. Hence we can easily choose parametric variations in regions of strong or weak chaos, or in regular regimes of the parameter space. An additional advantage is that we can solve the quantum problem exactly. One might argue that collective states form only a subset of the real spectrum as the excitation energy increases, so that the use of the IBM is not reasonable. This is not crucial, however, since the IBM provides a solvable theory with known spectral properties, which can be compared to those of the Gaussian Orthogonal Ensemble (GOE) throughout its parameter range. Certainly a more realistic description of the spectrum would embody the same features. For example, when broken pair states are added to the IBM model space, the spectrum becomes more GOE, as the interactions in the Hamiltonian become more complicated[@francoa]. This is certainly the case as one attempts to construct more realistic Hamiltonians. And as we are showing how [*model independent*]{} quantities emerge, the model we use is really not so important. Hence we use a simple form of the IBM Hamiltonian, known as the consistent-Q form: $$\label{eq:qham}
\hat{H}= E_0 + c_1\hat{n}_{d} + c_2 {\bf \hat{Q}}^{\chi}\cdot
{\bf \hat{Q}}^{\chi} + c_3 {\bf \hat{L}}\cdot {\bf \hat{L}},$$ where $$\hat{n}_{d}=d^\dagger\cdot\tilde d\; , \qquad \hat L_\mu =
\sqrt{10}[d^\dagger\times\tilde d]^{(1)}_\mu\; ,\qquad
\hat{Q}^{\chi}_{\mu}=d_{\mu}^{\dagger}s+s^{\dagger}\tilde{d}_{\mu}+\chi
[d^{\dagger}\times \tilde{d}]^{(2)}_{\mu}.$$ The parameters $c_i$ are defined by $c_1=\eta/4$ and $c_2=(1-\eta)/4N_b$, where $N_b$ is the number of bosons. Since the Hamiltonian is diagonalized in a basis of fixed angular momentum $L$, the constant $c_3$ does not play any role, and is hence omitted. Except when stated otherwise, we will use $N_b=25$, which will give optimal statistics for the quantities we consider. The resulting dimensions for $J^\pi=0^+,2^+,4^+,10^+$ states are 65,117,165,211. In this parameterization, one has the following limits: (a) $\eta=1$ corresponds to vibrational or $U(5)$ nuclei, (b) $\eta=0$ and $\chi=-\sqrt{7}/2$ corresponds to rotational or $SU(3)$ nuclei, and (c) $\eta=\chi=0$ describes $\gamma-$soft or $O(6)$ nuclei.
The interpretation of the Hamiltonian in terms of shape variables $\beta$ and $\gamma$ is possible using coherent states, in the large $N$ limit of $H$. The energy surfaces for the Hamiltonian in Eq. (1) is [@Joe] $${\cal E}(\beta,\gamma;\eta,\chi) = \beta^2\frac{4-3\eta}{2} +
\beta^4(1-\eta)(\frac{\chi^2}{14}-1) + \beta^3\cos
3\gamma\sqrt{1-\frac{\beta^2}{2}}(1-\eta)\frac{2\chi}{\sqrt{7}}.$$ For a particular value of $\eta$ and $\chi$, the energy ${\cal E}$ can be minimized to determine the quantities $\beta$ and $\gamma$. $\beta$ and $\gamma$ in turn define a deformed nuclear mean field. This can be made explicit by re-expressing the Hamiltonian in terms of excitations in a deformed mean field using boson condensate techniques[@ami]. This allows the interpretation of correlations of observables at different values of $\eta$ and $\chi$ in terms of the shape variable $\beta$ and $\gamma$. Correlations in observables at different values of the parameters are then precisely the correlations between properties of the nucleus in the presence of different mean field configurations. We will consider the behavior of the properties of the Hamiltonian under very general parametric deformation $z=z(\eta,\chi)$. For paths which lie entirely within the chaotic regime of the parameter space, the universal predictions we explore are path independent (up to effects due to Berry’s phase which we explore in Sec. 5); correlations in a nucleus changing from rotational to vibrational or vibrational to $\gamma-$soft are the same when properly interpreted.
Distributions and Correlations of Nuclear Matrix Elements
---------------------------------------------------------
One of the results presented in this article is that parametric nuclear Hamiltonians can be modelled by correlated, parametric gaussian random matrices. Recall that a gaussian random matrix has a distribution of matrix elements of the gaussian form $P(H_{ij})\propto
\exp(-H_{ij}^2/2\gamma(1+\delta_{ij}))$, where $\gamma$ a constant related to the level density. To implement random matrix theory does not imply that the actual nuclear Hamiltonian (1) have gaussian matrix elements. We note that the distributions of matrix elements of the interacting boson model Hamiltonian are not gaussian. At any given value of $(\eta,\chi)$, we find the distribution of matrix elements obeys roughly[@flam] $$P_{ibm}(|H_{ij}|) \propto \frac{1}{\sqrt{|H_{ij}|}} e^{-|H_{ij}|/V}$$ where the strength $V$ depends on whether one is in a chaotic or regular regime. Typical results are shown in Fig. 1 for both regular (crosses) and chaotic (boxes) choices of the parameters, together with the behavior (4) (solid curves). In the chaotic parameter regimes of the model, $V$ is of order unity, while in the regular regions, it is much smaller. But both regular and chaotic regimes display the same functional form of the distribution, suggesting that the functional form is due to the structure of the Hamiltonian, rather than to the presence of chaos. Similar distribution functions have been seen in parity non-conservation studies of the compound nucleus[@flam].
Another quantity of interest is the autocorrelation function $F_{ibm}$ computed from the IBM Hamiltonian: $$F_{ibm}(z-z') \equiv \left\langle H(z)H(z')\right\rangle =
\left\langle\frac{1}{N(N-1)}\sum_{i<j}
H_{ij}(z)H_{ij}(z')\right\rangle_{z-z'}.$$ The brackets $\langle \cdots\rangle_{z-z'}$ represent the averaging over a trajectory in parameter space $z(\eta,\chi)$, which remains either in a chaotic or in a regular region, keeping the difference $z-z'$ fixed. The results for the short distance behavior of the measured function $F_{ibm}$ are shown in Fig. 2, and are found to behave generically as $$F_{ibm}(z) \sim e^{-\gamma |z|} \sim 1 - \gamma |z| + \cdots$$ in both regular and chaotic regions. Again, the measured value of $F_{ibm}$ is not a good measure of the underlying chaos. If the averaging in (5) is restricted to a submatrix $N_1 \leq i,j\leq N_2$, there is no difference in the function $F_{ibm}$. One observation is that the apparent decorrelation, seen in the slow decay of $F_{ibm}$, is misleading. The actual decorrelation is much more rapid (when model specific dependencies are removed), as we will see below when we compute properties of wavefunctions and eigenvalues. Hence using Eq. (5) as actual input into a random matrix theory (e.g. into Eq. (16) below) is not meaningful.
Unfolded Parametric Energies
----------------------------
To study the statistical fluctuations of the nuclear energy levels, $E_n(z)$, where $z=z(\eta,\chi)$ is a general path in configuration space, we must separate out the average behavior of the energies which cannot be described by random matrix theory. This is done with the staircase function[@general] $$N(E;z)= {\rm Tr}\theta(E-\hat H)=\sum_n \theta[E-E_n(z)]$$ This function is computed along various paths in parameter space. On each path, 100-200 values of $z$ are taken, and the eigenvalues $\{E_n(z)\}$ determined. From this data, a polynomial fit is made to the staircase function using: $$N(E;z) = \sum_{i=0}^{k}\sum_{j=0}^{6} C_{ij} z^i E^j.$$ In the chaotic regions of parameter space, $k=2$ is sufficient, whereas higher values are needed is less chaotic regions. Once the coefficients $C_{ij}$ are determined, the [*unfolded energies*]{} are defined by $$\widetilde E_n(z) = N(E_n;z)$$ which results in a spectrum with a constant average level spacing. The degree of chaos in the energies can be measured through the Brody distribution of the level spacings $s= \widetilde E_n(z) - \widetilde E_{n-1}(z)$[@brody]: $$P(s) = A s^\omega e^{-\alpha s^{\omega+1}},\qquad\qquad
A=(1+\omega)\alpha,\quad \alpha=\left[\Gamma
\left(\frac{2+\omega}{1+\omega}\right)\right]^{1+\omega}.$$ When $\omega=1$, the distribution of level spacings is GOE, while for $\omega=0$, it is Poisson. In Fig. 3, we show the original (top) and unfolded (middle) parametric energies of the Hamiltonian for $\eta=0$ and $\chi$ as shown. The degree of ‘chaos’, measured in terms of the Brody parameter $\omega$, is shown in the bottom of the figure for this particular path.
For purposes of contrast, the parametric levels (not unfolded) in two regular regions are shown in Fig. 4. The right figure corresponds to the (regular) transition from $U(5)$ to $O(6)$, while the left corresponds to a path through the ‘valley of regularity’ recently studied[@Niall] (the two kinks in the parametric energies are artificial and only reflect the fact that the parameter path took a turn). In the figure, the average Brody parameters $\overline{\omega}$ are 0.25 and 0.23, respectively, with fluctuations up to $\omega=1.$ In the following studies, we will consider regular and chaotic regions of the parameter space. As chaos is a classical notion and we study quantum statistics here, let us be precise. The chaotic regions are general paths in parameter space $z(\eta,\chi)$ which stay in the areas of chaos in the classical limit of the Hamiltonian, as discussed in Ref. [@Niall]. The chaotic paths studied here are largely: $$z(\eta,\chi) = \left\{\begin{array}{l}
\eta=0,\quad \chi\in [-0.9,-0.4]\\
\eta=0.1,\quad \chi\in [-0.9,-0.4]\\
\chi=-0.66,\quad \eta\in [0.0,0.5]
\end{array}\right.$$ Similarly, the regular regions we study are: $$\label{eq:pathr}
z(\eta,\chi) = \left\{\begin{array}{ll}
\chi=-\frac{\sqrt{7}}{4}\eta\qquad \eta\in[0,1] & O(6) - U(5)\\
\chi=-0.661,\quad \eta\in [0.5,1.0] & \\
\end{array}\right.$$ Also included in the regular region is the path through the valley of regularity from $SU(3)$ to $U(5)$, as shown in the left half of Fig. 4. The first path in Eq. (\[eq:pathr\]) corresponds to the right half of Fig. 4, is not the direct path between $U(5)$ and $O(6)$ which would be $\chi=0$, $\eta\in
[0,1]$, and is entirely integrable. Rather we have chosen one that passes through a weakly chaotic regime. Hence, results in the following sections referred to as ‘regular’ are quantities which are averaged in the highest $\omega$ regions of these regular areas, such as $z\in[0.5,0.8]$ in Fig. 4.
Correlated Random Matrix Ensembles
==================================
In order to see what types of model independent quantities emerge from the IBM, we must construct a random matrix model which has an equivalent parametric dependence. There is no unique method to realize such an ensemble, for instance one might take: $$\begin{aligned}
H'(z) & = & H_1 + z H_2\\\label{eq:simple}
H''(z) & = & \sin z H_1 + \cos z H_2\\\label{eq:cosine}
H'''(z) & = & \int dy f(z-y) V(y),\qquad \overline{V(z)V(y)}=\delta(z-y)
.\label{eq:stoch}\end{aligned}$$ Here $H_1$, $H_2$ are constant $N\times N$ gaussian random matrices and $V_{ij}(y)$ is gaussian white noise for each $i,j$ and $y$. Each of these is a viable random matrix theory, however the stochastic integral of Eq. (\[eq:stoch\]), introduced by Wilkinson[@wilk], provides a more general framework and includes a broader class of processes[@caio]. One additional difference between $H'$ and $H''$, $H'''$ is that the former is not a translationally invariant theory. While translational invariance is not important to the results we derive here, its presence simplifies our constructions. We will only focus on constructions of the type (14)-(15) here.
The gaussian random matrix Hamiltonians are characterized by their first and second cumulants: $$\begin{aligned}
\label{eq:aver}
\overline {H_{ij} (z) } & = & 0 \nonumber \\
\overline{ H_{ij} (z) H_{kl} (z^\prime) } & = & {a^2 \over {2\nu}}
F(z-z^\prime) g^{(\nu)}_{ij,kl} \;,\end{aligned}$$ where $g^{(\nu =1)}_{ij,kl} = \delta_{ik} \delta_{jl} + \delta_{il}
\delta_{jk}$, $g^{(\nu =2)}_{ij,kl} = 2 \delta_{il} \delta_{jk}$, and $a$ determines the average level spacing $\Delta$ through the relation $a/\Delta =
\sqrt{2N}/ \pi$. Here $\nu=1$ (GOE) corresponds to real-symmetric matrices, or equivalently, to a system with time reversal symmetry, and $\nu=2$ (GUE) to complex hermitian matrices, or broken time reversal symmetry. From the definitions of $H(z)$, it is clear that $H(z)$ is GOE $(\nu=1)$ or GUE $(\nu=2)$ for any $z$.
In contrast to previous studies of chaos in nuclei, which deal with constant random matrices, we can introduce a measure for the parametric ensemble rather easily[@caio]: $$P\left[ H(z)\right] \propto
\exp \left\{-{\nu \over {2a^2}}\int dz dz^\prime
{\rm Tr} \left[ H(z) K(z-z^\prime) H(z^\prime) \right] \right\} \;,$$ where the measure $D\left[ H(z)\right] \equiv \prod_z{dH(z)} $ is a product over the continuous variable $z$ of the corresponding gaussian ensemble measure $dH(z)$. Here $K_{ij}(z)$ can be viewed in general as a banded matrix of bandwidth $\sigma$, connecting states $i$ and $j$ with $|i-j|\leq \sigma$. As we do not consider banded parametric matrices here, we will take $K_{ij}(z)=\delta_{ij} K(z)$, resulting in the measure: $$P\left[ H(z)\right] \propto
\exp \left\{-{\nu \over {2a^2}}\int dz dz^\prime K(z-z^\prime)
{\rm Tr} \left[ H(z) H(z^\prime) \right] \right\} \;,$$ These gaussian integrals are easily done to establish Eq. (\[eq:aver\]), providing $F$ is the inverse of $K$, $$\int dz^\prime K(z-z^\prime)F(z^\prime - z^{\prime\prime})
= \delta(z-z^{\prime\prime}).$$ The stochastic integral (\[eq:stoch\]) provides a direct method for constructing $H(z)$ with a desired $F(z)$. That is, we can choose $f$ to satisfy $$F(z-y) = \int dx f(z-x) f(y-x),$$ then $H$ is constructed as in (\[eq:stoch\]), and the desired covariance (\[eq:aver\]) is automatically satisfied. It is important to realize that the properties of the random matrix theory here are distinct from the observed properties of the Hamiltonian, both in the measured distributions of matrix elements $P_{ibm}(H_{ij})$ and their autocorrelation $F_{ibm}(z)$. The random matrix distribution of matrix elements is gaussian, and $F(z)$ is different from $F_{ibm}$, as we discuss below. In particular, $F(z)$ cannot be exponential as measured. Model independent results can be obtained from our random matrix constructions by a proper scaling of parameters. We discuss two approaches here to this scaling. The first is a general procedure based on Ref. [@caio], while the second is a more heuristic argument based on the Fokker-Planck equation and the original work of Dyson, and has been pursued in recent works[@dyson; @fokker; @walk].
Scaling from Anomalous Diffusion
--------------------------------
It was shown recently that universal (model independent) predictions can be obtained from the above translationally invariant random matrix theories if one introduces a proper scaling[@caio]. A general approach to do so is to view the parametric dependence of the energies as a diffusion process. Consider first the short distance behavior of the function $F(z)$: $$\label{eq:falpha}
F(z) \approx 1 - c_\alpha |z|^\alpha + \cdots\qquad .$$ From perturbation theory, one can see that $$\begin{aligned}
\label{eq:diff}
\delta E_n(x)&=& E_n(x')-E_n(x) = \delta H_{nn}
+ \sum_{m\not= n} \frac{|\delta H_{mn}|^2} {E_n-E_m} + ...\\
\mid\langle\Psi_n (x')|\Psi_n (x)\rangle\mid^2 &=&
1-\sum_{m\not= n}\frac{|\delta H_{mn}|^2}{(E_n-E_m)^2} + ... \, .\end{aligned}$$ By implementing the ensemble averages defined in (16), and following Dyson[@dyson], one easily finds that: $$\label{eq:ediff}
\overline{(\delta E_n(z))^2} \simeq \frac{4Nc_\alpha}{\nu\pi^2}\delta
z^\alpha \equiv D_\alpha \delta z^\alpha\quad,$$ and similarly $1-\overline{\mid\langle\Psi_n(z)\mid\Psi_n(z')\rangle\mid^2}\propto\delta
z^\alpha$. One can then view the parametric energy levels $\widetilde{E_n}(z)$, such as those in Fig. 2 (middle), as evolving diffusively on short distance scales according to Eq. (\[eq:ediff\]). This has been recently contrasted with the anomalous diffusion process of a particle in a chaotic or disordered medium, whose position obeys $<R^2(t)> = Dt^\alpha$, which although the physics is distinct, the formal treatment is similar[@caio]. For our random matrix model, the parameter $z(\eta,\chi)$ plays the role of time. The diffusion constant $D_\alpha$ contains both dimensional information ($N$) and model dependent data $(c_\alpha)$. Hence, by scaling the parameter $z$ by the diffusion constant $D$, all model and dimension dependence is removed. This is done by defining a new scaled parameter $$\widetilde{z} = [D_\alpha]^{1/\alpha} z
= \left(\frac{4Nc_\alpha}{\nu\pi^2}\right)^{1/\alpha} z.$$ For the case of $\alpha=2$, we have $D_2=C(0)$, where $C(0)$ is the scaling introduced in Ref. [@SA]. By computing observables in the scaled variable, one obtains model-independent predictions for desired quantities. Physically, $\alpha=2$ in Eq. (\[eq:falpha\]) corresponds to a Hamiltonian with a smooth dependence on the parameter $z$, while $\alpha<2$ corresponds to a theory with fractal parametric dependence, such as a parameter range taken from a Brownian trajectory. The computation of $D_\alpha$ is done through the definition (\[eq:ediff\]): $$D_\alpha = \overline{\frac{(\delta E_n(z))^2}{\delta z^\alpha}},\qquad\qquad
C(0)\equiv D_2 = \overline{\frac{(\partial E_n(z))^2}{\partial z^2}}.$$ In order to model the IBM Hamiltonian using parametric random matrix theory, we must choose a correlator $F(z)$ in Eq. (\[eq:aver\]) with $\alpha=2$ short distance behavior (see Eq. (\[eq:falpha\])). Otherwise the parametric dependence of the random matrix energies $E_n(z)$ would not be smooth. Hence, if we attempt to incorporate nuclear properties into the random matrix model by substituting the model specific, computed $F_{ibm}(z)$ in Eq. (6) into Eq. (\[eq:aver\]), we would end up with an energy spectrum $E_n(z)$ characterized by $\alpha=1$, resulting in non-smooth, brownian motion type paths for each $E_n(z)$. For the IBM, $\alpha=2$ is the proper result. Details can be found in Ref. [@caio].
Scaling and the Fokker-Planck Equation
--------------------------------------
A more heuristic argument can also be made with the Fokker-Planck equation. Fokker-Planck methods, first introduced by Dyson[@dyson], have been recently discussed with parametric correlations in mind[@fokker]-[@walk]. Consider first Dyson’s Brownian motion model for random matrices: $$\label{4}
\dot{H}_{ij} = - \gamma H_{ij} + f_{ij}(t) \quad .$$ The random force is white noise: $$\begin{aligned}
\label{5}
\overline{ f_{ij} (t) } & = & 0 \nonumber \\
\overline{ f_{ij}(t) f^*_{kl} (t') } & = & \Gamma g^{(\nu)}_{ij,kl}
\delta (t - t') \;.\end{aligned}$$ and $\gamma$ is a friction coefficient. The equilibrium solution can be found in the long time limit, by direct integration: $$\label{eq:integ}
H_{ij}(t) = \int^t e^{-\gamma|t-\tau|}f_{ij}(\tau)d\tau$$ which is same type of stochastic formulation as in Eq. (\[eq:stoch\]). It follows then from Eqs. (\[4\],\[5\],\[eq:integ\]) that: $$\label{eq:ave}
\left\langle H_{ij}(t)H_{kl}(t')\right\rangle
= \frac{\Gamma}{2\gamma} g^{(\nu)}_{ij,kl} e^{-\gamma |t-t'|}$$ This process can be formulated as well in terms of the Fokker-Planck equation for the distribution $P(H,t)$: $$\label{6}
\frac{\partial P}{\partial t} = \frac{\partial}{\partial H_{ij}}
(\gamma H_{ij} P) +
\frac{1}{2} g^{(\nu)}_{ij,ji} \Gamma \frac{\partial^2 P}
{\partial H_{ij} \partial H_{ij}^*} \;.$$ Since we are interested in a stationary process $P(H,t)=P(H)$ for $H(t)$, we can choose the initial distribution to be the equilibrium result $P(H)\propto \exp [- \nu {\rm Tr}H^2/2 a^2 ] $. The equilibrium solution is a solution of the Fokker-Planck equation providing the fluctuation dissipation theorem is satisfied:
$$g^{(\nu)}_{ij,ji}\Gamma/2 \gamma = \overline{\mid H_{ij} \mid^2 } =
\frac{a^2}{2\nu} g^{(\nu)}_{ij,ji}$$
or $a^2/\nu = \Gamma/\gamma$, where the last equality follows from our original construction in Eq. (\[eq:aver\]). Following Dyson’s argument that the eigenvalues of $H(t)$ behave as a diffusive coulomb gas, equilibrating (at the microscopic scale) on a timescale of $t\propto 1/(\gamma N)$, we see that in order to obtain $N-$independent correlations, we must have $\gamma\propto
1/N$, or $\gamma=\gamma'/N$. By equating the second cumulants of $H(t)$ for and Langevin process (\[eq:ave\]) and for our desired process (\[eq:aver\]), and implementing the fluctuation-dissipation theorem, we equate $$\label{eq:compare}
F(z-z^\prime) \leftrightarrow \exp{ \left( - \frac{\gamma ^\prime}{N}
\mid t-t^\prime \mid \right) } \;.$$ Expanding to leading order in the large $N$ limit, we associate: $$1-c|z|^n\simeq 1-\frac{\gamma'}{N}|t|$$ As the Langevin result is $N-$independent at the microscopic scale, $N$ independent results will also result if we define a new quantity $\hat{z}=(c/N)^{1/n} z$. Up to a factor of order unity ($4/\pi^2\nu$), this is precisely the same scaling we discussed earlier. Now to leading order, $$\label{12}
F\approx 1 - \nu \frac{\pi^2}{4} \frac{\mid\widetilde{z} -
\widetilde{z}^\prime \mid^n }{N} \;.$$ There is now no explicit model dependence $c$, and the $N-$dependence of correlation functions will be absent due to the explicit dependence on $N$, which was require to achieve the microscopic equilibrium condition discussed above.
Observables
===========
In this section we will see that many properties of parametric Hamiltonians have well defined model independent structure. There are two classes of observables we study here, those related to the energies $E_n(z)$ and those related to the instantaneous eigenfunctions $\mid\Psi_n(z)\rangle$. The relation of wavefunctions $\mid\Psi_n(z)\rangle$ at $z$ to those at $z'$ is given by the transformation matrix: $$U_{nm}(z-z') = \langle\Psi_n(z)\mid\Psi_m(z')\rangle$$ We will see below that correlation functions that depend on $U_{nm}(z)$ and $E_n(z)$ have universal predictions which generally agree well with the results of the IBM in the chaotic regions.
The universal predictions for an $\alpha=2$, GOE system are computed here with two different covariances. The first is the simple sum of two uncorrelated GOE matrices $$\label{eq:fone}
H'(z) = H_1 \cos{z} + H_2 \sin{z},\qquad
\overline{H'(z)H'(y)}=\cos(z-y)=F'(z-y).$$ The correlator is periodic with $F'(z)\approx 1-z^2/2 + \cdots$ , defining the scaling $z \rightarrow \widetilde{z}=\sqrt{D_2}z=(\sqrt{2N}/\pi)
z$ for universal correlations. The second construction is in terms of a stochastic integral, where one integrates over a continuous range of uncorrelated random matrices $V(y)$:
$$\label{eq:ftwo}
H''(z) = \int dy e^{-(z-y)^2/2} V(y),\qquad
\overline{H''(z)H''(y)}=e^{-(z-y)^2/4}=F''(z-y)$$
Here $\widetilde{z}=(\sqrt{N}/\pi)z$. We have computed various correlation functions described below with $N=50-300$. Generally $N=50$ is sufficiently large.
C(z)
----
The slope-slope correlation function of the unfolded parametric energies $\widetilde{E_n}(z)$ is defined as[@SA] $$C(z-z') = \left\langle \frac{\partial \widetilde{E_i}(z)} {\partial z}
\frac{ \partial \widetilde{E_i}(z')}{\partial z}\right\rangle_{E,z}$$ where the averaging is over energy and parameter. In Fig.5, the results for the IBM in the chaotic regions are shown for $J^\pi=0^+,2^+,4^+,10^+$. The averages are computed by averaging over the middle third of the spectrum and over the trajectory $z$. The two solid lines are the results of the random matrix simulations, Eqs. (\[eq:fone\]-\[eq:ftwo\]), for $N=50$ and 300. For comparison, a computation in the regular region of the IBM is shown indicating much slower decorrelation. As we will see in all computations here, the typical distance at which quantities decorrelate is $\widetilde{z}\sim 1$, which corresponds to the average separation between level crossings when the energies $\widetilde E_n$ are plotted as a function of the scaled parameter $\widetilde
z$. For regular systems, and the apparent level crossings in Fig. 4, this is not the case, and decorrelation happens over a much longer scale. In general the agreement with the universal functions is quite good. The $0^+$ states have the poorest agreement, which is in part statistical, having the smallest dimension.
N-Scaling of C(0)
-----------------
We have seen that the diffusion constant $C(0)=D_2$ scales linearly with $N$, the dimension of the space. This scaling can be tested in the IBM by modifying the boson number $N_b$. For $N_b=10,15,20,25$ we have dimensions of $J=10^+$ states of $N=16,56,121,211$. In Fig. 6 we plot $C(0)$ as a function of $N$, and see that this scaling is observed. At low $N_b$ (equivalently low $N$), the results are not as reliable due to statistics becoming increasingly poor, and the classical phase space becoming increasingly regular[@Niall].
Curvature Distribution P(k)
---------------------------
The distribution of curvatures of the parametric energies $E_n(z)$, or equally $\widetilde{E_n}(z)$, have a predicted distribution in the chaotic regime, given by[@delande]: $$P(k) = \frac{c_\nu}{(1+k^2)^{(\nu/2+1)}},\qquad\qquad k =
\frac{d^2 \widetilde{E_n}}{dz^2}\frac{1}{\pi\nu C_0}.$$ Here $c_\nu$ is the normalization, $\nu=1(2)$ for GOE(GUE), and $k$ is the scaled curvature of the parametric energy. This function is compared to our random matrix computation (Eq. (\[eq:ftwo\])) in Fig.7 (a). In Fig. 7 (b), the results for the chaotic region of the IBM (solid histogram) are seen to agree equally well. A similar calculation done in the regular region shows a much more strongly peaked function (dashed, and scaled by 1/5 vertically). This is expected since the regular regions have much fewer level crossings and are hence much flatter (see Fig. 4).
Diagonal Wavefunction Decorrelations: $P_{n}(|U_{nn}(z)|^2)$
------------------------------------------------------------
The adiabatic survival probability $|\langle\Psi_n(z)\mid\Psi_n(0)\rangle|^2$ measures how rapidly wavefunctions decorrelate. This was shown to be universal recently[@caio; @david], with a well defined Lorentzian shape: $$\label{eq:psis}
P_n(|U_{nn}(z)|^2) = \overline{|\langle \Psi_n(z)|\Psi_n(y)\rangle|^2}
= \left(\frac{1}{1+c|\widetilde{z}- \widetilde{y}|^\alpha}\right)^\nu$$ where $\nu=1(2)$ corresponds to GOE(GUE) eigenstates, $c$ is a constant, and $\alpha$ is given by the leading order behavior of $F(z)$. For the case at hand, $\nu=1$ and $\alpha=2$. In Fig. 8, comparisons are shown for selected spins, and in chaotic (symbols) and regular (dashed) regions. The regular regions indicate much longer correlations, while the GOE result provides the most rapid statistical decorrelation of states. The two solid lines are the random matrix predictions. There is no equivalent universal prediction for the non-chaotic regimes, and the dashed line is just a representative $P_n$ of the IBM in the regular region.
Off-Diagonal Wavefunction Decorrelation: $P_{k}(|U_{nm}(z)|^2)$
---------------------------------------------------------------
Wilkinson and Walker[@walk] have used perturbation theory to derive an approximate expression for the distribution of squared off-diagonal matrix elements, $|\langle \Psi_n(z)\mid\Psi_m(0)\rangle|^2$, in the limit of $|z|\rightarrow\infty$ and $k=|m-n|\gg 1$. They found that $$\label{eq:wilk}
P_{nm}(z) = \frac{\mu^2z^2}{ (E_n-E_m)^2 + (\pi\rho\mu^2
z^2)^2},\qquad\mu^2 = \left\langle |\frac{\partial H}{\partial
z}|^2\right\rangle_{E_n\sim E_m,n\not= m}$$ Here, the energies are not averaged over, $\mu^2\sim 1$, and $\rho$ is the average level density. In exploring the behavior of these quantities in the IBM, it is difficult to satisfy the validity conditions for Eq. (\[eq:wilk\]), since we largely study states in the middle portion of the spectrum, and $m\gg n$ is difficult to satisfy. If we use only two states separated by $k=|n-m|\gg 1$, and we do not average over energy, the statistics are very poor. In order to get good statistics, we have examined the equivalent distribution which is averaged over both coordinate $z$ and energy, with $k$ kept fixed. We define this distribution of off-diagonal matrix elements as $(k>0)$: $$P_k(\widetilde{z}) = \left\langle |U_{nm}(z)|^2\right\rangle
= \left\langle |\langle\Psi_n(z+z_o)|
\Psi_m(z_o)\rangle|^2\right\rangle_{E,z_o,k=|n-m|}$$ where the subscript indicates that it is averaged over the trajectory $z(\eta,\chi)$, as well as over energy, with the separation $|n-m|$ held fixed. We use the notation $P_{nm}$ for the quantity which is not energy averaged, and $P_k$ for the energy averaged result. In Fig. 9, we compare this function computed in random matrix theory (solid) and in the chaotic regime of the IBM (boxes). In general the agreement is quite good. The random matrix theory result was done using $N=50$, and averaging over the middle third of the spectrum. Hence as $k$ increases, the statistics get worse. The IBM results are for the $J^\pi=10^+$ states, with a dimension of 211, so that the statistics is better. In order to contrast our results with Eq. (\[eq:wilk\]), we have taken a rescaled form, which is not entirely justified, as the scaling by $\mu$ is distinct from $D_2=C(0)$. Nevertheless, we plot in Fig. 9, the following rescaled functions (whose regimes of validity are indicated): $$\begin{aligned}
P_k'(\widetilde{z}) & = & \frac{\widetilde{z}^2}{k^2 + \widetilde{z}^4},\qquad
k\gg 1, |z|\gg 1\\
P_k''(\widetilde{z}) & = & \int_0^\infty dx \cos (kx)
e^{-\widetilde{z}^2[1-\exp(-|x|)]},\qquad k\gg 1\end{aligned}$$ The function $P_k'$ (dots in Fig. 9) is an approximation of $P_k''$ (dot-dashed in Fig. 9) in the limit $|z|\gg 1$, which was derived in Ref. [@walk]. As the value of $k$ increases, there is better agreement with the exact results form the random matrix simulation and the IBM. We observe that a better overall fit can be found with the function (dashed line if Fig. 9) $$P_k (\widetilde{z}) =
\frac{\widetilde{z}^2}{k(k-c) + \widetilde{z}^4},\qquad c=3/4,$$ which trivially converges to the Wilkinson-Walker result in its regime $k\gg
1$, but better describes the results for all $k$. The results for the regular region $\chi=-0.661$, $\eta\in [0.5, 1.0]$, are given by the crosses. Once again, there is no universal result for the regular case. Further, as expected, all three analytic functions (42)-(44) agree in the large $k$ limit.
Diagonal Matrix Elements: $P_{\widetilde{z}}(U_{nn}(z))$
--------------------------------------------------------
The previous results have been averages over various matrix elements. We now show that the actual distributions of matrix elements can also be predicted by universal functions. Consider, for example, the distribution of the matrix elements $U_{nn}(z)=\langle\Psi_n(z)\mid\Psi_n(0)\rangle$. These can be seen to be described by a universal function for each $z$. The distribution $P(U_{nn}(\widetilde{z}))$ is shown in Fig. 10 for $J^\pi=10^+$ states at several values of $\widetilde{z}$. At $z=0$, the distribution is a $\delta-$function centered at 1. As the separation of the wavefunctions increases, the centroid shifts from 1 to 0, and the overlap evolves to an asymptotic Porter-Thomas distribution (an approximate gaussian) at large $z$:
$$P_{\widetilde{z}}(U_{nn}(z)) \rightarrow \left\{
\begin{array}{ll}
\delta(U_{nn}-1) & \widetilde{z} \rightarrow 0\\
\exp{(-N U_{nn}^2/2)} & \widetilde{z} > 1 \\
\end{array}\right.$$
The large $z$ limit is not universal, as can be seen by the explicit dimensional dependence of the result. The Porter-Thomas result lies beyond the range of universality which is largely $\widetilde{z}\leq 1$, but the precise decorrelation of matrix elements within this range has the model independent form shown in Fig. 10.
Off-Diagonal Matrix Elements: $P_{\widetilde{z}}(U_{nm}(z))$
------------------------------------------------------------
A similar result exists for the distribution of off-diagonal matrix elements $U_{nm}(z)=\langle\Psi_n(z)\mid\Psi_m(0)\rangle$ . The distribution $P_{\widetilde{z}}(U_{nm}(\widetilde{z}))$ is shown in Fig. 11 for $J^\pi=10^+$ states at several values of $\widetilde{z}$. At $z=0$, the distribution is also a $\delta-$function, but now centered at $z=0$. As the separation of the wavefunctions increases, the overlaps evolve to an asymptotic Porter-Thomas distribution (an approximate gaussian) at large $z$:
$$P_{\widetilde{z}}(U_{nm}(z)) \rightarrow \left\{
\begin{array}{ll}
\delta(U_{nm}) & \widetilde{z} \rightarrow 0\\
\exp{(-N U_{nm}^2/2)} & \widetilde{z} > 1 \\
\end{array}\right.$$
As before, the large $z$ limit is not universal, as can be seen by the explicit dimensional dependence of the result. The solid curve in Fig. 11 is the Porter-Thomas result.
Correlations of Mean Fields
---------------------------
Each value of $(\eta,\chi)$ corresponds to a deformed mean field characterized by $(\beta,\gamma)$ determined from the minimum of Eq. (3). Because wavefunctions decorrelate on order of $\widetilde{z}=\sqrt{D_2} z\sim 1$, the actual correlation length in terms of the parameters $\eta$ and $\chi$ depends on spin, and is given by $z\sim z_c\equiv 1/\sqrt{D_2(\Delta\beta,\Delta\gamma,J^\pi)}$. To explore the spin dependence of the correlation length, we compute $z_c$ in the IBM for $J^\pi=0^+,2^+,4^+,10^+$, and find typical values of $z_c=0.16,0.14,0.11,0.05$. This has a roughly behavior $$z_c \sim 1 - \gamma J$$ where $\gamma$ is a constant. How generic such an dependence might be in other nuclear models is unclear, but it does indicate how rapid states of different spin can decorrelate (n.b. the results can be corrected for the dependence of $z_c$ on $\sqrt{N}$, but this does not account entirely for the behavior). The equivalent values of $\Delta\beta$ and $\Delta\gamma$, which correspond to statistically decorrelated configurations, depend rather strongly on the parameter region. For example, in our calculations we can obtain a range from $\Delta\beta=0.01$ to $1.3$, for the same correlation length, depending on whether the shape is undergoing a rapid shape phase transition or not in the particular parameter regime. One can only conclude that near a shape phase transition, there can exist strong statistical correlations between very distinct nuclear shapes.
More Complicated Operators
--------------------------
It is clear that one can explore many classes of operators and establish the behavior of model-independent limits of those quantities. For instance in the study of the $E2$ decay of high-spin states, Aberg[@aberg] has introduced the matrix quantity $$T_{ij} = \mid\langle\Psi_i(J-2)\mid\Psi_j(J)\rangle\mid^2 (E_j(J)-E_i(J-2))^5$$ where the parameter $J$ is the angular momentum, and is equivalent to $z$. This matrix can be explored as a function of the correlation length, and has different results in the chaotic limit, depending on the spin dependent scaling $D_2$. One can consider other operators as well, and we would like to point out that additional quantities can be constructed using our universal predictions here, together with the analysis of Ref. [@walk], which discusses how to compute arbitrary correlation functions.
Multi-parameter Correlations: Topological Effects and Berry’s Phase
===================================================================
While formal studies of parametric correlations have been limited largely to single parameter systems (see Ref. [@walk] for some exceptions), nuclear deformation is usually described in terms of two or more shape parameters. When two or more parameters are involved, one finds that short distance correlations can be modified by topological effects, due to Berry’s phase. That is, the correlation between quantities at $\beta,\gamma$ and $\beta',\gamma'$ depend on the path used to connect these points. Generally, for correlation functions which are sensitive to phase information, we will show that interference terms can strongly modify the expected results. We explore the basic ideas here in the case of two parameters.
When a wavefunction undergoes parametric evolution on a closed circuit $C$, it is well known that the wavefunction can pick up a topological phase: $$\Psi_n(z) \longrightarrow e^{i\gamma(C)}\Psi_n(z),$$ where $C$ represents a loop in parameter space starting and ending at $z$. For real symmetric matrices, such as our GOE ensemble, $\gamma(C)$ is only $0$ or $\pi$ (mod $2\pi$)[@Berry]. Hence $$\Psi_n(z+C) = \pm \Psi_n(z)$$ where the sign depends upon the particular eigenstate and the path, and $z+C$ represents the same point $z$ after following the closed loop $C$. Of course, one does not have to follow a closed loop. A similar effect exists if one follows two distinct paths from $z$ to $z'$. Then phase differences result in interference. Whether or not paths are in a chaotic or regular regime does not change the flavor of the argument, but in the chaotic regime, more states are likely to pick up a negative phase due to the many avoided level crossings[@davida].
A 2-Parameter Random Matrix Model
---------------------------------
The simplest formulation of a two parameter correlated random matrix ensemble is $$H(X,Y) = H_1\cos X + H_2\sin X + H_3 \cos Y + H_4\sin Y$$ where the constant random matrices $H_i$ are uncorrelated: $\overline{H_iH_j}=\delta_{ij}$. It follows that $$\overline{H(X,Y) H(X',Y')} = \cos (X-X') + \cos ( Y-Y').$$ Generalizations to arbitrary dimensions have been discussed by Wilkinson[@walk]. We can now consider parametric excursions in the $(X,Y)$ plane, specifically two paths which connect $(0,0)$ to $(\Delta X,\Delta Y)$, one of shortest length, and the other a longer path enclosing an area $A$. Because the wavefunctions acquire a Berry’s phase around the closed loop, which can be $\pm 1$ for the GOE case, the area $A$ enclosed can modify expected short distance behaviors.
Correlations in the $\beta-\gamma$ plane
----------------------------------------
We can now explore some of the topological effects in our two parameter theory $H(\eta,\chi)$. Consider a rectangular loop $C$ in parameter space which encloses an area $A$. In analogy to scaled parameter $\widetilde{z}$, we define the scaled area of the loop as $\widetilde{A}=C(0)A\sim \widetilde{z}^2$. Then an area $\widetilde{A}\sim 1$ is a loop whose sides are approximately the decorrelation length of observables. Such a loop stays within the universal regime for all values of the parameter. In Fig. 12, we plot the distribution of matrix elements $P_z(U_{nn})$ (see Eq. (45)) for such a loop. Starting from the top of the figure, we have $\widetilde{z}=0$, and the distribution is a delta function. As $\widetilde{z}$ increases, the distribution spreads in accordance with universal predictions (cf. Fig. 10). At the farthest point of the loop, the distribution is given by the middle figure. As the trajectory returns to the initial point, approximately half the eigenfunctions develop a negative topological phase, and at the final point, which is precisely the initial point, the distribution is equally split. All of the results in Fig. 12 are within the universal regime, but one can see that topological effects can destroy the expected behavior discussed in Fig. 10. For smaller loops, the effect is smaller. The approximate behavior is[@davida] $$f = \left\{\begin{array}{ll}
\frac{\widetilde{A}}{2} & \widetilde{A} \leq 1\\
\frac{1}{2} & \widetilde{A} > 1\\
\end{array}\right.$$ Here $f$ is the fraction of the total states which split to $-1$, and $\widetilde{A}$ is the enclosed, scaled area. The fraction increases linearly with the area. Because saturation occurs near $\widetilde{A}=1=C(0)\Delta\chi\Delta\eta$, and $C(0)\propto N$, the size of the loop needed to see the maximal effect decreases like $1/N$: $\Delta\chi\Delta\eta\sim 1/N$. Hence we see that universal predictions can be modified in multi-parameter theories due to the topology of the parameter space. It is not sufficient to give only the metric distance in parameter space in order to provide all universal predictions. One must also consider the path taken to get to that metric separation.
So there are several aspects here to consider. Berry’s phase effects are independent of the underlying chaos along the chosen path, but depend more upon the nature of the parameter space enclosed by the path. One could imagine a loop in parameter space which is entirely regular, but encloses a chaotic regime. The expression for the fraction of total states $f$ above assumes one is always in a chaotic regime, and the enclosed area is also chaotic.
Conclusions
===========
In conclusion, we have explored the adiabatic behavior of collective nuclear excitations, and found that under the appropriate scaling of the parameter, correlation functions and distributions of matrix elements behave universally. Hence, if we wish to implement random matrix theory to study a complex nuclear situation, we specify immediately a multitude of model independent results related to the wavefunctions and energies. The results here indicate that a new universality exists in nuclei, related to the ‘deformation’ of the nucleus, which is quite robust. As the random matrix predictions are generic, they should be present in other classes of nuclear states, generated from the shell model, or other models. While we have focused on matrix element distributions and certain correlation functions, it is clear that the scaling provides a general type of approach to compute arbitrary correlation functions. This also establishes that the use of random matrix theory with covariances of the type (16) is quite reasonable.
We would like to thank R. Casten, C. Lewenkopf and V. Zamfir for useful discussions, and M.Wilkinson for providing an advance copy of Ref. [@walk]. This work was supported by DOE grant DE-FG02-91ER40608.
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[**Figures**]{}\
Figure 1. Distribution of matrix elements for the IBM in the chaotic region (boxes, $\eta=0,\chi=-0.7$), and in the regular region (crosses, $\eta=0.85,\chi=-0.661$). Both distributions are compared to the distribution $P(|H|)\propto 1\sqrt{|H|} \exp(-|H|/V)$, with values of $V=10$ and $V=0.3$, respectively.
Figure 2. Short distance behavior of the measured IBM autocorrelation function $\langle H(z)H(0)\rangle$ in the chaotic region (solid) and in the regular region (dots). The linear behavior at small $z$ suggests $\alpha=1$ in Eqs. (21) and (26), which is not consistent with the observed parametric energies $E_n(z)$. Hence $F_{ibm}$ cannot be used as physical input into the random matrix theory through Eq. (16).
Figure 3. Instantaneous eigenstates of the Hamiltonian (1) for the parameter range $\eta=0$ and $\chi$ as shown. (Top) Original energies; (Middle) unfolded eneries; (bottom) Brody parameter along this path, indicating a rather chaotic regime, $\omega=1$ being the GOE limit.
Figure 4. Instantaneous eigenstates of the Hamiltonian (1) for two largely regular regions. (Left) A path from rotational $(SU(3))$ to vibrational $(U(5))$ spectra through the regular region proposed recently[@Niall]. The energies have been scaled by 1/2. (Right) The transition from vibrational ($U(5)$) to $\gamma-$unstable $(O(6))$ choosing a path which is weakly chaotic. The average Brody parameters are $\overline{\omega}=0.23$ and $\overline{\omega}=0.25$, respectively.
Figure 5. Slope-slope autocorrelation function, $C(\widetilde{z})$, for the parametric energies $\widetilde{E_n}(\widetilde{z})$ in the chaotic (symbols) and regular (dots) regions. The solid lines are the random matrix predictions using (37) and (38) with $N=300$ and $N=50$, respectively. The $0^+$ states have the poorest statistics, due in part to the small dimension of the space (N=65).
Figure 6. Scaling of $D_2=C(0)$ with $N$, the dimension of the space. The boson number was varied as $N_b=10,15,20,25$, resulting in dimensions $N=16,56,121,211$ for $J^\pi = 10^+$ states. The anticipated scaling behavior, given by Eqs. (25)-(26) with $\alpha=2$, is linear, shown by the solid line. There are deviations at small boson number since the chaos is not as strong there, and the dimensions are small.
Figure 7. (a) Analytic level curvature distribution $P(k)$ (solid) compared to results of a GOE simulation (histogram). (b) Comparison of the analytic distribution to those for $2^+$ states in the IBM (solid histogram). The dashed histogram corresponds to $2^+$ states in the regular regime, and has been scaled vertically by 1/5.
Figure 8. Wavefunction decorrelation function $P_n(\widetilde{z})=\overline{|\langle\Psi_n(\widetilde{z})|
\Psi_n(0)\rangle|^2}$ for selected states in the chaotic regimes indicated (symbols), and one for $2^+$ states in a regular region (dashes). The solid curves are our universal predictions[@caio]. As there are no universal predictions in the regular regimes, the dashed curve is only representative.
Figure 9. Distributions of off-diagonal matrix elements $P_k(U_{nm}(\widetilde{z}))$ (where $k=|n-m|$). The IBM results for chaotic $J^\pi=10^+$ states (boxes) agree well with the random matrix predictions (solid) using $N=50$, as well as with a simple analytic function (dashes). In addition, we show the asymptotic results $P'_k(\widetilde{z})$ (dots) and $P''_k(\widetilde{z})$ (dot-dashed) of Ref. [@walk] which converge for low $k$ to the exact results. For reference, a similar calculation in the regular region for $10^+$ states is included (crosses).
Figure 10. Distributions of diagonal matrix elements $P_{\widetilde{z}}(U_{nn}(\widetilde{z}))$ at several values of $\widetilde{z}$ for $10^+$ state (solid) and the random matrix predictions (dashed). The distribution shifts from a delta function centered at one at $\widetilde{z}=0$, to an asymptotic, non-universal Porter-Thomas distribution for $\widetilde{z}\gg 1$
Figure 11. Distributions of off-diagonal matrix elements $P_{\widetilde{z}}(U_{nm}(\widetilde{z}))$ at several values of $\widetilde{z}$ for $10^+$ state (histogram). The distribution shifts from a delta function centered at zero at $\widetilde{z}=0$, to an asymptotic, non-universal Porter-Thomas distribution for $\widetilde{z}\gg 1$ (solid).
Figure 12. Effects of Berry’s phase on universal distribution functions. The parameter $\widetilde{z}$ undergoes motion on a closed loop, starting and ending at $\widetilde{z}=0$. Starting at the top, the distribution evolves and eventually bifurcates due to the presence of topological phases. Both results $\widetilde{z}=0.5$ are within the universal regime, but the lower figure shows that the path taken to get to the point of interest can be important. At the bottom, the fraction of matrix elements $f$ that change to $U_{nm}=-1$ depends on the area enclosed. The minimum area needed to achieve the maximum fraction of 1/2 scales as $1/N$, and is hence rapidly realized in large dimensional systems.
[^1]: E–mail: dimitri@nst.physics.yale.edu
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The mass of the axion and its decay rate are known to depend only on the scale of Peccei-Quinn symmetry breaking, which is constrained by astrophysics and cosmology to be between $10^9$ and $10^{12}$ GeV. We propose a new mechanism such that this effective scale is preserved and yet the fundamental breaking scale of $U(1)_{PQ}$ is very small (a kind of inverse seesaw) in the context of large extra dimensions with an anomalous U(1) gauge symmetry in our brane. Unlike any other (invisible) axion model, there are now possible collider signatures in this scenario.'
---
plus 1pt ‘@=12 -0.5in 0.0in 0.0in 8.5in 6.5in
UCRHEP-T283\
July 2000
[**Low-Scale Axion from Large Extra Dimensions\
**]{}
Although CP violation has been observed in weak interactions [@cp1; @cp2] and it is required for an explanation of the baryon asymmetry of the universe [@asym], it becomes a problem in strong interactions. The reason is that the multiple vacua of quantum chromodynamics (QCD) connected by instantons [@insta] require the existence of the CP violating $\theta$ term [@theta] $${\cal L}_\theta = \theta_{QCD} {g_s^2 \over 32 \pi^2}
G_{\mu \nu}^a \widetilde G^{a \mu \nu} ,$$ where $g_s$ is the strong coupling constant, $G^a_{\mu \nu}$ is the gluonic field strength and $\tilde G^a_{\mu \nu}$ is its dual. Nonobservation of the electric dipole moment of the neutron [@edm] implies that $$\bar \theta = \theta_{QCD} - Arg ~Det ~M_u ~M_d < 10^{-10},$$ instead of the theoretically expected order of unity. In the above, $M_u$ and $M_d$ are the respective mass matrices of the charge 2/3 and $-1/3$ quarks of the standard model of particle interactions. This is commonly known as the strong CP problem.
The first and best motivated solution to the strong CP problem was proposed by Peccei and Quinn [@pq], in which the quarks acquire a dynamical phase from the spontaneous breaking of a new global symmetry \[$U(1)_{PQ}$\] and relaxes $\bar \theta$ to its natural minimun value of zero. As a result, there appears a Goldstone boson called the axion but it is not strictly massless [@ww] because it couples to two gluons (like the neutral pion) through the axial triangle anomaly [@anomal].
The scale of $U(1)_{PQ}$ breaking (which is conventionally identified with the axion decay constant $f_a$) determines the axion coupling to gluons, which is proportional to $1/f_a$. If $f_a$ is the electroweak symmetry breaking scale as originally proposed [@pq], then the model is already ruled out by laboratory experiments [@expt]. In fact, $f_a$ is now known to be constrained by astrophysical and cosmological arguments [@astro] to be between $10^9$ and $10^{12}$ GeV. Hence the axion must be an electroweak singlet or predominantly so. It may couple to the usual quarks and leptons through a suppressed mixing with the standard Higgs doublet [@dfsz], or it may couple only to other unknown colored fermions [@ksvz], or it may couple to gluinos [@dms] as well as all other supersymmetric particles.
Because the axion must necessarily mix with the $\pi$ and $\eta$ mesons, it must have a two-photon decay mode. This is the basis of all experimental attempts [@expt] to discover its existence. On the other hand, the accompanying new particles in all viable axion models to date are very heavy, i.e. of order $f_a$; hence they are completely inaccessible to experimental verification.
In the following we consider instead the possiblily that the $U(1)_{PQ}$ breaking scale is actually very small, but that $f_a$ is large because of a kind of inverse seesaw mechanism. We show how this scenario may be realized in the context of large extra dimensions with an anomalous U(1) gauge symmetry in our brane. The associated new physics now exists at around 1 TeV, with a number of interesting observable consequences at future colliders.
We assume a singlet scalar field $\chi$ with a nonzero PQ charge existing in the bulk of large extra dimensions [@extra]. The $shining$ [@distant] of this field in our brane is the source of spontaneous $U(1)_{PQ}$ breaking in our world (called a 3-brane). The idea is that $\chi$ gets a large vacuum expectation value (VEV) in a distant brane, but its effect on our brane is small because we are far away from it. (In the case of lepton number, this mechanism has been used recently to obtain small Majorana neutrino masses [@extnu].) To convert this small $\langle \chi \rangle$ to a large $f_a$, we need to assume an anomalous U(1) gauge symmetry in our brane at the TeV energy scale, as explained below.
In a theory of large extra dimensions with quantum gravity at the TeV scale, there is no large scale available for the axion. Since the behavior of Goldstone bosons depends not on the coupling but only on the scale of symmetry breaking in general, it is a problem which is not easily resolved [@others]. Here we find a new and novel solution to this apparent contradiction in the case where there is an anomalous U(1) gauge symmetry, which is of course well studied [@u1] as a possible manifestation of string theory near the string scale (now considered also at around a few TeV) and has well-known applications in quark and lepton Yukawa textures and supersymmetry breaking.
We extend the standard model of particle interactions to include an extra $U(1)_A$ gauge symmetry and an extra $U(1)_{PQ}$ global symmetry. All standard-model particles are trivial under these two new symmetries. We then introduce a new heavy quark singlet $\psi$ and two scalar singlets $\sigma$ and $\eta$ with $U(1)_A$ and $U(1)_{PQ}$ charges as shown in Table 1. All fields except $\chi$ are confined to our brane.
-------------------- ---------------------------------------- ---------- -------------
Fields $SU(3)_C \times SU(2)_L \times U(1)_Y$ $U(1)_A$ $U(1)_{PQ}$
$(u_i, d_i)_L$ (3,2,1/6) 0 0
$u_{iR}$ (3,1,2/3) 0 0
$d_{iR}$ (3,1,$-$1/3) 0 0
$(\nu_i, e_i)_L$ (1,2,$-$1/2) 0 0
$e_{iR} $ (1,1,$-$1) 0 0
$\psi_L$ (3,1,–1/3) 1 $k$
$\psi_R$ (3,1,–1/3) –1 $-k$
$(\phi^+, \phi^0)$ (1,2,1/2) 0 0
$\sigma$ (1,1,0) 2 $2k$
$\eta $ (1,1,0) 2 $2k-2$
$\chi$ (1,1,0) 0 2
-------------------- ---------------------------------------- ---------- -------------
: Peccei-Quinn charges of the fermions and scalars
Because of our chosen charge assignments, only the field $\sigma$ couples to the colored fermion $\psi$, i.e. $${\cal L}_Y = f \sigma \bar \psi_L \psi_R + h.c.$$ Hence it also couples to two gluons through the usual triangular loop. As $\sigma$ acquires a VEV, say $u$, of order 1 TeV, both $U(1)_A$ and $U(1)_{PQ}$ are broken, whereas the latter is broken by $\langle \chi \rangle = z$, and it induces a VEV also for $\eta$, i.e. $\langle \eta \rangle = w$. We will show in the following that given $z$ is small from its origin in the bulk, $w$ is also small. Now the longitudinal component of the $Z_A$ boson is mostly given by Im$\sigma$, so the axion is excluded to be mostly a linear combination of Im$\eta$ and Im$\chi$, but the latter two fields do not couple to the colored fermion $\psi$. As a result, the axion’s coupling to two gluons is now effectively $${1 \over f_a} = {w^2 \over u^2 \sqrt {w^2 + z^2}},$$ which can be thought of as a kind of inverse seesaw, i.e. the largeness of $f_a$ is explained by the smallness of $w$. Details will be given later.
Our brane ${\cal P}$ is located at a point $y=0$ in the bulk. Peccei-Quinn symmetry is broken maximally in a distant brane ${\cal P}'$, located at a point $y=y_*$ in the bulk. We assume for simplicity that the separation of the two branes is of order the radius of compactification of the extra space dimensions, i.e. $|y_*|=r$, which is only a few $\mu$m in magnitude. The fundamental scale $M_*$ in this theory is then related to the reduced Planck scale $M_P = 2.4 \times 10^{18}$ GeV by the relation $$r^n M_*^{n + 2} \sim M_P^2 .$$ The $U(1)_{PQ}$ symmetry breaking in the distant brane acts as a point source $J$, which induces an effective VEV, i.e. $z$, to the singlet bulk field $\chi$. Other effects which may perturb the $shined$ value of $\langle
\chi \rangle$ in our world are all included as boundary conditions to the source $J$, so that the effect of the field $\chi$ in our brane always appears in the combination $z(y=0) e^{i\varphi}$, where $\varphi(x)$ is a dynamical phase which transforms under $U(1)_{PQ}$ to preserve its invariance. This formulation has also been used for the spontaneous breaking of lepton number in the case of neutrinos [@extnu].
In our brane, the profile of $\chi$ is given by the Yukawa potential in the transverse dimensions $$\langle \chi(y = 0) \rangle = J(y=y_*) \Delta_n(r),$$ where $$\Delta_n(r) = {1 \over (2 \pi )^{n \over 2}
M_*^{n- 3}} ~\left( {m_\chi \over r} \right)^{n-2 \over 2}
~K_{n - 2 \over 2} \left( m_\chi r \right),$$ $K$ being the modified Bessel function. We consider the source to be dimensionless, which we take to be $J=1$. For the interesting case of $n> 2$ and $m_\chi r \ll 1$, the $shined$ value of $\chi$ is given by $$\langle \chi \rangle \approx \displaystyle{
{ \Gamma ( {n -2 \over 2} ) \over
4 \pi^{n \over 2} }{M_* \over (M_* r)^{n-2} } }
= \displaystyle{
{ \Gamma ( {n -2 \over 2} ) \over
4 \pi^{n \over 2} }~ M_* ~\left({M_* \over M_P}\right)^{2 - (4/n)} }.$$ For $n=3$ and $M_* = 10$ TeV, we get $\langle \chi \rangle \sim 0.2$ keV. This is the smallest value possible with our assumptions. However, if the distant brane is located at $y_*$ less than $r$, larger values of $\langle \chi \rangle$ may be obtained. As we will show, the range 1 keV to 1 MeV corresponds nicely to the axion decay constant of $10^{12}$ to $10^9$ GeV.
We express the bulk field as $$\chi = {1 \over \sqrt 2} ( \rho + z \sqrt 2) e^{i \varphi} .$$ Its self-interaction terms are now given by $$V(\chi) = \lambda z(y)^2 \rho(x,y)^2 + {1 \over \sqrt 2} \lambda
z(y) \rho(x,y)^3 + {1 \over 8} \lambda \rho(x,y)^4 .$$ This Lagrangian has the virtue of universality, i.e., $\lambda$ is unchanged, but $z$ can change depending on where our brane is from the distant brane. The invariance under $U(1)_{PQ}$, i.e. $\rho \to \rho$ and $\varphi
\to \varphi + 2 \theta$, is also maintained in the other interactions, as described below. The parameters in the potential of $\chi$ are thus guaranteed to be independent of the parameters of our brane.
The scalar potential in our brane excluding $V(\chi)$ is now given by $$\begin{aligned}
V &=& m_1^2 \Phi^\dagger \Phi + m_2^2 \sigma^\dagger \sigma + m_3^2
\eta^\dagger \eta + {1 \over 2} \lambda_1 (\Phi^\dagger \Phi)^2 + {1 \over 2}
\lambda_2 (\sigma^\dagger \sigma)^2 + {1 \over 2} \lambda_3 (\eta^\dagger
\eta)^2 \nonumber \\ && + \lambda_4 (\Phi^\dagger \Phi)(\sigma^\dagger \sigma)
+ \lambda_5 (\Phi^\dagger \Phi)(\eta^\dagger \eta) + \lambda_6
(\sigma^\dagger \sigma)(\eta^\dagger \eta) + (\mu z e^{i \varphi}
\sigma^\dagger \eta + h.c.),\end{aligned}$$ where $\mu$ has the dimension of mass and we assume that all mass parameters are of the same order of magnitude, i.e. 1 TeV.
The minimum of $V$ satisfies the following conditions: $$\begin{aligned}
m_1^2 + \lambda_1 v^2 + \lambda_4 u^2 + \lambda_5 w^2 &=& 0, \\
u(m_2^2 + \lambda_2 u^2 + \lambda_4 v^2 + \lambda_6 w^2) + \mu z w &=& 0, \\
w(m_3 ^2 + \lambda_3 w^2 + \lambda_5 v^2 + \lambda_6 u^2) + \mu z u &=& 0,\end{aligned}$$ where $\langle \phi^0 \rangle = v$. Hence $$\begin{aligned}
v^2 &\simeq& {-\lambda_2 m_1^2 + \lambda_4 m_2^2 \over \lambda_1 \lambda_2 -
\lambda_4^2}, \\ u^2 &\simeq& {-\lambda_1 m_2^2 + \lambda_4 m_1^2 \over
\lambda_1 \lambda_2 - \lambda_4^2},\end{aligned}$$ and $$w \simeq {- \mu z u \over m_3^2 + \lambda_5 v^2 + \lambda_6 u^2},$$ which is indeed of order $z$ as mentioned earlier.
Whereas Im$\phi^0$ becomes the longitudinal component of the usual $Z$ boson, $(u {\rm Im} \sigma + w {\rm Im} \eta)/\sqrt {u^2 + w^2}$ becomes that of the new $Z_A$ boson. Since the $3 \times 3$ mass matrix in the basis \[Im$\sigma$, Im$\eta$, $z\varphi$\] is given by $$\pmatrix{-\mu z w / u
& \mu z& \mu w \cr
\mu z & - \mu z u / w &
-\mu u \cr \mu w & -\mu u & - \mu u w / z },$$ the axion $a$ is identified as the following: $$\begin{aligned}
{a \over \sqrt 2} &=& {1 \over {N}} \left[ uw^2 {\rm Im} \sigma
- w u^2 {\rm Im} \eta + z (u^2 + w^2) z \varphi \right] \nonumber \\
&\simeq& {w^2 \over u \left( w^2 + z^2 \right)^{1/2} } {\rm Im} \sigma -
{w \over \left( w^2 + z^2 \right)^{1/2} } {\rm Im} \eta +
{z \over \left( w^2 + z^2 \right)^{1/2} } z \varphi,\end{aligned}$$ where $N= \left\{ w^2 u^2 (w^2 + u^2) + z^2 (w^2 + u^2)^2 \right\}^{1/2}$ is the normalization. Since only $\sigma$ couples to the colored fermion $\psi$ and the component of Im$\sigma$ in the axion is $u$ times a phase, the axion coupling to the gluons through $\psi$ is effectively as given by Eq. (4) as mentioned earlier. Using $u \sim 1$ TeV and $w \sim z \sim 1$ keV to 1 MeV, we see that $f_a$ is indeed in the range $10^{12}$ to $10^9$ GeV.
In Table 1, we have not specified the value of $k$ for the PQ charge of $\psi$. This is intentional because our model is independent of it. This ambiguity also helps us to understand its pattern of symmetry breaking. For example, if $\langle \chi \rangle = 0$, then $\langle \eta \rangle = 0$ also. In that case, there is no axion and the Peccei-Quinn symmetry disappears, i.e. $k=0$. Hence the true scale of $U(1)_{PQ}$ breaking is indeed small, i.e. $z$ from the bulk, as asserted.
To understand why we have an exception to the general rule that the axion coupling is inversely proportional to the scale of $U(1)_{PQ}$ breaking, we point out that the anomalous nature of $U(1)_A$ is crucial. If we attempt to make it free of the axial triangle anomaly, we need to add colored fermions with opposite $U(1)_A$ charges to $\psi_{L,R}$. They must then acquire mass through a new scalar field with opposite $U(1)_A$ charge to $\sigma$. The longitudinal component of $Z_A$ takes up a linear combination of the two imaginary parts, leaving free the other to be the axion, which now couples to the colored fermions with the same scale as $U(1)_A$ symmetry breaking. The above is of course the analog of what happens in the well-known original Peccei-Quinn proposal [@pq].
All axion models to date have no accompanying verifiable new physics other than the $a \to \gamma \gamma$ decay, and that depends on the axion being a component of dark matter. In our scenario, the possibility exists for this new physics to be at the TeV scale and be observable at future colliders.
\(1) The stable heavy colored fermion $\psi$ may be produced in pairs, i.e. $gg \to \psi \bar \psi$. Both $\psi$ and $\bar \psi$ carry light quarks and gluons with them and appear as jets, but when these jets hit the hadron calorimeter in a typical detector, a large part (i.e. 2$m_\psi$) of the initial collision energy is “frozen” in the mass and appears “lost”.
\(2) There is mixing between the standard-model Higgs boson Re$\phi^0$ with the new scalar Re$\sigma$ of order $v/u$, i.e. 0.1 or so. This means that the lighter (call it $h$) of the two physical scalar bosons has a small component of Re$\sigma$, but that only modifies its (small) $gg$ and $\gamma \gamma$ decay amplitudes through the $\psi$ loop. Hence $h$ behaves almost exactly like the standard-model Higgs boson.
\(3) The $U(1)_A$ gauge boson $Z_A$ may be produced by $gg^*$ fusion through the $\psi$ loop. If kinematically allowed, it will decay into Re$\eta$ + Im$\eta$. Since Im$\eta$ is partly ($w/\sqrt{w^2+z^2}$) the axion $a$ which will escape detection, this event has a lot of possible missing transverse momentum. The subsequent decay of Re$\eta$ is into $a$ and a virtual $Z_A$ which turns into $gg$. This adds more missing transverse momentum. The end result of the production and subsequent decay of $Z_A$ is thus two gluon jets and two axions. This is a distinctive signature of our scenario [@note]. It predicts collider events with large missing energy without the existence of supersymmetry.
[*Acknowledgement.*]{} This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837. One of us (U.S.) would like to thank the Physics Department, University of California, Riverside for hospitality.
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Re$\eta$ will also decay (through its mixing with Re$\phi^0$) into standard-model final states such as $ZZ$, $WW$, etc. However, this mixing is very small, i.e. of order $wv/\mu u$. Details of the phenomenology of this model will be discussed in a forthcoming paper.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We analyze 15,000 spectra of 29 stellar-mass black hole candidates collected over the 16-year mission lifetime of [[*RXTE*]{}]{} using a simple phenomenological model. As these black holes vary widely in luminosity and progress through a sequence of spectral states, which we broadly refer to as hard and soft, we focus on two spectral components: The Compton power law and the reflection spectrum it generates by illuminating the accretion disk. Our proxy for the strength of reflection is the equivalent width of the Fe-K line as measured with respect to the power law. A key distinction of our work is that for [*all*]{} states we estimate the continuum under the line by excluding the thermal disk component and using only the component that is responsible for fluorescing the Fe-K line, namely the Compton power law. We find that reflection is several times more pronounced ($\sim 3$) in soft compared to hard spectral states. This is most readily caused by the dilution of the Fe line amplitude from Compton scattering in the corona, which has a higher optical depth in hard states. Alternatively, this could be explained by a more compact corona in soft (compared to hard) states, which would result in a higher reflection fraction.'
author:
- 'James F. Steiner, Ronald A. Remillard, Javier A. García, Jeffrey E. McClintock'
title: |
Stronger Reflection from Black Hole Accretion Disks\
in Soft X-ray States
---
Introduction {#section:intro}
============
During the course of its 16-year mission, the [*Rossi X-ray Timing Explorer*]{} ([[*RXTE*]{}]{}) detected far more photons (30 billion in PCU-2 alone) from accreting black holes than any other X-ray observatory. The sample of black holes (BHs) targeted by [*RXTE*]{} is chiefly comprised of nearby stellar-mass systems. While the total Galactic population of stellar BHs is believed to be many millions, only a tiny subset of approximately 50 are known to us, namely those located in X-ray binaries. A wondrous property of BHs, their utter simplicity, is the essence of the famous no-hair theorem: Each BH in nature is fully described by just its mass and spin. Roughly half of the known stellar BHs have a dynamically-determined mass. The measured masses range from $\sim5-20~{M_\odot}$ [@Ozel_2010; @Reid_2014; @Laycock_2015; @Wu_2016]. Meanwhile, estimates of spin have been obtained for many of them during the past decade, principally by modeling either the thermal continuum emission of the accretion disk [e.g.; @Zhang; @MNS14], or the relativistically-broadened reflection spectrum [e.g.; @Fabian_1989; @Reynolds_2014]. Our focus is primarily on transient BH systems that cycle between a minuscule fraction of the Eddington limit upward to the limit itself. During an outburst, a transient BH progresses through a sequence of spectral-timing states, which are broadly termed “hard” or “soft,” based on a measure of X-ray hardness [@Fender_2004]. As a source evolves over the course of months and its hardness varies, sweeping changes occur in many properties of the system including the composition of its spectrum, the intensity of Fourier flicker noise, and the presence or absence of quasi-periodic oscillations and jets [e.g.; @Homan_2005; @RM06; @Heil_2015]. Stellar BHs emit a complex multicomponent X-ray spectrum. A [ *thermal*]{} blackbody-like component is produced in the very inner accretion disk. The disk is truncated at a radius ${R_{\rm in}}$ before reaching the event horizon. A hard [*power-law*]{} component results from Compton scattering of the thermal disk photons in hot coronal gas that veils the disk. The third principal component is a [*reflection*]{} spectrum generated by illumination of the cold disk ($kT\sim0.1-1$keV) by the power-law component. The reflection component is a rich mix of radiative recombination continua, absorption edges and fluorescent lines [@Ross_1993; @Garcia_Kallman_2010]. An analysis of these three interacting spectral components provides constraints on the source properties including geometry (e.g., on ${R_{\rm in}}$ and the scale of the corona). The relationships between these components across the full range of behavior displayed by accreting stellar BHs is the focus of this paper. Our results are based on an analysis for 29 stellar BHs (10 dynamically-confirmed BHs and 19 BH candidates) of all the data collected using [[*RXTE*]{}]{}’s prime detector unit (PCU-2), some 15,000 spectra in all, with a total net exposure time of 30Ms. Importantly, we recalibrate the data using our tool [pcacorr]{}, which greatly reduces the level of systematic error [@pcacorr]. Given the scope of our study, relativistic reflection models are too complex and computationally slow for our purposes [e.g.; [reflionx, xillver, relxill;]{} @reflionx; @relxill2]. We therefore employ a simplistic, phenomenological model and estimate the strength of the reflection spectrum by determining the equivalent width with respect to the Compton continuum of its most prominent reflection feature, namely the $6.4-7.0$ keV Fe-K line. The paper is organized as follows: In Section \[section:data\] we describe the data sample and our approach to modeling the data. Our results are presented in Section \[section:results\], followed by a discussion in Section \[section:disc\] and our conclusions in Section \[section:conc\].
![([*top*]{}): Hardness-intensity diagrams for all data and ([ *bottom:*]{}) for six well-known BHs with abundant data (where for reference the gray background shows all data). For reference, the count rate of the Crab Nebula is $\approx
2600$ s$^{-1}$. Note that a HID does not allow one to compare the luminosities of sources because the intensity is in detector units.[]{data-label="fig:qdiag"}](fig_tmp_testplot_f1master1){width="1\columnwidth"}
Data {#section:data}
====
The [[*RXTE*]{}]{} archive provides the premier database for the synoptic study of stellar BHs. We exclusively use the data collected by PCU-2, one of the five proportional counter detectors that comprise [[*RXTE*]{}]{}’s principal instrument, the Proportional Counter Array (PCA). Throughout the mission, PCU-2 was the unit that was most often active, and it had the most reliable and stable calibration [@Jahoda_2006; @Shaposhnikov_2012]. Its area and energy resolution were 1300cm$^2$ and $\approx18$% at 6 keV. Table 1 summarizes our data sample. During an outburst, a BH was typically observed daily over a period of months as it systematically brightened and subsequently dimmed by orders of magnitude. We homogenized the data by segmenting it into continuous 300–5000s intervals, each of which was used to produce an energy spectrum and a power-density spectrum (PDS). Energy spectra were analyzed ignoring the lowest 4 channels, an effective lower bound $\approx
2.8$ keV, and an upper bound of 45keV was adopted. The effects of detector dead time were corrected as described in @McClintock_2006. We obtained an absolute calibration of the flux using the the standard @Toor_Seward spectrum of the Crab Nebula; our slope and normalization corrections are $\Delta\Gamma = 0.01$ and $f_{\rm TS} = 1.097$ [@Steiner_2010]. We computed the rms power, a measure of the flicker noise, by integrating the PDS over the band 0.1–10Hz. An unprecedented sensitivity to faint spectral features is achieved by employing the calibration tool [pcacorr]{} [@pcacorr], which improves the quality of the PCA’s spectral calibration by roughly an order of magnitude and results in a data precision of $\sim 0.1\%$. We include this small systematic uncertainty as a fractional error on each channel when conducting our analysis using [XSPEC]{} [@XSPEC]. The considerable increase in sensitivity [pcacorr]{} delivers is crucial for estimating the strength of line features. All PCU-2 data for 29 black holes are plotted in a hardness-intensity diagram (HID) [@Fender_2004; @RM06] in the top panel of Figure \[fig:qdiag\]. The normalized hard color (or hardness ratio ${\rm HR}$) is the ratio of count rates in the energy bands indicated in the upper panel, and is described in @Peris_2016. The data are color-coded to show the level of rms flicker noise. As is well-known and is evident here from the vertical striation, rms noise correlates with spectral state [e.g.; @Heil_2015; @RM06], with hard states showing several-times stronger rms than soft states. The six small panels are HIDs for selected sources. Note that transient sources characteristically trace a loop in the hardness-intensity diagram (HID), but that the persistent source Cyg X–1 is confined to a relatively narrow region. The other selected source showing stunted HID evolution is GRS 1915+105, which is an unusual transient system that has been in a protracted state of outburst since 1992.
Spectral Modeling {#subsec:model}
-----------------
We adopt a single simplistic spectral model that is applicable to both soft and hard spectra: [ phabs\*\[smedge(simpl$\otimes$diskbb)+gauss\]]{}. The disk and Compton components are modeled by [diskbb]{} [@DISKBB], and [simpl]{} [@Steiner_simpl], respectively. The reflection component is described by a Gaussian line with fixed energy of 6.5keV and an intrinsic width of 50eV. We note that owing to the broad detector resolution at the Fe line $\sim 1.2$ keV, the value adopted for the line width in our simplistic model is of minor consequence (as demonstrated in Section \[section:disc\]). Despite its coarse resolution, [[*RXTE*]{}]{} is very sensitive to the line flux, as measured by the normalization of the Gaussian feature. Accordingly, we adopt the line flux as our proxy for the intensity of the reflection component. We approximate the relativistically-broadened Fe-K absorption edge using [smedge]{} [@SMEDGE]; we fit for the peak depth $\tau_{\rm smedge}$ with the width fixed at 7keV and the shape index set to -2.67 [@Sobczak_2000]. Our adopted values of the column density ${N_{\rm H}}$ are summarized in Table1.
![([*top*]{}): Flux in the Fe line versus the adjacent flux in the power-law continuum. Color-mapping indicates the rms power. ([*bottom*]{}): Equivalent width of the Fe line computed using the power-law flux only (i.e., excluding the disk flux). The reflection is strongest in soft states, and it increases dramatically as the spectrum softens. Representative 1$\sigma$ error bars are shown in the upper-left corner in each panel. []{data-label="fig:ew"}](fig_fe_pl_fluxes){width="1\columnwidth"}
Results {#section:results}
=======
We have fitted all $\approx 15,000$ spectra with our model. In Figure \[fig:ew\], results are shown for approximately one-quarter of these spectra, those that meet three criteria: ${\chi^{2}/\nu}<
2$; Fe-line normalization significant at the $\ge1\sigma$ level; and power-law normalization ${f_{\rm SC}}$ significant at the $\ge3\sigma$ level. Plotted in the top panel is Gaussian line flux versus the Compton power law flux near the Fe edge computed by integrating the power-law component from 7–8keV. Since the fluorescent FeK line is produced preferentially by photons at energies above and near the FeK edge [@Reynolds_2009], i.e., at energies $E\gtrsim7$keV [@Kallman_2004], it is not surprising that the line and continuum fluxes are strongly correlated. Though notably, this strong correlation is a direct validation of the reflection paradigm (wherein power-law emission fluoresces strong line emission). What is surprising is that the soft states (red) appear to be much more efficient at producing reflection than hard states (blue), as evidenced by the vertical offset between the clouds of soft and hard data. This result runs counter to the conventional view that reflection is weak in soft states (e.g., @Ross_2007 [@Yuan_2014], but see the correlation between spectral index and reflection strength in @Zdziarski_1999, and related work), a view based on the weakness of reflection [*relative to the thermal disk component*]{}. Here instead, we appropriately relate the strength of the reflection signal to the power-law component that produces it, as isolated from the evolving thermal disk. In the lower panel of Figure \[fig:ew\], we plot (versus the hardness ratio ${\rm HR}$) our proxy for the strength of reflection: i.e., we plot the equivalent width (${{\rm EW}_{\rm PL}}$) of the line with respect to the coronal flux. That reflection is more pronounced for soft states is readily apparent. In soft states, ${{\rm EW}_{\rm PL}}$ decreases regularly by an order of magnitude as the spectrum hardens; then, for intermediate and hard states it plateaus at ${\rm HR}\gtrsim
0.7$. Given the inhomogeneity of our data sample, which includes both transient and persistent sources, the ordered quality of these data is striking.
[lllrrrrr]{} \[tab:data\] LMC X-3 & 05 38 56.3 & -64 05 03 & 704 & 1203.95 & 33.5 & 0.04 & \[1\]\
LMC X-1 & 05 39 38.8 & -69 44 36 & 1598 & 3120.72 & 106.4 & 0.7 & \[2\]\
XTE J1118+480 & 11 18 10.8 & +48 02 13 & 124 & 220.93 & 15.7 & 0.01 & \[3\]\
GS 1354-64 & 13 58 09.9 & -64 44 05 & 23 & 54.46 & 6.2 & 2 & \[4\]\
4U 1543-47 & 15 47 08.6 & -47 40 10 & 130 & 243.26 & 278.2 & 0.4 & \[3\]\
XTE J1550-564 & 15 50 58.8 & -56 28 35 & 517 & 970.78 & 1875.6 & 0.8 & \[5\]\
4U 1630-47 & 16 34 01.6 & -47 23 35 & 1194 & 2108.10 & 1034.0 & 11 & \[6\]\
XTE J1650-500 & 16 50 01.0 & -49 57 44 & 195 & 337.28 & 119.5 & 0.5 & \[3\]\
XTE J1652-453 & 16 52 20.3 & -45 20 40 & 61 & 99.84 & 9.2 & 6.7 & \[7\]\
GRO J1655-40 & 16 54 00.1 & -39 50 45 & 987 & 2483.87 & 5174.2 & 0.7 & \[3\]\
MAXI J1659-152 & 16 59 01.7 & -15 15 29 & 71 & 146.64 & 66.7 & 0.25 & \[8\]\
GX 339-4 & 17 02 49.4 & -48 47 23 & 1672 & 2868.46 & 894.2 & 0.3 & \[9\]\
IGR J17091-3624 & 17 09 08 & -36 24.4 & 256 & 479.60 & 46.8 & 1.2 & \[10\]\
XTE J1720-318 & 17 19 59.0 & -31 45 01 & 122 & 283.36 & 80.7 & 1.3 & \[4\]\
GRS 1739-278 & 17 42 40.0 & -27 44 53 & 12 & 26.45 & 15.2 & 3.7 & \[4\]\
H1743-322 & 17 46 15.6 & -32 14 01 & 649 & 1403.18 & 958.5 & 2.2 & \[11\]\
XTE J1748-288 & 17 48 05.1 & -28 28 26 & 39 & 98.58 & 46.6 & 7.5 & \[4\]\
SLX 1746-331 & 17 49 48.3 & -33 12 26 & 78 & 186.19 & 36.2 & 0.4 & \[3\]\
XTE J1752-223 & 17 52 15.1 & -22 20 33 & 234 & 435.52 & 147.0 & 0.6 & \[12\]\
Swift J1753.5-0127 &17 53 28.3 & -01 27 06 & 376 & 853.95 & 123.4 & 0.15 & \[3\]\
XTE J1817-330 & 18 17 43.5 & -33 01 08 & 191 & 430.11 & 270.7 & 0.15 & \[4\]\
XTE J1818-245 & 18 18 24.4 & -24 32 18 & 56 & 141.74 & 19.7 & 0.5 & \[13\]\
V4641 Sgr & 18 19 21.6 & -25 24 26 & 94 & 179.57 & 3.0 & 0.25 & \[3\]\
MAXI J1836-194 & 18 35 43.4 & -19 19 12 & 76 & 124.83 & 9.4 & 0.15 & \[3\]\
XTE J1859+226 & 18 58 41.6 & +22 39 29 & 170 & 336.24 & 270.3 & 0.2 & \[14\]\
GRS 1915+105 & 19 15 11.6 & +10 56 45 & 2566 & 5255.58 & 12520.1 & 6 & \[15\]\
Cyg X-1 & 19 58 21.7 & +35 12 06 & 2446 & 5413.02 & 6361.3 & 0.7 & \[3\]\
4U 1957+115 & 19 59 24.2 & +11 42 32 & 243 & 646.42 & 41.0 & 0.15 & \[3\]\
XTE J2012+381 & 20 12 37.7 & +38 11 01 & 30 & 53.98 & 13.5 & 1.3 & \[4\]\
Total & & & 14914 & 30207. & 30577. & &\
A Case Study in Reflection: XTE J1550–564 {#subsec:j1550}
-----------------------------------------
To more precisely study the behavior of reflection in soft spectral states, we examine one system in detail. We select an exceptionally bright transient with abundant data: XTE J1550–564 (hereafter, J1550; see the bottom-left subpanel of Fig. \[fig:qdiag\]). J1550 was discovered on 1998 September 6, and two weeks later it reached a peak intensity of 6.8Crab (2–10keV). Four additional fainter outbursts were observed during the following decade. We define ${\rm HR}= 0.7$ as the cut between soft and hard-state data, and we again (as in Figure \[fig:ew\]) examine the relationship between our proxy for reflection (namely the flux in the Fe K line) and the flux in the adjacent Compton continuum. The data are plotted in Figure \[fig:j1550\]. We begin with a simplistic assumption that the coronal flux and Fe line flux will scale together, i.e., $F_{\rm Fe} \equiv \alpha F_{\rm PL,7-8~{\rm keV}}^\beta$, and we proceed to fit for $\alpha$ and $\beta$. This [*scaling relation*]{} is termed SR-PL to emphasize that the line flux scales with the power law. We likewise pursue the SR-PL fit for hard spectra (i.e., ${\rm HR}> 0.7$), with the constants of soft and hard data determined independently.
For the soft data, we additionally investigate the possibility that disk self-irradiation may also contribute to the reflection emission. This is a motivated notion given that bright soft-states can produce appreciable thermal emission even above 5 keV, and further, some fraction of the thermal photons (the “returning radiation”) is bent back and strikes the disk rather than escaping to infinity. We consider returning radiation by allowing for an added scaling with the disk’s emission in the same 7–8 keV band proximate to the Fe-K edge, i.e., $F_{\rm Fe} \equiv \alpha F_{\rm PL,7-8~{\rm keV}}^\beta + \gamma F_{\rm disk,7-8~{\rm keV}}^\delta$ (this is the SR-PL&D scaling relation), and we likewise fit for its parameters.
In the top panel of Figure \[fig:j1550\], we show the Compton component’s contribution to the Fe-line flux for both the SR-PL (red) and SR-PL&D (green) fits. The contribution to the line flux from returning radiation is vanishingly small, particularly at the highest luminosities. In the same figure, for reference we also show the SR-PL fit to the hard data in blue. Note that the error bounds for both soft-data fits lie well above the hard correlation bounds. The best-fitting scale indexes and 1$\sigma$ errors are $\beta_{\rm SR-PL,soft}=0.90\pm0.01$, $\beta_{\rm
SR-PL,hard}=1.12\pm0.12$, and $\beta_{\rm SR-PL\&D,soft}=1.01\pm0.03$, $\delta_{\rm SR-PL\&D,soft}=0.43\pm0.03$. The SR-PL&D fit slightly outperformed the SR-PL fit ($\Delta\chi^2 \approx 20$ for 2 added degrees of freedom).
Having established that the effects of returning radiation are minor, we focus on comparing reflection for soft and hard data using solely the SR-PL curves. We produce Markov-Chain Monte Carlo realizations for both the hard and soft SR-PL correlations (top panel, Figure \[fig:j1550\]), and we compare their respective Fe-line fluxes. The results are shown in the bottom panel of Figure \[fig:j1550\]. On average, the Fe line is $>3$ times stronger for soft data, while the 95% confidence region ranges from $\sim 2-9$ times stronger.
![([*top*]{}): Fe-line flux versus 7–8 keV power-law flux for J1550. Our primary “SR-PL" scaling relation is shown in red with its associated 95% confidence region. The “SR-PL&D" curve in green includes the effects of returning radiation. For reference, the SR-PL fit to the hard data is shown in blue. At the level of detail, only those data with total flux $>3$mCrab and that yielded fits with ${\chi^{2}/\nu}< 2$ are shown. For clarity, data are marked with crosses and error bars omitted when $|N_{\rm line}|/\sigma_{\rm line} < 2$. Data consistent with absorption (i.e., $N_{\rm line} < 0$) are plotted in gray. ([*bottom*]{}): Ratio of the Fe-line flux for soft states to that for hard states computed using the SR-PL scaling relation. Our default curve is presented in orange with its associated 95% confidence intervals as a light-shaded region. Results for alternate spectral models with different values of the Gaussian line energy and line width, which produce modest systematic differences, are also shown for comparison.[]{data-label="fig:j1550"}](js_j1550_fek_2pan_xplot){width="1\columnwidth"}
Discussion {#section:disc}
==========
As illustrated in Figure \[fig:j1550\], our estimate of the ratio of the Fe-line flux in soft and hard states is only modestly sensitive to the values we adopted for the line energy (6.5keV) and width (50eV). Varying these values does not affect our conclusions. We have also explored other model formulations (e.g., including a power-law cutoff and replacing the Gaussian by a relativistic line profile) and have similarly found that our conclusions are unaffected. Varying the line shape systematically rescales the line flux, but it has a minor affect on the ratio of the fluxes in soft and hard states, which is our focus.
Consideration of two shortcomings of our simplistic model serve to strengthen our conclusion that soft states are more efficient in producing reflection emission. First, soft-state disks are hotter and more strongly ionized, and hence they generally produce more reflection continuum emission, which gets lumped together with the Compton power law. In the case of our simple model, this effect serves to boost the power-law continuum, thereby reducing ${{\rm EW}_{\rm PL}}$ for the soft state. Secondly, Fe-line absorption features in disk winds are preferentially and often observed in soft states [@Ponti_2012], and these absorption features act to weaken the emission line. Our finding of enhanced ${{\rm EW}_{\rm PL}}$ in soft states is thus contrary to these biases. Earlier work by @Petrucci_2001 which examined the effect of Comptonizing reflection spectra showed that at a fixed value of $\Gamma$, the fitted Fe-K equivalent width is very sensitive to the coronal temperature $kT_{\rm e}$, principally because as $kT_{\rm e}$ decreases, optical depth $\tau$ necessarily increases (a straightforward consequence of the equations governing thermal Comptonization). Accordingly, at high optical depth, the line is highly scattered in the corona. The scattered portion blends with the continuum, which results in a decrease in the Gaussian’s equivalent width. We have examined the evolution of equivalent width versus spectral hardness (changing $\Gamma$) and other spectral parameters. There is a strong anti-correlation between ${f_{\rm SC}}$ and ${{\rm EW}_{\rm PL}}$ in precisely the sense predicted by @Petrucci_2001, and is the most apparent explanation for the observed trend. This is shown in Figure \[fig:fsc\].
Alternatively, or in addition to the dilution of the line by Compton scattering in the corona, changes in the disk-coronal geometry can impact the reflection’s strength. In particular, a corona at very small scale-height ($h_{\rm corona}$) coupled with a close-in disk yields a higher reflection fraction than when either (1) the disk is truncated or (2) the corona is very large compared to the event horizon [@Dauser_2014]. Gradual evolution of accretion flow with more compact disk-coronal geometry (i.e., lower $h_{\rm corona}/{R_{\rm in}}$) in soft states would similarly contribute to the observed correlation.
We caution that the role of ionization is complex and varies nonlinearly as other coronal attributes change, including most notably, $\Gamma$. These differences are principally related to changes in the atmosphere’s temperature structure. For the range of ionization parameters observed around active stellar-mass BHs, i.e., log $\xi \approx 2-4$, we simulated PCU-2 data using the [[xillver]{}]{} reflection model [@Garcia_Kallman_2010], and applied our analysis method. We found that ${{\rm EW}_{\rm PL}}$ varies by a factor of $\sim 2-3$ over the span of $\Gamma$ ($1.4-3.4$) and $\xi$; this is insufficient to account for the full trend in Figure \[fig:ew\]. Moreover, for log $\xi \leq 3$, higher $\Gamma$ tends to produce lower ${{\rm EW}_{\rm PL}}$ whereas at log $\xi \gtrsim 3$, the trend reverses.
As summarized by @Zdziarski_2003, and also @Gilfanov_2014, there is abundant evidence for a strong positive correlation between spectral index and reflection strength (the “$R-\Gamma$" correlation). This has been seen in both BH X-ray binaries and in AGN. However, such work has been largely confined to examination of hard states. To our knowledge, our work is the first compelling evidence that the correlation between spectral softness (increasing $\Gamma$) and reflection fraction continues and is strongly amplified in soft, thermal states (i.e., $\Gamma \sim 2-3$). Further, from considering the [[*RXTE*]{}]{} archive of active BHs, as is readily apparent in Figure \[fig:ew\], this change is gradual and orderly amongst the full cast of stellar BHs.
Conclusions {#section:conc}
===========
We have examined the strength of reflection in a global study of stellar BHs using a simplistic, phenomenological spectral model. We directly validate the reflection paradigm, wherein power-law flux induces reflection emission. In separating possible contribution from disk self-irradiation, we demonstrate that the power law’s contribution is dominant. Most importantly, we show that the corona produces reflection features up to an order of magnitude more pronounced in soft rather than hard states. The data suggest an ordered transition in which the line-to-continuum strength declines gradually with spectral hardness. This is the first time the “$R-\Gamma$” correlation has been shown to extend through (and increase in) BH soft states. One possible explanation is that a more compact disk-coronal geometry in soft states would produce the observed trend. However, the most natural explanation for this trend is suggested by @Petrucci_2001, who describe the dilution of line features emitted by the disk due to Compton-scattering in the corona. In our case, because hard states have corona with higher optical depth than soft states, their line features are correspondingly weakened resulting in the observed anti-correlation between ${\rm HR}$ and reflection strength.
JFS has been supported by the NASA Einstein Fellowship grant PF5-160144.\
[*Facility:*]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The purpose of this paper is to develop a general existence theory for constrained minimization problems for functionals defined on function spaces on metric measure spaces $(\mathcal M, d, \mu)$. We apply this theory to functionals defined on metric graphs $\mathcal G$, in particular $L^2$-constrained minimization problems of the form $$E(u) = \frac{1}{2} a(u,u) - \frac{1}{q}\int_{\mathcal K} |u|^q \, \mathrm dx,$$ where $q>2$ and $a(\cdot, \cdot)$ is a suitable symmetric, sesquilinear form on some function space on $\mathcal G$ and $\mathcal K \subseteq \mathcal G$ is given. We show how the existence of solutions can be obtained via decomposition methods using spectral properties of the operator $A$ associated with the form $a(\cdot, \cdot)$ and discuss the spectral quantities involved. An example that we consider is the higher-order variant of the stationary NLS (nonlinear Schrödinger) energy functional with potential $V\in L^2+ L^\infty(\mathcal G)$ $$E^{(k)}(u)= \frac{1}{2} \int_{\mathcal G} |u^{(k)}|^2+ V(x) |u|^2 \, \mathrm dx - \frac{1}{p} \int_{\mathcal K} |u|^q \, \mathrm dx$$ defined on a class of higher-order Sobolev spaces $H^k(\mathcal G)$ that we introduce. When $\mathcal K$ is a bounded subgraph, one has localized nonlinearities, which we treat as a special case. When $k=1$ we also consider metric graphs with infinite edge set as well as magnetic potentials. Then the operator $A$ associated to the linear form is a Schrödinger operator, and in the $L^2$-subcritical case $2<q<6$, we obtain generalizations of existence results for the NLS functional as for instance obtained by Adami, Serra and Tilli \[JFA 271 (2016), 201-223\], and Cacciapuoti, Finco and Noja \[Nonlinearity 30 (2017), 3271–3303\], among others.'
address:
- 'University of Lisbon, Portugal'
- |
Grupo de Física Matemática\
Faculdade de Ciências da Universidade de Lisboa\
Campo Grande, Edifício C6\
P-1749-016 Lisboa, Portugal
author:
- Matthias Hofmann
title: An existence theory for nonlinear equations on metric graphs via energy methods
---
Introduction
============
In recent years, there has been a growth of interest in functionals on metric graphs $\mathcal G=(\mathcal V, \mathcal E)$ of the stationary NLS (Nonlinear Schrödinger) energy functional $$\label{eq:introminprobfunc}
E_{\text{NLS}}(u, \mathcal G)= \frac{1}{2} \int_{\mathcal G} |u'|^2\, \mathrm dx - \frac{\mu}{q} \int_{\mathcal G} |u|^q \, \mathrm dx, \qquad u\in H^1(\mathcal G),\; \|u\|_{L^2}^2 =1,\; q>2, \; \mu >0$$ and associated ground states of the stationary NLS energy functional, i.e. minimizers for the constrained minimization problem $$\label{eq:introminprob}
E_{\text{NLS}}(\mathcal G):= \inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_{L^2}^2=1}} E_{\text{NLS}}(u, \mathcal G), \qquad 2<q< 6.$$ Such minimizers are solutions to the stationary nonlinear Schrödinger equation on $\mathcal G$ given by $$\begin{cases}
-u'' + \lambda u = \mu |u|^{q-2} u \qquad \text{edgewise,}\vspace{.5em}\\
\begin{gathered}u \text{ is continuous on }\mathcal G \text{ and satisfies the Kirchhoff condition}\\
\sum_{e\in \mathcal E:e\succ \mathsf v} \frac{\partial u}{\partial \nu} \Big |_e(\mathsf v)=0, \qquad \forall \mathsf v\in \mathcal V,
\end{gathered}
\end{cases}$$ where we recall that $e\succ \mathsf v$ denotes the relation that the edge $e$ is adjacent to the vertex $\mathsf v\in \mathcal V$ and $\frac{\partial u}{\partial \nu}|_e(\mathsf v)$ denotes the inward pointing derivative at $\mathsf v$ towards the interior of the edge $e$. While in the simplest case of the real line, existence of minimizers in can be deduced by standard techniques, on general noncompact graphs existence results are not as easy to obtain due to the lack of a concept of translation invariance. In [@adami2015nls] it was shown on the one hand that under certain topological configurations the problem does not admit a minimizer; on the other, in a later paper the same authors derive an existence principle based on a comparison inequality:
\[thm:introast2016\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges and $2<q<6$. Assume $$\label{eq:introadamiestimate}
E_{\text{NLS}}(\mathcal G) < E_{\text{NLS}}(\mathbb R),$$ then there exists a minimizer for $E_{\text{NLS}}(\mathcal G)$.
This result can be used to obtain existence results on concrete graphs $\mathcal G$ via construction of so called competitors, i.e. test functions $u\in H^1(\mathcal G)$ for which $E_{\text{NLS}}(u, \mathcal G) < E_{\text{NLS}}(\mathbb R)$. This allows to deduce existence of minimizers in certain situations as shown in [@tentarelli2016nls] and [@adami2017negative].
A variant of this problem with potential was considered in [@cacciapuoti2017ground] and [@cacciapuoti2018existence], where the energy functional was given by $$\label{eq:introfunctionaltocons}
E_{\text{NLS}}^{V}(u) = \frac{1}{2} \int_{\mathcal G} \left | u'\right |^2+ V|u|^2 \, \mathrm dx -\frac{\mu}{q} \int_{\mathcal G} |u|^q \, \mathrm dx, \qquad \|u\|_{L^2}^2=1.$$ In [@cacciapuoti2017ground] the existence of minimizers of was related to the existence of eigenvalues of the Schrödinger operator $-\Delta+V$ below the essential spectrum:
\[thm:introcfn2017\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges and $V\in L^1+ L^\infty(\mathcal G)$ with $V_- \in L^r(\mathcal G)$ for $r\in[1,1+ \frac{2}{q-2}]$ and $2<q\le 6$. Assume $$\label{eq:introJameswants}
\inf \sigma(-\Delta+V) < \inf \sigma_{\text{ess}}(-\Delta+V).$$ Then there exists $\mu^*>0$ such that for $\mu \in (0, \mu^*)$ the functional is bounded below and the associated constrained minimization problem $$E_{\text{NLS}}^{V}:= \inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_{L^2}^2=1}} E_{\text{NLS}}^V(u)$$ admits a minimizer.
\[thm:introcac2018\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges and $V\in L^1+L^\infty(\mathcal G)$ satisfying the assumptions in Theorem \[thm:introcfn2017\]. Let $$\Sigma_0 := \inf \sigma(-\Delta+V)<0, \qquad \gamma_q := \inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_{L^2}^2=1}}\frac{1}{2} \int_{\mathbb R} \left | u'\right |^2+ V|u|^2 \, \mathrm dx -\frac{1}{q} \int_{\mathcal G} |u|^q \, \mathrm dx<0.$$ Then we have existence of minimizers of $$E_{\text{NLS}}^{V}= \inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_{L^2}^2=1}} E_{\text{NLS}}^V(u)$$ for $0<\mu\le (\Sigma_0/\gamma_q)^{\frac{3}{2}-\frac{q}{4}}$.
Our goal in this paper is threefold. Firstly, we develop a general existence theory in a far more abstract setting which can be applied to a variety of problems as for example $E_{\text{NLS}}$ and $E_{\text{NLS}}^{V}$ but which is not limited to metric graphs. For example, the existence theory may be also applied to functionals defined on function spaces on combinatorial graphs or general domains in $\mathbb R^n$. We then use this existence theory to obtain generalizations of the results in [@adami2016threshold] and [@cacciapuoti2017ground] by considering more general graphs, higher-order derivatives in the functionals, as well as different variants of the problems. We will also discuss the spectral quantities appearing in , which replaces the inequality in Theorem \[thm:introast2016\] for the existence results. Variants we will consider include the case of decaying potentials and localized nonlinearities, i.e. we replace the set of integration in the term corresponding to the nonlinearity by a bounded subgraph $\mathcal K \subset \mathcal G$. Unlike [@adami2016threshold] our proof of Theorem \[thm:introast2016\] does not involve rearrangement inequalities, which allows us to consider potentials and deduce a similar existence result in the case of decaying potentials.
Let us be now more precise about the abstract setting we will consider. Let $(\mathcal M, d, \mu)$ be a nonempty metric measure space. Assume $p\in [1,\infty]$ and let $X(\mathcal M)\subset L^p(\mathcal M)$ be a Banach space continuously and locally compactly imbedded in $L^p(\mathcal M)$, i.e. for any relatively compact, connected subset $K$, the restriction $X(K)$ is compactly imbedded in $L^p(K)$. In the case of metric graphs a prototype would be $H^1(\mathcal G)$, but we will also apply this to higher-order Sobolev spaces $H^k(\mathcal G)$ with $k\in \mathbb N$. We will consider functionals $E\in C(X(\mathcal M), \mathbb R)$ with $E(0)=0$ for which the mapping $$t\mapsto E_t := \inf_{\substack{u\in X(\mathcal M)\\ \|u\|_p^p =t}}E(u)$$ is continuous for $t\ge 0$. It is easy to verify that these properties are satisfied by and . Our first result is a dichotomy result for a class of such functionals:
\[thm:introdich\] Let $c>0$ and $(\mathcal M, d, \mu)$ be a metric measure space. Let $X(\mathcal M)$ and $E$ be as above. Assume additionally
(1) $E$ is *strictly subadditive*, i.e. $$E_{t_1+t_2} < E_{t_1} + E_{t_2}, \quad \forall t_1,t_2 >0,$$
(2) $E$ is *weak limit superadditive*, i.e. for any weakly convergent sequence $u_n \rightharpoonup u$ in $X(\mathcal M)$ there exists a subsequence such that $$\limsup_{n\to \infty} E(u_n) \ge E(u) + \limsup_{n\to \infty} E(u_n-u)$$
If there exists a weakly convergent minimizing sequence $u_n \rightharpoonup u$ in $X(\mathcal M)$ for $$\label{eq:introdichminprob}
E_c= \inf_{\substack{u\in X(\mathcal M)\\ \|u\|_p^p =c}}E(u),$$ then either
(i) $u= 0$ or
(ii) $u_n \to u$ in $L^p(\mathcal M)$ and $u\neq 0$ is a minimizer of $E_c$.
Observe that under our assumptions, if $u_n \rightharpoonup 0$ in $X(\mathcal M)$, then $$u_n \to 0 \qquad \text{ on } L^p_{\text{loc}}(\mathcal G).$$ For functionals as considered in Theorem \[thm:introdich\] this means that either minimizing sequences of vanish to infinity, or converge towards minimizers in an appropriate sense. In the first case, we will say that minimizing sequences are vanishing. In this case one cannot deduce existence of minimizers without any additional information. This gives rise to our second result, which gives a sufficient condition for the existence of minimizers. Since the mass of vanishing sequences moves outside of any precompact set of $\mathcal M$ it is natural to separate the supports of the functions $u_n$ into an expanding part $O_n^{(1)}$ around some fixed precompact set and a part $O_n^{(2)}$ retreating to infinity via sequences of partitions of unity $$\{\Psi_{O_n^{(1)}}, \Psi_{O_n^{(2)}}\}.$$ If we then require that such a sequence of partitions of unity does not increase the energy functional $E$ under decomposition for vanishing sequences $u_n \rightharpoonup 0$ in Theorem \[thm:introdich\], i.e. $$\limsup_{n\to \infty} E(v_n) \ge \limsup_{n\to \infty} E(\Psi_{O_n^{(1)}} v_n)+ \limsup_{n\to \infty} E(\Psi_{O_n^{(2)}} v_n),$$ then we say $E$ is superadditive with respect to this *vanishing-compatible sequence of partitions of unity* and obtain the existence principle:
\[thm:introextenceprinciple\] Let $c>0$ and $K$ be a precompact subset of $\mathcal M$. Assume $E\in C(X(\mathcal M), \mathbb R)$ is
- strictly subadditive,
- weak limit superadditive,
- superadditive with respect to a vanishing-compatible sequence of partitions of unity, and
$$E_c < \widetilde E_c:= \lim_{n\to \infty} \inf_{\substack{u\in X(\mathcal M)\\ \operatorname{supp} u\subset \mathcal M\setminus K_n, \; \|u\|_{L^p}^p=c}}E(u),$$ where $K_n:= \{x\in \mathcal G | d(x, K)<n\}$ is the expanding ball around $K$. Assume there exists a weakly convergent minimizing sequence $u_n \rightharpoonup u$, then $u\neq 0$ is a minimizer of $E_c$.
It will turn out that the functionals and as considered in [@adami2016threshold] and [@cacciapuoti2017ground] satisfy the prerequisites of this theory. In fact, we will apply Theorem \[thm:introextenceprinciple\] to a natural generalization of , namely the higher-order stationary NLS energy functional
$$\label{eq:introkfunc}
E^{(k)}(u) = \frac{1}{2} \int_{\mathcal G} |u^{(k)}|^2 + V|u|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal G} |u|^q\, \mathrm dx, \qquad \begin{multlined}\mu>0,\quad 2<q<4k+2,\\
V\in L^2+L^\infty(\mathcal G)\end{multlined}$$
and consider the ground state problem $$\label{eq:introkground}
E^{(k)} = \inf_{\substack{u\in H^k(\mathcal G)\\ \|u\|_{L^2}^2 =1}} \frac{1}{2} \int_{\mathcal G} |u^{(k)}|^2 + V|u|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal G} |u|^q\, \mathrm dx.$$ Here, we write $V\in L^2+ L^\infty$ to mean that $V$ admits a decomposition $V=V_2+V_\infty$ such that $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$. When $k=1$ the energy functional reduces to the stationary NLS energy functional and we derive conditions for which the theory is applicable. Minimizers of satisfy the stationary higher-order nonlinear Schrödinger equation $$\begin{cases}
(-1)^k u_e^{(2k)} + \left ( V+\lambda\right ) u_e = \mu |u_e|^{q-1} u_e, \qquad \forall e \in \mathcal E\vspace{1em}\\
\begin{multlined}
u^{(i)} \in C(\mathcal G)\quad \text{ for all } i\le 2k-1 \text{ even} \qquad \text{\it (Continuity)}\\
\qquad \land \quad\sum_{e:e\succ \mathsf v} u^{(k)}_e(\mathsf v) =0 \quad \forall i\le 2k-1 \text{ odd } \;\forall \mathsf v\in V\\
\text{\it (Kirchhoff condition)}.
\end{multlined}
\end{cases}$$ While to the best of our knowledge this functional has not yet been considered on metric graphs, the stationary higher-order nonlinear Schrödinger equation on the real line of 4^th^ order is for instance related to traveling wave solutions of the nonlinear higher-order Schrödinger equation for the pulse envelope with higher-order dispersion as shown in [@kruglov2019exact §II]. For combinatorial locally finite graphs a discussion on the existence of solutions of the nonlinear higher-order Schrödinger equation of 4^th^ order was for instance considered very recently in [@hanshaozhao2019].
A minor difficulty in defining is that one needs to define higher-order Sobolev spaces $H^k(\mathcal G)$, as to date no standard way to define these spaces has emerged. We will define them in such a way that the formal Polylaplacian $$\label{eq:operators1}
\begin{gathered}
A=(-\Delta)^k+V\\ D(A) = H^{2k}(\mathcal G)
\end{gathered}$$ is a self-adjoint operator on $L^2(\mathcal G)$. We remark that the choice is not necessarily unique. A discussion of self-adjoint realizations for the Bilaplacian on metric graphs can be for instance found in [@gregorio2017bi]. One major result on the existence of minimizers in is as follows:
\[thm:intromainnowcloser\] Let $\mathcal G$ be a noncompact metric graph with finitely many edges. Assume that either
(i) there exists $V=V_2+V_\infty$ such that $V_2\in L^2(\mathcal G)$ and $V_\infty \in L^\infty (\mathcal G)$ and $$V_\infty(x)\to 0\qquad (x\to \infty)$$ on all edges of infinite length, or
(ii) $A=(-\Delta)^k+V$ admits a ground state, i.e. $\inf \sigma(A)$ is an eigenvalue.
Then $E^{(k)}$ is strictly subadditive, and if additionally $$\label{eq:intromainnowcloser}E^{(k)} < \widetilde{E^{(k)}} := \lim_{n\to \infty}\inf_{\substack{u\in H^k(\mathcal G)\\\|u\|_{L^2}^2=1,\; \operatorname{supp} u \subset \mathcal G\setminus K_n}} E^{(k)}(u),$$ then $E^{(k)}$ admits a minimizer.
Analogously to and , we will refer to the minimizers as ground states. Theorem \[thm:intromainnowcloser\] generalizes Theorem \[thm:introast2016\] since satisfies the prerequisites of Theorem \[thm:intromainnowcloser\]. Indeed, one can show with a test function argument (see Example \[ex:NLSclassic\]) that if $\mathcal G$ is a metric graph with finitely many edges then $$\widetilde{E_{\text{NLS}}}(\mathcal G) := \lim_{n\to \infty}\inf_{\substack{u\in H^1(\mathcal G)\\\|u\|_{L^2}^2=1,\; \operatorname{supp} u \subset \mathcal G\setminus K_n}} E_{\text{NLS}}(u, \mathcal G)= E_{\text{NLS}}(\mathbb R)$$ and we recover Theorem \[thm:introast2016\].
Under the assumption that eigenvalues exist below the essential spectrum, i.e. $$\inf \sigma((-\Delta)^k+ V) <\inf \sigma_{\text{ess}}((-\Delta)^k+V),$$ by a perturbation argument one can ensure that is satisfied for small nonlinearities and deduce a generalization of Theorem \[thm:introcfn2017\]:
\[thm:intromain1\] Let $\mathcal G$ be a noncompact metric graph with finite edge set. If $$\label{eq:introimportantinequality}
\inf \sigma ( (-\Delta)^k +V) < \inf \sigma_{\text{ess}}((-\Delta)^k +V)$$ then admits a ground state for sufficiently small $\mu>0$.
For $\mathcal G=\mathbb R$ and $V\equiv 0$, due to translation invariance the inequalities in Theorem \[thm:intromainnowcloser\] and Theorem \[thm:intromain1\] cannot be satisfied. However, here one can exploit the translation invariance to obtain a similar existence result, for which we are unaware of any reference in the literature:
\[thm:intromain11\] Let $V\equiv 0$ and $\mathcal G=\mathbb R$, then admits a ground state for all $\mu >0$. In particular, there exists a solution $u\in C^\infty(\mathbb R)$ to the stationary higher-order NLS equation $$(-1)^k u^{(2k)} + \lambda u = \mu |u_e|^{q-1} u,$$ with $\|u\|_{L^2(\mathbb R)}^2=1$ for some $\lambda \in \mathbb R$.
![An illustration for the classes of graphs that are considered. To the left a finite graph, sometimes referred to as starlike, consisting of a core graph and attached rays and to the right an infinite tree graph as an example for a locally finite graph, i.e. finite on any precompact set.[]{data-label="fig:introstuff"}](graph_example.pdf "fig:") ![An illustration for the classes of graphs that are considered. To the left a finite graph, sometimes referred to as starlike, consisting of a core graph and attached rays and to the right an infinite tree graph as an example for a locally finite graph, i.e. finite on any precompact set.[]{data-label="fig:introstuff"}](infinite_tree_more.pdf "fig:")
The results in Theorem \[thm:introast2016\] and Theorem \[thm:introcfn2017\] were shown for metric graphs with finitely many edges, which we refer to as finite graphs throughout the paper. Such graphs consist of a finite number (possibly zero) of edges of infinite length, i.e. half-lines, whch we call rays, and a complement, which is compact, and which we will call the core of the graph. [@cacciapuoti2017ground], [@cacciapuoti2018existence] call such graphs starlike graphs (see also Figure \[fig:introstuff\]). Our theory also allows us to handle more general graphs. For the next result we will consider a class of graphs with countable edge set, which is finite when restricted to any precompact subset. We will to refer to such graphs as locally finite graphs in the following. On locally finite graphs we consider the following variant of the NLS energy functional $$\label{eq:introNLSfunc}
E_{\text{NLS}}^{(\mathcal K)}(u) = \frac{1}{2} \int_{\mathcal G} \left |\left (i \frac{\mathrm d}{\mathrm dx} + M\right )u\right |^2+ V|u|^2 \, \mathrm dx - \frac{\mu}{p} \int_{\mathcal K} |u|^q \, \mathrm dx,\qquad \|u\|_{L^2}^2=1,$$ where $\mathcal K\subseteq \mathcal G$ is a subgraph of $\mathcal G$. In this context, we consider the magnetic Schrödinger operator with external potential $$\label{eq:operators2}
\begin{gathered}
A^M= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2+ V\end{gathered}$$ with its natural domain of definition, which we describe in detail in §5.1.
The following theorem is an analog of Theorem \[thm:intromain1\]. Interestingly, if one considers localized nonlinearities, i.e. $\mathcal K$ is a bounded subgraph of $\mathcal G$, then the existence result can be shown independent of the parameter $\mu>0$ in the nonlinearity:
\[thm:intromain2\] Let $\mathcal G$ be a noncompact locally finite graph and $\mathcal K\subseteq \mathcal G$ a connected subgraph. Suppose $A^M=(i \frac{\mathrm d}{\mathrm dx}+ M)^2+V$ admits a ground state that does not vanish identically on $\mathcal K$.
(i) If $\inf \sigma(A^M) < \inf \sigma_{\text{ess}}(A^M)$, then $$E_{\text{NLS}}^{(\mathcal K)}:= \inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_{L^2}^2=1}}\frac{1}{2} \int_{\mathcal G} \left |\left (i \frac{\mathrm d}{\mathrm dx} + M\right )u\right |^2+ V|u|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal K} |u|^q \, \mathrm dx$$ admits a minimizer for sufficiently small $\mu>0$.
(ii) If $\mathcal K$ is a bounded subgraph of $\mathcal G$, then minimizers exist for all $\mu >0$.
In §7 we are going to show that for a tree graph $\mathcal G$ the ground states of Schrödinger operators with magnetic potential do not vanish anywhere on $\mathcal G$. Then, given a *decaying potential* $V\in L^2+ L^\infty(\mathcal G)$ with $V=V_2+V_\infty$, such that $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$ satisfying $$\label{eq:introdecaying}
\sup_{x\in \mathcal G\setminus K} |V_\infty(x)| \to 0 \qquad (n\to \infty),$$ we show:
\[thm:introlastlastlast\] Let $\mathcal G$ be a noncompact locally finite tree graph with finitely many vertices of degree $1$ and suppose that $V\in L^2+ L^\infty$ satisfies . Then admits a minimizer if $$E_{\text{NLS}}^{(\mathcal K)} =\inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_{L^2}^2=1}}\frac{1}{2} \int_{\mathcal G} \left |\left (i \frac{\mathrm d}{\mathrm dx} + M\right )u\right |^2+ V|u|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal K} |u|^q \, \mathrm dx< E_{\text{NLS}}(\mathbb R).$$ In particular, if $$\inf \sigma\left ( \left ( i \frac{\mathrm d}{\mathrm dx} +M\right )^2 +V\right )<0,$$ then we have existence of minimizers of $E_{\text{NLS}}^{(\mathcal K)}$ for $0 < \mu \le (\Sigma_0/\gamma_q)^{\frac{3}{2}-\frac{p}{4}}$ as in Theorem \[thm:introcac2018\].
This paper is organized as follows. In §2 we introduce higher-order Sobolev spaces and obtain inequalities on Sobolev spaces on metric graphs including variants of Sobolev and Gagliardo–Nirenberg inequalities. We also discuss basic properties of these spaces such as density results and a characterization of $W^{1,\infty}$ via uniformly bounded Lipschitz functions. In §3 we build the existence theory that is the foundation for all of our results. Theorem \[thm:introdich\] is shown in §3.1 and Theorem \[thm:introextenceprinciple\] is shown in §3.2. In §3.3 we develop an existence result for translation-invariant functionals. In §4 we discuss the application of this existence theory to $E^{(k)}$ on finite graphs and prove Theorem \[thm:intromainnowcloser\] and Theorem \[thm:intromain1\]. In §4.1 we formalize the problem and show basic properties of the functional. In §4.2 we construct suitable partitions of unity and prove a decomposition formula in §4.3, which we use to show that the existence theory is applicable. In §4.4 we prove Theorem \[thm:intromain11\] and show in §4.5 that the existence theory is applicable for decaying potentials. In §5 we discuss existence results for ground states of $E_{\text{NLS}}^{(\mathcal K)}$, where $\mathcal K\subseteq \mathcal G$, on locally finite graphs. In §6 we discuss the energy inequality that is essential for the existence theory and relate it to spectral estimates by developing a Persson type theory for the operators in and . This will also conclude the proof of Theorem \[thm:intromain2\]. In particular, we discuss sufficient conditions for the potential $V$ such that is satisfied. In §7 we finish the paper with an application of the existence results to infinite metric trees via reduction of the problem to one without magnetic potential and prove Theorem \[thm:introlastlastlast\].
Let us finish the introduction by mentioning a few other recent results on related topics. For a general reference on metric graphs we refer to [@berkolaiko2013introduction]. For a broad overview of spectral theory of operators we refer to [@reedmethods]. We refer to [@exner2018spectral] for a recent article on spectral theory for metric graphs with infinitely many edges. The stationary energy functional $$E_{\text{NLS}}^{(\mathcal K)} (u) = \frac{1}{2} \int_{\mathcal G} |u'|^2 \, \mathrm dx - \frac{\mu}{q} \int_{\mathcal K} |u|^q\, \mathrm dx, \qquad \|u\|_{L^2}^2 =1.$$ with $\mathcal K=\mathcal G$ was considered in [@adami2012stationary], [@adami2015nls], [@adami2016threshold], [@adami2017negative] among others. A variant of the problem with localized nonlinearities in the $L^2$-subcritical case was considered in [@tentarelli2016nls] and for the $L^2$-critical case extended in [@dovetta2018ground] and [@dovetta20182], where the domain of integration in the nonlinearity is taken to be a bounded subgraph $\mathcal K$. A very recent survey on results on the stationary NLS energy functional with localized nonlinearity can be found in [@borrelli2019overview]. The main tool for the existence theorems regarding existence of ground states of the stationary NLS energy functional is the constructions of so called competitors, i.e. test functions or a sequence of test functions that establish the energy inequality of Theorem \[thm:introast2016\]. Recently, classes of graphs that do not necessarily consist of finitely many edges have also been considered. For instance, [@dovetta2019nls] deals with a certain class of infinite tree graphs, which fall into the category of the locally finite graphs that we consider here. We would also like to mention the results obtained by [@Akduman2019] for the NLS energy functional with growing potentials for a class of general metric graphs satisfying certain volume growth assumptions using a generalized Nehari approach.
**Acknowledgments.** I want to thank James Kennedy, who helped me shape the article and for all the helpful discussions with him. I thank Hugo Tavares, who suggested studying the NLS energy functional and provided me with references and overall knowledge on the topic and giving a few helpful suggestions. I thank Marcel Griesemer for helpful discussions as the approach on the real line is loosely based on discussions we had in the past in the case of the NLS energy functional on the real line. The work was supported by the Fundação para a Ciência e a Tecnologia, Portugal, via the program “Bolseiro de Investigação”, reference PD/BD/128412/2017, and FCT project UID/MAT/00208/2019.
Sobolev spaces on graphs
========================
In this section we define metric graphs and define Sobolev spaces on metric graphs and prove Sobolev inequalities on these spaces.
Metric graphs
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Let $\mathcal G=(\mathcal V,\mathcal E)$ be a metric graph, where each edge $e\in \mathcal E$ is associated with an interval $I_e$ of length $l_e\in (0,\infty]$, where $I_e=[0,\infty)$ if $l_e=\infty$ and $I_e=[0, l_e]$ otherwise. We assume every vertex to be at least of degree one. For every $e\in {{\mathcal E}}$ joining two vertices we associate each of the vertices to $0$ and $l_e<\infty$ respectively on the interval $I_e$. However, we always assume that the half-line $I_e=[0,\infty)$, which we also call a ray, is attached to the remaining part of the graph at $x_e=0$, and the vertex of the graph corresponding to $x_e=\infty$ is called a vertex at infinity. In particular, there are no edges between vertices at infinity. We denote by $\mathcal E_\infty\subset \mathcal E$ the set of all rays. A connected metric graph $\mathcal G$ admits a natural structure of a metric space. The metric is the shortest distance measured along the edges of the graph. A more detailed introduction to metric graphs can be found for instance in [@berkolaiko2013introduction].
We consider two classes of noncompact graphs:
\[df:localfin\] Let $\mathcal G$ be a connected metric graph. Then we say
(1) $\mathcal G$ is a *finite graph* if there are at most finitely many edges.
(2) $\mathcal G$ is a *locally finite graph* if ${{\mathcal E}}$ is a countable set such that any bounded subset of the graph intersects at most finitely many edges
We will always take our graphs to be connected, locally finite noncompact metric graphs. Note that a finite graph $\mathcal G$ is compact if and only if the graph $\mathcal G$ does not admit any rays, that is, there are no edges of $\mathcal G$ that are half-lines. Note that a locally finite graph is compact, if and only if the graph is bounded. We see immediately that all compact locally finite metric graphs are finite, and all finite graphs are locally finite. In particular for compact graphs the notions in Definition \[df:localfin\] coincide.
First-order Sobolev spaces
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Let $\mathcal G$ be any locally finite graph. Here and in the rest of the article denote $$\label{eq:short}
u_e:=u\big |_e.$$ We denote by $C(\mathcal G)$ the set of continuous complex-valued functions in $\mathcal G$ and define for $1\le p <\infty$ $$\begin{aligned}
L^p(\mathcal G) &= \left \{u\in \bigoplus_{e\in {{\mathcal E}}} L^p(I_e)\bigg | \|u\|_p^p:=\sum_{e\in {{\mathcal E}}} \left \|u_e \right \|_{p}^p<\infty\right \}, \\
W^{1,p}(\mathcal G)&=\left \{u\in C(\mathcal G) \bigg| u_e \in W^{1,p}(I_e)\; \land\; \|u\|_{W^{1,p}}^p:=\sum_{e\in {{\mathcal E}}} \|u_e \|_{W^{1,p}(I_e)}^p <\infty \right \}.
\end{aligned}$$ Then we set $H^1(\mathcal G) = W^{1,2}(\mathcal G)$ as usual.
For $q=\infty$ we need to adapt the definition above slightly: $$\begin{aligned}
L^\infty(\mathcal G) &= \left \{u\in \bigoplus_{e\in {{\mathcal E}}} L^\infty(I_e)\bigg | \|u\|_\infty:=\max_{e\in {{\mathcal E}}} \left \|u_e \right \|_{\infty}<\infty\right \}, \\
W^{1,\infty}(\mathcal G)&=\bigg \{u\in C(\mathcal G) | u_e \in W^{1,\infty}(I_e)\; \\
&\qquad\qquad \land\; \|u\|_{W^{1,\infty}}:=\max_{e\in {{\mathcal E}}} \|u_e \|_{W^{1,\infty}(I_e)} <\infty \bigg \}.
\end{aligned}$$
Gagliardo–Nirenberg inequality with magnetic potential
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The goal of this section is to show a Gagliardo–Nirenberg inequality for locally finite, connected metric graphs. With future applications in mind we actually consider a modified Gagliardo–Nirenberg inequality:
\[cor:Gagliardo-nirenberg\] Let $\mathcal G$ be a locally finite, connected metric graph and $M\in C(\mathcal G)$. For $p\in [2,\infty)$ there exists a constant $C>0$ independent of $M$ such that $$\label{eq:Gagliardo-nirenberg}
\|u\|_{p}^p \le C \left \|\left ( i \frac{\mathrm d}{\mathrm dx}+M\right )u\right \|_{2}^{\frac{p-2}{2}} \| u\|_{2}^{\frac{p+2}{2}},$$ for all $u\in H^1(\mathcal G)$.
The inequality reduces to the usual Gagliardo–Nirenberg inequality when $M\equiv 0$.
Suppose $\mathcal G$ is a tree graph at first. Then using the unitary gauge transform $G: H^1(\mathcal G) \to H^1(\mathcal G)$ (see also §7 for details) we deduce that is equivalent to $$\label{eq:Gagliardo-nirenberg2}
\|u\|_{p}^p \le C \left \|u'\right \|_{2}^{\frac{p-2}{2}} \| u\|_{2}^{\frac{p+2}{2}},$$ which can be shown via symmetrization methods as shown[^1] in [@adami2015nls]. In particular, the constant $C>0$ can be chosen independent of $M$. Cutting the graph at a discrete set of points on the metric graph, i.e. we can find a tree graph $\widetilde{\mathcal G}$ such that identifying a discrete set of points on the graph results in a graph isometric isomorph to $\mathcal G$. Moreover, since $H^1(\mathcal G)$ imbeds isometrically into $H^1(\overline{\mathcal G})$, also holds for $H^1(\widetilde{\mathcal G})$ and the constant $C>0$ can be chosen independent of $M\in C(\mathcal G)$.
Similarly, one can also show the following Sobolev inequality.
Let $\mathcal G$ be a locally finite, connected metric graph and $M\in C(\mathcal G)$. Let $p\in [2,\infty]$ then there exists a constant $C>0$ independent of $M$ such that $$\label{eq:Sobolevinequality}
\|u\|_{p} \le C \left (\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) u\right |^2\, \mathrm dx+\int_{\mathcal G} |u|^2\, \mathrm dx\right )^{1/2},$$ for all $u\in H^1(\mathcal G)$.
The aproach is similar as before, indeeed we can use the known result in absence of $M$ and use a gauge transform to show that holds with a constant $C>0$ independent of the potential $M\in C(\mathcal G)$.
Higher-order Sobolev spaces
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In this section we introduce the notion of higher-order Sobolev spaces on graphs for $p\in [1,\infty)$. Let $\mathcal G$ be a locally finite metric graph. One naive way of doing so is simply defining it analogously as in $W^{1,p}(\mathcal G)$ $$\widetilde {W^{k,p}}(\mathcal G) := \left \{u\in C(\mathcal G) \bigg | u_e \in W^{k,p} (I_e)\quad \forall e\in E\; \land\; \|u \|_{W^{k,p}}^p:=\sum_{e\in E} \|u_e \|_{W^{k,p}(I_e)}^p <\infty \right \}$$ Then for $u\in \widetilde{W^{k,p}}(\mathcal G)$ we always have $u_e \in C^{k-1}(I_e)$ for all $e\in \mathcal E$. However, we also want to specify a condition on the higher-order derivatives. We define $$\begin{gathered}
W^{k,p}(\mathcal G) = \{u\in \widetilde{W^{k,p}}(\mathcal G)| u^{(j)} \in C(\mathcal G)\quad \forall j\le k-1 \text{ even}\\
\land \quad\sum_{e:e\succ \mathsf v} \frac{\partial^j}{\partial\nu^j}u_e(\mathsf v) =0 \quad \forall j\le k-1 \text{ odd }\, \forall \mathsf v\in V\},\end{gathered}$$ where $e:e\succ \mathsf v$ denotes the set of edges $e$ adjacent to a vertex $\mathsf v$. This definition is natural in the sense that if we consider a dummy vertex $\hat v$ of degree $2$, i.e. subdividing an edge $e\in \mathcal E$ connecting two vertices $v_1, v_2$ into two edges $e_1, e_2$ connecting $v_1, \hat v$ and $\hat v, v_2$ respectively such that the total length of the graph is preserved, then the Kirchhoff condition simply reduces to a continuity statement of the derivatives. As usual we define $\widetilde{H^k}(\mathcal G)= \widetilde{W^{k,2}}(\mathcal G)$ and $H^k(\mathcal G)= W^{k,2}(\mathcal G)$.
While the Sobolev spaces as defined here are domains of self-adjoint realizations of differential operators on $L^2(\mathcal G)$, the definitions are not necessarily canonical. We refer to [@gregorio2017bi] for a discussion on self-adjoint extension of the Bilaplacian, and a discussion for $W^{2,p}$ spaces on graphs.
In this context, we are going to define some useful related spaces: $$\label{eq:sobolev0}
\begin{gathered}
\widetilde{W^{k,p}_{0}}(\mathcal G) := \{u\in \widetilde{W^{k,p}}(\mathcal G)| u^{(l)}(\mathsf v)=0, \quad \forall 1\le l\le k-1, \quad \forall \mathsf v\in \mathcal V\}\\
\widetilde{W^{k,p}_{c}} (\mathcal G):= \left \{u\in \widetilde{W^{k,p}_0}(\mathcal G)| \operatorname{supp}(u) \text{ compact}\right \}.
\end{gathered}$$ Of special importantance will be the following test function spaces: $$\begin{split}
\widetilde{C^\infty(\mathcal G)} &:= \bigcap_{k\in \mathbb N}\widetilde{W^{k, \infty}}(\mathcal G)\\
\widetilde{C^\infty_b(\mathcal G)} &:= \bigcap_{k\in \mathbb N}\widetilde{W_0^{k, \infty}}(\mathcal G)\\
\widetilde{C^\infty_c(\mathcal G)} &:= \bigcap_{k\in \mathbb N}\widetilde{W_c^{k, \infty}}(\mathcal G).
\end{split}$$
Higher-order Gagliardo–Nirenberg inequality for finite metric graphs
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Let $\mathcal G$ be a finite metric graph. Consider the norm on $H^k(\mathcal G)$ defined as $$|u|_{H^k} := \left (\int_{\mathcal G} |u^{(k)}|^2+ |u|^2\, \mathrm dx \right )^{1/2}.$$ Then due to the Gagliardo–Nirenberg interpolation inequality on intervals (see e.g. [@leoni2017first Theorem 7.41]) applied edgewise $$\label{eq:ineedthis}
|u|_{H^k}^2 \le \|u\|_{H^k}^2\le C |u|_{H^k}^2$$ and we conclude that $\|\cdot\|_{H^k}$ and $|\cdot|_{H^k}$ are equivalent norms in $H^k(\mathcal G)$.
\[prop:higherordergag\] Let $\mathcal G$ be a finite metric graph. Then $$\label{eq:propgagk}
\|u\|_{p}^p \le C \|u\|_2^{\frac{(2k-1)p+2}{2k}} |u|_{H^k}^{\frac{p-2}{2k}}$$
From the Gagliardo–Nirenberg inequality on metric graphs and Gagliardo–Nirenberg interpolation inequality on intervals we compute $$\begin{aligned}
\|u\|_{p}^p &\le C_1 \|u\|_{2}^{\frac{p}{2}+1}\|u'\|_{2}^{\frac{p}{2}-1}\\
&\le C_k \left \|u\right \|_{2}^{\frac{(2k-1)p+2}{2k}} \big |u\big |_{H^{k}}^{\frac{p-2}{2k}}.
\end{aligned}$$
On the density of Sobolev spaces
--------------------------------
\[prop:density\] Let $\mathcal G$ be a finite, connected metric graph and $p\in [1,\infty)$, then $W^{m,p}(\mathcal G)$ is dense in $W^{n,p}(\mathcal G)$ for $m\ge n\ge 1$.
For $n=0$ this corresponds to the fact that $W^{m,p}$ is dense in $L^p$ for $m\ge 1$.
It suffices to prove that $W^{k+1,p}(\mathcal G)$ is dense in $W^{k,p}(\mathcal G)$. To this end, let $u\in W^{k}(\mathcal G)$ arbitrary and $u_n$ be an edgewise approximating sequence in $\oplus_{e\in \mathcal E} C^\infty(I_e)\cap W^{k+1,p}(I_e)$ such that $$\label{eq:ineedthisnow}
\sum_{e\in \mathcal E} \left \|u_n -u\big |_e\right \|_{W^{k,p}} \le \frac{1}{2^n}$$ for all $n\in \mathbb N$. The general idea is to construct sequences $v_n \in \oplus_{e\in \mathcal E} W^{k+1,p}(I_e)$ such that $u_n + v_n \in W^{k+1,p}$ and $$u_n + v_n \stackrel{W^{k,p}}\longrightarrow u\qquad (n\to \infty).$$ For fixed $\mathsf v\in \mathcal V$ and for $n$ satisfying $$\frac{2}{n} \le \min_{e\in \mathcal E} |I_e|$$ for all $e\succ \mathsf v$ and $\hat k \in \{0, \ldots, k\}$ we define $$\label{eq:testfunctions}
v_{n,\hat k, \mathsf v}(x)\big |_e=\begin{cases}
\frac{c_{n, \hat k, \mathsf v}}{\hat k!} x_{\mathsf v}^{\hat k} \left (1-nx_{\mathsf v}\right )^{k+1}, &\quad \text{for }x\in e \text{ with } x_{\mathsf v}:=\operatorname{dist}(x,\mathsf v)\le \frac{1}{n} \\
0, &\quad \text{otherwise}.
\end{cases}$$ where $c_{n, \hat k, \mathsf v}$ is given by $$\begin{aligned}
\text{for }\hat k=0:\qquad c_{n, 0, \mathsf v}&= u-u_n \big |_e (0_{\mathsf v})\\
\text{for } 1\le \hat k \le k-1:\qquad c_{n, \hat k, \mathsf v} &= u^{(\hat k)}-u_n^{(\hat k)} \big |_e(0_{\mathsf v}) - \sum_{\ell=0}^{\hat k-1} (k+1)_{\ell} (-n)^{\ell} c_{n,\ell, \mathsf v}.
\end{aligned}$$ We can extend the functions $v_{n,\hat k, \mathsf v}$ by zero on the rest of the graph. With the Leibniz rule for $1\le \ell \le k+1$ we compute $$\label{eq:leibnizcomp}
v_{n,\hat k, v}^{(\ell)} (x)\big |_e= \chi_{\{x_{\mathsf v}\le \frac{1}{n}\}}\sum_{m=0}^\ell \frac 1{\hat k!}{\binom{\ell}{m}} c_{n, \hat k, \mathsf v} (-n)^{\ell-m}(\hat k)_m (k)_{\ell-m} x_{\mathsf v}^{\hat k - m}(1-n x_{\mathsf v})^{k+m+1-\ell}$$ Then $$\tilde v_n := \sum_{\ell=0}^{k-1}\sum_{\mathsf v\in \mathcal V} v_{n,\ell, \mathsf v}$$ satisfies $\tilde v_n \in \oplus_{e\in \mathcal E} W^{k+1,p}(I_e)$ and $u_n+\tilde v_n \in W^{k,p}(\mathcal G)$ since $u_n+\tilde v_n$ coincides in all $k-1$ derivatives with $u$ by construction. Indeed, the restrictions of the $k$^th^ derivatives at the vertices are of rank $\le 2|\mathcal E|$. Then we can find $c_{n,k,\mathsf v}$ for all $\mathsf v\in \mathcal V$ $$v_{n,k,\mathsf v}\big |_e (x) =\begin{cases}
\frac{c_{n, k, \mathsf v}}{k!} x_{\mathsf v}^{k} \left (1-\max\{n, c_{n,k, \mathsf v}^{2}\}x_{\mathsf v}\right )^{k+1}, &\quad \text{for }x\in e \text{ with } \\
&x_{\mathsf v}\le \min\{n^{-1}, c_{n,k, \mathsf v}^{-2}\} \\
0, &\quad \text{otherwise}.
\end{cases}$$ such that $$u_n + \tilde v_n + \sum_{\mathsf v\in \mathcal V} v_{n,k,\mathsf v}\in W^{k+1,p}(\mathcal G).$$ By assumption we deduce by applying the Sobolev imbedding edgewise $$\label{eq:additionalinequ}
\sum_{e\in \mathcal E} \left \|u_n-u\big |_e\right \|_{C^{k-1}} \le \frac{C}{2^n}$$ for all $n\in \mathbb N$ and some $C>0$ and satisfies by construction $$\label{eq:trickher}
(-n)^\ell c_{n,\hat k, \mathsf v} \to 0 \qquad (n\to \infty)$$ for all $1\le \ell\le k$ and $\mathsf v\in \mathcal V$. By a change of variables we then compute for $0\le m \le \ell \le k$ $$\begin{gathered}
n^{\ell-m} \int_{I_e} x_{\mathsf v}^{\hat k-m} (1-n x_{\mathsf v})^{k+m+1-\ell}\, \mathrm dx_{\mathsf v}= n^{\ell-1 -\hat k} \int_0^1 t^{\hat k-m} (1-t)^{k+m+1-\ell}\, \mathrm dt\\
\begin{multlined}
c_{n,k, \mathsf v}\max\{n, c_{n,k, \mathsf v}^2\}^{\ell-m} \int_{I_e} x_{\mathsf v}^{k-m} (1-\max\{n, c_{n,k, \mathsf v}^2\} x_{\mathsf v})^{k+m+1-\ell}\, \mathrm dx_{\mathsf v}\\
= c_{n,k, \mathsf v}\min\{n^{-1}, c_{n,k, \mathsf v}^{-2}\}^{k+1-\ell} \int_0^1 t^{k-m} (1-t)^{k+m+1-\ell}\, \mathrm dt \longrightarrow 0 \\
(n\to\infty)
\end{multlined}\end{gathered}$$ and with and we conclude $$\begin{aligned}
&\left \|u- \left [u_n + \tilde v_n + \sum_{\mathsf v\in \mathcal V} v_{n,k,\mathsf v} \right ]\right \|_{W^{k,p}}\\
&\qquad \qquad\le \left \|u- u_n\right \|_{W^{k,p}} + \left \|\tilde v_n + \sum_{\mathsf v\in \mathcal V} v_{n,k,\mathsf v}\right \|_{W^{k,p}}\longrightarrow 0 \qquad (n\to \infty).
\end{aligned}$$
\[prop:densitycompact\] Let $\mathcal G$ be a locally finite, connected metric graph and $p\in [1,\infty)$. Then $$W^{1,p}_c(\mathcal G)= \{u\in W^{1,p}(\mathcal G)| \operatorname{supp} u \text{ is bounded}\}$$ is dense in $W^{1,p}(\mathcal G)$.
Let $K$ be a bounded, connected subgraph of $\mathcal G$. For $R>0$ set $$K_R :=\{x\in \mathcal G| \operatorname{dist}(x, K)<R\}.$$ We define the cut-off functions $\psi_n$ via $$\widetilde{\psi_n} := \frac{1}{n}\max \{n, \operatorname{dist}(x,K_n)\},\quad \psi_n:= 1- \widetilde{\psi_n}$$ For all $u\in W^{1,p}(\mathcal G)$ one then computes $$\begin{aligned}
&\limsup_{n\to \infty}\|u- \psi_n u\|_{W^{1,p}}^p = \limsup_{n\to \infty}\left [\int_{\mathcal G} \left |\frac{\mathrm d}{\mathrm dx} \widetilde{\psi_n} u\right |^p\, \mathrm dx+ \int_{\mathcal G} \left |\widetilde {\psi_n} u\right |^p\, \mathrm dx\right ]\\
&\qquad\le \limsup_{n\to \infty} \left [\frac{2^p}{n^p} \int_{\mathcal G\setminus K_n} |u|^p\, \mathrm dx +2^p \int_{\mathcal G\setminus K_n} \left |\widetilde \psi_n u \right |^p\, \mathrm dx + \int_{\mathcal G \setminus K_n} \left |\widetilde \psi_n u\right |^p\, \mathrm dx\right]=0,
\end{aligned}$$ where in the equation we used $$\int_{\mathcal G\setminus K_n} |\widetilde{\psi_n} u|^p \, \mathrm dx \le \int_{\mathcal G\setminus K_n} |u|^p \, \mathrm dx \to 0 \qquad (n\to \infty).$$ As such $\psi_n u \to u$ in $W^{1,p}(\mathcal G)$ as $n\to \infty$.
\[prop:densitylocfinite\] Let $\mathcal G$ be a locally finite, connected metric graph and $p\in [1,\infty)$. Then $$W^{2,p}_c(\mathcal G)= \{u\in W^{2,p}(\mathcal G)| \operatorname{supp} u \text{ is bounded}\}$$ is dense in $W^{1,p}$.
Let $u\in W^{1,p}(\mathcal G)$. By Lemma \[prop:densitycompact\] we can find a sequence $u_n\in W^{1,p}_c(\mathcal G)$ with $u_n \to u$ in $W^{1,p}$. Then by Proposition \[prop:density\] for each $n$ we find a sequence $u_{n,m} \in W^{2,p}(\mathcal G)$, after extending by zero on the whole graph, converging towards $u_n$ in $W^{1,p}(\mathcal G)$ as $m\to \infty$. Then one can construct a sequence in $W^{2,p}_c(\mathcal G)$ converging to $u$ in $W^{1,p}$ by a diagonal argument.
Characterization of $W^{1,\infty}$
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We give a characterization of $W^{1, \infty}$ on locally finite, connected metric graphs in the following:
\[prop:lipschitz\] Let $\mathcal G$ be a locally finite, connected metric graph. Then $W^{1,\infty}(\mathcal G)= C^{0,1}_b(\mathcal G)$ is the set of uniformly bounded, Lipschitz continuous functions.
Assume $u\in W^{1,\infty}(\mathcal G)$. Let $x,y\in \mathcal G$ and $\gamma$ be a path of length $L(\gamma)$ connecting $x,y$, parametrized by arc length. In the first step let us assume $u\in C^1$ edgewise, then using the continuity of $u$ we have $$|u(x)-u(y)| \le \int_0^{L(\gamma)} |u'(\gamma)|\, \mathrm d|\gamma| \le \max_{t} |u'(\gamma(t)| L(\gamma).$$ Due to density this holds also for $W^{1,\infty}(\mathcal G)$. Taking the infimum over all paths connecting $x,y$ we conclude $$|u(x)-u(y)|\le \|u'\|_{\infty} \operatorname{dist}(x,y)$$ and thus $u\in C^{0,1}_b(\mathcal G)$. On the other hand, let $f\in C^{0,1}_b(\mathcal G)$, then $$\frac{|u(x)-u(y)|}{\operatorname{dist}(x,y)} \le L$$ for some constant $L>0$. On each edge $e\in \mathcal E$ then $u\in W^{1,\infty}(I_e)$ and $u'$ exists a.e. and $$\|u'\|_{\infty} \le L.$$ We conclude $u\in W^{1,\infty}(\mathcal G)$ since $u$ is also uniformly bounded by assumption.
A general Existence Principle
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In this section we derive an existence theory for ground states of functionals as in and . To do so, we derive a more general existence principle for functionals on function spaces on metric measure spaces, which we will apply later to the functionals introduced before to discuss the existence of minimizers.
A Dichotomy Result
------------------
In the following we work with a slightly more abstract space $X(\mathcal M)$, namely a function space on a metric measure space $\mathcal M$.
\[as:assumption1\] Let $p\in [1,\infty)$. Let $(\mathcal M, d)$ be a metric space with a locally finite Borel measure $\mu$ on $\mathcal M$. Assume $X= X(\mathcal M)\subset L^p(\mathcal M)$ is a nontrivial Banach function space continuously and locally compactly imbedded in $L^p(\mathcal M)$, i.e. $\mathcal M$ restricted to $$K_R(y) := \{x\in \mathcal M|\operatorname{dist}(x,y) \le R\}$$ is compactly imbedded in $L^p(K_R(y))$ for all $R>0$ and $y\in \mathcal M$.
Our prototype to satisfy Assumption \[as:assumption1\] is $X(\mathcal G)= H^1(\mathcal G)$ where $\mathcal G$ is a connected, locally finite metric graph. However, it is for instance also satisfied by $X(\Omega) = H^1(\Omega)$ for a domain $\Omega \subset \mathbb R^n$ with $n\in \mathbb N$.
Let $p\ge 2$ and let $\mathcal M$ and $X=X(\mathcal M)$ be as in Assumption \[as:assumption1\]. Let $E \in C(X(\mathcal M), \mathbb R)$ and $$E_t := \inf_{\substack{u\in X(\mathcal M)\\ \|u\|_{p}^p=t}} E(u)$$ be bounded from below for each $t\ge 0$ and $E(0)=0$. We say:
(1) $t\mapsto E_t$ is *strictly subadditive* if $$E_{t_1+t_2} < E_{t_1} + E_{t_2}, \qquad \forall t_1, t_2 >0.$$
(2) $E$ is *weak limit superadditive* in $X$ if for all $c>0$ any weakly convergent minimizing sequence $u_n \rightharpoonup u$ in $X(\mathcal M)$ of $E_c$ satisfies up to a subsequence $$\limsup_{n\to \infty} E(u_n) \ge E(u) + \limsup_{n\to \infty} E(u_n -u).$$
\[thm:main1\] Let $p\in [2,\infty)$, $c>0$ and let $\mathcal M$, $X=X(\mathcal M)$ be as in Assumption \[as:assumption1\]. Let $E\in C(X(\mathcal M), \mathbb R)$ be a weak limit superadditive functional in $X$. Let $$t\mapsto E_t = \inf_{\substack{u\in X(\mathcal M)\\ \|u\|_{p}^p=t}} E(u)$$ be a strictly subadditive, continuous function of $t\in [0,c]$. Let $u_n$ be a minimizing sequence of $E_c$, and assume there exists $u\in X$ such that up to a subsequence $u_n \rightharpoonup u$. Then either $u\equiv 0$ or $u_n \to u$ in $L^p(\mathcal M)$ and $u\neq 0$ is a minimizer.
Theorem \[thm:main1\] gives rise to a dichotomy. If the requirements of Theorem \[thm:main1\] are satisfied, then a minimizing sequence satisfies either $u_n \rightharpoonup 0$ in $X$ or there exists an $L^p$ convergent subsequence converging to a minimizer of $E_c$. Up to a subsequence, this implies pointwise convergence almost everywhere. On the other hand, $u_n \rightharpoonup 0$ in $X(\mathcal M)$ implies $\|u_n\|_{L^p(K)}\to 0$ on any bounded subset $K$ of $\mathcal G$, but since $\|u_n\|_{p}^p=c$ for all $n\in \mathbb N$ the mass needs to move outside any compact set.
In virtue of Theorem \[thm:main1\] we say a minimizing sequence of $E_c$ is *vanishing* if $u_n \rightharpoonup 0$ in $X$ and *non-vanishing* otherwise.
Let $u_n \rightharpoonup u$ in $X$ with $u\neq 0$. Then since $u_n \to u$ in $L^{p}_{\text{loc}}$ we deduce $u\neq 0$ and $$c\ge \|u\|_{p}^p>0.$$ Up to a subsequence $u_n \to u$ pointwise almost everywhere, and from the Brezis-Lieb Lemma (see [@brezis1983relation]) we conclude $$\|u\|_p^p + \limsup_{n\to \infty} \|u_n-u\|_p^p =c.$$ By weak limit superadditivity and strict subadditivity of $t\mapsto E_t$ we deduce that up to a subsequence $$\begin{aligned}
E_c &\ge E(u) + \limsup_{n\to \infty} E(u-u_n)\\
&\ge E_{\|u\|_p^p} + \limsup_{n\to \infty} E_{\|u-u_n\|_p^p}\\
&\ge E_{\|u\|_p^p} + E_{\limsup_{n\to \infty} \|u_n-u\|_p^p} \ge E_c.
\end{aligned}$$ where equality is only attained when $\|u\|_p^p=c$ and $\limsup_{n\to \infty} \|u_n-u\|_p^p=0$. Thus $\|u\|_p^p=c$ and we conclude $$E_c = E(u)$$ and $u$ is a minimizer of $E_c$.
Vanishing sequences and Ionization Energies
-------------------------------------------
As in the previous section we consider $\mathcal M$ to be a metric measure space and $X(\mathcal M)\subset L^p(\mathcal M)$ to be a Banach space which is locally compactly imbedded in $L^p(\mathcal M)$. In the following we want to introduce partitions of unity and therefore assume the following:
\[as:assumption2\] Let $(\mathcal M, d)$ be a metric space with locally finite Borel measure $\mu$ on $\mathcal M$. We assume $X(\mathcal M)$ to be a subspace of the space of $\mu$-measurable functions on $\mathcal M$ and $Y(\mathcal M)$ to be a set of real-valued functions on $\mathcal M$ such that $X(\mathcal M)$ is invariant with respect to multiplication of elements in $Y(\mathcal M)$.
Let $\mathcal G$ be a locally finite metric graph. If $X(\mathcal G)= H^1(\mathcal G)$ and $Y=W^{1,\infty}(\mathcal G)$, then $X(\mathcal G), Y(\mathcal G)$ are imbedded in $C(\mathcal G)$ and $X(\mathcal G)$ is invariant by multiplications of $Y(\mathcal G)$, i.e. elements in $X(\mathcal G)$ are invariant by pointwise multiplication of elements in $Y(\mathcal G)$, and for $f\in W^{1,\infty}(\mathcal G)$ and $g\in H^1(\mathcal G)$ $$(fg)'= f' g+ g'f.$$
In this section we show that the existence of vanishing sequences gives a bound from below on the ground state energy $E_c$, which allows us, under stronger assumptions, to deduce an existence result from Theorem \[thm:main1\]. Let us first introduce partitions of unity on metric spaces.
Let $Y$ be as in Assumption \[as:assumption2\]. Assume $\cup_{O\in \mathcal O} O=\mathcal G$ is a locally finite open covering $\mathcal O$ of $\mathcal M$. Then we say a family of nonnegative functions $\psi_O\in Y(\mathcal G)$ is a partition of unity subordinate to $\mathcal O$ if $$\operatorname{supp} \psi_O \subset O, \quad \forall O\in \mathcal O \quad\land \quad \bigcup_{O\in \mathcal O} \operatorname{supp} \psi_O = \mathcal G \quad \land \quad 0 \le \psi_O \le 1$$ and $\sum_{O\in \mathcal O} \psi_O(x)\neq 0$ for all $x\in \mathcal M$ and $$\Psi_O(x)= 1, \qquad \forall x\in \operatorname{supp} \Psi_O \setminus \bigcup_{\widehat O\in \mathcal O \setminus \{O\}} \operatorname{supp} \Psi_{\widehat{O}}.$$
In the following we define for $R> 0$ the open and closed $R$-neighborhoods of a subset $K\subset \mathcal M$ by $$\label{eq:core}
\begin{aligned}
K_{R} &:= \{x\in \mathcal M| \operatorname{dist} (x, K)< R\}\\
\overline{K_{R}} &:= \{x\in \mathcal M|\operatorname{dist} (x, K) \le R\}.
\end{aligned}$$
\[ex:first\] Let $\mathcal G$ be a locally finite graph. Let $K$ be some compact, connected subgraph. A simple example for a partition of unity in $W^{1,\infty}(\mathcal G)= C^{0,1}_b(\mathcal G)$ on a locally finite metric graph subordinate to $K_2$, $\mathcal G\setminus K_1$ is given by $$\psi(x) = \max\{\operatorname{dist}(\mathcal G\setminus K_{2}, x), 1\}, \qquad \widetilde{\psi}(x) = 1-\psi.$$
In the following we will characterize a property of functionals with regards to partitions of unity.
Let $k \in \mathbb N$ and let $\mathcal M$ be a metric space. Let $K$ be a bounded subset of $\mathcal M$ and $K_n$ be defined by for $n\in \mathbb N$. We say a sequence of open coverings $\mathcal O_n=\{O_n^{(1)}, \ldots, O_n^{(k)}\}$ consisting of $k$ open subsets (not necessarily connected) is *vanishing-compatible*, if $$K_n\cap O_n^{(i)}= \emptyset, \qquad \forall i\in \{2,\ldots, k\}$$ and $O_n^{(1)}$ is bounded for all $n$.
In particular, $K_n \subset O_n^{(1)}$. That is, for a sequence of open coverings $\mathcal O_n=\{O_n^{(1)}, \ldots, O_n^{(k)}\}$ all its members except $O_n^{(1)}$ move away from $K$. Furthermore, this notion does not depend on the choice of $K$, i.e. up to a subsequence any sequence of open coverings is vanishing-compatible for any other $K$.
Let $k\in \mathbb N$ and $\mathcal O_n=\{O_n^{(1)}, \ldots, O_n^{(k)}\}$ be a vanishing-compatible sequence of open coverings. Then we say $E\in C(X(\mathcal M), \mathbb R)$ is $k$-superadditive with respect to a sequence of partitions of unity $$\{\psi_{O}\}_{O\in \mathcal O_n}=\left \{\psi_{O_n^{(1)}},\ldots, \psi_{O_n^{(k)}}\right \}$$ if for any vanishing sequence $(v_n)$ up to a subsequence $$\limsup_{n\to \infty} E(v_n) \ge \sum_{i=1}^k \limsup_{n\to \infty} E(\psi_{O_{n}^{(i)}} v_n).$$
\[ex:unity\] Let $\mathcal G$ be a locally finite metric graph and let $K$ be a compact subgraph of $\mathcal G$. Recall on $\mathcal G$ the open covering $\mathcal O$ as in Example \[ex:first\] with partition of unity $\psi, \widetilde{\psi}$. Similarly, we define the sequence of partitions of unity $$\psi_n= \frac{1}{n} \max \{\operatorname{dist}(\mathcal G \setminus K_{2n}, x), n\}, \quad \widetilde{\psi_n}= 1- \psi_n$$ and gives a sequence of partitions of unity with regards to a vanishing-compatible sequence of open coverings.
This gives rise to our second main result:
\[thm:main2\] Let $p\in [2,\infty)$, $c>0$ and let $(\mathcal M, \mu)$, $X(\mathcal M)$ and $Y(\mathcal M)$ satisfy Assumption \[as:assumption1\] and Assumption \[as:assumption2\]. Let $K$ be a bounded, connected, nonempty set in $\mathcal G$. Let $E\in C(X(\mathcal M), \mathbb R)$, such that $$t\mapsto E_t = \inf_{\substack{u\in X(\mathcal M)\\ \|u\|_{p}^p=t}} E(u)$$ is continuous and assume $E$ to be $2$-superadditive with respect to a sequence of partitions of unity $\{\psi_{O}\}_{O\in \mathcal O_n}$ in $Y(\mathcal M)$ subordinate to a vanishing-compatible sequence of open coverings $\mathcal O_n=(O_1^{(n)}, O_2^{(n)})$. If there exists a minimizing sequence which is vanishing, then $$E_c = \lim_{R\to \infty} \inf_{\substack{u\in X(\mathcal M), \|u\|_{p}^p=c\\ \operatorname{supp} u \subset \mathcal M \setminus K_R}} E(u) =: \widetilde{E_c}.$$
Let $u_n$ be a vanishing sequence. Assume $(O_n^{(1)}, O_n^{(2)})$ to be such that $$K \subset O_n^{(1)}$$ and $O_n^{(1)}$ is bounded.
For each fixed $m\in \mathbb N$ we have $$\int_{O_m^{(1)}} |u_n|^p\, \mathrm d\mu \to 0 \qquad (n\to \infty).$$ Then for any $m\in \mathbb N$ we find an $n_m$, such that for $n>n_m$ $$\int_{O_m^{(1)}} |u_n|^p \, \mathrm d\mu \le \frac{1}{m}.$$ Using a diagonal argument we deduce the existence of a subsequence of $u_n$, still denoted by $u_n$, such that $$\int_{O_n^{(1)}} |u_n|^p \, \mathrm d\mu \to 0 \qquad (n\to \infty).$$ In particular, $$\begin{aligned}
0\le \int_{O_n^{(1)}} |\psi_{O_n^{(1)}}u_n|^p\, \mathrm d\mu &\le \int_{O_n^{(1)}} |u_n|^p\, \mathrm d\mu\\
c -\int_{O_n^{(1)}} |u_n|^p\, \mathrm d\mu \le \int_{O_n^{(2)}} |\psi_{O_n^{(2)}}u_n|^p \, \mathrm d\mu&\le \int_{\mathcal M} |u_n|^p\, \mathrm d\mu =c
\end{aligned}$$ and we obtain $$\begin{aligned}
\int_{O_n^{(1)}} |\psi_{O_n^{(1)}}u_n|^p \, \mathrm d\mu &\to 0 \qquad (n\to \infty)\\
\int_{O_n^{(2)}} |\psi_{O_n^{(2)}}u_n|^p \, \mathrm d\mu &\to c \qquad (n\to \infty).
\end{aligned}$$ Then by superadditivity we have $$\begin{aligned}
E_c &= \lim_{n\to \infty} E(u_n) \\
&\ge \limsup_{n\to \infty} E\left (\psi_{O_{n}^{(2)}} u_n\right )\ge \widetilde {E_c}.
\end{aligned}$$ This concludes the inequality $E_c \ge \widetilde{E_c}$. The reverse inequality is trivial since $$E_c \le \inf_{\substack{u\in X(\mathcal M), \|u\|_{p}^p=c\\ \operatorname{supp} u \subset \mathcal M \setminus K_R}} E(u)$$ for all $R>0$.
\[cor:existence\] Suppose the assumptions in Theorem \[thm:main1\] and Theorem \[thm:main2\] are satisfied and $$E_c <\widetilde{E_c},$$ then a minimizer of $E_c$ exists, and any minimizing sequence for $E_c$ admits a subsequence converging in $L^p$ towards a minimizer of $E_c$.
Throughout the rest of the paper given a functional $E\in C(X(\mathcal G), \mathbb R)$ we define the corresponding threshold energy $$\label{eq:defion}
\widetilde{E_c} := \lim_{R\to \infty} \inf_{\substack{u\in X(\mathcal G), \, \|u\|_{p}^p=c\\ \operatorname{supp} u \subset \mathcal G\setminus K_R}}E(u).$$ In the case of many-body quantum particle systems, this quantity refers to the ionization energy (see [@griesemer2004exponential] and [@simon1983semiclassical]). For this reason, throughout of the rest of the paper we are going to refer to the quantity in also as the *ionization threshold* or *ionization energy*.
\[ex:NLSclassic\] Let $\mathcal G$ be a locally finite, connected noncompact metric graph with at least one ray. Consider the NLS energy functional as considered in [@adami2016threshold] $$E_{\text{NLS}}(u,\mathcal G) = \int_{\mathcal G} |u'|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathcal G} |u|^p\, \mathrm dx.$$ Consider the minimization problem $$\label{eq:exeq}
E_{\text{NLS}}(\mathcal G) := \inf_{\substack{u\in H^1(\mathcal G) \\\|u\|_2^2=1}} E_{\text{NLS}}(u, \mathcal G).$$ Then in the context of decaying potentials, we will see in §4.5 that the functional $E_{\text{NLS}}$ satisfies the prerequisites of Theorem \[thm:main1\] and Theorem \[thm:main2\]. Moreover, we will characterize the ionization threshold of $\widetilde{E_{\text{NLS}}}(\mathcal G)$. As discussed in Remark \[rmk:decayingpotential\] we will show $$\label{eq:excomp}
\widetilde{E_{\text{NLS}}}(\mathcal G):=\lim_{n\to \infty}\inf_{\substack{u\in H^1(\mathcal G) \\\|u\|_2^2=1,\; \operatorname{supp} u \subset \mathcal G\setminus K_n}} E_{\text{NLS}}(u, \mathcal G) \le E_{\text{NLS}}(\mathbb R).$$ If $\mathcal G$ is a finite graph we have equality in due to Lemma \[lem:decayingpotential\]. In particular Corollary \[cor:existence\] gives a generalization of the result in [@adami2016threshold] where it was shown that if $\mathcal G$ is finite then minimizers of exist if $$\label{eq:inequality}
E_{\text{NLS}}(\mathcal G) < E_{\text{NLS}}(\mathbb R).$$ Since does not guarantee existence by Corollary \[cor:existence\] under the assumption one cannot necessarily extend this result to general locally finite graphs. But as shown in Example \[ex:unrootedtrees\], for a class of infinite tree graphs, one can show the reverse inequality $$\label{eq:excomp2}
\widetilde{E_{\text{NLS}}}(\mathcal G) \ge E_{\text{NLS}}(\mathbb R).$$ In particular one can derive for such graphs satisfying existence of minimizers of $E_{\text{NLS}}(\mathcal G)$ under assumption .
An existence result on the real line for translation-invariant functionals
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Assume $X(\mathbb R)\subset L^p(\mathbb R)$ is a Banach space on $\mathbb R$. Let $E \in C(X, \mathbb R)$ be a translation invariant functional, i.e. if $T_{\lambda} u(x)= u(x-\lambda)$ then $$E(u)= E(T_\lambda u)$$ for all $\lambda \in \mathbb R$. The space $\mathbb R$ can be understood as a graph consisting of two half-lines. In this context we consider $K=\{0\}$ to be the compact core of this particular graph.
\[thm:main3\] Let $p\in [2,\infty)$, $c>0$, let $E\in C(X(\mathbb R), \mathbb R)$ be translation invariant and satisfy Assumption \[as:assumption1\] and Assumption \[as:assumption2\]. Let $$t\mapsto E_t = \inf_{\substack{u\in X(\mathbb R)\\ \|u\|_p^p =t}} E(u)$$ be a strictly subadditive functional in $X(\mathbb R)$. Assume $E$ to be superadditive with respect to a sequence of $3$-partitions of unity $\{\psi_O\}_{O\in \mathcal O_n}$ subordinate to the vanishing-compatible sequence of open coverings $$\mathcal O_n = \{(-2n,2n), (n, \infty), (-\infty, -n)\},$$ then it admits a minimizer.
By Theorem \[thm:main2\] we only need to construct non-vanishing minimizing sequences. Assume $u_n$ to be a minimizing sequence of the functional $E_c$. Then we may construct such a sequence by using the translation invariance of the functional. Indeed, we may assume up to translation invariance $$\begin{aligned}
\int_{0}^\infty |u_n|^p\, \mathrm dx &\to \frac{c}{2} \quad (n\to \infty)\\
\int_{-\infty}^0 |u_n|^p\, \mathrm dx &\to \frac{c}{2} \quad (n\to \infty).
\end{aligned}$$ Assume $u_n$ is vanishing. Then since $u_n \to 0$ in $L^\infty_{\text{loc}}$ (up to a subsequence) due to a diagonal argument, we have that$$\begin{aligned}
\int_{\mathbb R} |\psi_{(-2n,2n)} u_n|^p \, \mathrm dx &\to 0 \qquad (n\to \infty) \\
\int_{\mathbb R} |\psi_{(n, \infty)} u_n|^p \, \mathrm dx &\to \frac{c}{2}\qquad (n\to \infty) \\
\int_{\mathbb R} |\psi_{(-n, -\infty)} u_n|^p \, \mathrm dx &\to \frac{c}{2} \qquad (n\to \infty)
\end{aligned}
$$ Then using the subadditivity of the functional and the strict subadditivity of $t\mapsto E_t$ we conclude $$\begin{aligned}
E_c &= \lim_{n\to \infty} E(u_n) \\
&\ge \limsup_{n\to \infty} E\left ( \psi_{(-\infty, -n)} u_n\right ) + \limsup_{n\to \infty} E\left (\psi_{(n, \infty)} u_n\right ) \\
&\ge E_{c/2} + E_{c/2} > E_c.
\end{aligned}$$ By contradiction after translating the $u_n$ if necessary we can find a non-vanishing subsequence. Passing to a further subsequence there exists a weakly convergent subsequence in $H^1$ that converges up to a further subsequence to a minimizer by Theorem \[thm:main1\].
\[ex:NLSclassic2\] Recall the NLS energy functional from Example \[ex:NLSclassic\] on the real line $$E_{\text{NLS}}(\mathbb R) = \inf_{\substack{u\in H^1(\mathbb R) \\\|u\|_2^2=1}} E_{\text{NLS}}(u, \mathbb R).$$ Then as discussed in Example \[ex:NLSclassic\], we have $$\widetilde{E_{\text{NLS}}}(\mathbb R) = E_{\text{NLS}}(\mathbb R).$$ In particular inequality cannot be satisfied and we cannot apply Corollary \[cor:existence\]. On the other hand, the prerequisites of Theorem \[thm:main3\] are satisfied. In particular this ensures existence of ground states for all $\mu >0$. More explicitly, one can show that ground states of the NLS energy functional on the real line are unique up to translation invariance of the functional and are the well-known soliton solutions.
Existence of Ground states of a class of Nonlinear Equations with Polylaplacian on Finite Graphs
================================================================================================
In this section, we give a first application of the results derived in §3 on finite graphs. In this context we show a decomposition formula for the Polylaplacian.
Formulation of the problem {#sec:existence}
--------------------------
Let $\mathcal G$ be a connected, finite metric graph and let $K$ be a connected subgraph of $\mathcal G$. For $k\in \mathbb N$ consider the energy functional $$E^{(k)} = \frac{1}{2}\int_{\mathcal G} |u^{(k)}|^2 + V |u|^2 \, \mathrm dx - \frac{\mu}{p} \int_{\mathcal G} |u|^p\, \mathrm dx,$$ with $2<p< 4k+2$ and $V\in L^2+ L^\infty$ and for $c>0$ consider the minimization problem $$\label{eq:minimizationeq}
E_{c}^{(k)} := \inf_{\substack{u\in H^k\\ \|u\|_{2}^2=c}} E^{(k)}(u).$$
\[lem:energyestimateneu\] Let $\mathcal G$ be a finite connected metric graph. The functional $E^{(k)}$ under the $L^2$-constraint $\|\cdot\|_{L^2}^2=c$ is bounded below for $2<p<4k+2$ and $c>0$. Moreover, for each $0<\varepsilon <1$ there exists a $C_\varepsilon>0$, such that $$E^{(k)}(u) \ge \frac{1-\varepsilon}{2} \int_{\mathcal G} |u^{(k)}|^2 + V |u|^2 \, \mathrm dx - C_\varepsilon.$$
Let $\varepsilon>0$. Consider a decomposition of $V\in L^2+ L^\infty$ such that $$V = V_2+ V_\infty, \quad \|V_2\|_{2}\le\varepsilon.$$ Then $$\begin{gathered}
\int_{\mathcal G} \left |u^{(k)}\right |^2- \left \|V_\infty\right \|_\infty |u|^2\, \mathrm dx- \varepsilon \|u\|_{4}^2\\
\le \int_{\mathcal G} \left |u^{(k)}\right |^2+ V |u|^2\, \mathrm dx\\
\le \int_{\mathcal G} \left |u^{(k)}\right |^2+ \left \|V_\infty\right \|_\infty |u|^2\, \mathrm dx+ \varepsilon \|u\|_{4}^2.\end{gathered}$$ By the Sobolev inequality we infer $$\|u\|_4^2 \le C_1 \|u\|_{H^k}^2 \le C_2 \left (\left |u^{(k)}\right |_2^2+ \left |u\right |_2^2 \right ).$$ Adding a constant to the potential if necessary we have that $$\|u\| :=\left (\int_{\mathcal G} \left |u^{(k)}\right |^2+ V |u|^2\, \mathrm dx\right )^{1/2}$$ defines an equivalent norm on $H^k$.
From Proposition \[prop:higherordergag\] we have $$\|u\|_{L^p}^p\le C \left \|u\right \|_{L^2(\mathcal G)}^{\frac{(2k-1)p+2}{2k}} \big \|u\big \|^{\frac{p-2}{2k}}$$ for some $C>0$. Let $0<\varepsilon <1$, then with Young’s inequality we infer for all $u\in H^k(\mathcal G)$ with $\|u\|_2^2=c$$$\frac{\mu}{p}\|u\|_{L^p}^p \le \frac{\varepsilon}{2} \|u\|^2 + C_{\varepsilon,c}$$ for some $C_{\varepsilon, c} >0$ and we obtain $$E^{(k)} \ge \frac{1-\varepsilon}{2} \int_{\mathcal G} |u^{(k)}|^2 + V |u|^2 \, \mathrm dx - C_{\varepsilon,c}$$ for $2< p<4k+2$.
\[prop:NLSmult\]Let $\mathcal G$ be a finite, connected metric graph. Assume $u\in H^k$ is a minimizer of $E^{(k)}_c$, then $u\in H^{2k}$ and there exists $\lambda\in \mathbb R$ such that $$\label{eq:NLSELG}
(-1)^k u_e^{(2k)} + \left ( V+\lambda\right ) u_e = \mu |u_e|^{p-1} u_e$$ for all $e\in {{\mathcal E}}$.
Since $E^{(k)}\in C^1(H^k, \mathbb R)$ and the $L^2$ constraint is also $C^1$, and $u$ is a constrained critical point we can compute the Gâteaux derivative $$\int_{\mathcal G} \left ( u^{(k)} \eta^{(k)}- u |u|^{p-2} \eta\right )\, \mathrm dx+ \int_{\mathcal G} \left ( V+\lambda\right ) u\eta\, \mathrm dx=0, \qquad \forall \eta\in H^k(\mathcal G)$$ where $\lambda$ is a Lagrange multiplier. Fixing an edge $e$, then with $\eta \in C_c^\infty(I_e)$ and integration by parts we deduce for each $e\in {{\mathcal E}}$ and by elliptic regularity $u\in \widetilde{H^{2k}}$. Fixing now $\mathsf v\in V$ and taking $\eta\in H^k$ to be locally supported near $\mathsf v$ and not supported at any other vertex, then by integration by parts we deduce $$\sum_{j=1}^{k}(-1)^{j} \sum_{e\succ \mathsf v}\tfrac{\partial^{(k+j-1)}}{\partial^{(k+j-1)}\nu}u_e \tfrac{\partial^{(k-j)}}{\partial^{(k-j)}\nu}\eta_e(\mathsf v)=0.$$ Since the choice $\eta\in H^{k}$ is arbitrary we deduce $$\begin{cases}
\sum_{e\succ \mathsf v} \frac{\partial^\ell}{\partial \nu^\ell}u_e(\mathsf v)=0, \qquad \forall k\le \ell \le 2k-1 \text{ odd},\\
u_{e_1}^{(\ell)}(\mathsf v) = u_{e_2}^{(\ell)}(\mathsf v), \qquad \forall k \le \ell \le 2k-1 \text{ even and } \forall e_1, e_2 \text{ adjacent at } \mathsf v
\end{cases}$$ for all $\mathsf v\in \mathcal V$.
Partitions of unity in $\widetilde{C_b^\infty}$
-----------------------------------------------
Let $\mathcal G$ be any locally finite, connected graph and $\mathcal O$ be a finite covering of $\mathcal O$. We construct a partition of unity in $\widetilde{C_b^\infty}(\mathcal G)$ by choosing appropriate partitions of unities subordinate to the covering. One rather different “normalization” of the usual partition of unity will be especially useful in applications:
\[lem:unity2\] Let $\mathcal G$ be a metric graph. Consider any finite open covering $\mathcal O$ of $\mathcal G$. There exists a partition of unity subordinate to $\mathcal O$ in $\widetilde {C_b^\infty}$ satisfying $$\label{eq:normalizationt}
\sum_{O\in \mathcal O} \Psi_O^2 \equiv 1.$$
Consider any smooth partition of unity $\{\psi_O\}_{O\in \mathcal O}$ on the graph subordinate to the open covering $\mathcal O$ satisfying $$\sum_{O\in \mathcal O} \Psi_O \equiv 1.$$ Then we may define $$\Psi_O:= \frac{\psi_O}{\sqrt{\sum_{O\in \mathcal O} \psi_O^2}}$$ for all $O\in \mathcal O$, which is smooth restricted as functions on all edges since $\sum_{O\in \mathcal O} \psi_O^2(y)\neq 0$ for all $y\in\mathcal G$. Furthermore, it is constant in a neighborhood of any vertex and we infer $\Psi_O\in \widetilde{C_b^\infty}$. By construction we conclude $$\sum_{O\in \mathcal O} \Psi_O^2 \equiv 1.$$
The normalization in replaces in this context the typical normalization, where one assumes $$\sum_{O\in \mathcal O } \psi_O \equiv 1.$$ Throughout the rest of the paper we will only work with partitions of unity that satisfy the normalization .
\[ex:unity2\] Let $\mathcal G$ be a finite, connected metric graph with core $K= \mathcal G\setminus \mathcal E_\infty$. Consider on $\mathcal G$ the open covering $ \mathcal O$ given by $K_2$ and $\mathcal G\setminus K_1$, where $K_1$ and $K_2$ are the neighborhoods of $K$ given as in , such that $\mathcal G\setminus K_1$ only consists of disjoint rays. Consider the partition of unity subordinate to $\mathcal O$ from Lemma \[lem:unity2\] given by $\psi_K, \{\psi_e\}_{e\in \mathcal E_\infty}$ respective to $K_2$ and $\mathcal G\setminus K_1$, then we define slight modifications $$\begin{aligned}
\psi_{K,R}(x) &=\begin{cases}
1, &\qquad \text{on K}\\
\psi_{K}(x/R) &\qquad \text{on all rays } e\in \mathcal E_\infty;
\end{cases}\\
\psi_{e,R}(x) &=\begin{cases}
0, &\qquad \text{on }\mathcal G\setminus \{e\}\\
\psi_{e}(x/R) &\qquad \text{on } e\in \mathcal E_\infty.
\end{cases}\end{aligned}$$ Then by definition, $\{\psi_{O,n}\}_{O\in \mathcal O}$ is a vanishing-compatible sequence of partitions of unity subordinate to the open coverings given by $K_{2n}$ and $\mathcal G\setminus K_{n}$. By Lemma \[lem:unity2\] there exists a sequence of partitions of unity $$\Psi_n:=\Psi_{K,n},\qquad \widetilde{\Psi_n}:=\sum_{e\in \mathcal E_\infty}\Psi_{e,n}$$ in $\widetilde{C_b^\infty}$ satisfying $$\Psi_{n}^2+ \widetilde{\Psi_n}^2\equiv 1.$$
A decomposition formula {#sec:decompositionpoly}
-----------------------
In the following we identify a given function $f\in \widetilde{C_b^\infty}(\mathcal G)$ with its corresponding multiplication operator $\mathcal M_f \phi:= f\phi$. Let $A$ be an operator such that $fD(A) \subset D(A)$, then we can define the commutator $[A,f]=Af-fA$ and $$\begin{aligned}
fAf&= f^2 A + f [A, f]\\
fAf&= Af^2 + [A,f]f.\end{aligned}$$ Averaging the two preceding equations we conclude $$\label{eq:decompositionn}
fAf = \frac{1}{2} ( f^2 A + A f^2) + \frac{1}{2} ( f [A,f] - [A,f]f).$$
\[lem:IMSunity\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $\{\psi_k\}_{k=1}^N$ is a family of function in $\widetilde{C_b^\infty}(\mathcal G)$ with $0 \le \psi_k\le 1$ for all $k\in \{1,\ldots, k\}$ and $$\sum_{k=1}^N \psi_k^2 \equiv 1.$$ Assume $D(A)$ is invariant under multiplication by elements in $\widetilde{C_b^\infty}(\mathcal G)$, then $$\label{eq:decompositionadv}
A= \sum_{k=1}^N \psi_k A \psi_k -\frac{1}{2} ( \psi_k [A,\psi_k] - [A,\psi_k]\psi_k).$$
Follows immediately with .
We refer to as a decomposition formula for $A$. In the following, we develop a decomposition formula for the Polylaplacian $A=(-\Delta)^k$.
Let $\mathcal G$ be a finite metric graph and let $k\in \mathbb N$. In the following, we define the Polylaplacian $A=(-\Delta)^k$ on $\mathcal G$ as an operator $A: D(A) \subset L^2(\mathcal G) \to L^2(\mathcal G)$ given by $$\begin{gathered}
\left ((-\Delta)^k u\right )_e := (-\Delta)^k u_e:= (-1)^k u_e^{(2k)}\\
D(A) = H^{2k}(\mathcal G).\end{gathered}$$
\[lem:decomposition2\] Let $\mathcal G$ be a locally finite connected graph. Let $A=(-\Delta)^k$ with $D(A)= H^{2k}$, then
(i) $fD(A)\subset D(A)$ for all $f\in \widetilde{C_b^\infty}(\mathcal G)$.
(ii) Let $f\in \widetilde{C_b^\infty}(\mathcal G)$, then the operator $fAf$ is given by $$\label{eq:decomposition2}
\begin{aligned}
\left (fAf\right ) \phi &= \frac{1}{2} ( f^2 A + A f^2)\phi \\
&\qquad+ \frac{(-1)^{k+1}}{2} \sum_{m=1}^{2k-1} \sum_{n=1}^{2k-m} \frac{(2k)_{m+n}}{m! n!} f^{(m)} f^{(n)} \phi^{(2k-m-n)}
\end{aligned}$$ for all $\phi \in D(A)$.
We apply Leibniz’ formula and compute $$\begin{aligned}
[A,f]\phi &= (-\Delta)^k f \phi - f(-\Delta)^k \phi\\
&= (-1)^k \sum_{m=1}^{2k} \binom{2k}{l} f^{(m)} \phi^{(2k-m)}.
\end{aligned}$$ Then we apply Leibniz’ formula again and compute $$\begin{aligned}
\left ([A,f]f\right )\phi &= (-1)^k \sum_{m=1}^{2k}\sum_{n=0}^{2k-m} \binom{2k}{m}\binom{2k-m}{n} f^{(m)} f^{(n)} \phi^{2k-m-n}\\
&= (-1)^k \sum_{m=1}^{2k} \sum_{n=0}^{2k-m} \frac{(2k)_{m+n}}{m! n!} f^{(m)} f^{(n)} \phi^{(2k-m-n)}
\end{aligned}$$ and we conclude $$\frac{1}{2} ( f [A,f] - [A,f]f)\phi=\frac{(-1)^{k+1}}{2} \sum_{m=1}^{2k-1} \sum_{n=1}^{2k-m} \frac{(2k)_{m+n}}{m! n!} f^{(m)} f^{(n)} \phi^{(2k-m-n)}.$$ The statement follows upon combining this with .
Given the core $K= \mathcal G\setminus {{\mathcal E}}_\infty$ of $\mathcal G$ and $R>0$ we define $$\label{eq:numerateR}
\begin{aligned}
D_R&:= \{\phi \in D(A) |\operatorname{supp}(\phi) \subset \mathcal G \setminus K_R\}\\
\Sigma_R&:= \inf\{\langle \phi, A\phi\rangle|\phi \in D_R, \|\phi\|_{2}^2=1\},
\end{aligned}$$ where $K_R$ was defined in .
For $R=0$ we set $$\label{eq:numerate0}
\begin{aligned}
D_0&:= D(A)\\
\Sigma_0&:= \inf\{\langle \phi, A\phi \rangle |\phi \in D(A), \|\phi\|_{2}^2=1\}.
\end{aligned}$$ The last relevant quantity, which we will discuss later in §\[sec:onthreshold\] is $$\label{eq:numerate1}
\Sigma := \lim_{R\to \infty} \Sigma_R=\sup_{R>0} \Sigma_R.$$
\[lem:preconditions\] Let $\mathcal G$ be a finite, connected metric graph and let $V\in L^2+ L^\infty(\mathcal G)$. $E^{(k)}$ is weak limit superadditive, superadditive with respect to the partition of unity in Example \[ex:unity2\]. Assume $A= (-\Delta)^k + V$ admits a ground state, then $t\mapsto E_t$ as defined in is strictly subadditive.
*Weak limit superadditivity.* We showed in the proof of Lemma \[lem:energyestimateneu\] that $$\|u\| =\left (\int_{\mathcal G} \left |u^{(k)}\right |^2+ V |u|^2\, \mathrm dx\right )^{1/2}$$ defines an equivalent norm on $H^k$ upon adding a constant to $V$.
Assume $u_n \rightharpoonup u$ in $H^k$ weakly, then up to a subsequence, which we still denote by $u_n$, by the Brezis–Lieb Lemma and weak convergence $$\begin{gathered}
\limsup_{n\to \infty} \|u_n\|^2 = \|u\|^2 + \limsup_{n\to \infty} \|u-u_n\|,\\
\limsup_{n\to \infty} \int_{\mathcal G} |u_n|^p \, \mathrm dx= \int_{\mathcal G} |u|^p\, \mathrm dx + \limsup_{n\to \infty}\int_{\mathcal G} |u-u_n|^p \, \mathrm dx.
\end{gathered}$$ Then $$\limsup_{n\to \infty} E^{(k)} (u_n) =E^{(k)} (u)+ \limsup_{n\to \infty} E^{(k)} (u-u_n)$$ and $E^{(k)}$ is weak limit superadditive.
*Superadditivity with respect to a sequence of partitions of unity.* Finally we need to show superadditivity with respect to the partition of unity $\{\Psi_n, \widetilde {\Psi_n}\}$ in Example \[ex:unity2\]. Let $u_n\rightharpoonup 0$ be a vanishing sequence with $\|u_n\|_{L^2}^2=c$, then $$\|\Psi_n u_n\|_{L^2}^2 +\|\widetilde{\Psi_n} u_n\|_{L^2}^2= c$$ and up to a subsequence, still denoted by $u_n$, $$\begin{gathered}
\lim_{n\to \infty} \int_{K_{2n}} |u_n|^p\, \mathrm dx =0, \qquad \lim_{n\to \infty} \int_{K_{2n}} |u_n|^2\, \mathrm dx = 0\\
\lim_{n\to \infty} \int_{\mathcal G} |u_n|^p=\liminf_{n\to \infty} \int_{\mathcal G} |u_n|^p = \limsup_{n\to \infty} \int_{\mathcal G} |u_n|^p \, \mathrm dx.
$$ Then from $$\begin{aligned}
0&\le \liminf_{n\to \infty} \int_{K_{2n}} |\Psi_n u_n|^p\, \mathrm dx \le \limsup_{n\to \infty} \int_{K_{2n}} |\Psi_n u_n|^p\, \mathrm dx \le \lim_{n\to \infty} \int_{K_{2n}} |u_n|^p\, \mathrm dx\\
0&\le \liminf_{n\to \infty} \int_{K_{2n}} |\widetilde{\Psi_n} u_n|^p\, \mathrm dx \le \limsup_{n\to \infty} \int_{K_{2n}} |\widetilde{\Psi_n} u_n|^p\, \mathrm dx \le \lim_{n\to \infty} \int_{K_{2n}} |u_n|^p\, \mathrm dx
\end{aligned}$$ we deduce $$\begin{gathered}
\lim_{n\to \infty} \int_{\mathcal G} |\Psi_n u_n|^p\, \mathrm dx=\lim_{n\to \infty} \int_{\mathcal K_{2n}} |\Psi_n u_n|^p =0\\
\lim_{n\to \infty} \int_{K_{2n}} |\widetilde{\Psi_n} u_n|^p\, \mathrm dx=0.\end{gathered}$$ and in particular $$\lim_{n\to \infty} \int_{\mathcal G} |u_n|^p{\mathrm dx} = \lim_{n \to \infty} \int_{\mathcal G\setminus K_{2n}} |\widetilde{\Psi_n} u_n|^p \, \mathrm dx = \lim_{n\to \infty} \int_{\mathcal G} |\widetilde{\Psi_n} u_n|^p.$$
$H^{2k}$ is dense in $H^k$ by Proposition \[prop:density\] and we may assume that there exists a minimizing sequence in $H^{2k}$. Let $u_n$ be a minimizing sequence in $H^{2k}$, then by Lemma \[lem:decomposition2\] we conclude passing to a subsequence, still denoted by $u_n$, using integration by parts and Young’s inequality $$\begin{aligned}
&\sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \left |\left \langle \Psi_n^{(i)} \Psi_n^{(j)} u_n^{(2k-i-j)}, \phi\right \rangle_{L^2} \right | \\
&\qquad= \sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \left |\left\langle \frac{\mathrm d^{k-j}}{\mathrm dx^{k-j}} \Psi_n^{(i)} \Psi_n^{(j)}u_n, u_n^{(k-i)}\right \rangle_{L^2}\right |\\
&\qquad\le \sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \left \| \frac{\mathrm d^{k-j}}{\mathrm dx^{k-j}} \Psi_n^{(i)} \Psi_n^{(j)}u_n \right \|_{L^2} \left \|u_n^{(k-i)}\right \|_{L^2} \\ &\qquad \le \frac{C }{n^2}\sum_{i=1}^{2k-1} \sum_{j=1}^{2k-m} \frac{(2k)_{m+n}}{i! j!} \|u_n\|_{H^k}\to 0 \qquad (n\to \infty),
\end{aligned}$$ and we infer $$\begin{aligned}
E^{(k)}&= \lim_{n\to \infty} \frac{1}{2}\langle u_n, Au_n\rangle + \frac{\mu}{p}\|u_n\|_{p}^p\\
&= \limsup_{n\to \infty} \frac{1}{2}\langle \Psi_n u_n, A \Psi_nu_n \rangle + \frac{\mu}{p}\|\Psi_n u_n \|_p^p \\
&\qquad + \limsup_{n\to \infty}\frac{1}{2}\langle \widetilde{\Psi_n} u_n, A \widetilde{\Psi_n} u_n \rangle+ \frac{\mu}{p}\|\widetilde{\Psi_n} u_n \|_p^p\\
&= \limsup_{n\to \infty} E^{(k)} (\Psi_n u_n) + \limsup_{n\to \infty} E^{(k)} (\widetilde{\Psi_n} u_n)
\end{aligned}$$ and $E^{(k)}$ is (super-)additive with respect to the partition of unity $\{\Psi_n, \widetilde{\Psi_n}\}$ in Example \[ex:unity2\].
*Subadditivity.* To show the subadditivity, note that $$\label{eq:scalingarg2}
E_t^{(k)} = t \inf_{\substack{u\in H^1\\ \|u\|_{L^2}^2=1}} \left \{\frac{1}{2}\int_{\mathcal G} \left |u^{(k)}\right |^2+ V |u|^2\, \mathrm dx - t^{\frac{p-2}{2}} \frac{\mu}{p} \int_{\mathcal G} |u|^p\, \mathrm dx \right \}.$$ We deduce the property by showing that $t\mapsto E_t^{(k)}$ is a concave function. Indeed, the scaling defines a concave function and passing to the limit we deduce concavity of the functional. Hence, $$\label{eq:concavityarg2}
E_t^{(k)} \ge t E_1^{(k)}, \qquad t\in [0,1],$$ so that $$E_t^{(k)} + E_{1-t}^{(k)} \ge E_1^{(k)}, \qquad t\in [0,1].$$ For the strictness in the inequality it suffices to show strictness in the inequality . Assume $$E_t^{(k)} = t E_1^{(k)}$$ for some $t\in (0,1)$ and let $u_n$ be a minimizing sequence for $E_t^{(k)}$, then in particular due to $$\label{eq:recipe1}
\int_{\mathcal G} |u_n|^p\, \mathrm dx \to 0 \qquad (n\to \infty).$$ By density we may assume $u_n \in D(A)$ and we deduce $$\begin{aligned}
E_t^{(k)} &= \lim_{n\to \infty} \frac{1}{2} \langle A u_n, u_n\rangle - \frac{\mu}{p} \int_{\mathcal G} |u|^p\, \mathrm dx \\
&= \lim_{n\to \infty} \frac{1}{2} \langle A u_n, u_n \rangle \ge \frac{\Sigma_0 t}{2}.
\end{aligned}$$ Now suppose that a ground state to $E^{(k)}$ exists, i.e. there exists $u\in D(A)$ with $\|u\|_2^2=t$ and $\langle Au, u\rangle = \Sigma_0 t$, then $$\label{eq:recipe2}
E^{(k)}_t \le \frac{1}{2}\langle Au, u\rangle - \frac{\mu}{p}\int_{\mathcal G} |u|^p\, \mathrm dx < \frac{\Sigma_0 t}{2}.$$ This is a contradiction, and we conclude strict subadditivity in this case.
\[prop:second\] Assume $\mathcal G$ is a finite, connected metric graph and $\Sigma_0 < \Sigma$ as defined in and . Then there exists $\hat \mu >0$, such that for all $\mu \in (0, \hat \mu)$ $$\widetilde \Sigma_0^{(\mu,k)} := \inf_{\substack{u \in D(A)\\ \|\phi\|_2^2=1}} E^{(k)}(u) < \inf_{\substack{u \in D_R(A)\\\|\phi\|_2^2=1}} E^{(k)}(u) =: \widetilde \Sigma^{(\mu,k)}$$ for some $R>0$.
Without loss of generality we may assume $\Sigma_0>0$ as we otherwise can simply add a constant to the functional. In particular, $$\|u\|= \left ( \int_{\mathcal G} |u^{(k)}|^2 + V |u|^2\, \mathrm dx \right )^{1/2}$$ defines an equivalent norm on $H^k$ as shown in Lemma \[lem:energyestimateneu\]. Let $\varepsilon>0$. By Proposition \[prop:higherordergag\] $$\|u\|_{L^p}^p \le C \|u\|_{L^2(\mathcal G)}^{\frac{(2k-1)p+2}{2k}} \|u\|^{\frac{p-2}{2k}}$$ for some $C>0$. In particular, for sufficiently small $\mu$ with Young’s inequality we have $$\frac{\mu}{p} \|u\|_{L^p}^p\le \frac{\varepsilon}{2} \int_{\mathcal G} |u^{(k)}|^2+ V|u|^2\, \mathrm dx + \frac{\widetilde{C} \varepsilon}{2}$$ for sufficiently small $\mu>0$ we deduce $$E^{(k)}(u) \ge \frac{1- \varepsilon}{2} \left ( \int_{\mathcal G} |u^{(k)}|^2+ V|u|^2\, \mathrm dx\right )- \frac{\widetilde{C} \varepsilon}{2}.$$ Hence, $$\widetilde{\Sigma}^{(\mu,k)} -\widetilde{\Sigma}_0^{(\mu,k)} \ge \frac{1-\varepsilon}{2} \Sigma-\frac{\varepsilon}{2} - \frac{1}{2}\Sigma_0 = \frac{1}{2} \left (\Sigma - \Sigma_0\right ) -\frac{\varepsilon}{2} \left (\widetilde{C}+ \Sigma\right ).$$ Since $\varepsilon$ can be chosen arbitrarily small and $\Sigma > \Sigma_0>0$, we have for sufficiently small $\mu$ $$\widetilde \Sigma^{(\mu,k)} > \widetilde \Sigma^{(\mu,k)}_0.$$
\[thm:bigresult2\] Let $\mathcal G$ be a finite, connected graph and let $c>0$. Assume $\Sigma_0 < \Sigma$ as defined in and , then $E_c^{(k)}(\mathcal G)$ admits a minimizer for $\mu \in (0, \hat \mu)$ as in Proposition \[prop:second\] and let $\hat \mu$ be as in Proposition \[prop:second\]. Then $E_c^{(k)}$ admits a minimizer for any $\mu \in (0,\hat \mu)$.
By Lemma \[lem:energyestimateneu\] any minimizing sequence admits a weakly convergent subsequence. $E^{(k)}$ satisfies the prerequisites of Theorem \[thm:main1\] and Theorem \[thm:main2\] with $X(\mathcal G)= H^k(\mathcal G)$ and $Y(\mathcal G) = \widetilde{C_b^\infty}(\mathcal G)$ by Lemma \[lem:preconditions\]. Then due to Proposition \[prop:second\] the energy inequality in Corollary \[cor:existence\] is satisfied. In particular we deduce existence of a minimizer of $E_c^{(k)}(\mathcal G)$ under the assumptions of the statement.
The case $V\equiv 0$ on the real line
-------------------------------------
Consider the infimization problem on the real line $$E^{(k)}(\mathbb R) = \inf_{\substack{u\in H^1(\mathbb R)\\ \|u\|_2^2=1}} \frac{1}{2} \int_{\mathbb R} |u^{(k)}|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathbb R} |u|^p \, \mathrm dx.$$ This is the special case when $V\equiv 0$ and $\mathcal G=\mathbb R$ in . In this case $E_c$ admits a minimizer due to Theorem \[thm:main3\]:
\[thm:bigresult3\] Let $V\equiv 0$ and $\mathcal G=\mathbb R$. The minimization problem $$E^{(k)}(\mathbb R) = \inf_{\substack{u\in H^1(\mathbb R)\\ \left \|u\right \|_2^2=1}} E^{(k)}(u)$$ admits a minimizer for all $\mu >0$. Furthermore, any minimizer of $E_c(\mathbb R)$ is $C^\infty_b$ and satisfies the Euler–Lagrange equation $$(-1)^k u^{(2k)} + \lambda u = \mu |u|^{p-2}u,$$ where $\lambda \in \mathbb R$ is a Lagrange multiplier.
Due to Lemma \[lem:preconditions\] it suffices to show that $$t\mapsto E_{t}^{(k)}= \inf_{\substack{u\in H^1(\mathbb R)\\ \|u\|_2^2=t}}\frac{1}{2} \int_{\mathbb R} |u^{(k)}|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathbb R} |u|^p \, \mathrm dx.$$ is strictly subadditive, then the prerequisites of Theorem \[thm:main3\] are satisfied. The Euler–Lagrange equation is satisfied because of Proposition \[prop:NLSmult\]. The regularity is due to elliptic regularity and a bootstrap argument.
For a contradiction, assume as in the proof of Lemma \[lem:preconditions\] $$E_t = t E_1$$ for some $t\in [0,c]$. Assume $u_n$ is a minimizing sequence for $E_t$, then as in the proof of Lemma \[lem:preconditions\] we have $$\label{eq:rcase}
\lim_{n\to \infty} \int_{\mathbb R} |u_n|^p\, \mathrm dx = 0.$$ Given a test function $\phi \in C_c^\infty(\mathbb R)$ with $\|\phi\|_{L^2}^2=1$, we define the rescaling for $\lambda >0$ $$\phi_{\lambda} := \lambda^{1/2} \phi( \lambda x).$$ Then $\|\phi_{\lambda}\|_{L^2}^2=1$ for all $\lambda >0$. Then $$E^{(k)}(\phi_{\lambda}) = \lambda^{2k} \int_{\mathbb R} |\phi^{(k)}|^2\, \mathrm dx - \lambda^{\frac{p}{2}-1} \int_{\mathbb R} |\phi|^p \, \mathrm dx.$$ In particular for sufficiently small $\lambda>0$ $$E^{(k)} \le E^{(k)}(\phi_{\lambda}) <0.$$ On the other hand with $$E^{(k)} = \lim_{n\to \infty} E^{(k)}(u_n) = \lim_{n\to \infty} \int_{\mathcal G} |u_n^{(k)}|^2\, \mathrm dx\ge 0$$ and we conclude subadditivity due to the contradiction. This concludes the proof.
\[cor:mainresult3\] Let $V\equiv 0$. Then $$E^{(k)}(\mathbb R)<0.$$
Decaying potentials
-------------------
In the following we study the minimization problem on finite metric graphs $\mathcal G$ under the assumption that $V=V_2+V_\infty$ with $V_2\in L^2(\mathcal G), V_\infty\in L^\infty(\mathcal G)$ such that $$\label{eq:decayingpotential}
V_\infty(x) \to 0 \qquad (x\to \infty)$$ on all of the rays. Consider the quantitities $$\begin{aligned}
\widetilde{\Sigma}_0^{(\mu,k)} &= \inf_{\substack{\phi \in D(A)\\ \|\phi\|_{L^2}^2 =1}} E^{(k)}(u)\\
\widetilde{\Sigma}^{(\mu,k)} &= \lim_{R\to \infty} \inf_{\substack{\phi \in D_R(A)\\ \|\phi\|_{L^2}^2 =1}} E^{(k)}(u)\\
E^{(k)}(\mathbb R) &= \inf_{\substack{u\in H^1(\mathbb R)\\ \|u\|_{L^2(\mathbb R)}^2=1}} \frac{1}{2} \int_{\mathbb R} |u^{(k)}|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathbb R} |u|^p \, \mathrm dx.
\end{aligned}$$
\[lem:decayingpotential\] Let $\mathcal G$ be a finite metric graph and assume that $V\in L^2+L^\infty(\mathcal G)$ satisfies . Then $$\widetilde{\Sigma}^{(\mu,k)} = E^{(k)}(\mathbb R).$$
Assume $\phi$ is a minimizer of $E^{(k)}(\mathbb R)$, which exists due to Theorem \[thm:bigresult3\]. Due to density, we can consider a minimizing sequence $u_n$ for $E^{(k)}(\mathbb R)$ in $C_c^\infty(\mathbb R)$ satisfying $\|u_n\|_{2}^2=1$, such that $u_n \to \phi$ strongly in $H^1$ as $n\to \infty$. Now by translation invariance we may assume that $u_n$ is supported in $[n,\infty)$ for $n\in \mathbb N$. identifying the half-line with one of the rays of $\mathcal G$, we may consider $u_n$ as a function in $H^1(\mathcal G)$. Then $$\begin{aligned}
\left |\int_{\mathcal G} V |u_n|^2\, \mathrm dx \right | &= \left |\int_{\mathcal G\setminus K_n} V |u_n|^2\, \mathrm dx \right |\\
&\le C \left (\sup_{x\in \mathcal G\setminus K_n} |V_\infty(x)|^2 +\int_{\mathcal G\setminus K_n} |V_2|^2\, \mathrm dx\right )\to 0 \qquad (n\to \infty)
\end{aligned}$$ and we compute $$\begin{aligned}
E^{(k)}(\mathbb R)&=\lim_{n\to \infty}E^{(k)}(u_n)\\
&= \lim_{n\to \infty} \frac{1}{2}\int_{\mathcal G} |u^{(k)}_n|^2 \, \mathrm dx - \frac{\mu}{p} \int_{\mathcal G} |u|^p\, \mathrm dx\\
&= \lim_{n\to \infty} E^{(k)}(u_n)\ge \widetilde{\Sigma}^{(\mu,k)}.
\end{aligned}$$ On the other hand given a minimizing sequence $u_n$ for $\widetilde{\Sigma}^{(\mu,k)}$, such that $\operatorname{supp} u_n \subset \mathcal G \setminus K_n$ then the functions in the sequence are supported on each of the rays and $$\label{eq:goesto0}
\begin{aligned}
&\left |\int_{\mathcal G} V |u_n|^2\, \mathrm dx \right | = \left |\int_{\mathcal G\setminus K_n} V |u_n|^2\, \mathrm dx \right |\\
&\qquad\le C \left (\sup_{x\in \mathcal G\setminus K_n} |V_\infty(x)|+\left (\int_{\mathcal G\setminus K_n} |V_2|^2\, \mathrm dx\right )^{1/2}\right )\to 0 \qquad (n\to \infty).
\end{aligned}$$ By density we can consider a collection of sequences $u_n^{(1)}, \ldots, u_n^{(|\mathcal E_\infty|)}$ in $C_c^\infty(\mathbb R)$ and choose them to have disjoint supports. Then if we define $$\widetilde{u}_n := \sum_{i=1}^{|\mathcal E_\infty|} u_n^{(i)}.$$ Then with we compute $$\begin{aligned}
\widetilde{\Sigma}^{(\mu,k)} &= \lim_{n\to \infty} \sum_{i=1}^{\mathcal E_\infty} E^{(k)}(u_n^{(i)})\\
&= \lim_{n\to \infty} E^{(k)}\left ( \widetilde{u}_n \right ) \ge E^{(k)}(\mathbb R). \\
\end{aligned}$$
\[rmk:decayingpotential\] Suppose $\mathcal G$ is a locally finite metric graph with at least one ray. Then the inequality $$\widetilde{\Sigma}^{(\mu,k)}\le E^{(k)}(\mathbb R)$$ can still be shown as in the proof of Lemma \[lem:decayingpotential\] using the test function argument on the half-line.
\[thm:decayingpotential\] Let $\mathcal G$ be a finite metric graph. Assume $V\in L^2+L^\infty(\mathcal G)$ satisfies , then $E^{(k)}$ is strictly subadditive and if $$\widetilde{\Sigma}_0^{(\mu,k)} < E^{(k)}(\mathbb R)$$ then there exists a minimizer to the minimization problem.
By Lemma \[lem:energyestimateneu\] any minimizing sequence admits a weakly convergent subsequence. By Lemma \[lem:preconditions\] and Lemma \[lem:decayingpotential\] it suffices to prove the strict subadditivity of $E^{(k)}$. As in Lemma \[lem:preconditions\] we can argue by contradiction. Assume namely that $$E_t^{(k)}= t E_1^{(k)}$$ for some $t\in (0,1)$ and let $u_n$ be a minimizing sequence for $E_t^{(k)}$, then in particular $$\int_{\mathcal G} |u_n|^p \, \mathrm dx \to 0 \qquad (n\to \infty).$$ But then $u_n$ is a vanishing sequence and passing to a subsequence still denoted by $u_n$, we deduce with superaddditivity with respect to a sequence of partitions of unity as defined in Example \[ex:unity2\] $$\label{eq:reipse2}
\begin{aligned}
E_t^{(k)} &= \limsup_{n\to \infty} E^{(k)}(\Psi_n u_n) +\limsup_{n\to \infty} E^{(k)}(\widetilde{\Psi_n} u_n) \\
&\ge \frac{1}{2}\lim_{n\to \infty}\inf_{\substack{\phi\in H^1\\ \|u\|_{L^p}^p=t, \operatorname{supp} u \subset \mathcal G\setminus K_n}}\langle A u, u\rangle\\
&\ge -\frac{C}{2} \lim_{n\to \infty} \left (\sup_{x\in \mathcal G\setminus K_n} |V_\infty(x)|+ \left (\int_{\mathcal G\setminus K_n} |V_2|^2\, \mathrm dx\right )^{1/2}\right ) =0,
\end{aligned}$$ since $\|u_n\|_{H^1}\le C$ for some $C>0$ by Lemma \[lem:energyestimateneu\]. On the other hand, by Lemma \[lem:decayingpotential\] and Corollary \[cor:mainresult3\] $$\widetilde{\Sigma}^{(\mu,k)} = E^{(k)}(\mathbb R)<0$$ and by contradiction we deduce strict subadditivity.
Hence, the prerequisites of Theorem \[thm:main1\] and Theorem \[thm:main2\] are satisfied with $X(\mathcal G)= H^k(\mathcal G)$ and $Y(\mathcal G) = \widetilde{C_b^\infty}(\mathcal G)$. Then due to Proposition \[prop:second\] the energy inequality in Corollary \[cor:existence\] is satisfied. In particular we deduce existence of a minimizer of $E^{(k)}(\mathcal G)$ under the stated assumptions.
\[ex:shouldbeputinintroduction\] Let $\mathcal G$ be a finite metric graph and let $V\in L^2+L^\infty$ satisfy . Similarly as in Lemma \[lem:decayingpotential\] we can show $$\Sigma=\lim_{R\to \infty} \inf_{\substack{\phi \in D_R(A)\\ \|\phi\|_{L^2}^2 =1}} \langle \phi, A\phi\rangle_{L^2}=0.$$ In particular if $\Sigma_0<0$, then by Theorem \[thm:bigresult2\] there exists $\hat \mu>0$, such that for $\mu \in (0,\hat\mu]$ there exists a minimizer to $E^{(1)}$. As in [@cacciapuoti2018existence] one can show due to scaling properties that $$\Sigma^{(\mu,1)}_0 < \Sigma_0 \le \gamma_p \mu^{\frac{4}{6-p}}= E^{(1)}(\mathbb R)$$ for some $\gamma_p<0$ and $0<\mu\le (\Sigma_0/\gamma_p)^{\frac{3}{2}-\frac{p}{4}}$. In particular, we can deduce existence of minimizers for $E^{(1)}$ and $0<\mu\le(\Sigma_0/\gamma_p)^{\frac{6-p}{4}}$ by Theorem \[thm:decayingpotential\].
Locally Finite Graphs
=====================
In this section, we study the NLS energy functional with potentials on more general graphs. We show a decomposition formula for the form associated with the magnetic Schrödinger operator and adapt previous arguments by introducing a suitable sequence of partitions of unity in the case of locally finite graphs.
Formulation of the problem {#formulation-of-the-problem}
--------------------------
Consider the Schödinger operator with potentials $M\in H^1+W^{1,\infty}(\mathcal G)$ and $V\in L^2+L^\infty(\mathcal G)$ satisfying natural vertex conditions on $\mathcal G$: $$\label{eq:schroedinger2}
\begin{aligned}
A&=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^2+V \\
D(A)&= \bigg \{u\in C(\mathcal G)\bigg| u_e\in H^2(e), \qquad\forall e\in {{\mathcal E}}\\
&\qquad\qquad\qquad \land\qquad \sum_{e \succ \mathsf v} \left ( i \frac{\partial}{\partial\nu} + M \right ) u_e (\mathsf v)=0\bigg \}.
\end{aligned}$$
Consider the NLS functional $$E_\text{NLS}^{(\mathcal K)} (u) := \frac{1}{2}\int_{\mathcal G} \left |\left (i \frac{\mathrm d}{\mathrm dx}+M\right ) u\right |^2 + V |u|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal K}|u|^p\, \mathrm dx$$ where $V\in L^2+ L^\infty$ and $\mathcal K$ is a not necessarily bounded subgraph of $\mathcal G$. Define the corresponding minimization problem $$E_{\text{NLS}}^{(\mathcal K)}:=\inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_2^2=1}} E_{\text{NLS}}^{(\mathcal K)} (u)$$ similarly as in Section \[sec:existence\]. ses:
- *The localized case,* when $\mathcal K$ is a bounded subgraph of $\mathcal G$;
- *The global case,* when $\mathcal K= \mathcal G$ is the whole graph. In this case, we drop the argument and simply define $$E_\text{NLS} (u) := \frac{1}{2}\int_{\mathcal G} \left |\left (i \frac{\mathrm d}{\mathrm dx}+M\right ) u\right |^2 + V |u|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal G}|u|^p\, \mathrm dx$$ and $$E_{\text{NLS}}:=\inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_2^2=1}} E_{\text{NLS}} (u).$$
We define quantities analogous to , and . Given a bounded subgraph of $\mathcal G$ and $R>0$ we define $$\begin{aligned}\label{eq:numerateR2}
D_R&:= \{\phi \in D(A) |\operatorname{supp}(\phi) \subset \mathcal G \setminus K_R\}\\
\Sigma_R&:= \inf\{\langle \phi, A\phi\rangle|\phi \in D_R, \|\phi\|_{2}^2=1\},
\end{aligned}$$ where $K_R$ was defined in , $$\label{eq:numerate02}
\begin{aligned}
D_0&:= D(A)\\
\Sigma_0&:= \inf\{\langle \phi, A\phi \rangle |\phi \in D(A), \|\phi\|_{2}^2=1\}
\end{aligned}$$ and $$\label{eq:numerate12}
\Sigma := \lim_{R\to \infty} \Sigma_R=\sup_{R>0} \Sigma_R.$$ Most results can be extended simply to this case following the previous proofs. The principal difficulty lies in establishing superadditivity with respect to a suitable sequence of partitions of unity. We give a construction of such a sequence of partitions of unity in the following.
Partitions of unity in $W^{1,\infty}(\mathcal G)$
-------------------------------------------------
Here we give an important example for a partition of unity in $W^{1,\infty}(\mathcal G)= C^{0,1}(\mathcal G)$. Given any partition of unity in $W^{1,\infty}(\mathcal G)$ one can always find a renormalization as in Lemma \[lem:unity2\]:
\[lem:unity\] Let $\mathcal G$ be a connected, locally finite metric graph. Consider any finite open covering $\mathcal O$ of $\mathcal G$. Then there exists a partition of unity in $W^{1,\infty}(\mathcal G)$ subordinate to $\mathcal O$ satisfying $$\sum_{O\in \mathcal O} \Psi^2_O \equiv 1.$$
Consider a partition of unity $\{\psi_O\}_{O\in \mathcal O}$ on the graph subordinate to the open covering $\mathcal O$. Then we define $$\Psi_O := \frac{\psi_O}{\sqrt{\sum_{O\in \mathcal O} \psi_O^2}}$$ for all $O\in \mathcal O$. As a product of uniformly bounded Lipschitz continuous functions, $\Psi_O$ is also one; and by Proposition \[prop:lipschitz\] we conclude $\psi_O\in W^{1,\infty}(\mathcal G)$. Moreover, $\sum_{O\in \mathcal O} \psi_O^2 \equiv 1$ by construction.
\[ex:first2\] Let $\mathcal G$ be a locally finite graph and let $K$ be some bounded, connected subgraph. Let $X(\mathcal G)= H^1(\mathcal G)$ and $Y(\mathcal G)= W^{1,\infty}(\mathcal G)$. Then $X(\mathcal G),Y(\mathcal G)$ satisfy Assumption \[as:assumption1\] and Assumption \[as:assumption2\]. Recall the partition of unity in $W^{1,\infty}(\mathcal G)$ in Example \[ex:first\] $$\psi(x) = \max\{\operatorname{dist}(\mathcal G\setminus K_{2}, x), 1\}, \qquad \widetilde {\psi}(x) = 1-\psi.$$ We construct a sequence of partitions of unity via $$\psi_n(x)= \frac{1}{n} \max\{\operatorname{dist}(\mathcal G\setminus K_{2n}, x), n\}, \qquad \widetilde {\psi_n}(x) = 1-\psi_n.$$ By Lemma \[lem:unity\] we can rescale them in such a way that $$\Psi_n^2 + \widetilde{\Psi_n}^2 \equiv 1.$$ By definition the partitions $K_{2n}, \mathcal K\setminus K_n$ are vanishing-compatible and $\Psi_n, \widetilde{\Psi_n}$ is a vanishing-compatible sequence of partitions of unity.
\[df:kirchhoff\] Let $f\in C^{0,1}(\mathcal G)$. We call a point $x\in \mathcal G$ a Kirchhoff point of $f$ if one of the following holds:
(1) $x\in \mathcal V$ is a vertex of degree $d_x\neq 2$, the derivatives $f_e'(x)$ exist for all $e\succ x$, and $f$ satisfies the Kirchhoff condition $$\sum_{e\succ x} \tfrac{\partial}{\partial\nu}f_e(x) =0,$$
(2) $x\in\mathcal G$ is an interior point of an edge (equivalently, a dummy vertex of degree $2$), and $f$ is differentiable at $x$.
We call the set $$\mathcal N_f = \mathcal G\setminus \{x\in \mathcal G: x \text{ is a Kirchhoff point of }f\}$$ the non-Kirchoff set of $f$.
\[rmk:important\] The sequence constructed in Example \[ex:unity2\] do not work here, since the functions are not in $W^{1,\infty}(\mathcal G)$. We are going to consider the sequence of partitions of unity in Example \[ex:first2\] instead. This concrete sequence has some interesting properties, such that for all $n\in \mathbb N$ $$\label{eq:interestingprop1}
\|\psi_n'\|_{L^\infty} = \frac{1}{n} \qquad \|\widetilde{\psi_n}'\|_{L^\infty} = \frac{1}{n}$$ and in particular $$\label{eq:interestingprop2}
\|\Psi_n'\|_{L^\infty} \le \frac{C}{n} \qquad \|\widetilde{\Psi_n}'\|_{L^\infty} \le \frac{C}{n}$$ for a $C=C(\mathcal G)$ only dependent on the graph.
A decomposition formula {#a-decomposition-formula}
-----------------------
For the Schrödinger operator with magnetic potential $$\label{eq:test}
\begin{gathered}
\widetilde{A}=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^{2}\\
D(\widetilde A)= \widetilde{H^2}
\end{gathered}$$ one can show as in Section \[sec:decompositionpoly\], see Lemma \[lem:decomposition2\]:
\[lem:decompositionmagnetic\] Let $\mathcal G$ be a locally finite connected metric graph. Let $$\begin{gathered}
\widetilde A:=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^{2}\\
D(\widetilde A):= \widetilde{H^2}(\mathcal G)
\end{gathered}$$ edgewise defined, i.e. $$\left (\widetilde A \phi\right )_e = \widetilde A \phi_e.$$ Then $\widetilde A$ defines an unbounded operator on $L^2(\mathcal G)$ and satisfies
(i) $fD(\widetilde A)\subset D(\widetilde A)$ for all $f\in \widetilde{C^\infty}(\mathcal G)$.
(ii) Let $f\in \widetilde{C^\infty}(\mathcal G)$, then the operator $f\widetilde Af$ is given by $$\label{eq:formulareduced}
f\widetilde A f= \frac{1}{2} \left (f^2\widetilde A + \widetilde A f^2\right ) + |f'|^2$$
The proof is analogous to the one in Lemma \[lem:decomposition2\].
does not uniquely determine an operator. Indeed is the special case of when $k=1$. In particular, formula in the case $k=1$ holds for all self-adjoint realizations of the magnetic Schrödinger operators (e.g. ) and independent of the choice of $M\in H^1+ W^{1,\infty}(\mathcal G)$.
We will be interested in a decomposition lemma on the form associated to $A$ as given in .
\[lem:lemma5.7\] Let $\mathcal G$ be a locally finite, connected metric graph and $a(\cdot, \cdot)$ be the symmetric bilinearform given by $$a(u,v):= \int_{\mathcal G} \overline{\left ( i \frac{\mathrm d}{\mathrm dx}+M\right )u} \left ( i \frac{\mathrm d}{\mathrm dx}+M\right ) v \, \mathrm dx$$ for $u,v\in H^1(\mathcal G)$ Then for $f\in W^{1,\infty}(\mathcal G)\cap \widetilde{C^\infty}(\mathcal G)$ we have $$\label{eq:IMSformulapre}
a(fu, fv) = \frac{1}{2} \left (a_{\mathcal G}(u, f^2 v)+ a(f^2 u, v)\right )+\langle |f'|^2 u, v\rangle_{L^2(\mathcal G)}$$
By Proposition \[prop:densitylocfinite\] we may assume $u,v\in H^2_{c}(\mathcal G)$ and $fu, fv\in \widetilde{H^2}(\mathcal G)\cap H^1_c(\mathcal G)$. Integrating by parts on an arbitrary bounded subgraph $K$ containing $\operatorname{supp}u$ and $\operatorname{supp}{v}$ we compute $$\begin{aligned}
a(fu,fv)&= -\int_{K} \overline {\left (fAf\right ) u} v \, \mathrm dx + \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) fu}\right ]_e fv(\mathsf v)\\
&=-\int_{K} \left (\overline{\frac{1}{2}\left (f^2\widetilde A + \widetilde A f^2\right )u + |f'|^2 u}\right ) v\, \mathrm dx\\
&\qquad \qquad\quad \qquad\qquad + \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) fu}\right ]_e fv(\mathsf v)\\&= \frac{1}{2} \left (a(u, f^2 v)+ a(f^2 u, v)\right )+ \int_{\mathcal G} |f'|^2 \overline u v\, \mathrm dx\\
&\qquad\qquad - \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \frac{1}{2}\left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) f^2 u}\right ]_e v(\mathsf v)\\
&\qquad \qquad \qquad-\sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \frac{1}{2}\left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) u}\right ]_e f^2v(\mathsf v)\\
&\qquad \qquad \qquad \qquad + \sum_{\mathsf v\in \mathcal N_f\cap K} \sum_{e\succ \mathsf v} \left [\overline{\left ( i \frac{\mathrm d}{\mathrm dx} + M\right ) fu}\right ]_e fv(\mathsf v)\\
&= \frac{1}{2} \left (a(u, f^2 v)+ a(f^2 u, v)\right )+ \int_{\mathcal G} |f'|^2 \overline u v\, \mathrm dx
\end{aligned}$$ and the statement follows by density.
Existence of NLS ground state for a class of Schrödinger operators
------------------------------------------------------------------
### The localized setting
In the following we study the localized case. We also remark that some of the lemmas will also apply to the global case. For $t>0$ we define $$\label{eq:minimizationdefloc}
E_t^{(\mathcal K)}:= \inf_{\substack{u\in H^1(\mathcal G)\\\|u\|_{L^2}^2=t}} E_{\text{NLS}}^{(\mathcal K)}.$$
\[eq:NLSboundedbelow\] Let $\mathcal G$ be a connected locally finite metric graph. Let $\mathcal K$ be a not necessarily bounded subset of $\mathcal G$. The functional $E_{\text{NLS}}^{(\mathcal K)}$ under $L^2$-constraint $\|\cdot\|_{L^2}^2=1$ is bounded below for $2<p<6$.
From the Gagliardo–Nirenberg inequality we have $$\begin{aligned}
\int_{\mathcal K} |u|^p\, \mathrm dx &\le \int_{\mathcal G} |u|^p\, \mathrm dx \\
&\le \varepsilon \int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right |^2+ V |u|^2\, \mathrm dx + C_{\varepsilon} \int_{\mathcal G} |u|^2\, \mathrm dx
\end{aligned}$$ and therefore $$E_{\text{NLS}}^{(\mathcal K)}(u) \ge(1-\varepsilon) \int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right |^2+ V |u|^2\, \mathrm dx - C_{\varepsilon} \ge -C_{\varepsilon}.$$ for all $u\in H^1(\mathcal G)$ satisfying $\|u\|_2^2=1$.
\[lem:energyinequality\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state, then $$E_t^{(\mathcal K)}=\inf_{\substack{u\in H^1(\mathcal G)\\\|u\|_{L^2}^2=t}} E_{\text{NLS}}^{(\mathcal K)} \le \frac{\Sigma_0t}{2}.$$ The inequality is strict if the ground state does not vanish identically on $\mathcal K$
Assume $u$ is a ground state of $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ with $\|u\|_{L^2}^2=t$, then $$E_{\text{NLS}}^{(\mathcal K)}(u) = \frac{\Sigma_0t}{2} - \frac{\mu}{p} \int_{\mathcal K} |u|^{p}\, \mathrm dx\le \frac{\Sigma_0}{2} t$$ and the inequality is strict if $u$ is not identically vanishing on $\mathcal K$. In particular $$\inf_{\substack{u\in H^1\\\|u\|_{L^2}^2=t}} E_{\text{NLS}}^{(\mathcal K)} \le \frac{\Sigma_0t}{2}$$ with strictness in the inequality if there exists a ground state, which is not identically vanishing on $\mathcal K$.
\[lem:prework\] Let $\mathcal G$ be a locally finite, connected metric graph and let $\mathcal K$ be any subgraph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state that is not identically vanishing on $\mathcal K$, then the functional $E_{\text{NLS}}^{(\mathcal K)}$ is weak limit superadditive, superadditive with respect to the partition of unity in Example \[ex:first2\] and $t\mapsto E_t$ as defined in is strictly subadditive.
With the Minkowski inequality we have $$\begin{aligned}
&\left ( \int_{\mathcal G}|u'|^2\mathrm dx\right )^{1/2}- \left (\int_{\mathcal G} |M|^2 |u|^2\, \mathrm dx\right )^{1/2}\\
&\qquad \qquad \le \left (\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right |^2\, \mathrm dx\right )^{1/2}\\
&\qquad \qquad \qquad \qquad\le \left ( \int_{\mathcal G}|u'|^2\mathrm dx\right )^{1/2}+ \left (\int_{\mathcal G} |M|^2 |u|^2\, \mathrm dx\right )^{1/2}.
\end{aligned}$$ Adding a constant to the potential similarly as in the proof of Lemma \[lem:preconditions\] we may assume $$\|u\|_{2, M,V} =\left (\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right |^2+ V |u|^2\, \mathrm dx\right )^{1/2}$$ to define an equivalent norm on $H^1$.
*Weak limit superadditivity.* Assume $u_n \rightharpoonup u$ weakly in $H^1$, then up to a subsequence by the Brezis–Lieb Lemma and weak limit superadditivity (similarly as in the proof of Lemma \[lem:preconditions\]) $$\limsup_{n\to \infty} E_{\text{NLS}}^{(\mathcal K)} (u_n) =E_{\text{NLS}}^{(\mathcal K)}(u)+ \limsup_{n\to \infty} E_{\text{NLS}}^{(\mathcal K)} (u-u_n)$$ and $E_{\text{NLS}}$ is weak limit superadditive.
*Superaddivity with respect to a sequence of partitions of unity.* For the superadditivity, since $u_n$ is vanishing, up to a subsequence $$\|\widetilde{\Psi_n} u_n\|_p^p- \|u_n\|_p^p\to 0 \qquad (n\to \infty).$$ Then using the decomposition formula we compute, similarly as in the proof of Lemma \[lem:preconditions\]: $$\begin{aligned}
\limsup_{n\to \infty}E_{\text{NLS}}^{(\mathcal K)}(u_n) &= \limsup_{n\to \infty} \frac{1}{2} a(u_n, u_n) - \frac{\mu}{p} \|u_n\|_p^p\\
&\ge \limsup_{n\to \infty} a(\widetilde{\Psi_n} u_n, \widetilde{\Psi_n}u_n) - \frac{\mu}{p} \|\widetilde{\Psi_n} u_n\|_p^p\\
&\qquad \qquad \qquad + a(\Psi_n u_n, u_n) - \frac{\mu}{p} \|\Psi_n u_n\|_p^p\\
&= \limsup_{n\to \infty} E_{\text{NLS}}^{(\mathcal K)}(\widetilde{\Psi_n}u_n)+ E_{\text{NLS}}^{(\mathcal K)}(\Psi_n u_n).
\end{aligned}$$
*Subadditivity.* To show the subadditivity, note that $$\label{eq:scalingarg}
E_t^{(\mathcal K)} = t \inf_{\substack{u\in H^1\\ \|u\|_{L^2}^2=1}} \left \{\frac{1}{2}\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right |^2+ V |u|^2\, \mathrm dx - t^{\frac{p-2}{2}} \frac{\mu}{p} \int_{K} |u|^p\, \mathrm dx \right \}.$$ We deduce the property by showing that $t\mapsto E_t^{(\mathcal K)}$ is a concave function. Indeed, the scaling defines a concave function and in the limit we deduce concavity of the functional. In particular $$\label{eq:concavityarg}
E_t^{(\mathcal K)} \ge t E_1^{(\mathcal K)}, \qquad t\in [0,1].$$ Then $$E_t^{(\mathcal K)} + E_{1-t}^{(\mathcal K)} \ge E_1^{(\mathcal K)}, \qquad t\in [0,1].$$ For the strictness in the inequality it suffices to show strictness in the inequality . Assume $$E_t^{(\mathcal K)} = t E_1^{(\mathcal K)}$$ for some $t\in (0,1)$ and let $u_n$ be a minimizing sequence for $E_t$, then in particular due to $$\int_{\mathcal K} |u_n|^p\, \mathrm dx \to 0 \qquad (n\to \infty).$$ Then by density we may assume $u_n \in D(A)$ and we infer $$\begin{aligned}
E_t^{(\mathcal K)} &= \lim_{n\to \infty} E_{NLS}^{(\mathcal K)} (u_n)\\
&\ge \frac{1}{2} \limsup_{n\to \infty} \langle Au_n, u_n\rangle \ge \frac{\Sigma_0 t}{2},
\end{aligned}$$ which is a contradiction to the inequality in Lemma \[lem:energyinequality\].
\[thm:bigresulttt\] Let $\mathcal G$ be a connected, locally finite metric graph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state, which is not identically vanishing on $\mathcal K$, then $E_{\text{NLS}}^{(\mathcal K)}$ admits a minimizer for all $\mu >0$.
For $R>0$ sufficiently large since $\mathcal K$ is considered to be bounded $$\begin{aligned}
\inf_{\substack{u\in D_R(A) \\ \|u\|_{L^2}^2=1}} E_{\text{NLS}}^{(\mathcal K)}(u)&= \inf_{\substack{u\in D_R(A)\\\|u\|_{L^2}^2=1}} \frac{1}{2}\langle Au, u\rangle\\
&\ge \inf_{\substack{u\in D(A) \\ \|u\|_{L^2}^2=1}} \frac{1}{2}\langle Au, u\rangle= \frac{\Sigma_0}{2}
\end{aligned}$$ In particular with Lemma \[lem:energyinequality\] we have $$E_{\text{NLS}}^{(\mathcal K)} < \lim_{R\to \infty} \inf_{\substack{u\in D_R(A) \\ \|u\|_{L^2}^2=1}} E_{\text{NLS}}^{(\mathcal K)}(u)=: \widetilde{E_{\text{NLS}}^{(\mathcal K)}}.$$ Due to Lemma \[lem:prework\] the requirements of Theorem \[thm:main1\] and \[thm:main2\] are satisfied and up to a subsequence any minimizing sequence admits a strong limit in $L^p$ such that the limit achieves the minimum in $E_{\text{NLS}}^{(\mathcal K)}$.
\[rem:importantvanishing\] If $M\equiv 0$ then we can assume that a ground state of $A$ is nonnegative and in fact by Hopf’s maximum principle positive everywhere. In particular, any ground state of $A$ is not identically vanishing on any subset of $\mathcal G$. We will see in §6 that $\Sigma_0<\Sigma$ implies the existence of ground states of $A$. In particular, this condition becomes obsolete in this case.
### The global setting $\mathcal K=\mathcal G$
Consider now the global case, where we consider the functional $$E_\text{NLS} (\phi) = \frac{1}{2}\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right |^2+ V |\phi|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal G}|\phi|^p\, \mathrm dx, \qquad \|\phi\|_{L^2}=1.$$ In the global case Lemma \[lem:prework\] applies since any ground state of the magnetic Schrödinger operator $A=\left ( i \frac{\mathrm d}{\mathrm dx}+M\right )^2+V$ is not identically zero. In the following we give a criterion for existence of ground states with regards to these quantities.
\[prop:first\] Assume $\mathcal G$ is a locally finite, connected metric graph and $\Sigma_0 < \Sigma$. Then there exists $\hat \mu >0$, such that for all $\mu \in (0, \hat \mu)$. $$\widetilde \Sigma_0^{(\mu)} := \inf_{\phi \in D(A)} E_{\text{NLS}}(\phi) < \lim_{R\to \infty}\inf_{\phi \in D_R(A)} E_{\text{NLS}}(\phi) =: \widetilde \Sigma^{(\mu)}.$$
W.l.o.g. $\Sigma_0>0$; otherwise we simply add a constant to the potential $V$. Let $0<\varepsilon<1$ arbitrary, which we will only fix later. With Proposition \[cor:Gagliardo-nirenberg\] we deduce (similarly as in Proposition \[prop:second\]) that for sufficiently small $\mu>0$ $$E_\text{NLS}(\phi) \ge \frac{1- \varepsilon}{2} \left ( \int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right |^2+ V|\phi|^2\, \mathrm dx\right )- \frac{C\varepsilon}{2}.$$ Then $$\widetilde{\Sigma}^{(\mu)} -\widetilde{\Sigma}_0^{(\mu)} \ge \frac{1-\varepsilon}{2} \Sigma-\frac{\varepsilon}{2} - \frac{1}{2}\Sigma_0 = \frac{1}{2} \left (\Sigma - \Sigma_0\right ) -\frac{\varepsilon}{2} \left (\widetilde{C}+ \Sigma\right ).$$ Since $\varepsilon$ can be chosen arbitrarily small, we have for sufficiently small $\mu$ $$\widetilde \Sigma^{(\mu)} > \widetilde \Sigma^{(\mu)}_0.$$
\[lem:energyinequality2\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $A= \left ( i \frac{\mathrm d}{\mathrm dx} + M\right )^2 + V$ admits a ground state, then $$E_t=\inf_{\substack{u\in H^1(\mathcal G)\\\|u\|_{L^2}^2=t}} E_{\text{NLS}}(u) < \frac{\Sigma_0t}{2}.$$
In the nonlocalized case, we can proceed analogously to before. Given a ground state $u\in H^2$ we simply compute analogously as in Lemma \[lem:energyinequality\] $$E_t < \frac{\Sigma_0 t}{2}.$$
\[lem:essential\] Assume $\mathcal G$ is a locally finite, connected metric graph and $\Sigma_0 < \Sigma$. Then $ E_{\text{NLS}}$ is weak limit superadditive, superadditive with respect to the sequence of partitions of unity in Example \[ex:unity2\] and $t\mapsto E_t$ defines a strictly subadditive functional.
The proof is analogous to the one in Lemma \[lem:prework\] by simply replacing $\mathcal K$ with the whole graph. $\Sigma_0 < \Sigma$, as we will see later, implies by Theorem \[thm:persson2\] $$\inf \sigma\left ( \left ( i \frac{\mathrm d}{\mathrm dx} +M\right )^2 +V\right ) < \inf \sigma_{\text{ess}}\left ( \left ( i \frac{\mathrm d}{\mathrm dx} +M\right )^2 +V\right ).$$ In particular, there exist discrete eigenvalues below the essential spectrum and $A$ admits a ground state.
\[thm:bigresultt\] Let $\mathcal G$ be a locally finite, connected metric graph. Assume $\Sigma_0 < \Sigma$, then for $\mu \in (0, \hat \mu)$ as in Proposition \[prop:second\] $$E_\text{NLS} (\phi) = \frac{1}{2}\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right |^2+ V |\phi|^2\, \mathrm dx- \frac{\mu}{p} \int_{\mathcal G}|\phi|^p\, \mathrm dx, \qquad \|\phi\|_2=1.$$ admits a minimizer.
By Lemma \[lem:essential\] the requirements of Theorem \[thm:main2\] are satisfied. Furthermore the energy inequality in Corollary \[cor:existence\] is satisfied by Proposition \[prop:first\] and we infer the statement.
On the Threshold condition {#sec:onthreshold}
==========================
In this section we study the quantities $\Sigma_0$ and $\Sigma$ that appeared in the applications of the previous sections.
Relation to essential spectrum
------------------------------
For details on the definitions and characterizations of the essential spectrum we refer to [@reedmethods §VII].
### Polylaplacians
Let $\mathcal G= (\mathcal V, \mathcal E)$ be a connected metric finite graph in this first part of the section. In particular $\mathcal G$ consists of a compact core $K$ with rays $\mathcal E_\infty\subset \mathcal E$ attached to $K$. Consider the Polylaplacian on $\mathcal G$ defined edgewise on $H^{2k}(\mathcal G)$: $$A= (-\Delta)^k,\qquad D(A) =H^{2k}(\mathcal G)$$ Combining Lemma \[lem:IMSunity\] and the abstract decomposition formula in Lemma \[lem:decomposition2\] we have the decomposition formula for the Polylaplacian:
\[lem:decomposition3\] Let $\mathcal G= (V, \mathcal E)$ be a connected finite graph and assume $\{\Psi_1, \ldots, \Psi_N\}$ to be a partition of unity subordinate to an open covering $\mathcal O=\{O_1, \ldots, O_n\}$ satisfying $$\sum_{k=1}^N \Psi_k^2 \equiv 1.$$ Then $$\label{eq:decomposition3}
\begin{aligned}
A\phi&= \sum_{j=1}^k \Psi_j A \Psi_j\phi+ \frac{(-1)^k}{2} \sum_{m=1}^{2k} \sum_{n=1}^{2k-m} \frac{(2k)_{m+n}}{m! n!} \Psi_j^{(m)} \Psi_j^{(n)} \phi^{(2k-m-n)}
\end{aligned}$$ for all $\phi \in D(A)$.
We recall that $K=\mathcal G\setminus \mathcal E_\infty$ is the core of the graph and that for $R>0$ $$\begin{aligned}
D_R&= \{\phi \in D(A) |\operatorname{supp}(\phi) \subset \mathcal G \setminus K_R\}\\
\Sigma_R&= \inf\{\langle \phi, A\phi\rangle|\phi \in D_R, \|\phi\|_{2}^2=1\}.\end{aligned}$$
Since $D(A)$ is nontrivial and invariant under multiplication by test functions in $\widetilde{C_c^\infty}$ the set $D_R$ is nonempty.
For $R=0$ we set $$\begin{aligned}
D_0&= D(A)\\
\Sigma_0&= \inf\{\langle \phi, A\phi \rangle |\phi \in D(A), \|\phi\|_{2}^2=1\}\end{aligned}$$ and recall that $$\Sigma = \lim_{R\to \infty} \Sigma_R=\sup_{R>0} \Sigma_R.$$ In the following we characterize the quantities that were central to the existence theorems in the existence results before. Since $A$ is self-adjoint one can show $$\Sigma_0 = \inf \sigma(A).$$
\[thm:persson\] Assume $\mathcal G$ is a finite metric graph. Let $A$ be a self-adjoint, nonnegative operator on $L^2(\mathcal G)$ that satisfies the decomposition formula . Additionally let $f(A+i)^{-1}$ be compact for all $f\in\widetilde{C_c^\infty}(\mathcal G)$. Then $$\Sigma= \inf \sigma_{\text{ess}} (A).$$
*$\inf \sigma_{\text{ess}}(A)\ge \Sigma$.* Let $\lambda \in \sigma_{\text{ess}}(A)$ and let $(\phi_n)$ be an associated Weyl sequence satisfying $\|\phi_n\|_2^2=1$. Consider the vanishing-compatible sequence of partitions of unity $\Psi_{n}, \widetilde{\Psi_n}$ from Example \[ex:unity2\].
Since $\Psi_R^2 (A+i)^{-1}$ is compact for all $R>0$, and since $(A+i)\phi_n\rightharpoonup 0$ as $n\to \infty$ we deduce that $$\|\Psi_R \phi_n\|_2= \|\Psi_R (A+i)^{-1} (A+i) \phi_n\|_2\to 0\qquad (n\to \infty)$$ and passing to a subsequence, still denoted by $\phi_n$, we may assume $$\|\Psi_n \phi_n\|_2= \|\Psi_n (A+i)^{-1} (A+i) \phi_n\|_2\to 0.$$ Furthermore, with we deduce that $$\|\phi_n\|_{H^{2k}} \le C |\phi_n|_{H^{2k}}= C \left ( \|A \phi_n\|_2^2+ \|\phi_n\|_2^2\right )^{1/2}$$ is uniformly bounded. Since $\phi_n$ is a Weyl sequence for $\lambda \in \sigma_{\text{ess}}(A)$ with the decomposition formula in Lemma \[lem:decomposition3\] we then compute $$\begin{aligned}
\lambda &= \lim_{n\to \infty} \langle \phi_n, A \phi_n\rangle_{L^2} \\ &= \lim_{n\to \infty} \langle \Psi_n\phi_n, A \Psi_n \phi_n\rangle_{L^2} + \langle \widetilde{\Psi_n} \phi_n, A \widetilde{\Psi_n} \phi_n\rangle_{L^2} + O\left (\frac{1}{n^2}\right ) \\
&\ge \lim_{n\to \infty} \sum_{e\in \mathcal E_\infty} \langle \widetilde{\Psi_n} \phi_n, A\widetilde{\Psi_n} \phi_n\rangle_{L^2}\ge \lim_{n\to \infty} \Sigma_n = \Sigma.
\end{aligned}$$ Since $\lambda \in \sigma_{\text{ess}}(A)$ was arbitrary, we conclude $\inf \sigma_{\text{ess}}(A)\ge \Sigma$.
*$\inf \sigma_{\text{ess}}(A)\le \Sigma$.* Assume for a contradiction that $\inf \sigma_{\text{ess}}(A)\ge \Sigma+ 3\varepsilon$ with $\varepsilon >0$. Then $\sigma(A) \cap (-\infty, \Sigma+ 2\varepsilon]$ is discrete and since $A$ is bounded from below, the spectral projector $P_\Sigma:= P_{(-\infty, \Sigma+2\varepsilon]}$ is of finite rank. Assume $\phi_n \in D_n(A)$ is a sequence such that $$\langle \phi_n, A\phi_n\rangle \le \Sigma + \varepsilon$$ and $\phi_n \rightharpoonup 0$ in $L^2$. Then since $\left (A +\Sigma + 2\varepsilon\right )P_\Sigma $ is a compact operator and $$\left (A +\Sigma + 2\varepsilon\right )P_\Sigma \phi_n\to 0 \qquad (n\to \infty).$$ Hence $$\begin{aligned}
\langle \phi_n, A\phi_n\rangle_{L^2} &= \langle \phi_n, A(1-P_{\Sigma})\phi_n\rangle + \langle \phi_n, A P_\Sigma \phi_n\rangle_{L^2}\\
&\ge (\Sigma + 2\varepsilon) \langle \phi_n, (1-P_{\Sigma})\phi_n\rangle_{L^2} + \langle \phi_n, AP_\Sigma \phi_n\rangle_{L^2} \\
&\ge \Sigma+2\varepsilon + \left \langle \phi_n, \left (A +\Sigma + 2\varepsilon\right )P_\Sigma \phi_n \right \rangle_{L^2}.
\end{aligned}$$ Passing to the limit we conclude $$\liminf_{n\to \infty} \langle \phi_n, A\phi_n\rangle_{L^2} \ge \Sigma+2\varepsilon$$ and the statement follows by contradiction.
### Schrödinger operators and IMS formula on locally finite graphs
Let $\mathcal G=(\mathcal V, \mathcal E)$ be a locally finite graph throughout the rest of the section. Consider the Schrödinger operators with potentials $M$ and $V$ satisfying natural vertex conditions on $\mathcal G$: $$\label{eq:schroedinger}
\begin{aligned}
A&=\left (i \frac{\mathrm d}{\mathrm dx}+M\right )^2+V \\
D(A)&= \bigg \{u\in C(\mathcal G)\bigg| u_e\in H^2(e), \qquad\forall e\in {{\mathcal E}}\\
&\qquad\qquad\qquad \land\qquad \sum_{e \succ \mathsf v} \left ( i \frac{\mathrm d}{\mathrm dx} + M \right ) u_e (\mathsf v)=0,\qquad \forall \mathsf v \in \mathcal V\bigg \}.
\end{aligned}$$ In the case of magnetic Schrödinger operators $$A= \left ( i \frac{\mathrm d}{\mathrm dx}+M\right )^2 + V$$ on domains $\Omega \subset \mathbb R^N$ the decomposition formula can be obtained via the IMS formula[^2] $$A = \sum_{j=1}^k \Psi_k A \Psi_k + |\Psi_k'|^2 \quad \text{where}\quad \sum_{j=1}^k \Psi_k^2 \equiv 1.$$ When considering locally finite graphs, we may not use the approach as before because general locally finite graphs do not necessarily contain a core and we need to adapt the theory using sequences of partitions of unity as in Example \[ex:first2\].
\[df:IMS\] Let $\mathcal G$ be a locally finite, connected metric graph. Let $A: D(A) \subset L^2(\mathcal G) \to L^2(\mathcal G)$ be a densely defined, self-adjoint operator and assume $a(\cdot, \cdot)$ is the associated symmetric, sesquilinear form, defined on $H^1(\mathcal G)$. We say $A$ satisfies the IMS formula if for all $f\in W^{1,\infty}(\mathcal G)\cap \widetilde{C^\infty}(\mathcal G)$ $$\begin{gathered}
\label{eq:IMSformula}
a(fu,fv) = \frac{1}{2} \left (a(u, f^2 v)+ a(f^2 u, v)\right )+\langle |f'|^2 u, v\rangle_{L^2}, \qquad \forall u,v\in D(A).\end{gathered}$$
We showed in Lemma \[lem:lemma5.7\] that the magnetic Schrödinger operator in satisfies the IMS formula ; and in particular the following result applies:
\[thm:persson2\] Assume $\mathcal G$ is a locally finite, connected metric graph. Let $A$ be a self-adjoint, nonnegative operator on $L^2(\mathcal G)$ that satisfies the IMS formula . Additionally let $f(A+i)^{-1}$ be compact for all $f\in C_c^{0,1}\cap \widetilde{C^\infty}$ then $$\Sigma= \inf \sigma_{\text{ess}} (A).$$
$\inf \sigma_{\text{ess}}(A)\le \Sigma$ follows by an abstract argument analogous to the argument in Theorem \[thm:persson\]. To establish $$\inf \sigma_{\text{ess}}(A)\le \Sigma$$ consider $\lambda \in \sigma_{\text{ess}}(A)$ and let $(\phi_n)$ be an associated Weyl sequence satisfying $$\|\phi_n\|_2^2=1.$$ Consider the vanishing-compatible sequence of partitions of unity $\Psi_{n}, \widetilde{\Psi_n}$ from Example \[ex:first2\].
Since $\Psi_R^2 (A+i)^{-1}$ is compact for all $R>0$, and since $(A+i)\phi_n\rightharpoonup 0$ as $n\to \infty$ we deduce $$\|\Psi_R \phi_n\|_2= \|\Psi_R (A+i)^{-1} (A+i) \phi_n\|_2\to 0\qquad (n\to \infty)$$ and passing to a subsequence, still denoted by $\phi_n$, we may assume $$\|\Psi_n \phi_n\|_2= \|\Psi_n (A+i)^{-1} (A+i) \phi_n\|_2\to 0.$$
Since $\phi_n$ is a Weyl sequence for $\lambda \in \sigma_{\text{ess}}(A)$, with the decomposition formula in Lemma \[lem:decomposition3\] we then compute $$\begin{aligned}
\lambda &= \lim_{n\to \infty} \langle \phi_n, A \phi_n\rangle_{L^2} \\ &= \lim_{n\to \infty} a(\Psi_n \phi_n, \Psi_n \phi_n) + a(\widetilde{\Psi_n} \phi_n, \widetilde{\Psi_n}\phi_n) + O\left (\frac{1}{n^2}\right ) \\
&\ge \lim_{n\to \infty} \sum_{e\in \mathcal E_\infty} a(\widetilde{\Psi_n}\phi_n, \widetilde{\Psi_n}\phi_n)\ge \lim_{n\to \infty} \Sigma_n = \Sigma.
\end{aligned}$$ Since $\lambda \in \sigma_{\text{ess}}(A)$ was arbitrary, we conclude $\inf \sigma_{\text{ess}}(A)\ge \Sigma$.
Sufficient conditions for the threshold condition for the Polylaplacian {#sec:subsecsuff}
-----------------------------------------------------------------------
In this section we obtain criteria for the threshold condition for the operator $$\begin{gathered}
A= (-\Delta)^k + V\\
D(A)=H^{2k}
\end{gathered}$$ We start with the case $k=1$ for general locally finite graphs satisfying a volume growth assumption. For $k\ge 2$ we restrict ourselves to finite graphs.
\[thm:lastresultt1\] Let $\mathcal G$ be a locally finite, connected metric graph and let $K$ be a connected, precompact subgraph. We suppose additionally the volume assumption $$\label{eq:volumegrowthassumpt}
\left |K_{2n} \setminus K_{n}\right |=o(n^2) \qquad (n\to \infty).$$ Assume $\sigma_\text{ess}(-\Delta+V) \subset [0, \infty)$ and assume additionally either
(i) $V\in L^1(\mathcal G)\cap L^2(\mathcal G)$ and $$\int_{\mathcal G} V \, \mathrm dx <0$$
(ii) or $V<0$ on $\mathcal G$.
Then $\Sigma_0 < \Sigma$ (as defined in and ) and there exists $\hat \mu>0$ such that the minimization problem $$\label{eq:minimizerlast}
E^{(1)}=\inf_{\substack{u\in H^1\\\|u\|_{L^2}^2=1}} E^{(1)}(u)$$ admits a minimizer for $\mu \in (0,\hat \mu)$.
Consider as a test function $\Psi_n$ as defined in Example \[ex:first2\], then we only need to show that for $n$ sufficiently high, the Rayleigh quotient $$\mathcal R[\Psi_n] := \frac{\int_{\mathcal G} |\Psi'_n|^2 + V |\Psi_n|^2\, \mathrm dx }{\int_{\mathcal G} |\Psi_n|^2\, \mathrm dx}<0.$$ Indeed, since $\|\Psi'_n\|_{\infty}^2\le O(\frac{1}{n^2})$ as $n\to \infty$ we deduce $$\|\Psi'_n\|_2^2\le \|\Psi'_n\|_{\infty}^2 |K_{2n}\setminus K_n|\to 0 \qquad (n\to \infty).$$ If $V<0$ then for sufficiently large $n$ and $\varepsilon>0$ sufficiently small $$\int_{\mathcal G} V |\Psi_n|^2\, \mathrm dx\le -\left |\left \{x\in \mathcal G: V(x)\le -\varepsilon\right \}\right | \varepsilon<0.$$ If $\int_{\mathcal G} V \, \mathrm dx <0$, then $$\liminf_{n\to \infty}\int_{\mathcal G} V |\Psi_n|^2 \, \mathrm dx= \int_{\mathcal G} V\, \mathrm dx<0$$ by dominated convergence. In particular for $n$ large enough $$\int_{\mathcal G} V|\Psi_n|^2 \le \frac{1}{2} \int_{\mathcal G} V\, \mathrm dx <0.$$ We deduce $R[\Psi_n]<0$ and thus $\inf \sigma((-\Delta)^k+V)<0$. Then $\Sigma_0 < \Sigma$ and we conclude the existence of minimizers of by Theorem \[thm:bigresultt\].
Similarly, for $k\ge 2$ we have:
\[thm:lastresultt2\] Let $\mathcal G$ be a finite, connected metric graph and $k\ge 1$. Assume $\sigma_\text{ess}((-\Delta)^k+V) \subset [0,\infty)$ and assume additionally either
(i) $V\in L^1(\mathcal G)\cap L^2(\mathcal G)$ and $$\int_{\mathcal G} V \, \mathrm dx <0$$
(ii) or $V<0$ on $\mathcal G$.
Then $\Sigma_0 < \Sigma$ (as defined in and ) and there exists $\hat \mu>0$, such that the minimization problem $$\label{eq:minimizerklast}
E^{(k)}=\inf_{\substack{u\in H^1\\\|u\|_{L^2}^2=1}} E^{(k)}(u)$$ admits a minimizer for $\mu \in (0,\hat \mu)$.
The proof is analagous to the proof in Proposition \[thm:lastresultt1\]. We only need to replace the test functions $\Psi_n$ with the ones in Example \[ex:unity2\]. Then $$\|\Psi_n^{(k)}\|_\infty \le \frac{1}{n^{2k}}C$$ with $C$ independent of $n$. We infer the result as in the proof of Proposition \[thm:lastresultt1\].
If $V$ is a relativly compact perturbation of $(-\Delta)^k$, i.e. $$V\left ((-\Delta)^k+i\right )^{-1}$$ is compact, then $\inf \sigma_\text{ess}\left ((-\Delta)^k+V\right ) =0$ and we deduce $$\inf \sigma_\text{ess}\left ((-\Delta)^k+V\right )\subset [0,\infty).$$
We finish the section by giving a criterion for the potential $V$ such that $$\Sigma = \lim_{n\to \infty} \inf_{\substack{u\in D(A) \\ \|u\|_2^2=1, \; \operatorname{supp} u\subset \mathcal G\setminus K_n}} \langle u, Au\rangle \ge 0.$$ In the situations considered in §6.1 this in particular implies $$\sigma_\text{ess}((-\Delta)^k+V) \subset [0,\infty).$$ Consider decaying potentials $V=V_2+ V_\infty$ with $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$ such that $$\label{eq:newdecayingpotentials}
\sup_{x\in \mathcal G\setminus K_n} |V_\infty(x)|\to 0 \qquad (n\to \infty).$$
\[prop:Iwantthistoend\] Let $\mathcal G$ be a locally finite metric graph. Assume $V\in L^2+ L^\infty$ satisfying . Let $A=(-\Delta)^k+V$, then $$\Sigma = \lim_{n\to \infty} \inf_{\substack{u\in H^k(\mathcal G)\\ \|u\|_2^2=1, \; \operatorname{supp} u\subset \mathcal G\setminus K_n}} \langle u, Au\rangle_{L^2} \ge 0.$$
Assume $u_n$ is a minimizing sequence, such that $\|u_n\|_{L^2}^2$, $\operatorname{supp} u\subset \mathcal G\setminus K_n$ and $$\langle u_n, Au_n\rangle_{L^2}\to \Sigma.$$ With we deduce that $$\label{eq:ineedthis2}
\|u_n\|_{H^k} \le C \left (\langle u_n, Au_n\rangle_{L^2}^2 + \|u_n\|_2^2\right )$$ is uniformly bounded. Integrating by parts and using we infer $$\begin{aligned}
\int_{\mathcal G} \left |u^{(k)}_n\right |^2 + V|u_n|^2\, \mathrm dx&\ge \int_{\mathcal G} \left |u^{(k)}_n\right |^2\, \mathrm dx \\
&\qquad - \widetilde{C} \left ( \left (\int_{\mathcal G\setminus K_n} |V|^2\, \mathrm dx\right )^{1/2} + \sup_{x\in \mathcal G\setminus K_n} |V_\infty(x)|\right ).
\end{aligned}$$ We have $$\left ( \left (\int_{\mathcal G\setminus K_n} |V|^2\, \mathrm dx\right )^{1/2} + \sup_{x\in \mathcal G\setminus K_n} |V_\infty(x)|\right )\to 0 \qquad (n\to \infty).$$ Thus, $$\Sigma = \lim_{n\to \infty} \langle u_n, Au_n\rangle_{L^2} \ge 0.\qedhere$$
Application: Schrödinger operators with magnetic potentials on infinite tree graphs
===================================================================================
In certain cases as discussed in [@berkolaiko2013introduction §2.6] the gauge transform $G$, as defined below, unitarily transforms the Schrödinger operator with magnetic potential into a Schrödinger operator without magnetic potential, and the NLS functional under gauge transform reduces to a problem without magnetic potential. We may use the results from Section \[sec:subsecsuff\] to show existence of minimizers of the NLS functional with magnetic potential.
For infinite tree graphs in the context of locally finite, connected metric graphs it is particularly easy to see this. In this context, let $\mathcal G$ be an infinite tree graph. Given a vertex $\mathsf v$ we can define the gauge transform $G$ radially. For any $x\in \mathcal G$, let $\gamma$ be a simple path from $\mathsf v$ to $x$ parametrized by arc length, then $$G:u(x)\mapsto e^{i \int_{\operatorname{im} \gamma} M\, \mathrm d\gamma} u(x).$$
Assume $A^{M}= (i \frac{\mathrm d}{\mathrm dx} + M)^2+V$ admits a ground state. In this particular case since $$G^{-1} A^{M} G = -\Delta + V= A^{0},$$ this is equivalent to the assertion that $A^{0}$ admits a ground state. Indeed, let $u_M$ be a ground state to $A^{M}$, then $$A^{0} G^{-1} u_0 =G^{-1} A^{M} u_M = \Sigma G^{-1} u_M$$ and $G^{-1} u_M$ is a ground state of $A^{0}$. Then we may assume $u_0>0$ by phase invariance and the maximum principle. Then $u_M$ does not vanish anywhere. In particular independent of $M\in H^1+W^{1,\infty}(\mathcal G)$ $$\label{eq:unitaryequivalence}
\begin{aligned}
\Sigma_0^{M} &= \inf_{\substack{u\in D(A^{M})\\\|u\|_{2}^2=1}}\left \langle A^{M}u, u\right \rangle= \inf_{\substack{u\in D(A^{0})\\\|u\|_{2}^2=1}}\left \langle A^{0}u, u\right \rangle= \Sigma_0\\
\Sigma_{R}^{M} &= \inf_{\substack{u\in D_R(A^{M})\\\|u\|_{2}^2=1}}\left \langle A^{M}u, u\right \rangle= \inf_{\substack{u\in D_R(A^{0})\\\|u\|_{2}^2=1}}\left \langle A^{0}u, u\right \rangle= \Sigma_R\\
\Sigma^{M}&= \lim_{R\to \infty} \Sigma_R^M = \lim_{R\to \infty} \Sigma_R= \Sigma
\end{aligned}$$ and in Section \[sec:subsecsuff\] we gave sufficient conditions for $\Sigma_0 < \Sigma$.
\[prop:app1\] Assume $\mathcal G$ is an infinite tree graph, connected and locally finite. Assume $\mathcal K$ is a bounded subgraph of $\mathcal G$ and $-\Delta+V$ admits a ground state, then the infimization problem $$E_{\text{NLS}}^{(\mathcal K)}=\inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_2^2=1}} \frac{1}{2}\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right |^2+ V |u|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathcal K} |u|^p\, \mathrm dx$$ admits a minimizer for all $\mu \in \mathbb R$.
This follows immediately from Theorem \[thm:bigresulttt\] and the unitary equivalence of the problem in absence of a magnetic potential under the gauge transform.
\[prop:app2\] Assume $\mathcal G$ is a infinite tree graph, locally finite and connected. Assume $\mathcal K$ is any unbounded subgraph and $\Sigma_0 < \Sigma$ then there exists $\hat \mu>0$, such that the infimization problem $$E_{\text{NLS}}=\inf_{\substack{\phi\in H^1(\mathcal G)\\ \|\phi \|_2^2=1}} \frac{1}{2}\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )\phi\right |^2+ V |\phi|^2\, \mathrm dx- \int_{\mathcal K} \frac{\mu}{p} |\phi|^p\, \mathrm dx.$$ admits a minimizer for all $\mu\in (0,\hat \mu)$.
This follows immediately from Theorem \[thm:bigresultt\] and the unitary equivalence of the problem in absence of a magnetic potential under the gauge transform
For decaying potentials in §6.2 we discussed criteria such that $\Sigma_0< \Sigma$ is satisfied. Indeed, for any given locally finite graph, one can construct decaying potentials in the following way:
Let $\mathcal G$ be a locally finite, connected graph and $K$ a bounded, connected subgraph. Consider the higher-order Schrödinger operator $A=(-\Delta)^k+V$ with potential $V$. We define a potential $V$ a.e. via $$\begin{aligned}
V\bigg |_{K_1} &\equiv -\frac{1}{2}\\
V\bigg |_{K_{2n}\setminus K_n} &\equiv -\frac{1}{2^n |K_{2n} \setminus K_n|}, \quad n\ge 2
\end{aligned}$$ on each “annulus” $K_{2n}\setminus K_n$. Then $V\in L^2 \cap L^1(\mathcal G)$, $$\int_{\mathcal G} V\, \mathrm d\mu= -\sum_{n=0}^\infty \frac{1}{2^n} <0$$ and by Proposition \[prop:Iwantthistoend\] we infer $\inf \sigma_{\text{ess}}(A)\ge 0$. In particular, if $\mathcal G$ is an infinite tree graph satisfying the volume growth assumption , then the prerequisites in Proposition \[thm:lastresultt1\] are satisfied as well and we have $$\Sigma_0 < \Sigma.$$ In particular Proposition \[prop:app1\] and Proposition \[prop:app2\] are applicable and there exists $\hat \mu >0$ such that $$E_{\text{NLS}}^{(\mathcal K)}=\inf_{\substack{u\in H^1(\mathcal G)\\ \|u\|_2^2=1}} \frac{1}{2}\int_{\mathcal G} \left |\left ( i \frac{\mathrm d}{\mathrm dx} + M \right )u\right |^2+ V |u|^2\, \mathrm dx - \frac{\mu}{p} \int_{\mathcal K} |u|^p\, \mathrm dx$$ admits a minimizer for $\mu \in (0, \hat \mu)$. If $\mathcal K\subset \mathcal G$ is precompact, then minimizers exist for all $\mu>0$.
For a certain class of infinite tree graphs we can in a similar way as in Example \[ex:shouldbeputinintroduction\] give an explicit $\hat \mu$ such that for $\mu\in (0,\hat \mu]$ the minimization problem $E_{NLS}$ admits a minimizer.
\[ex:unrootedtrees\] Consider an *unrooted tree graph* $\mathcal G$ as considered for instance in [@dovetta2019nls], i.e. there are no vertices of degree $1$ apart of vertices at infinity. Such trees in particular satisfy the **(H)**-condition formulated in [@adami2015nls] in the special case of finite graphs:
- For every point $x\in \mathcal G$, there exist two injective curves $\gamma_1, \gamma_2:[0,+\infty) \to \mathcal G$ parametrized by arc length, with disjoint images except on a discrete set of points, and such that $\gamma_1(0)=\gamma_2(0)=x$.
and by rearrangement methods as in [@dovetta2019nls] one can show for decaying potentials $V= V_2+V_\infty$ with $V_2\in L^2(\mathcal G)$ and $V_\infty\in L^\infty(\mathcal G)$ satisfying $$\sup_{x\in \mathcal G\setminus K_n} |V_\infty(x)|\to 0 \qquad (n\to \infty)$$ that $$\begin{aligned}
\widetilde{\Sigma}^{(\mu)} &= \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\|u\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^V(u)\\
&\ge \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\|u\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^0(u) \ge E_{\text{NLS}}(\mathbb R),
\end{aligned}$$ where by Remark \[rmk:decayingpotential\] one has equality if $\mathcal G$ contains a half line.
When $V\equiv 0$, by strictness in the rearrangement inequality one can prove nonexistence results similarly as in [@adami2015nls]. On the other hand, under the assumption $$\Sigma_0 = \inf \sigma(-\Delta +V) <0,$$ as discussed in Example \[ex:shouldbeputinintroduction\] we have thus the existence of minimizers of $E_{\text{NLS}}$ for $$\mu \in \left [ 0, \left ( \frac{\Sigma_0}{\gamma_p} \right )^{\frac{6-p}{4}} \right ].$$
\[rmk:unrooted\] The arguments in Example \[ex:unrootedtrees\] can be applied to all graphs that satisfy the **(H)**-condition. One can even consider more general graphs as long they satisfy the following weaker version of the **(H)**-condition:
- There exists a precompact set $K\subset \mathcal G$, such that for every point $x\in \mathcal G\setminus K$, there exist two injective curves $\gamma_1, \gamma_2:[0,+\infty) \to \mathcal G$ parametrized by arc length, with disjoint images except on a discrete set of points, and such that $\gamma_1(0)=\gamma_2(0)=x$.
Consider the graph consisting of two half-lines and a pendant edge joined at a single vertex (see also Figure \[fig:donedone\]), then the graph satisfies the **(H)**-condition but not the **(H)**-condition and the existence result from Example \[ex:unrootedtrees\] as discussed in Remark \[rmk:unrooted\] is still applicable.
![The graph consisting of two half-lines and a pendant edge as an example of a graph that satisfies the the **(H)**-condition but not the **(H)**-condition.[]{data-label="fig:donedone"}](simple3.pdf)
We finish this section by proving Theorem \[thm:introlastlastlast\]:
Let $\mathcal G$ be a locally finite metric tree graph that contains at most finitely many vertices of degree $1$. Then there exists a connected, precompact set $K\subset \mathcal G$ that contains all vertices of degree $1$ by assumption. Consider the set $\overline{\mathcal G}$ of points $x\in \mathcal G$, such that there exist two injective curves $\gamma_1, \gamma_2:[0,+\infty)\to \mathcal G$ parametrized by arc length, with disjoint images except on a discrete set of points, and such that $\gamma_1(0)= \gamma_2(0)=x$. In particular, if $x\in \overline{\mathcal G}$, then $$\operatorname{im} \gamma_1,\; \operatorname{im} \gamma_2\subset \overline{\mathcal G}.$$
*1st Case: $\overline{\mathcal G}\neq \emptyset$.* Then by assumption $\mathcal G\setminus \overline{\mathcal G}$ contains at most finitely many connected components. Moreover of the connected components is precompact. Otherwise one could construct an injective curve $\gamma_1:[0, +\infty)\to \mathcal G\setminus \overline{\mathcal G}$ for all $x\in \mathcal G\setminus \overline{\mathcal G}$ and since we assumed $\overline{\mathcal G}\neq \emptyset$, we can construct $\gamma_2:[0,+\infty)\to \mathcal G\setminus \overline{\mathcal G}$. This would then imply that $\mathcal G\setminus \overline{\mathcal G}$ is necessarily precompact. Since $\mathcal G$ is a tree graph, this also implies that each connected component of $\mathcal G\setminus \overline{\mathcal G}$ contains necessarily a vertex of degree $1$. In particular, $\mathcal G\setminus \overline{\mathcal G}$ admits at most finitely many connected components and is precompact. By construction, $\overline{\mathcal G}$ satisfies the **(H)**-condition and hence $\mathcal G$ satisfies the **(H)**-condition. Then as in Example \[ex:unrootedtrees\] we have $$\begin{aligned}
\widetilde{\Sigma}^{(\mu)} &= \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\|u\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^V(u)\\
&\ge \lim_{n\to \infty} \inf_{\substack{u\in H^1(\mathcal G\\\|u\|_2^2=1, \, \operatorname{supp} u\subset \mathcal G\setminus K_n}} E_{\text{NLS}}^0(u) \ge E_{\text{NLS}}(\mathbb R)
\end{aligned}$$ and we obtain existence of minimizers of $E_{\text{NLS}}$ for $$\mu \in \left [ 0, \left ( \frac{\Sigma_0}{\gamma_p} \right )^{\frac{6-p}{4}} \right ].$$
*2nd Case: $\overline{\mathcal G}=\emptyset$.* In particular for each $x\in \mathcal G$ there exists only one connected component of $\mathcal G$ that contains a vertex at infinity. Assume $K$ is a precompact set that contains all vertices of degree $1$, then by assumption for any $x\in \mathcal G\setminus K$ the connected components of $\mathcal G\setminus \{x\}$ consist of a compact core graph containing all vertices of degree $1$ and a half-line. In particular, $\mathcal G$ is a finite graph and Example \[ex:shouldbeputinintroduction\] yields the existence of minimizers of $E_{\text{NLS}}$ for $$\mu \in \left [ 0, \left ( \frac{\Sigma_0}{\gamma_p} \right )^{\frac{6-p}{4}} \right ].$$
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The charmless bottom meson decays are systematically investigated based on an approximate six quark operator effective Hamiltonian from perturbative QCD. It is shown that within this framework the naive QCD factorization method provides a simple way to evaluate the hadronic matrix elements of two body mesonic decays. The singularities caused by on mass-shell quark propagator and gluon exchanging interaction are appropriately treated. Such a simple framework allows us to make theoretical predictions for the decay amplitudes with reasonable input parameters. The resulting theoretical predictions for all the branching ratios and $CP$ asymmetries in the charmless $B^0,\ B^+,\ B_s\to \pi\pi,\ \pi K,\
KK$ decays are found to be consistent with the current experimental data except for a few decay modes. The observed large branching ratio in $B\to \pi^0\pi^0$ decay remains a puzzle though the predicted branching ratio may be significantly improved by considering the large vertex corrections in the effective Wilson coefficients. More precise measurements of charmless bottom meson decays, especially on CP-violations in $B\to K K$ and $B_s\to
\pi\pi, \pi K, KK$ decay modes, will provide a useful test and guide us to a better understanding on perturbative and nonperturbative QCD.
author:
- 'Fang Su$^{\ast \dagger}$, Yue-Liang Wu$^{\ast}$, Yi-Bo Yang$^{\ast\ddagger}$ and Ci Zhuang$^{\ast}$'
title: |
QCD Factorization Based on Six-Quark Operator Effective Hamiltonian from Perturbative QCD and Charmless Bottom Meson Decays $B_{(s)}\to
\pi\pi,\pi K, KK$
---
Introduction
============
Hadronic B-meson decays play importance role not only for understanding the dynamical scheme of hadronic decays and testing the flavor structure of the Standard Model(SM), but also for probing the origin of CP violation and new physics signals beyond the SM. In particular, the precise measurement and systematic study for hadronic charmless B decays may provide a window for such purposes. The branching ratios of $B \rightarrow \pi \pi$ and $\pi K$ modes have been measured with a good accuracy[@HFAG] and a large direct CP violation has been established in $\pi^+ K^-$ mode [@HFAG]. The most severe discrepancies between the experimental data and theoretical predictions come from the unexpected large branch ratio of $B \to \pi^0\pi^0$ and some unclear CP violations in $B\to \pi^0 K$ decays, which are called $\pi\pi, \pi K$ puzzles[@pikpuzzle; @WZ]. Theoretically, to predict consistently those decays, it needs to deal with the short-distance contributions in a complete and systematic way from the high energy scale to a proper low energy scale at which the perturbative calculations remain reliable, and treat the long-distance contributions which contain the non-perturbative strong interactions involved in those decays. The main task is to reliably compute the hadronic matrix elements between the initial and final hadron states. Several novel methods based on the naive factorization approach (FA) and four quark operator effective Hamiltonian have been developed to evaluate the hadronic matrix elements, such as the QCD factorization approach (QCDF)[@Beneke:1999br], the perturbation QCD method (pQCD) [@Keum:2000ph], and the soft-collinear effective theory (SCET)[@Bauer:2000ew]. These methods have been widely used in analyzing hadronic B-meson decays and made great progresses in understanding the hadronic structure and properties of strong interactions. To understand the puzzles whether they are due to the unknown new physics or it is because of the lack of our knowledge on the hadronic properties of strong interactions, it still needs to investigate further the various approaches within the framework of QCD and to check the validity of assumptions and approximations made in the practical calculations.
The widely used theoretical framework of weak decays is based on the current-current four fermion operator effective Hamiltonian derived via operator product expansion and renormalization group evolution. In hadronic weak decays, the short-distance contributions of QCD are characterized by the Wilson coefficient functions of four quark operators and the long-distance contributions are in principle obtained by evaluating the hadronic matrix elements of four quark operators. The Wilson coefficient functions are in general calculated by perturbative QCD which is well developed, while the evaluation of hadronic matrix elements remains a hard task as it involves non-perturbative effects of QCD. To deepen our insights into the hadronic decays, we shall first reinvestigate the four quark operator effective Hamiltonian whether it is always suitable as a basic framework for all hadronic weak decays. In fact, for the mesonic two body decays of B meson, it concerns three quark-antiquark pairs once each meson is regarded as the quark-antiquark bound state at the quark level structure. This fact then naturally motivates us to consider six-quark operator effective Hamiltonian instead of four-quark operator effective Hamiltonian. Namely, we shall begin with six quark diagrams of weak decays with both W-boson exchange and gluon exchange, and derive formally the six-quark operator effective Hamiltonian based on operator product expansion and renormalization group evolution when including loop corrections of six quark diagrams. We shall show how this approach allows us to figure out what are the assumptions and approximations made in effective four quark operator approach, and how the simple QCD factorization scheme can reliably be applied to evaluate the hadronic matrix elements with the six quark operator effective Hamiltonian. For the infrared singularity caused by the gluon exchanging interaction when evaluating the hadronic matrix elements of effective six quark operators, it is shown to be simply treated by the introduction of a mass scale motivated from the gauge invariant loop regularization method [@LRC], where the energy scale $\mu_g$ is introduced to play the role of infrared cut-off energy scale without violating gauge invariance.
The paper is organized as follows. In section \[sec:sqeh\], after briefly reviewing the four quark operator effective Hamiltonian, we begin with the primary six quark diagrams with a single W-boson exchange and a single gluon exchange, and the corresponding initial six-quark operator. It is shown that a complete six quark operator effective Hamiltonian is in general necessary to include all contributions from both perturbative and non-perturbative QCD corrections, especially the non-pertubative QCD corrections at low energy scale $\mu < m_c\sim 1.5$ GeV could be sizable. To demonstrate how the six quark operator effective Hamiltonian provides a reliable framework for hadronic two body decays of B meson, we will focus, as a good approximation, on the dominant QCD loop diagrams of six quarks so as to avoid the tedious calculations. In section \[sec:QCDF\], it is demonstrated how the QCD factorization approach becomes a simple and natural tool to evaluate the hadronic matrix elements of mesonic two body decays based on the six quark operator effective Hamiltonian. In particular, the so-called factorizable and non-factorizable, emission and annihilation diagram contributions are automatically the consequences of QCD factorization for the hadronic matrix elements of effective six quark operators. The treatment on the singularities caused by the gluon exchanging interactions and the on mass-shell fermion propagator is presented in Section \[sec:TOD\]. In Section \[sec:Amplitude\], all the amplitudes of charmless bottom meson decays are completely obtained by using the QCD factorization approach based on the approximate six quark operator effective Hamiltonian. Our numerical results with appropriate input parameters are presented in section \[sec:nrcpe\], as a good approximation, the resulting predictions on branching ratios and CP violations of charmless bottom meson decays are much improved and also more closed to the current experimental data. The conclusions and remarks are given in last section. The detailed calculations involved in the evaluation of various decay amplitudes are presented in the Appendix.
Effective Hamiltonian of Six Quark Operators {#sec:sqeh}
============================================
Four Quark Operator Effective Hamiltonian
-----------------------------------------
Let us start from the four-quark effective operators in the effective weak Hamiltonian. The initial four quark operator due to weak interaction via W-boson exchange is given as follows for B decays $$O_{1}=(\bar{q}^u_{i}b_{i})_{V-A}(\bar{q}^d_{j}u_{j})_{V-A}, \qquad
q^u=u,\ c, \quad q^d = d,\ s$$ The complete set of four quark operators are obtained from QCD and QED corrections which contain the gluon exchange diagrams, strong penguin diagrams and electroweak penguin diagrams. The resulting effective Hamiltonian(for $b\to s$ transition) with four quark operators is known to be as follows $$\begin{aligned}
H_{\rm eff}\, =\, {G_F\over\sqrt{2}} \sum_{q=u,c}
\lambda_q^{s} \left[C_1(\mu)O_1^{(q)}(\mu) +C_2(\mu)O_2^{(q)}(\mu)+
\sum_{i=3}^{10}C_i(\mu)O_i(\mu)\right]+h.c.\;,\label{eq:hpk}\end{aligned}$$ with $\lambda_q^{s} = V_{qb}V^*_{qs}$ and $V_{ij}$ the CKM matrix elements, $C_i(\mu)$ the Wilson coefficient functions[@4qham] and $O_i(\mu)$ the four quark operators $$\begin{aligned}
\begin{array}{ll}
\displaystyle O_1^{(q)}\, =\,
(\bar{q}_ib_i)_{V-A}(\bar{s}_jq_j)_{V-A}\;, & \displaystyle
O_2^{(q)}\, =\,(\bar{s}_ib_i)_{V-A}(\bar{q}_jq_j)_{V-A}\;,
\\
\displaystyle O_3\,
=\,(\bar{s}_ib_i)_{V-A}\sum_{q'}(\bar{q}'_jq'_j)_{V-A}\;,
&\displaystyle O_4\,
=\,\sum_{q'}(\bar{q}'_ib_i)_{V-A}(\bar{s}_jq'_j)_{V-A}\;,
\\
\displaystyle O_5\,
=\,(\bar{s}_ib_i)_{V-A}\sum_{q'}(\bar{q}'_jq'_j)_{V+A}\;,
&\displaystyle O_6\,
=\,-2\sum_{q'}(\bar{q}'_ib_i)_{S-P}(\bar{s}_jq'_j)_{S+P}\;,
\\
\displaystyle O_7\,
=\,\frac{3}{2}(\bar{s}_ib_i)_{V-A}\sum_{q'}e_{q'}(\bar{q}'_jq'_j)_{V+A}\;,&
\displaystyle O_8\, =\,
-3\sum_{q'}e_{q'}(\bar{q}'_ib_i)_{S-P}(\bar{s}_jq'_j)_{S+P}\;,
\\
\displaystyle O_9\,
=\,\frac{3}{2}(\bar{s}_ib_i)_{V-A}\sum_{q'}e_{q'}(\bar{q}'_jq'_j)_{V-A}\;,&
\displaystyle O_{10}\, =\,
\frac{3}{2}\sum_{q'}e_{q'}(\bar{q}'_ib_i)_{V-A}(\bar{s}_jq'_j)_{V-A}\;,\\
\end{array}
\label{eq:o}\end{aligned}$$ Here the Fermi constant $G_F=1.16639\times
10^{-5}\;{\rm GeV}^{-2}$, and the color indices $i, \ j$, and the notations $(\bar{q}'q')_{V\pm A} = \bar q' \gamma_\mu (1\pm
\gamma_5)q'$. The index $q'$ in the summation of the above operators runs through $u,\;d,\;s$, $c$, and $b$. The effective Hamiltonian for the $b\to d$ transition can be obtained by changing $s$ into $d$ in Eqs. (\[eq:hpk\])and (\[eq:o\]).
Six Quark Diagrams and Effective Operators {#sec:sqd}
------------------------------------------
As mesons are regarded as quark and anti-quark bound states, the mesonic two body decays actually involve three quark-antiquark pairs. It is then natural to consider the six quark Feynman diagrams which lead to three effective currents of quark-antiquark. The initial six quark diagrams of weak decays contain one W-boson exchange and one gluon exchange, thus there are four different diagrams as the gluon exchange interaction can occur for each of four quarks in the W-boson exchange diagram, see Fig. \[pic:4insert\].
![Four different six quark diagrams with a single W-boson exchange and a single gluon exchange[]{data-label="pic:4insert"}](pic1.eps "fig:")\
The resulting initial effective operators contain four terms corresponding to the four diagrams, respectively. In a good approximation, the four quarks via W-boson exchange can be regarded as a local four quark interaction at the energy scale much below the W-boson mass, while two QCD vertexes due to gluon exchange are at the independent space-time points, the resulting effective six quark operators are hence in general nonlocal. The six-quark operators corresponding to the four diagrams in Fig. \[pic:4insert\] are found to be $$\begin{aligned}
O^{(6)}_{q_1}\, &=&\, 4\pi\alpha_s \int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\, \frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
(\bar{q'}(x_3)\gamma_{\nu}T^{a} q'(x_3))\frac{1}{k^2+i\epsilon}\nonumber\\
&&(\bar{q}_{2}(x_1)\Gamma_{1}\frac{p\!\!\!/+m_b}{p^2-m_b^2+i\epsilon}\gamma^{\nu}T^{a} q_{1}(x_2))*
(\bar{q}_{4}(x_1) \Gamma_{2} q_{3}(x_1)),\nonumber\\
O^{(6)}_{q_2}\, &=&\, 4\pi\alpha_s \int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\, \frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
(\bar{q'}(x_3)\gamma_{\nu}T^{a} q'(x_3))\frac{1}{k^2+i\epsilon}\nonumber\\
&&(\bar{q}_{2}(x_2)\frac{p\!\!\!/+m_{q_1}}{p^2-m_{q_1}^2+i\epsilon}\gamma^{\nu}T^{a}\Gamma_{1} q_{1}(x_1))*
(\bar{q}_{4}(x_1) \Gamma_{2} q_{3}(x_1)),\nonumber\\
O^{(6)}_{q_3}\, &=&\, 4\pi\alpha_s \int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\, \frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
(\bar{q'}(x_3)\gamma_{\nu}T^{a} q'(x_3))\frac{1}{k^2+i\epsilon}\nonumber\\
&&(\bar{q}_{2}(x_1)\Gamma_{1} q_{1}(x_1))*
(\bar{q}_{4}(x_1) \Gamma_{2} \frac{p\!\!\!/+m_{q_3}}{p^2-m_{q_3}^2+i\epsilon}\gamma^{\nu}T^{a} q_{3}(x_2)),\nonumber\\
O^{(6)}_{q_4}\, &=&\, 4\pi\alpha_s \int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\, \frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
(\bar{q'}(x_3)\gamma_{\nu}T^{a} q'(x_3))\frac{1}{k^2+i\epsilon}\nonumber\\
&&(\bar{q}_{2}(x_1)\Gamma_{1} q_{1}(x_1))*
(\bar{q}_{4}(x_2)\frac{p\!\!\!/+m_{q_2}}{p^2-m_{q_2}^2+i\epsilon}\gamma^{\nu}T^{a} \Gamma_{2} q_{3}(x_1)),
\label{eq:six}\end{aligned}$$
where $k$ and $p$ correspond to the momenta of gluon and quark in their propagators. $q_1$ is usually set to be heavy quark like b quark. $x_1$, $x_2$ and $x_3$ are space-time points corresponding to three vertexes. The color index is summed between $q_1,q_2$ and $q_3,q_4$. Note that all the six quark operators are proportional to the QCD coupling constant $\alpha_s$ due to gluon exchange. Thus the initial six quark operator is given by summing over the above four operators $$\begin{aligned}
O^{(6)}=\sum_{j=1}^4 O^{(6)}_{q_j}.\end{aligned}$$
Actually, the initial six quark operators $O^{(6)}_{q_j}$ ($j=1,2,3,4$) can be obtained from the following initial four quark operator via a single gluon exchange $$\begin{aligned}
O \equiv (\bar{q}_{2} \Gamma_{1} q_{1})*(\bar{q}_{4} \Gamma_{2} q_{3}).\label{eq:any}\end{aligned}$$
Six Quark Operator Effective Hamiltonian via Perturbative QCD {#sec:sqehbd}
--------------------------------------------------------------
Based on the above considerations with the introduction of six quark operators, in this section we shall specify the initial six quark operator $O_1^{(q)(6)}$ $(q=u,\, c)$ to the case of nonleptonic bottom hadron decays and show how to obtain six quark operator effective Hamiltonian. The initial six quark operator in b-decay with $\Delta S \neq 0$ is as follows (for the $b\to d$ transition with $\Delta S = 0$, just replacing $s$ by $d$) $$\begin{aligned}
O_1^{(q)(6)}\,&=&\sum_{l=1}^4 O^{(q)(6)}_{1q_l} \nonumber\\
&=&4\pi\alpha_s(m_W)\int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\, \frac{\emph{d}^4p}{(2\pi)}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}\nonumber\\
&&\{ \ (\bar{q}_i(x_1)\gamma^{\mu}(1-\gamma^{5})\frac{p\!\!\!/+m_b}{p^2-m_b^2+i\epsilon}\gamma^{\nu}T_{ik}^{a}b_k(x_2))
(\bar{s}_j(x_1)\gamma_{\mu}(1-\gamma^{5})q_j(x_1))\nonumber\\
&&+(\bar{q}_k(x_2)\frac{p\!\!\!/+m_q}{p^2-m_q^2+i\epsilon}\gamma^{\nu}T_{ki}^{a}\gamma^{\mu}(1-\gamma^{5})b_i(x_1))
(\bar{s}_j(x_1)\gamma_{\mu}(1-\gamma^{5})q_j(x_1))\nonumber\\
&&+(\bar{q}_i(x_1)\gamma^{\mu}(1-\gamma^{5})b_i(x_1))
(\bar{s}_j(x_1)\gamma_{\mu}(1-\gamma^{5})\frac{p\!\!\!/+m_q}{p^2-m_q^2+i\epsilon}\gamma^{\nu}T_{jk}^{a}q_k(x_2))\nonumber\\
&&+(\bar{q}_i(x_1)\gamma^{\mu}(1-\gamma^{5})b_i(x_1))
(\bar{s}_k(x_2)\frac{p\!\!\!/+m_s}{p^2-m_s^2+i\epsilon}\gamma^{\nu}T_{kj}^{a}\gamma_{\mu}(1-\gamma^{5})q_j(x_1)) \ \}\nonumber\\
&&\frac{1}{k^2+i\epsilon}(\bar{q'}_m(x_3)\gamma_{\nu}T_{mn}^{a} q'_n(x_3)),
\label{eq:ot6}\end{aligned}$$ which can be regarded as an effective operator resulting from the corresponding initial four-quark operator with a single gluon exchange $$\begin{aligned}
O^{(q)}_1\, &=&\,(\bar{q}_ib_i)_{V-A}(\bar{s}_jq_j)_{V-A}\nonumber\\
&=&(\bar{q}_i\gamma^{\mu}(1-\gamma^{5})b_i)(\bar{s}_j\gamma_{\mu}(1-\gamma^{5})q_j)\label{eq:ot}\end{aligned}$$ with $q=u,\, c$.
Similar to the procedure of obtaining the four quark operator effective Hamiltonian from the initial four quark operator $O_1^{(q)}$ of weak interaction, one should evaluate the six quark operator effective Hamiltonian from the initial six quark operator $O_1^{(q)(6)}$ when running the energy scale from $m_{W}$ to the low energy scale $\mu \sim m_b$. As the first step for finding out the complete set of independent effective six quark operators, one needs to evaluate all possible one loop diagrams based on the initial six quark diagrams (Fig. \[pic:4insert\]). The possible six quark diagrams at one loop level are plotted in Fig. \[pic:oneloop\].
![ The diagrams in (a) are loop contributions only to the effective weak vertex (type I), and diagrams in (b) are loop contributions only to the gluon vertexes (type II). The diagrams in (c) are loop contributions for both weak and strong vertexes (type III). []{data-label="pic:oneloop"}](pic2.eps "fig:")\
It is useful to classify those diagrams into three types: type I is the loop diagrams in which only the effective four quark vertex of weak interaction receives loop corrections including the penguin type loops, the single gluon exchanging interaction for six quark operators remains mediating between one of four external quark lines of loops and a spectator quark line (see Fig. \[pic:oneloop\]a); type II is the loop diagrams where only the single gluon exchanging vertexes receive loop corrections (see Fig. \[pic:oneloop\]b); the remaining loop diagrams are regarded as type III in which one of the gluon exchanging vertexes touches to the internal quark/gluon line of loops (see Fig. \[pic:oneloop\]c). Note that in Fig. \[pic:oneloop\]a and Fig. \[pic:oneloop\]b we only plot, for an illustration, the six quark diagrams with a gluon exchanging between one of the four external quark lines of effective weak vertex and a spectator quark line, while for each of them, there are actually three additional different diagrams corresponding to other three choices of external quark lines, they are omitted just for simplicity.
To evaluate all the diagrams is a hard task, as a good approximation, we shall pay attention to the type I and type II diagrams. The type III diagrams are in general suppressed at the perturbative region with energy scale around $m_b$ as they involve more internal quark lines and contain no large logarithmic enhancements. From the evaluation of four quark operator effective Hamiltonian, it is known that when the energy scale runs via the renormalization group evolution from the high energy scale at $\mu \simeq m_W$ to the low energy scale around $\mu \sim m_b$, the loop corrections of type I diagrams should result in the six quark operators with all effective four quark operators and the corresponding Wilson coefficient functions, meanwhile the loop corrections of type II diagrams will lead the strong coupling constant $\alpha_s$ of the gluon exchanging interaction to run from high energy scale at $m_W$ to the low energy scale at $\mu$. Thus, when ignoring the type III diagrams, we arrive at an approximate six quark operator effective Hamiltonian as follows $$\begin{aligned}
H_{\rm eff}^{(6)}\, &=&\, \frac{G_F}{\sqrt{2}}\sum_{j=1}^4\{
\sum_{q=u,c}\lambda_q^{s(d)}[C_1(\mu)O_{1q_j}^{(q)(6)}(\mu)
+C_2(\mu)O_{2q_j}^{(q)(6)}(\mu)]\nonumber\\
&& + \sum_{i=3}^{10}\lambda_t^{s(d)} C_i(\mu)O^{(6)}_{i\
q_j}(\mu)\}+h.c.+\dots, \label{eq:hpk6}\end{aligned}$$ with the CKM factor $\lambda_q^{s(d)} = V_{qb}V^*_{q s(d)}$. The dots represent other possible terms that have been neglected in our present considerations. $O^{(6)}_{i\ q_j}(\mu)$ ($j=1,2,3,4$) are six quark operators which may effectively be obtained from the corresponding four quark operators $ O_{i}(\mu)$ (in Eq. (\[eq:o\])) at the scale $\mu$ via the effective gluon exchanging interactions between one of the external quark lines of four quark operators and a spectator quark line at the same scale $\mu$. The general forms and definitions of $O^{(6)}_{i\ q_j}(\mu)$ ($j=1,2,3,4$) for the corresponding four quark operators $O_i(\mu)$ are similar to the ones of $O^{(6)}_{q_j}$ ($j=1,2,3,4$) given in Eq. (\[eq:six\]) but with replacing $\alpha_s(m_W)$ by $\alpha_s(\mu)$ due to QCD corrections of type II diagrams.
Before proceeding, we would like to point out that a complete six quark operator effective Hamiltonian may involve more effective operators from the type III diagrams and lead to a non-negligible contribution to hadronic B meson decays when evaluating the hadronic matrix elements of six quark operator effective Hamiltonian around the energy scale $\mu \sim \sqrt{2\Lambda_{QCD} m_b} \sim m_c \sim
1.5$ GeV where the nonperturbative effects may play the role. We shall keep this in mind and regard the above six quark operator effective Hamiltonian as an approximate one.
QCD Factorization Based on Effective Six Quark Operators {#sec:QCDF}
========================================================
We shall apply the above effective Hamiltonian with six quark operators to the nonleptonic two body decays of bottom mesons. The evaluation of hadronic matrix elements is the most hard task in the calculations of the decay amplitudes. In this section, we are going to demonstrate how the factorization approach naturally works for evaluating the hadronic matrix elements of nonleptonic two body decays of B meson with six quark operators.
To be explicit, we here examine the hadronic matrix element of $B\to
\pi^0\pi^0$ decay for a typical six quark operator $O^{(6)}_{LL}$ $$\begin{aligned}
O^{(6)}_{LL} & = & \int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\,
\frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}\frac{1}
{k^2}\ \frac{1}{p^2-m_d^2} \nonumber \\
& & [\bar{d}_k(x_2) (p\!\!\!/+m_d)
\gamma^{\nu}T_{ki}^{a}\gamma^{\mu}(1-\gamma^{5})b_i(x_1)]
[\bar{d}_j(x_1)\gamma_{\mu}(1-\gamma^{5})d_j(x_1)][\bar{d}_m(x_3)\gamma_{\nu}T_{mn}^{a}
d_n(x_3)],\end{aligned}$$ which is actually a part of the six quark operator $O^{(6)}_{4q_2}$ in the effective Hamiltonian. Its hadronic matrix element for $B\to
\pi^0\pi^0$ decay leads to the following most general terms in the QCD factorization approach
$$\begin{aligned}
& & M_{LL}^{O}(B\pi\pi) = <\pi^{0} \pi^{0}\mid O^{(6)}_{LL}\mid\bar{B_0}>\,\nonumber\\
& & = \int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\,
\frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}\frac{1}
{k^2} \ \frac{1}{p^2-m_d^2} \nonumber \\
& & <\pi^{0} \pi^{0}\mid [\bar{d}_k(x_2) (p\!\!\!/+m_d)
\gamma^{\nu}T_{ki}^{a}\gamma^{\mu}(1-\gamma^{5})b_i(x_1)]
[\bar{d}_j(x_1)\gamma_{\mu}(1-\gamma^{5})d_j(x_1)][\bar{d}_m(x_3)\gamma_{\nu}T_{mn}^{a}
d_n(x_3)] \mid\bar{B_0}> \nonumber\\
& &\equiv M_{LL}^{O(1)}+M_{LL}^{O(2)}+M_{LL}^{O(3)}+M_{LL}^{O(4)}\label{eq:loop?},\end{aligned}$$
with $$\begin{aligned}
&& M_{LL}^{O(1)}=\int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\,
\frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
\frac{1}{k^2(p^2-m_d^2)} T_{ki}^{a}T_{mn}^{a} \nonumber \\
&&\ \,
[(p\!\!\!/+m_d)\gamma^{\nu}\gamma^{\mu}(1-\gamma^{5})]_{\rho\sigma}
[\gamma_{\mu}(1-\gamma^{5})]_{\alpha\beta}
[\gamma_{\nu}]_{\gamma\delta} M_{Bim}^{\ \,\sigma\gamma}(x_1,x_3)M_{\pi nk}^{\
\delta\rho}(x_3,x_2)M_{\pi jj}^{\ \beta\alpha}(x_1,x_1),
\nonumber\\
& & M_{LL}^{O(2)}=\int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\,
\frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
\frac{1}{k^2(p^2-m_d^2)} T_{ki}^{a}T_{mn}^{a}
\nonumber \\
&&\ \,
[(p\!\!\!/+m_d)\gamma^{\nu}\gamma^{\mu}(1-\gamma^{5})]_{\rho\sigma}
[\gamma_{\mu}(1-\gamma^{5})]_{\alpha\beta}
[\gamma_{\nu}]_{\gamma\delta} M_{Bim}^{\ \,\sigma\gamma}(x_1,x_3)M_{\pi nj}^{\
\delta\alpha}(x_3,x_1)M_{\pi jk}^{\ \beta\rho}(x_1,x_2),
\nonumber\\
& & M_{LL}^{O(3)}=\int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\,
\frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
\frac{1}{k^2(p^2-m_d^2)} T_{ki}^{a}T_{mn}^{a}
\nonumber \\
&&\ \,
[(p\!\!\!/+m_d)\gamma^{\nu}\gamma^{\mu}(1-\gamma^{5})]_{\rho\sigma}
[\gamma_{\mu}(1-\gamma^{5})]_{\alpha\beta}
[\gamma_{\nu}]_{\gamma\delta} M_{Bij}^{\ \,\sigma\alpha}(x_1,x_1)M_{\pi jm}^{\
\beta\gamma}(x_1,x_3)M_{\pi nk}^{\ \delta\rho}(x_3,x_2),
\nonumber\\
& & M_{LL}^{O(4)}=\int\!\!\int \frac{\emph{d}^4k}{(2\pi)^4}\,
\frac{\emph{d}^4p}{(2\pi)^4}\,e^{-i((x_1-x_2)p+(x_2-x_3)k)}
\frac{1}{k^2(p^2-m_d^2)} T_{ki}^{a}T_{mn}^{a}
\nonumber \\
& &\ \, [(p\!\!\!/+m_d)\gamma^{\nu}\gamma^{\mu}(1-\gamma^{5})]_{\rho\sigma}
[\gamma_{\mu}(1-\gamma^{5})]_{\alpha\beta}
[\gamma_{\nu}]_{\gamma\delta}M_{Bik}^{\ \,\sigma\rho}(x_1,x_2)M_{\pi jm}^{\
\beta\gamma}(x_1,x_3)M_{\pi nj}^{\ \delta\alpha}(x_3,x_1),\end{aligned}$$ where $M_{X nm}^{\ \, \beta\alpha}(x_i,x_j)\equiv
[M_{X}(x_i,x_j)]_{nm}^{\beta\alpha}$ ($X=B,\pi$) with $n,m$ the color indices and $\alpha,\beta$ the spinor indices, is the hadronic matrix element of two quark operators for a single meson $X$. In light-cone QCD approach, it is found to be [@NPB.592.003] $$\begin{aligned}
M_{B nm}^{\ \,\beta\alpha}(x_i,x_j)&=&< 0\mid \bar{d}^{\alpha}_{m}(x_j)b^{\beta}_{n}(x_i)\mid \bar{B}^0(P_B)>
=-\frac{i F_B}{4}\frac{\delta_{mn}}{N_c}\int_0^1 \emph{d}u
\,e^{-i(u\,P_B^+\,x_j+(P_B-u\,P_B^+)\,x_i)} M_{\text{B}}^{\beta\alpha}(u,P_B),\nonumber\\
M_{\pi nm}^{\ \beta\alpha}(x_i,x_j)&=&< \pi^{0}(P)\mid \bar{d}^{\alpha}_{m}(x_j) d^{\beta}_{m}(x_i) \mid 0>
=\frac{iF_{\pi}}{4}\frac{\delta_{mn}}{N_c}\int_0^1 \emph{d}x
\,e^{-i(x\,P\,x_j+(1-x)P\,x_i)}M_{\pi}^{\beta\alpha}(x,P),
\label{eq:lcda}\end{aligned}$$ with $F_{M}$ ($M=B,\pi$) the decay constants. Here $M_{\text{B}}^{\beta\alpha}(u,P_B)$ and $M_{\pi}^{\beta\alpha}(x,P)$ are the spin structures for the bottom meson and light meson $\pi$ and characterized by the corresponding distribution amplitudes $$\begin{aligned}
M_{\text{B}}^{\beta\alpha}(u,P_B)&=&-[m_B+P_B\!\!{\!\!\!\!\!/\,}\ \ \gamma^5 \phi_{B}(u)]_{\beta\alpha},\nonumber\\
M_{\pi}^{\beta\alpha}(x,P)&=&[P\,{\!\!\!\!\!/\,}\,
\gamma^5 \phi_{\pi}(x)-\mu_{\pi}\gamma^5(\phi_{\pi}^{p}(x)-
i\sigma_{\mu \nu}n^\mu{\!\!\!\!\!/\,}v^\nu{\!\!\!\!\!/\,}\phi^{T}_{\pi}(x)+i\sigma _{\mu \nu}P^{\mu}
\frac{\phi^\sigma(u)}{6} \frac{\partial}{\partial
k_{\bot\nu}})]_{\beta\alpha}\label{eq:daoM},\end{aligned}$$ with $v=\frac{P}{\sqrt{2}|\overrightarrow{P}|}$, $n=n^+ +n^- -v$ and $\phi^{T}\equiv \phi^{\sigma\prime}/6$. The light-cone distribution amplitudes $\phi_M^X(u)$ ($M=B,\pi$, $X=-,p,T$) are given in [@NPB.592.003] up to twist-3. The definition of momentum for quarks and mesons is explicitly shown in Fig.\[pic:definition\]. As a good approximation, both the light quarks and light mesons are taken to be massless, i.e., $P^2=0$.
It is interesting to note that the four amplitudes $M_{LL}^{O (i)}$ ($i=1,2,3,4$) are corresponding to four diagrams (1)-(4) in Fig.\[pic:Hadronic\]. The first diagram is known as the factorizable one, the second is the non-factorizable one and color suppressed. The third is the factorizable annihilation diagram and color suppressed, and the fourth is an annihilation diagram and its matrix element vanishes.
![ Different ways of reducing hadronic matrix element of effective six quark operator by QCD factorization approach. []{data-label="pic:Hadronic"}](pic3.eps "fig:")\
After performing the integration over space-time and momentum, the above amplitude is simplified to be $$\begin{aligned}
&& M_{LL}^{O}(B\pi\pi) = <\pi^{0} \pi^{0}\mid O_{LL}^{(6)}\mid\bar{B_0}> =
\int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,
\frac{1}{(u\,P_B^+ -(1-x)P_1)^2}\nonumber\\
&&[\ \frac{M_{LL}^{(1)}}{(P_{1}-u\,P_B^+)^2-m_d^2}
+\frac{M_{LL}^{(2)}}{((1-x)P_{1}+y\,P_{2}-u\,P_B^+)^2-m_d^2} \nonumber\\
&& +\frac{M_{LL}^{(3)}}{(x\,P_{1}+P_{2})^2-m_d^2}
+\frac{M_{LL}^{(4)}}{(x\,P_{1}+(1-y)P_{2}-u\,P_B^+)^2-m_d^2}\ ], \label{eq:ampl}\end{aligned}$$ with $$\begin{aligned}
M_{LL}^{(1)}&=&\frac{C_{F}}{N_c}*F_B\ F_{\pi}^2\text{Tr}[M_{\text{B}}(u,P_B)\gamma_{\nu}
M_{\pi}(x,P_1)\gamma^{\nu}(P_{1}\!{\!\!\!\!\!/\,}-u\,P_B^+\!\!{\!\!\!\!\!/\,}+m_d)\gamma_{\mu}(1-\gamma^{5})] \nonumber \\
& & \text{Tr}[M_{\pi}(y,P_2)\gamma^{\mu}(1-\gamma^{5})]
= i\frac{C_{F}}{4N_c}F_B\ F_{\pi}^2 \phi_{B}(u)m_B^3 \mu_{\pi}\phi_{\pi}(y)\phi^{p}_{\pi}(x),\nonumber\\
M_{LL}^{(2)}&=&\frac{C_{F}}{N_c^2}*F_B\ F_{\pi}^2\text{Tr}[M_{\text{B}}(u,P_B)\gamma_{\nu}
M_{\pi}(x,P_1)\gamma_{\mu}(1-\gamma^{5})M_{\pi}(y,P_2)\gamma^{\nu}\nonumber\\&&
((1-x)P_{1}\!{\!\!\!\!\!/\,}+y\,P_{2}\!{\!\!\!\!\!/\,}-u\,P_B^+\!\!{\!\!\!\!\!/\,}+m_d)\gamma^{\mu}(1-\gamma^{5})]
\nonumber\\
&=&i\frac{C_{F}}{4N_c^2}F_B\ F_{\pi}^2 \phi_{B}(u)m_B^3(m_B (u+x+y-2) \phi_{\pi}(x)+\mu_{\pi} (1-x)
(\phi^{p}_{\pi}(x)-\phi^{T}_{\pi}(x)))\phi_{\pi}(y),\nonumber\\
M_{LL}^{(3)}&=&\frac{C_{F}}{N_c^2}*F_B\ F_{\pi}^2\text{Tr}[M_{\pi}(x,P_1)\gamma_{\nu}
M_{\pi}(y,P_2)\gamma^{\nu}(x\,P_{1}\!{\!\!\!\!\!/\,}+P_{2}\!{\!\!\!\!\!/\,}+m_d)\gamma_{\mu}(1-\gamma^{5})]\text{Tr}
[M_{\text{B}}(u,P_B)\gamma^{\mu}(1-\gamma^{5})]
\nonumber\\
&=&i\frac{C_{F}}{4N_c^2}F_B\ F_{\pi}^2 \phi_{B}(u)m_B^2(x m_B^2\phi_{\pi}(y) \phi_{\pi}(x)+2
\mu_{\pi}^2((1+x)\phi^{p}_{\pi}(x)-
(1-x)\phi^{T}_{\pi}(x))\phi^{p}_{\pi}(y),)\nonumber \\
M_{LL}^{(4)}&=&0*F_B\ F_{\pi}^2\text{Tr}[M_{\pi}(x,P_1)\gamma_{\nu}M_{\pi}(y,P_2)\gamma_{\mu}(1-\gamma^{5})\nonumber \\
&& M_{\text{B}}(u,P_B)(x\,P_{1}\!{\!\!\!\!\!/\,}+(1-y)P_{2}\!{\!\!\!\!\!/\,}-u\,P_B^+\!\!{\!\!\!\!\!/\,}+m_d)\gamma^{\nu}\gamma^{\mu}(1-\gamma^{5})] = 0,\end{aligned}$$ where $M_{LL}^{(i)}$ ($i=1,2,3,4$) are obtained by performing the trace of matrices and determined by the distribution amplitudes. $C_F=\frac{N_c^2-1}{2N_c}$ is resulted from summing over the color indices. It can be seen that $M_{LL}^{(1)}$ corresponding to Fig.\[pic:Hadronic\].(1) is color allowed, $M_{LL}^{(2)}$ and $M_{LL}^{(3)}$ corresponding to Fig.\[pic:Hadronic\].(2) and Fig.\[pic:Hadronic\].(3) are color suppressed, while $M_{LL}^{(4)}$ corresponding to Fig.\[pic:Hadronic\].(4) vanishes as it is not allowed for colorless mesons.
From the above explicit demonstration, it can be seen that the simple QCD factorization approach becomes a natural tool to evaluate the hadronic matrix element of effective six quark operators in the mesonic two body decays. For a given effective six quark operator, its hadronic matrix element for mesonic two body decays gets four different combinations in the QCD factorization approach, namely it consists of four different amplitudes corresponding to four topologically different diagrams. From the above example, it is noticed that the amplitude $M_{LL}^{O (1)}$ is a color-allowed factorizable one in an emission diagram, $M_{LL}^{O (2)}$ is a color-suppressed non-factorizable one in an emission diagram, $M_{LL}^{O (3)}$ is a color-suppressed factorizable one in an annihilation diagram, while $M_{LL}^{O (4)}$ vanishes as it cannot match to a colorless meson.
When generalizing the above analysis to the present framework based on the approximate six quark operator effective Hamiltonian, there are in general four types of six quark diagrams corresponding to four types of effective six quark operators, their hadronic matrix elements for two body mesonic decays lead to sixteen kinds of diagrams (see Fig.\[pic:b-kpi\].$(ai)$-$(di)$, $i=1,2,3,4)$ as each of the effective six quark operators leads to four kinds of amplitudes in the QCD factorization approach.
![Four types of effective six quark diagrams lead to sixteen diagrams for hadronic two body decays of heavy meson via QCD factorization.[]{data-label="pic:b-kpi"}](pic4.eps "fig:")\
It is known that the effective four quark vertexes concern three types of current-current interactions: $(V-A)\times (V-A)$ or $(LL)$, $(V-A)\times (V+A)$ or $(LR)$, $(S-P)\times (S+P)$ or $(SP)$, thus each of the diagrams in Fig.\[pic:b-kpi\] actually contains three kinds of diagrams corresponding to three types of current-current interactions. Therefore, there are totally 48 kinds of hadronic matrix elements involved in the QCD factorization approach, while it is easy to check that only half of them are independent with the following relations: $$\begin{aligned}
\begin{array}{cccccccccccc}
M^{a1}_{LL}&=&T^F_{LLa};&M^{a2}_{LL}&=&T^F_{LLa}/N_c;&M^{a3}_{LL}&=&A^N_{LLa}/N_c;&M^{a4}_{LL}&=&0;\\
M^{a1}_{LR}&=&T^F_{LRa};&M^{a2}_{LR}&=&T^F_{SPa}/N_c;&M^{a3}_{LR}&=&A^N_{SPa}/N_c;&M^{a4}_{LR}&=&0;\\
M^{a1}_{SP}&=&T^F_{SPa};&M^{a2}_{SP}&=&T^F_{LRa}/N_c;&M^{a3}_{SP}&=&A^N_{LRa}/N_c;&M^{a4}_{SP}&=&0;\\
M^{b1}_{LL}&=&T^F_{LLb};&M^{b2}_{LL}&=&T^N_{LLb}/N_c;&M^{b3}_{LL}&=&A^F_{LLb}/N_c;&M^{b4}_{LL}&=&0;\\
M^{b1}_{LR}&=&T^F_{LRb};&M^{b2}_{LR}&=&T^N_{SPb}/N_c;&M^{b3}_{LR}&=&A^F_{SPb}/N_c;&M^{b4}_{LR}&=&0;\\
M^{b1}_{SP}&=&T^F_{LLb};&M^{b2}_{SP}&=&T^N_{LRb}/N_c;&M^{b3}_{SP}&=&A^F_{LRb}/N_c;&M^{b4}_{SP}&=&0;\\
M^{c1}_{LL}&=&0;&M^{c2}_{LL}&=&T^N_{LLa}/N_c;&M^{c3}_{LL}&=&A^F_{LLa}/N_c;&M^{c4}_{LL}&=&A^F_{LLa};\\
M^{c1}_{LR}&=&0;&M^{c2}_{LR}&=&T^N_{SPa}/N_c;&M^{c3}_{LR}&=&A^F_{SPa}/N_c;&M^{c4}_{LR}&=&A^F_{LRa};\\
M^{c1}_{SP}&=&0;&M^{c2}_{SP}&=&T^N_{LRa}/N_c;&M^{c3}_{SP}&=&A^F_{LRa}/N_c;&M^{c4}_{SP}&=&A^F_{SPa};\\
M^{d1}_{LL}&=&0;&M^{d2}_{LL}&=&T^F_{LLb}/N_c;&M^{d3}_{LL}&=&A^N_{LLb}/N_c;&M^{d4}_{LL}&=&A^F_{LLb};\\
M^{d1}_{LR}&=&0;&M^{d2}_{LR}&=&T^F_{SPb}/N_c;&M^{d3}_{LL}&=&A^N_{SPb}/N_c;&M^{d4}_{LL}&=&A^F_{LRb};\\
M^{d1}_{SP}&=&0;&M^{d2}_{SP}&=&T^F_{LRb}/N_c;&M^{d3}_{LL}&=&A^N_{LRb}/N_c;&M^{d4}_{LL}&=&A^F_{SPb}.
\end{array}\end{aligned}$$ where $T^F_{Xa}$ and $T^F_{Xb}$ ($X=LL,LR,SP$) represent the factorizable emission diagram contributions, $T^N_{Xa}$ and $T^N_{Xb}$ ($X=LL,LR,SP$) are the non-factorizable emission diagram contributions. $A^F_{Xa}$, $A^F_{Xb}$ and $A^N_{Xa}$, $A^N_{Xb}$ ($X=LL,LR,SP$) denote the so-called factorizable and non-factorizable annihilation diagram contributions respectively. Their detailed definitions and general formalisms are presented in the Appendix.
Treatment of Singularities {#sec:TOD}
===========================
In the evaluation of hadronic matrix elements, there are two kinds of singularities, one is caused by the infrared divergence of gluon exchanging interaction, and the other arises from the on-mass shell divergence of internal quark propagator. As the quark propagator singularity is a physical-region singularity, one can simply add $i\epsilon$ to the denominator of quark propagator and apply the Cutkosky rule [@cutkosky] to avoid such a singularity. It then allows us to obtain the virtual part of amplitudes as the Cutkosky rule gives a compact expression for the discontinuity across the cut arising from a physical-region singularity. In general, a Feynman diagram will yield an imaginary part for the decay amplitudes when the virtual particles in the diagram become on mass-shell, and the resulting diagram can be considered as a genuine physical process. It is well-known that when applying the Cutkosky rule to deal with a physical-region singularity of all propagators, the following formula holds $$\begin{aligned}
\frac{1}{p^2-m_b^2+i\epsilon}&=&P\biggl[\frac{1}{p^2-m_b^2}
\biggl]-i\pi\delta[p^2-m_b^2],\nonumber\\
\frac{1}{p^2-m_q^2+i\epsilon}&=&P\biggl[\frac{1}{p^2-m_q^2}
\biggl]-i\pi\delta[p^2-m_q^2],\label{quarkd}\end{aligned}$$ which is known as the principal integration method. Where the first integration with the notation of capital letter $P$ is the so-called principal integration.
For the infrared divergence of gluon exchanging interactions, only adding $i\epsilon$ to the gluon propagator is not enough as such an infrared divergence is not a physical-region singularity, one cannot simply apply the Cutkosky rule. To regulate such an infrared divergence, we may apply the prescription used in the symmetry-preserving loop regularization[@LRC] which allows us to introduce an intrinsic energy scale without destroying the non-abelian gauge invariance and translational invariance. The description of the loop regularization is simple: evaluating the Feynman integrals to an irreducible integrals, replacing the integration variable $k^2$ and integration measure $\int\frac{d^4k}{(2\pi)^4}$ by the regularized ones via [@LRC] $$\begin{aligned}
& & k^2\rightarrow[k^2]_l\equiv k^2-M_l^2, \nonumber \\
& & \int\frac{d^4k}{(2\pi)^4}\rightarrow
\int[\frac{d^4k}{(2\pi)^4}]_l\equiv\lim_{N,M_i^2\to \infty}
\sum_{l=0}^Nc_l^N\int\frac{d^4k}{(2\pi)^4}\label{precedure},\end{aligned}$$ with conditions $$\begin{aligned}
& & \lim_{N,M_i^2}\sum_{l=0}^Nc_l^N(M_l^2)^n=0, \quad c_0^N=1 ~~~~(
i=0,1,\cdots,N~~\mbox{and} ~~n=0,1,\cdots),\label{condition}\end{aligned}$$ where $c_l^N$ are the coefficients determined by the above conditions. With a simple form for the regulator masses $M_l = \mu_g
+ l\, M_R$ ($l= 0,1, \cdots$), the coefficients $c_l^N$ is found to be $ c_l^N = (-1)^l \frac{N!}{(N-l)!\ l!}$, so that $$\begin{aligned}
k^2~\Rightarrow~k^2-\mu_g^2-l\,M_R^2, \qquad \int
\frac{\emph{d}^4k}{(2\pi)^4}~\Rightarrow~ \lim_{N,M_R\to\infty}
\sum^{N}_{l=0} (-1)^l\frac{N!}{l!(N-l)!}\int
\frac{\emph{d}^4k}{(2\pi)^4},\end{aligned}$$ which leads the regularized integrals to be independent of the regulators. Here the energy scale $M_0 = \mu_g$ plays the role of infrared cut-off but preserving gauge symmetry and translational symmetry of original theory.
In the present case, there is no ultraviolet divergence for the integral over $k$ as it is constrained by the finite momentum of hadrons, so all the terms with $l\neq 0$ in the summation over $l$ vanish in the limit $M_R\to \infty$. As a consequence, it is equivalent to add an intrinsic regulator energy scale $\mu_g$ in the denominator $k^2$ in Eq. (\[eq:loop?\]), thus one can use the usual principal integration method to avoid such a singularity, i.e., $$\frac{1}{k^{2}}~\Rightarrow~\frac{1}{k^{2}-\mu_g^2+i\epsilon}
=P\biggl[\frac{1}{k^{2}-\mu_g^2} \biggl]-i\pi\delta[k^{2}-\mu_g^2].$$ With the above considerations, the singularities appearing in the integrations over $k$ and $p$ can simply be avoided by the following prescription $$\begin{aligned}
\frac{1}{k^2}\frac{1}{(p^2-m^2)} \to
\frac{1}{(k^2-\mu_g^2+i\epsilon)}\frac{1}{(p^2-m^2+i\epsilon)},\end{aligned}$$
Note that as the gauge depending team $k_{\mu}k_{\nu}$ can be transformed to the momentum $p\,\,{\!\!\!\!\!/\,}$ on exterior line of spectator quark, they are all on mass shell in our present consideration (as defined in Fig.\[pic:definition\] in the appendix), their contributions equal to zero, thus our results are gauge independent.
Applying this prescription to the amplitude illustrated in previous section, we have $$\begin{aligned}
&& M_{LL}^{O}(B\pi\pi) = <\pi^{0} \pi^{0}\mid O_{LL}^{(6)}\mid\bar{B_0}> =
\int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,
\frac{1}{(u\,P_B^+ -(1-x)P_1)^2-\mu_g^2+i\epsilon}\nonumber\\
&&[\ \frac{M_{LL}^{(1)}}{(P_{1}-u\,P_B^+)^2-m_{q}^2+i\epsilon}
+\frac{M_{LL}^{(2)}}{((1-x)P_{1}+y\,P_{2}-u\,P_B^+)^2-m_{q}^2+i\epsilon} \nonumber\\
&& +\frac{M_{LL}^{(3)}}{(x\,P_{1}+P_{2})^2-m_{q}^2+i\epsilon}
+\frac{M_{LL}^{(4)}}{(x\,P_{1}+(1-y)P_{2}-u\,P_B^+)^2-m_{q}^2+i\epsilon}\ ] .\label{eq:ampl}\end{aligned}$$
Amplitudes of Charmless Bottom Meson Decays {#sec:Amplitude}
===========================================
With the above considerations and analyses, the QCD factorization approach enables us to evaluate all the hadronic matrix elements of nonleptonic two body decays of B meson based on the approximate six quark operator effective Hamiltonian. The amplitudes of charmless B meson decays can be expressed as follows: $$\begin{aligned}
\label{eq:pptopologyT}
A(B^0\to \pi^+\pi^-)&=&V_{td}V^*_{tb}[P_T^{\pi\pi}(B)+\frac{2}{3}P_{EW}^{C\pi\pi}(B)+P_E^{\pi\pi}(B)+2P_A^{\pi\pi}(B)+\frac{1}{3}P_{EW}^{A\pi\pi}(B)-\frac{1}{3}A_{EW}^{E\pi\pi}(B)]
\nonumber\\&&-V_{ud}V^*_{ub}[T^{\pi\pi}(B)+E^{\pi\pi}(B)],\nonumber\\
A(B^+\to \pi^+\pi^0)&=&\frac{1}{\sqrt{2}}\{V_{td}V^*_{tb}[P_{EW}^{\pi\pi}(B)+P_{EW}^{C\pi\pi}(B)]-V_{ud}V^*_{ub}[T^{\pi\pi}(B)+C^{\pi\pi}(B)]\},\nonumber\\
A(B^0\to \pi^0\pi^0)&=&\frac{1}{\sqrt{2}}\{-V_{td}V^*_{tb}[P_T^{\pi\pi}(B)-P_{EW}^{\pi\pi}(B)-\frac{1}{3}P_{EW}^{C\pi\pi}(B)+P_E^{\pi\pi}(B)+2P_A^{\pi\pi}(B)
\nonumber\\&&+\frac{1}{3}P_{EW}^{A\pi\pi}(B)-\frac{1}{3}P_{EW}^{E\pi\pi}(B)]+V_{ud}V^*_{ub}[-C^{\pi\pi}(B)+E^{\pi\pi}(B)]\},\end{aligned}$$ for $B\to \pi\pi$ decay amplitudes, and $$\begin{aligned}
A(B^+\to \pi^+ K^0)&=&-V_{ts}V^*_{tb}[P_T^{\pi K}(B)-\frac{1}{3} P_{EW}^{C\pi K}(B)+P_E^{\pi K}(B)+\frac{2}{3}P_{EW}^{E\pi K}(B)]+V_{us}V^*_{ub}A^{\pi K}(B),\nonumber\\
A(B^+\to \pi^0 K^+)&=&\frac{1}{\sqrt{2}}\{V_{td}V^*_{tb}[P_T^{\pi K}(B)+P_{EW}^{K\pi}(B)+\frac{2}{3}P_{EW}^{C\pi K}(B)+P_E^{\pi K}(B)+\frac{2}{3}P_{EW}^{E\pi K}(B)]
\nonumber\\&&-V_{us}V^*_{ub}[T^{\pi K}(B)+C^{K\pi}(B)+A^{\pi K}(B)]\},\nonumber\\
A(B^0\to \pi^- K^+)&=&V_{td}V^*_{tb}[P_T^{\pi K}(B)+\frac{2}{3}P_{EW}^{C\pi K}(B)+P_E^{\pi K}(B)-\frac{1}{3}P_{EW}^{E\pi K}(B)]-V_{us}V^*_{ub}T^{\pi K}(B),\nonumber\\
A(B^0\to \pi^0 K^0)&=&-\frac{1}{\sqrt{2}}\{V_{td}V^*_{tb}[P_T^{\pi K}(B)-P_{EW}^{K\pi}(B)-\frac{1}{3}P_{EW}^{C\pi K}(B)+P_E^{\pi K}(B)\nonumber\\&&
-\frac{1}{3}P_{EW}^{E\pi K}(B)]+V_{us}V^*_{ub}C^{K\pi}(B)\},\end{aligned}$$ for $B\to \pi K$ decay amplitudes, and $$\begin{aligned}
A(B^0\to K^+K^-)&=&-V_{td}V^*_{tb}*[P_A^{K\bar{K}}(B)+P_A^{\bar{K}K}(B)+\frac{2}{3}P_{EW}^{AK\bar{K}}(B)-\frac{1}{3}P_{EW}^{A\bar{K}K}(B)]+V_{ud}V^*_{ub}E^{K\bar{K}}(B),
\nonumber\\
A(B^+\to K^+\bar{K}^0)&=&-V_{td}V^*_{tb}[P_T^{K\bar{K}}(B)-\frac{1}{3}P_{EW}^{C\pi\pi}(B)+P_E^{K\bar{K}}(B)+\frac{2}{3}P_{EW}^{E\bar{K}K}(B)]+V_{ud}V^*_{ub}A^{K\bar{K}}(B),
\nonumber\\
A(B^0\to K^0\bar{K}^0)&=&-V_{td}V^*_{tb}[P_T^{K\bar{K}}(B)-\frac{1}{3}P_{EW}^{C\pi\pi}(B)+P_E^{K\bar{K}}(B)+P_A^{K\bar{K}}(B)+P_A^{\bar{K}K}(B)
\nonumber\\&&-\frac{1}{3}P_{EW}^{AK\bar{K}}(B)-\frac{1}{3}P_{EW}^{A\bar{K}K}(B)-\frac{1}{3}P_{EW}^{EK\bar{K}}(B)],\end{aligned}$$ for $B\to K K$ decay amplitudes, and $$\begin{aligned}
A(B_s^0\to \pi^+\pi^-)&=&-V_{us}V^*_{ub}E^{\pi\pi}(B_s)+V_{ts}V^*_{tb}[2P_{A}^{\pi\pi}(B_s)+\frac{1}{3}P_{EW}^{A\pi\pi}(B_s)],\nonumber\\
A(B_s^0\to \pi^0\pi^0)&=&\frac{1}{\sqrt{2}}A(B_s^0\to \pi^+\pi^-),\nonumber\\
A(B_s^0\to \pi^+ K^-)&=&V_{td}V^*_{tb}[P_T^{\bar{K}\pi}(B_s)+\frac{2}{3}P_{EW}^{C\bar{K}\pi}(B_s)+P_{E}^{\bar{K}\pi}(B_s)-\frac{1}{3}P_{EW}^{E\bar{K}\pi}(B_s)]-V_{us}V^*_{ub}T^{\bar{K}\pi}(B_s),\nonumber\\
A(B_s^0\to \pi^0 K^0)&=&-\frac{1}{\sqrt{2}}\{V_{td}V^*_{tb}[P_T^{\bar{K}\pi}(B_s)-P_{EW}^{\bar{K}\pi}(B_s)-\frac{1}{3}P_{EW}^{C\bar{K}\pi}(B_s)+P_{E}^{\bar{K}\pi}(B_s)-\frac{1}{3}P_{EW}^{E\bar{K}\pi}(B_s)]
\nonumber\\&&+V_{us}V^*_{ub}C^{\bar{K}\pi}(B_s)\},\nonumber\\
A(B_s^0\to K^+K^-)&=&-V_{ts}V^*_{tb}[P_T^{\bar{K}K}(B_s)+\frac{2}{3}P_{EW}^{C\bar{K}K}(B_s)+P_E^{\bar{K}K}(B_s)+
P_{A}^{\bar{K}K}(B_s)+P_{A}^{K\bar{K}}(B_s)\nonumber\\&&+\frac{2}{3}P_{EW}^{AK\bar{K}}(B_s)-\frac{1}{3}P_{EW}^{A\bar{K}K}(B_s)
-\frac{1}{3}P_{EW}^{E\bar{K}K}(B_s)]+V_{us}V^*_{ub}[T^{\bar{K}K}(B_s)+E^{\bar{K}K}(B_s)],\nonumber\\
A(B_s^0\to K^0K^0)&=&-V_{ts}V^*_{tb}[P_T^{\bar{K}K}(B_s)-\frac{1}{3}P_{EW}^{C\bar{K}K}(B_s)+P_E^{\bar{K}K}(B_s)+
P_{A}^{\bar{K}K}(B_s)+P_{A}^{K\bar{K}}(B_s)\nonumber\\&&-\frac{1}{3}P_{EW}^{AK\bar{K}}(B_s)-\frac{1}{3}P_{EW}^{A\bar{K}K}(B_s)
-\frac{1}{3}P_{EW}^{E\bar{K}K}(B_s)],\end{aligned}$$ for $B_s\to \pi\pi, \ \pi K,\ KK$ decay amplitudes. The eleven types of amplitudes $T^{M_1M_2}(M)$, $C^{M_1M_2}(M)$, $P_T^{M_1M_2}(M)$, $P_{EW}^{M_1M_2}(M)$, $A^{M_1M_2}(M)$, $E^{M_1M_2}(M)$, $P_E^{M_1M_2}(M)$, $P_A^{M_1M_2}(M)$, $P_{EW}^{CM_1M_2}(M)$, $P_{EW}^{EM_1M_2}(M)$, $P_{EW}^{AM_1M_2}(M)$, with $M_1M_2 = \pi\pi, \pi K, K\pi, K\bar{K}, \bar{K} K$ are defined as follows $$\begin{aligned}
\label{eq:pptopologyT}
T^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\big\{[C_1(\mu)+\frac{1}{N_c}C_2(\mu)]T_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_2(\mu)T_{LL}^{NM_1M_2}(M)\big\},\nonumber\\
C^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\big\{[C_2(\mu)+\frac{1}{N_c}C_1(\mu)]T_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_1(\mu)T_{LL}^{NM_1M_2}(M)\big\},\nonumber\\
P_T^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\big\{[C_4(\mu)+\frac{1}{N_c}C_3(\mu)]T_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_3(\mu)T_{LL}^{NM_1M_2}(M)
\nonumber\\&&+[C_6(\mu)+\frac{1}{N_c}C_5(\mu)]T_{SP}^{FM_1M_2}(M)+\frac{1}{N_c}C_5(\mu)T_{LR}^{NM_1M_2}(M)\big\},\nonumber\\
P_{EW}^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\frac{3}{2}\big\{[C_9(\mu)+\frac{1}{N_c}C_{10}(\mu)]T_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_{10}(\mu)T_{LL}^{NM_1M_2}(M)
\nonumber\\&&+[C_7(\mu)+\frac{1}{N_c}C_8(\mu)]T_{LR}^{FM_1M_2}(M)+\frac{1}{N_c}C_8(\mu)T_{SP}^{NM_1M_2}(M)\},
\nonumber\\
P_{EW}^{CM_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\frac{3}{2}\big\{[C_{10}(\mu)+\frac{1}{N_c}C_9(\mu)]T_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_9(\mu)T_{LL}^{NM_1M_2}(M)
\nonumber\\&&+[C_8(\mu)+\frac{1}{N_c}C_7(\mu)]T_{SP}^{FM_1M_2}(M)+\frac{1}{N_c}C_7(\mu)T_{LR}^{NM_1M_2}(M)\big\},\end{aligned}$$ for the so-called emission diagrams, and $$\begin{aligned}
\label{eq:pptopologyA}
A^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\big\{[C_1(\mu)+\frac{1}{N_c}C_2(\mu)]A_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_2(\mu)A_{LL}^{NM_1M_2}(M)\}
.\nonumber\\
E^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\big\{[C_2(\mu)+\frac{1}{N_c}C_1(\mu)]A_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_1(\mu)A_{LL}^{NM_1M_2}(M)\big\},\nonumber\\
P_{E}^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\big\{[C_4(\mu)+\frac{1}{N_c}C_3(\mu)]A_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_3(\mu)A_{LL}^{NM_1M_2}(M)
\nonumber\\&&+[C_6(\mu)+\frac{1}{N_c}C_5(\mu)]A_{SP}^{FM_1M_2}(M)+\frac{1}{N_c}C_5(\mu)A_{LR}^{NM_1M_2}(M)\}
,\nonumber\\
P_{A}^{M_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\big\{[C_3(\mu)+\frac{1}{N_c}C_4(\mu)]A_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_4(\mu)A_{LL}^{NM_1M_2}(M)
\nonumber\\&&+[C_5(\mu)+\frac{1}{N_c}C_6(\mu)]A_{LR}^{FM_1M_2}(M)+\frac{1}{N_c}C_6(\mu)A_{SP}^{NM_1M_2}(M)\}
,\nonumber\\
P_{EW}^{AM_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\frac{3}{2}\big\{[C_9(\mu)+\frac{1}{N_c}C_{10}(\mu)]A_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_{10}(\mu)A_{LL}^{NM_1M_2}(M)
\nonumber\\&&+[C_7(\mu)+\frac{1}{N_c}C_8(\mu)]A_{LR}^{FM_1M_2}(M)+\frac{1}{N_c}C_8(\mu)A_{SP}^{NM_1M_2}(M)\}
,\nonumber\\
P_{EW}^{EM_1M_2}(M)&=&4\pi\alpha_s(\mu)\frac{G_F}{\sqrt{2}}\frac{3}{2}\big\{[C_{10}(\mu)+\frac{1}{N_c}C_9(\mu)]A_{LL}^{FM_1M_2}(M)+\frac{1}{N_c}C_9(\mu)A_{LL}^{NM_1M_2}(M)
\nonumber\\&&+[C_8(\mu)+\frac{1}{N_c}C_7(\mu)]A_{SP}^{FM_1M_2}(M)+\frac{1}{N_c}C_7(\mu)A_{LR}^{NM_1M_2}(M)\},\end{aligned}$$ for the so-called annihilation diagrams. Where $T^F_{XA}$, $T^N_{XA}$, $A^F_{XA}$, $A^N_{XA}$ ($X=LL,LR,SP$, $A=a,b$) arise from the hadronic matrix elements and their detailed expressions are given in Appendix. Note that $T_B^{FK\bar{K}}$ and $T_B^{F\bar{K}K}$ are slightly different as the wave functions of $K$ meson and $\bar{K}$ meson are not equal at high order in the twist expansion.
When redefining the above amplitudes to the widely used diagrammatic amplitudes in the phenomenological analysis, $$\begin{aligned}
&&T=T^{\pi\pi}(B),\ C=C^{\pi\pi}(B),\ E=E^{\pi\pi}(B),\ P=P_T^{\pi\pi}(B)+P_E^{\pi\pi}(B),\ P_A=2P_A^{\pi\pi},\nonumber\\
&&P_{EW}=P_{EW}^{\pi\pi}(B),\ P_{EW}^C=P_{EW}^{C\pi\pi}(B),\
P_{EW}^A=P_{EW}^{A\pi\pi}(B),\ P_{EW}^E=P_{EW}^{E\pi\pi}(B),\nonumber\\
&&T'=T^{\pi K}(B),\ C=C^{K\pi}(B),\ A'=A^{\pi K}(B),\ P'=P_T^{\pi K}(B)+P_E^{\pi K}(B),\ P_A'=2P_A^{K\pi},\nonumber\\
&&P_{EW}'=P_{EW}^{K\pi}(B),\ P_{EW}'^C=P_{EW}^{C\pi K}(B),\
P_{EW}'^A=P_{EW}^{AK\pi}(B),\ P_{EW}'^E=P_{EW}^{E\pi K}(B),\\
&&P''=P_T^{K\bar{K}}(B)+P_E^{K\bar{K}}(B),\
P_A''=P_A^{K\bar{K}}(B)+P_A^{\bar{K}K}(B),\ P_{EW}''^C=P_{EW}^{CK\bar{K}}(B),\ A''=A^{K\bar{K}}(B),\nonumber\\
&&P_{EW}''^A=[P_{EW}^{AK\bar{K}}(B) + P_{EW}^{A\bar{K}K}(B)]/2,\
\tilde{P}_{EW}''^A=[P_{EW}^{AK\bar{K}}(B) - P_{EW}^{A\bar{K}K}(B)]/2,\
P_{EW}''^E=P_{EW}^{EK\bar{K}}(B).\nonumber\end{aligned}$$ the decay amplitudes can be reexpressed in terms of the familiar forms in the diagrammatic decomposition approach $$\begin{aligned}
\label{eq:pptopologyT}
A(B^0\to \pi^+\pi^-)&=&V_{td}V^*_{tb}(P+P_A+\frac{2}{3}P_{EW}^C+\frac{1}{3}P_{EW}^A-\frac{1}{3}P_{EW}^E)-V_{ud}V^*_{ub}(T+E),\nonumber\\
A(B^+\to \pi^+\pi^0)&=&\frac{1}{\sqrt{2}}[V_{td}V^*_{tb}(P_{EW}+P_{EW}^C)-V_{ud}V^*_{ub}(T+C)],\nonumber\\
A(B^0\to \pi^0\pi^0)&=&\frac{1}{\sqrt{2}}[V_{td}V^*_{tb}(-P-P_A+P_{EW}+\frac{1}{3}P_{EW}^C+\frac{1}{3}P_{EW}^A-\frac{1}{3}P_{EW}^E)-V_{ud}V^*_{ub}(C-E)],\nonumber\\
A(B^+\to \pi^+ K^0)&=&V_{ts}V^*_{tb}(P'-\frac{1}{3}P_{EW}'^C+\frac{2}{3}P_{EW}'^E)+V_{us}V^*_{ub}A',\nonumber\\
A(B^+\to \pi^0 K^+)&=&\frac{1}{\sqrt{2}}[V_{ts}V^*_{tb}(P'+\frac{2}{3}P_{EW}'^C+P_{EW}'+\frac{2}{3}P_{EW}'^E)+V_{us}V^*_{ub}(T'+C'+A')],\nonumber\\
A(B^0\to \pi^- K^+)&=&V_{ts}V^*_{tb}(P'+\frac{2}{3}P_{EW}'^C-\frac{1}{3}P_{EW}'^E)+V_{us}V^*_{ub}T',\nonumber\\
A(B^0\to \pi^0
K^0)&=&\frac{1}{\sqrt{2}}[V_{ts}V^*_{tb}(P'-\frac{1}{3}P_{EW}'^C-P_{EW}'-\frac{1}{3}P_{EW}'^E)+V_{us}V^*_{ub}C'],
\nonumber \\
A(B^0\to K^+K^-)&=&V_{td}V^*_{tb}(P_A''+\frac{1}{3}P_{EW}''^A +\tilde{P}_{EW}''^A)+V_{ud}V^*_{ub}E'',\nonumber\\
A(B^+\to K^+\bar{K}^0)&=&V_{td}V^*_{tb}(P''-\frac{1}{3}P_{EW}''^C+\frac{2}{3}P_{EW}''^E)+V_{ud}V^*_{ub}A'',\nonumber\\
A(B^0\to K^0\bar{K}^0)&=&V_{td}V^*_{tb}(P''-\frac{1}{3}P_{EW}''^C+P_A''-\frac{2}{3}P_{EW}''^A-\frac{1}{3}P_{EW}''^E).\end{aligned}$$ It is noticed that there is a slight difference to the usual diagrammatic decomposition approach with the extra contributions from the annihilation electro-weak diagrammatic amplitudes $P_{EW}^{AM_1M_2}$ and $P_{EW}^{EM_1M_2}$, which are actually small and neglected in the usual diagrammatic decomposition approach.
A similar redefinition can be made for $B_s$ decays, $$\begin{aligned}
&&E_s=E^{\pi\pi}(B_s),P_{sA}=2P_A^{\pi\pi}(B_s),P_{sEW}^A=P_{EW}^{A\pi\pi}(B_s),\nonumber\\
&&T'_s=T^{\bar{K}\pi}(B_s),C'_s=C^{\bar{K}\pi}(B_s),P'_s=P_T^{\bar{K}\pi}(B)+P_E^{\bar{K}\pi}(B_s),P_{sEW}'=P_{EW}^{\bar{K}\pi}(B_s),\nonumber\\
&&P_{sEW}'^C=P_{EW}^{C\bar{K}\pi}(B_s),P_{sEW}'^E=P_{EW}^{E\bar{K}\pi}(B_s),\nonumber\\
&&T''_s=T^{\bar{K}K}(B_s),P''_s=P_T^{\bar{K}K}(B_s)+P_E^{\bar{K}K}(B_s),
P_{sA}''=P_A^{K\bar{K}}(B_s)+P_A^{\bar{K}K}(B_s),P_{sEW}''^C=P_{EW}^{C\bar{K}K}(B),\nonumber\\
&&E_s''=E^{\bar{K}K}(B_s),P_{sEW}''^A=[P_{EW}^{AK\bar{K}}(B_s) + P_{EW}^{A\bar{K}K}(B_s)]/2,\
\tilde{P}_{sEW}''^A=[P_{EW}^{AK\bar{K}}(B_s) - P_{EW}^{A\bar{K}K}(B_s)]/2,\nonumber\\
&&P_{sEW}''^E=P_{EW}^{E\bar{K}K}(B_s).\end{aligned}$$ Then the decay amplitudes can be reexpressed as follows $$\begin{aligned}
\label{eq:pptopologyT}
A(B_s\to \pi^+\pi^-)&=&V_{td}V^*_{tb}(P_{sA}+\frac{1}{3}P_{sEW}^A)-V_{ud}V^*_{ub}E_s,\nonumber\\
A(B_s\to \pi^0\pi^0)&=&\frac{1}{\sqrt{2}}A(B_s\to \pi^+\pi^-),\nonumber\\
A(B_s\to \pi^- K^+)&=&V_{ts}V^*_{tb}(P'_s+\frac{2}{3}P_{sEW}'^C-\frac{1}{3}P_{sEW}'^E)+V_{us}V^*_{ub}T'_s,\nonumber\\
A(B_s\to \pi^0 K^0)&=&-\frac{1}{\sqrt{2}}[V_{ts}V^*_{tb}(P'_s-\frac{1}{3}P_{sEW}'^C-P_{sEW}'-\frac{1}{3}P_{sEW}'^E)+V_{us}V^*_{ub}C'_s],\nonumber\\
A(B_s\to K^+K^-)&=&V_{td}V^*_{tb}(P_{sA}''+\frac{1}{3}P_{sEW}''^A+\tilde{P}_{sEW}''^A)+V_{ud}V^*_{ub}E''_s,\nonumber\\
A(B_s\to K^0\bar{K}^0)&=&V_{td}V^*_{tb}(P''_s-\frac{1}{3}P_{sEW}''^C+P_{sA}''-\frac{2}{3}P_{sEW}''^A-\frac{1}{3}P_{sEW}''^E).\end{aligned}$$
Numerical Calculations {#sec:nrcpe}
======================
We are now in the position to make numerical calculations.
Theoretical Input Parameters
----------------------------
The short distance contributions characterized by the Wilson coefficient functions for the effective four quark operators were calculated by several groups at the leading order(LO) and next-to-leading order(NLO) [@hep-ph/9806471], their values mainly depend on the choice for the running scale $\mu$. In our numerical calculations, it is taken to be $$\begin{aligned}
\mu = \sqrt{2\Lambda_{QCD}m_b} \simeq (1.5\pm 0.1){\rm GeV}.\end{aligned}$$ The $\alpha_{s}$ value in the six quark operator effective Hamiltonian is also taken at $\mu = (1.5\pm 0.1)$ GeV.
When considering the NLO Wilson coefficient functions and $\alpha_{s}$, one needs to include the magnetic penguin-like operator $O_{8g}$ which is defined as [@4qham] $$\begin{aligned}
O_{8g}\, =\,
\frac{g}{8\pi^2}m_b{\bar{q}}_i\sigma_{\mu\nu}(1+\gamma_5)T_{ij}^aG^{a\mu\nu}b_j\;,\end{aligned}$$ where $i$, $j$ are the color indices. The magnetic-penguin contribution to the $B\to\pi K$, $\pi\pi$ decays leads to the modification for the Wilson coefficients corresponding to the penguin operators, $$\begin{aligned}
a_{4,6}(\mu)&\to& a_{4,6}(\mu) - \frac{\alpha_s(\mu)}{9\pi}
\frac{2m_B}{\sqrt{|l^2|}}C_{8g}^{\rm eff}(\mu)\end{aligned}$$ with $C_{8g}^{\rm eff}=C_{8g}+C_5$ and $|l^2| = m_B^2/4$. Where $a_{4,6}$ are known to be defined as $a_{4,6} =
C_{4,6}+\frac{C_{3,5}}{N_c}$ which appear in the factorizable diagrams.
For other parameters, we take the following typical values $$\begin{aligned}
&&m_B = 5.28{\rm GeV},\ m_{\pi^+} = 139.6{\rm MeV},\ m_{\pi^0} = 135{\rm MeV},\ m_b = 4.4{\rm GeV},\ m_c = 1.5{\rm GeV},m_s = 0.1{\rm GeV},\nonumber\\
&&m_u=m_d = 5{\rm MeV},
f_{B} = 216.19{\rm MeV},\ f_{\pi} = 130.1{\rm MeV},\ F_K = 159.8{\rm MeV},
\mu_{\pi} \simeq 1.7{\rm GeV},\nonumber \\& &\mu_K \simeq 1.8{\rm GeV},
\tau_{B^0} = 1.536 ps,\ \tau_{B^+} = 1.638 ps,\ \lambda =
0.2272,\ A = 0.806,\ \bar{\rho}= 0.195,\ \bar{\eta}=
0.326.\label{def:num}\end{aligned}$$ Especially, for the infrared energy scale $\mu_g$ introduced in this paper to regulate the infrared divergence of gluon exchanging interactions, we take the typical value of $\mu_g$ to be a universal one around the hadronic bounding energy scale of non-perturbative QCD $$\begin{aligned}
\mu_g = (400\pm 50){\rm MeV}.\end{aligned}$$
To evaluate numerically the hadronic matrix elements of effective six quark operators based on the QCD factorization, it needs to know the twist wave functions of mesons. For the wave function of $B$ meson, we take the following form [@prd64074004] in our numerical calculations: $$\begin{aligned}
\phi_B(x)&=&N_Bx^2(1-x)^2
\exp\left[-\frac{1}{2}\left(\frac{xm_B}{\omega_B}\right)^2 \right]
\;, \label{def:lcda1}\end{aligned}$$
For the light meson wave functions, it needs to know the twist distribution amplitudes which contains the twist-2 pion (kaon) distribution amplitude $\phi_{\pi(K)}$, and the twist-3 ones $\phi_{\pi(K)}^p$ and $\phi_{\pi(K)}^T$, they are parameterized as [@0508041]: $$\begin{aligned}
\phi_{\pi(K)}(x) &=& 6x(1-x) (1 + a_1^{\pi(K)} 3 (2x - 1) +
a_2^{\pi(K)}\frac{3}{2}(5(2x-1)^2-1)\nonumber\\
& &+a_4^{\pi(K)}\frac{15}{8}(21(2x-1)^4-14(2x-1)^2+1)) \;,
\\
\phi^p_{\pi(K)}(x) &=& 1 +\left(30\eta_3
-\frac{5}{2}\rho_{\pi(K)}^2\right)
\frac{1}{2}\left(3(2x-1)^2-1\right)
\nonumber\\
&& -\, 3\left\{ \eta_3\omega_3 +
\frac{9}{20}\rho_{\pi(K)}^2(1+6a_2^{\pi(K)}) \right\}
\frac{1}{8}\left(35(2x-1)^4-30(2x-1)^2+3\right) \;,
\\
\phi^T_{\pi(K)}(x) &=& (1-2x)\bigg[ 1 + 6\left(5\eta_3
-\frac{1}{2}\eta_3\omega_3 - \frac{7}{20}
\rho_{\pi(K)}^2 - \frac{3}{5}\rho_{\pi(K)}^2 a_2^{\pi(K)} \right)\nonumber\\
&&(1-10x+10x^2) \bigg]\;,\label{def:lcda2}\end{aligned}$$
In our numerical calculations, the shape parameters in the distribution amplitudes are taken the following typical values: $$\begin{aligned}
& &\omega_B\, =\, 0.25\; {\rm GeV}\;\;,\omega_{B_s}\, =\, 0.33\; {\rm GeV}\;\;,\;
\eta_3\, =\, 0.015\;\;,\;\omega_3\, =\, -3\;\;, \nonumber\\
& & a_1^\pi\, =\, 0\;,\;\;\;\;a_1^K\, =\, 0.06\;\;,\;\;\;a_2^K\,=\, 0.10\pm0.10\;,\;\;\;\;
a_2^\pi\, =\, 0.15\pm0.15\;,\;\;\;\; a_4^K\, =a_4^\pi\, =\,0\pm0.10\;.
\label{def:lcda3}\end{aligned}$$ where the shape parameters for the bottom mesons are taken from [@prd64074004], and other shape parameters are taken to fit the data. Since those shape parameters can vary by 100%, they agree with the ones in Refs.[@0508041; @BsPV], All parameters for the light mesons are taken at the energy scale 1 GeV [@0605112], run to our calculation scale. Note that they may vary significantly when the scale runs to different values.
Numerical Results and Discussions
---------------------------------
The numerical results for the CP averaged branching ratios and CP violations of charmless B meson decays are presented in Table I for $B\to \pi\pi, \pi K$ decay channels and in Table II for $B\to K
\bar{K}$ channels. In Table III, we give the results for the branching ratios and CP violations for $B_s \to \pi\pi, \pi K, KK$ decay channels. The LO and NLO are corresponding to the leading order hadronic matrix elements with the leading order and next-to-leading order Wilson coefficients(which include the magnetic penguin-like operator $O_{8g}$). For comparison, the numerical predictions from the QCDF approach and pQCD approach are also listed in the Tables. It is seen that most resulting predictions in our present calculations are in good agreement with experimental data within the possible uncertainties from both experiments and theories, while it remains unclear how to understand the puzzles in the decay channel $B^0\to \pi^0\pi^0$ for its large branching ratio and possible positive CP violation, and in the decay channel $B\to
\pi^0 K^+$ for the unexpected large positive CP violation.
As shown in[@0508041], adding vertex corrections may improve the CP violation. The vertex corrections [@NPB.606.245] were proposed to improve the scale dependence of Wilson coefficient functions. The ref.[@NPB.606.245] has considered vertex corrections that only influence the Wilson coefficients of factorizable emission amplitudes. Those coefficients are always combined as $C_{2n-1}+\frac{C_{2n}}{N_c}$ and $C_{2n}+\frac{C_{2n-1}}{N_c}$ and modified by $$\begin{aligned}
C_{2n-1}(\mu)+\frac{C_{2n}}{N_c}(\mu) \to
C_{2n-1}(\mu)+\frac{C_{2n}}{N_c}(\mu)
+\frac{\alpha_s(\mu)}{4\pi}C_F\frac{C_{2n}(\mu)}{N_c} V_{2n-1}(M_2)
\;,&&
\nonumber\\
C_{2n}(\mu)+\frac{C_{2n-1}}{N_c}(\mu) \to
C_{2n}(\mu)+\frac{C_{2n-1}}{N_c}(\mu)
+\frac{\alpha_s(\mu)}{4\pi}C_F\frac{C_{2n-1}(\mu)}{N_c} V_{2n}(M_2) \;,&&\nonumber\\
n=1 - 5\;\;&&\end{aligned}$$ with $M_2$ being the meson emitted from the weak vertex. In the NDR scheme, $V_i(M)$ are given by [@NPB.606.245] $$\begin{aligned}
V_i(M) &=&\left\{
\begin{array}{ll}
12\ln(\frac{m_b}{\mu})-18+\int_0^1 dx\, \phi_M(x)\, g(x)\;, &
\mbox{\rm for }i=1-4,9,10\;,
\\
-12\ln(\frac{m_b}{\mu})+6-\int_0^1dx\, \phi_M(x)\, g(1-x)\;, &
\mbox{\rm for }i=5,7\;,
\\
-6 +\int_0^1 dx\,\phi_M^{p}(x)\, h(x) \;, &
\mbox{\rm for }i=6,8\;,
\end{array}
\right.\label{vim}\end{aligned}$$ $\phi_M(x)$/$\phi_M^p(x)$ is the twist-2/twist-3 meson distribution amplitudes defined in Eq. \[eq:daoM\]. The functions $g(x)/h(x)$ used in the integration are: $$\begin{aligned}
g(x) &=& 3\left( \frac{1-2x}{1-x}\ln{x} -i\,\pi \right)\nonumber\\
& & +\left[ 2\,{\rm Li}_2(x)-\ln^2 x +\frac{2\ln
x}{1-x}-(3+2i\,\pi)\ln x - (x\leftrightarrow 1-x) \right] \;,
\\
h(x) &=& 2\,{\rm Li}_2(x)-\ln^2 x -(1+2i\,\pi)\ln x -
(x\leftrightarrow 1-x) \;.\end{aligned}$$
Such a correction does not include the contributions of the first two diagrams in Fig.2a which are considered as a part of form factor or meson amplitude. It is interesting to notice that the vertex corrections do improve the predictions for CP violations and bring CP violations in the decay channels $B^0\to \pi^0\pi^0$ and $B^+\to
\pi^0 K^+$ to be more close to the experimental data.
To enlarge the branching ratio of $B \to \pi^0 \pi^0$, we shall examine an interesting case that only two vertexes concerning the operators $O_1$ and $O_2$ receive additional large nonperturbative contributions, namely the Wilson coefficients $a_1 =
C_1+\frac{C_2}{N_c}$ and $a_2 = C_2+\frac{C_1}{N_c}$ are modified to be the effective ones: $$\begin{aligned}
a_1 \to a_1^{eff} = C_1(\mu)+\frac{C_2}{N_c}(\mu)
+\frac{\alpha_s(\mu)}{4\pi}C_F\frac{C_2(\mu)}{N_c}( V_1(M_2)+V_0)
\;,&&
\nonumber\\
a_2 \to a_2^{eff} = C_2(\mu)+\frac{C_1}{N_c}(\mu)
+\frac{\alpha_s(\mu)}{4\pi}C_F\frac{C_1(\mu)}{N_c} (V_2(M_2)+V_0)
\;,&&\end{aligned}$$ Taking the value $V_0 = 25$, the resulting branching ratio for $B\to
\pi^0 \pi^0$ becomes consistent with the experimental data.
It is more interesting to consider the possible nonperturbative effects by taking the effective color number $N_c^{eff}$ in color-suppressed diagrams. The numerical results with $N_c^{eff}=1.7$ are presented in Table IV-VI, which provides an alternative explanation to the puzzle of observed large branching ratio $B\to \pi^0\pi^0$. For comparison, we also list in Table \[tab:br11\]-\[tab:br13\] the predicted results via the S4 scenario in QCDF [@QCDF].
The method allows us to calculate the relevant transition form factors at maximal recoil (with NLO Wilson coefficients including magnetic penguin contribution), $$\begin{aligned}
F_0^{B\to\pi}=0.262^{+0.029+0.10}_{-0.024-0.010},\ F_0^{B\to K}=0.322^{+0.034+0.013}_{-0.029-0.011},\ F_0^{B_s\to K}=0.274^{+0.023+0.013}_{-0.013+0.0005},
\label{eq:formfac}\end{aligned}$$ with input parameters $\mu_g$=400MeV, $\mu$=1.5GeV, $\mu_{\pi}$=1.7GeV, $\mu_{K}$=1.8GeV.The first error arises from the range for $\mu_g =
350 \sim 450$ MeV, the second error is caused by the running scale $\mu = 1.4\sim 1.6 $ GeV.
The resulting form factors agree with the ones obtained from the light-cone QCD sum rule of heavy quark effective field theory[@0604007] $$\begin{aligned}
F_0^{B\to\pi}=0.285^{+0.016}_{-0.015},\ F_0^{B\to K}= 0.345\pm 0.021,\
F_0^{B_s\to K}=0.296\pm 0.018, \nonumber\end{aligned}$$ and from the full QCD sum rule[@SR] $$\begin{aligned}
F_0^{B\to\pi}=0.258\pm 0.031,\ F_0^{B\to K}= 0.331 \pm 0.041.
\nonumber\end{aligned}$$
To know the relative contributions from various diagrams and hadronic matrix elements of effective six quark operators, we present in the Table \[tab:Ttopo\] and Table \[tab:Ptopo\] the numerical results for different kinds of topology amplitudes, the predictions for the strong phases are all relative to the leading order tree amplitude phase $\delta_T\simeq 1.93$ in $B\to \pi\pi$ decay. It is interesting to see that the amplitudes from the annihilation diagrams are significant in comparison with the color suppressed emission diagrams.
The predictions of $S_{\pi^0 K_S}$ in each method are almost the same and obviously larger than averaged data in PDG. But some new data in [@belle; @babar] prefer a larger prediction.
----------------------- --------------------------------- ---------------------------------------------------------------- -------------- ------------ -------- ---------------------------------------- --
Mode Data [@HFAG] QCDF[@QCDF]
LO NLO(+MP) LO NLO(+MP)
$B^+ \to \pi^+ K^0$ $ 23.1 \pm 1.0 $ $19.3^{+1.9+11.3+1.9+13.2}_{-1.9-7.8-2.1-5.6}$ $17.0$ $24.1$ 16.46 $21.60^{+7.33+4.36}_{-4.86-3.29}$
$B^+\to \pi^0 K^+$ $ 12.8 \pm 0.6 $ $11.1^{+1.8+5.8+0.9+6.9}_{-1.7-4.0-1.0-3.0}$ $10.2$ $14.0$ 9.12 $11.78^{+3.81+2.22}_{-2.53-1.69}$
$B^0 \to \pi^- K^+$ $ 19.4 \pm 0.6 $ $16.3^{+2.6+9.6+1.4+11.4}_{-2.3-6.5-1.4-4.8}$ $14.2$ $20.5$ 14.42 $19.03^{+6.60+3.86}_{-4.39-2.93}$
$B^0 \to \pi^0 K^0 $ $ 10.0 \pm 0.6 $ $\,\,7^{+0.7+4.7+0.7+5.4}_{-0.7-3.2-0.7-2.3}$ $\,\,5.7$ $\,\,8.7$ 6.61 $\,\,8.84^{+3.22+1.89}_{-2.13-1.44}$
$B^0\to\pi^- \pi^+ $ $\,\,5.16\pm0.22$ $\,\,8.9^{+4.0+3.6+0.6+1.2}_{-3.4-3.0-1.0-0.8}$ $\,\,7.0$ $\,\,6.7$ 6.63 $\,\,6.71^{+1.69+0.70}_{-1.24-0.57}$
$B^+\to\pi^+\pi^0$ $\,\,5.7\,\,\pm 0.4$ $\,\,6.0^{+3.0+2.1+1.0+0.4}_{-2.4-1.8-0.5-0.4}$ $\,\,3.5$ $\,\,4.1$ 4.43 $\,\,4.69^{+1.03+0.30}_{-0.71-0.26}$
$B^0\to\pi^0\pi^0$ $\,\,1.31\pm0.21$ $\,\,0.3^{+0.2+0.2+0.3+0.2}_{-0.2-0.1-0.1-0.1}$ $\,\,\,0.12$ $\,\,0.29$ 0.11 $\,\,0.16^{+0.05+0.02}_{-0.05-0.03}$
$A_{CP}(\pi^+ K^0)$ $ 0.009 \pm0.025$ $0.009^{+0.002+0.003+0.001+0.006}_{-0.003-0.003-0.001-0.005}$ $-0.01$ $-0.01$ 0.016 $+0.016^{-0.002-0.000}_{+0.003+0.001}$
$A_{CP}(\pi^0 K^+)$ $ 0.047\pm 0.026$ $0.071^{+0.017+0.020+0.008+0.090}_{-0.018-0.020-0.006-0.097}$ $-0.08$ $-0.08$ -0.093 $-0.080^{+0.008+0.006}_{-0.004-0.003}$
$A_{CP}(\pi^- K^+)$ $ -0.095\pm0.013$ $0.043^{+0.011+0.022+0.005+0.087}_{-0.011-0.025-0.006-0.095}$ $-0.12$ $-0.10$ -0.150 $-0.124^{+0.014+0.008}_{-0.014-0.007}$
$A_{CP}(\pi^0 K^0) $ $ -0.12\pm 0.11 $ $-0.033^{+0.010+0.013+0.005+0.034}_{-0.008-0.016-0.010-0.033}$ $-0.02$ $\,\,0.00$ -0.006 $-0.001^{-0.001-0.003}_{+0.000+0.002}$
$S_{\pi^0 K_S}$ $ 0.58\pm 0.17$[@belle; @babar] **–** $0.70$ $0.73$ 0.711 $ 0.715^{-0.012-0.003}_{+0.002+0.003}$
$A_{CP}(\pi^- \pi^+)$ $0.38\pm 0.07$ $-0.065^{+0.021+0.030+0.001+0.132}_{-0.021-0.028-0.003-0.128}$ $0.14$ $0.20$ 0.178 $0.187^{+0.002+0.014}_{-0.001-0.011}$
$A_{CP}(\pi^+\pi^0)$ $0.04\pm0.05$ $-0.000^{+0.000+0.000+0.000+0.00}_{-0.000-0.000-0.000-0.000}$ $0.00$ $0.00$ 0.000 $0.000^{+0.000+0.000}_{-0.000-0.000}$
$A_{CP}(\pi^0\pi^0)$ $0.36\pm0.32$ $0.451^{+0.184+0.151+0.043+0.465}_{-0.128-0.138-0.141-0.616}$ $-0.04$ $-0.43$ -0.571 $-0.547^{+0.018+0.046}_{-0.025+0.033}$
$S_{\pi\pi}$ $-0.61\pm0.08$ **–** $-0.34$ $-0.41$ -0.528 $-0.561^{-0.011-0.010}_{+0.011+0.009}$
----------------------- --------------------------------- ---------------------------------------------------------------- -------------- ------------ -------- ---------------------------------------- --
: CP averaged branching ratios and CP violations for $B\to
\pi\pi, \pi K$ decay channels. The central values are obtained with parameters: $\mu_g$=400MeV, $\mu$=1.5GeV, $\mu_{\pi}$=1.7GeV, $\mu_{K}$=1.8GeV. The first error arises from the range for $\mu_g =
350 \sim 450$ MeV, the second error stems from the running scale $\mu = 1.4\sim 1.6 $ GeV.[]{data-label="tab:BR1"}
------------------------- ---------------- ---------------------------------------------------------------- ---------------- ------- ---------------------------------------
Mode Data [@HFAG] QCDF[@QCDF] pQCD[@0411146]
LO NLO(+MP)
$B^+\to K^+ \bar{K}^0$ $1.36\pm 0.28$ $1.36^{+0.45+0.72+0.14+0.91}_{-0.39-0.49-0.15-0.40}$ 1.65 0.85 $1.09^{+0.26+0.18}_{-0.17-0.14}$
$B^0\to K^0 \bar{K}^0$ $0.96\pm0.20$ $1.35^{+0.41+0.71+0.13+1.09}_{-0.36-0.48-0.15-0.45}$ 1.75 0.65 $0.84^{+0.22+0.15}_{-0.15-0.12}$
$B^0\to K^+\bar{K}^- $ $0.15\pm0.10$ $0.013^{+0.005+0.008+0.000+0.087}_{-0.005-0.005-0.000-0.011}$ **–** 0.07 $0.07^{+0.03+0.01}_{-0.03-0.01}$
$A_{CP}(K^+ \bar{K}^0)$ $0.12\pm 0.17$ $-0.163^{+0.047+0.050+0.016+0.113}_{-0.037-0.057-0.017-0.133}$ **–** 0.096 $0.078^{+0.013+0.001}_{-0.013-0.001}$
$A_{CP}(K^0 \bar{K}^0)$ $-0.58\pm0.7$ $-0.167^{+0.047+0.045+0.015+0.046}_{-0.037-0.051-0.017-0.036}$ **–** 0.000 $0.000^{+0.000+0.000}_{-0.000-0.000}$
$A_{CP}(K^+ \bar{K}^-)$ **–** **–** **–** 0.807 $0.842^{-0.006-0.000}_{+0.042+0.000}$
------------------------- ---------------- ---------------------------------------------------------------- ---------------- ------- ---------------------------------------
: $B\to KK$ modes with the same input parameters as Table I.[]{data-label="tab:BR2"}
--------------------------- ---------------- ---------------------------------------------------------------- ----------------------------------------------------- -------- ----------------------------------------
Mode Data [@HFAG] QCDF[@QCDF] pQCD[@BsPV]
LO NLO(+MP)
$B_s \to \pi^+ \pi^-$ $0.5\pm 0.5$ $0.024^{+0.003+0.025+0.000+0.163}_{-0.003-0.012-0.000-0.021}$ $0.57^{+0.16+0.09+0.01}_{-0.13-0.10-0.00}$ 0.19 $0.23^{+0.01+0.07}_{-0.01-0.05}$
$B_s \to \pi^0 \pi^0$ **–** $0.012^{+0.001+0.013+0.000+0.082}_{-0.001-0.006-0.000-0.011}$ $0.28^{+0.08+0.04+0.01}_{-0.07-0.05-0.00}$ 0.10 $0.11^{+0.01+0.03}_{-0.01-0.02}$
$B_s \to \pi^+ \bar{K}^-$ $5.0\pm1.25$ $10.2^{+4.5+3.8+0.7+0.8}_{-3.9-3.2-1.2-0.7}$ $7.6^{+3.2+0.7+0.5}_{-2.3-0.7-0.5}$ 6.96 $7.02^{+1.11+0.63}_{-0.91-0.51}$
$B_s \to \pi^0 \bar{K}^0$ **–** $0.49^{+0.28+0.22+0.40+0.33}_{-0.24-0.14-0.14-0.17}$ $0.16^{+0.05+0.10+0.02}_{-0.04-0.05-0.01}$ 0.07 $0.09^{+0.04+0.03}_{-0.03-0.02}$
$B_s \to K^+\bar{K}^- $ $24.4\pm4.8$ $22.7^{+3.5+12.7+2.0+24.1}_{-3.2-8.4-2.0-9.1}$ $13.6^{+4.2+7.5+0.7}_{-3.2-4.1-0.2}$ 13.26 $16.68^{+5.37+4.32}_{-3.71-3.24}$
$B_s \to K^0\bar{K}^0 $ **–** $24.7^{+2.5+13.7+2.6+25.6}_{-2.4-9.2-2.9-9.8}$ $15.6^{+5.0+8.3+0.0}_{-3.8-4.7-0.0}$ 15.25 $18.94^{+5.80+4.56}_{-3.96-3.42}$
$A_{CP}(\pi^+\pi^- )$ **–** **–** $-0.012^{+0.001+0.012+0.001}_{-0.004-0.012-0.001}$ 0.018 $0.015^{+0.028-0.003}_{-0.020+0.002}$
$A_{CP}(\pi^0\pi^0 )$ **–** **–** $-0.012^{+0.001+0.012+0.001}_{-0.004-0.012-0.001}$ 0.018 $0.015^{+0.028-0.003}_{-0.020+0.002}$
$A_{CP}(\pi^+\bar{K}^-)$ $0.39\pm 0.17$ $-0.067^{+0.021+0.031+0.002+0.155}_{-0.022-0.029-0.004-0.152}$ $0.241^{+0.039+0.033+0.023}_{-0.036-0.030-0.012}$ 0.182 $0.183^{+0.012+0.018}_{-0.009-0.015}$
$A_{CP}(\pi^0\bar{K}^0)$ **–** $0.416^{+0.166+0.143+0.078+0.409}_{-0.120-0.133-0.145-0.510}$ $0.594^{+0.018+0.074+0.022}_{-0.040-0.113-0.035}$ 0.128 $-0.054^{+0.014+0.089}_{-0.014-0.081}$
$A_{CP}(K^+\bar{K}^- )$ **–** $0.040^{+0.010+0.020+0.005+0.104}_{-0.010-0.023-0.005-0.113}$ $-0.23.3^{+0.009+0.049+0.008}_{-0.002-0.044-0.011}$ -0.218 $-0.185^{+0.014+0.007}_{-0.010-0.009}$
$A_{CP}(K^0\bar{K}^0 )$ **–** $0.009^{+0.002+0.002+0.001+0.002}_{-0.002-0.002-0.001-0.003}$ $0$ 0.000 $0.000^{+0.000+0.000}_{-0.000-0.000}$
--------------------------- ---------------- ---------------------------------------------------------------- ----------------------------------------------------- -------- ----------------------------------------
: $B_s\to \pi\pi, \pi K, KK$ modes with the same input parameters as Table I.[]{data-label="tab:BR3"}
----------------------- --------------------------------- ---------------- ------------ ------------------------------------------------- -------- ---------------------------------------- ----------------- -------------
Mode Data [@HFAG] QCDF S4[@QCDF]
LO NLO+Vertex LO NLO+Vertex $a_{1,2}^{eff}$ $N_c^{eff}$
$B^+ \to \pi^+ K^0$ $ 23.1 \pm 1.0 $ $20.3$ $17.0$ $24.5^{+13.6\,(+12.9)}_{-\ 8.1\,(-\ 7.8)}$ 16.45 $22.06^{+7.39+4.25}_{-4.86-3.21}$ 22.06 19.50
$B^+\to \pi^0 K^+$ $ 12.8 \pm 0.6 $ $11.7$ $10.2$ $13.9^{+10.0\,(+\ 7.0)}_{-\ 5.6\,(-\ 4.2)}$ 9.12 $12.00^{+3.84+2.19}_{-2.54-1.65}$ 11.66 10.95
$B^0 \to \pi^- K^+$ $ 19.4 \pm 0.6 $ $18.4$ $14.2$ $20.9^{+15.6\,(+11.0)}_{-\ 8.3\,(-\ 6.5)}$ 14.41 $19.32^{+6.67+3.84}_{-4.41-2.91}$ 19.62 18.68
$B^0 \to \pi^0 K^0 $ $ 10.0 \pm 0.6 $ $\,\,8.0$ $\,\,5.7$ $\,\,9.1^{+\ 5.6\,(+\ 5.1)}_{-\ 3.3\,(-\ 2.9)}$ 6.61 $\,\,8.98^{+3.25+1.88}_{-2.14-1.42}$ 9.70 8.71
$B^0\to\pi^- \pi^+ $ $\,\,5.16\pm0.22$ $\,\,5.2$ $\,\,7.0$ $\,\,6.5^{+\ 6.7\,(+\ 2.7)}_{-\ 3.8\,(-\ 1.8)}$ 6.62 $\,\,7.07^{+1.67+0.71}_{-1.29-0.58}$ 5.38 4.89
$B^+\to\pi^+\pi^0$ $\,\,5.7\pm 0.4$ $\,\,5.1$ $\,\,3.5$ $\,\,4.0^{+\ 3.4\,(+\ 1.7)}_{-\ 1.9\,(-\ 1.2)}$ 4.43 $\,\,4.27^{+0.96+0.33}_{-0.73-0.29}$ 6.98 6.43
$B^0\to\pi^0\pi^0$ $\,\,1.31\pm0.21$ $\,\,0.7$ $\,\,0.12$ $\,\,0.29^{+0.50\,(+0.13)}_{-0.20\,(-0.08)}$ 0.11 $\,\,0.18^{+0.07+0.05}_{-0.04-0.03}$ 1.03 0.98
$A_{CP}(\pi^+ K^0)$ $ 0.009 \pm0.025$ $0.003$ $-0.01$ $-0.01\pm 0.00\,(\pm 0.00)$ +0.016 $0.020^{-0.003-0.001}_{+0.003+0.001}$ 0.020 0.018
$A_{CP}(\pi^0 K^+)$ $ 0.047\pm 0.026$ $-0.036$ $-0.08$ $-0.01^{+0.03\,(+0.03)}_{-0.05\,(-0.05)}$ -0.093 $-0.035^{+0.006+0.004}_{-0.002-0.002}$ -0.068 -0.0529
$A_{CP}(\pi^- K^+)$ $ -0.095\pm0.013$ $-0.041$ $-0.12$ $-0.09^{+0.06\,(+0.04)}_{-0.08\,(-0.06)}$ -0.150 $-0.133^{+0.015+0.008}_{-0.011-0.007}$ -0.117 -0.131
$A_{CP}(\pi^0 K^0) $ $ -0.12\pm 0.11 $ $0.008$ $-0.02$ $-0.07^{+0.03\,(+0.01)}_{-0.03\,(-0.01)}$ -0.006 $-0.051^{+0.003+0.000}_{-0.002-0.000}$ 0.002 -0.029
$S_{\pi^0 K_S}$ $ 0.58\pm 0.17$[@belle; @babar] **–** $0.70$ $0.73^{+0.03\,(+0.01)}_{-0.02\,(-0.01)}$ 0.710 $ 0.710^{-0.002-0.001}_{-0.002+0.002}$ 0.789 0.781
$A_{CP}(\pi^- \pi^+)$ $0.38\pm 0.07$ $0.103$ $0.14$ $0.18^{+0.20\,(+0.07)}_{-0.12\,(-0.06)}$ 0.178 $0.186^{+0.002+0.015}_{-0.002-0.014}$ 0.214 0.223
$A_{CP}(\pi^+\pi^0)$ $0.04\pm0.05$ $-0.0002$ $0.00$ $0.00\pm 0.00\,(\pm 0.00)$ 0.000 $0.000^{+0.000+0.000}_{-0.000-0.0000}$ 0.000 0.000
$A_{CP}(\pi^0\pi^0)$ $0.36\pm0.32$ $-0.19$ $-0.04$ $0.63^{+0.35\,(+0.09)}_{-0.34\,(-0.15)}$ -0.571 $0.470^{+0.010+0.032}_{-0.011-0.018}$ -0.174 -0.208
$S_{\pi\pi}$ $-0.61\pm0.08$ **–** $-0.34$ $-0.43^{+1.00\,(+0.05)}_{-0.56\,(-0.05)}$ -0.528 $-0.556^{-0.010-0.009}_{+0.004+0.008}$ -0.586 -0.479
----------------------- --------------------------------- ---------------- ------------ ------------------------------------------------- -------- ---------------------------------------- ----------------- -------------
: The same as Table I but including the vertex contributions and compare with QCDF S4.[]{data-label="tab:br11"}
------------------------- ---------------- ---------------- ---------------- ------- --------------------------------------- ----------------- ------------- --
Mode Data [@HFAG] QCDF S4[@QCDF] pQCD[@0411146]
LO NLO+Vertex $a_{1,2}^{eff}$ $N_c^{eff}$
$B^+\to K^+ \bar{K}^0$ $1.36\pm 0.28$ $1.46$ 1.65 0.85 $1.13^{+0.26+0.18}_{-0.17-0.14}$ 1.13 0.85
$B^0\to K^0 \bar{K}^0$ $0.96\pm0.20$ $1.58$ 1.75 0.65 $0.87^{+0.22+0.16}_{-0.14-0.11}$ 0.87 0.608
$B^0\to K^+\bar{K}^- $ $0.15\pm0.10$ $0.070$ **–** 0.07 $0.07^{+0.03+0.01}_{-0.10-0.01}$ 0.07 0.29
$A_{CP}(K^+ \bar{K}^0)$ $0.12\pm 0.17$ $-0.043$ **–** 0.096 $0.080^{+0.014+0.002}_{-0.009-0.001}$ 0.080 0.207
$A_{CP}(K^0 \bar{K}^0)$ $-0.58\pm0.7$ $-0.115$ **–** 0.000 $0.000^{+0.000+0.000}_{-0.000-0.000}$ 0.000 0.000
$A_{CP}(K^+ \bar{K}^-)$ **–** **–** **–** 0.807 $0.842^{-0.005-0.000}_{+0.041+0.000}$ 0.84 0.78
------------------------- ---------------- ---------------- ---------------- ------- --------------------------------------- ----------------- ------------- --
: The same as Table II but including the vertex contributions and compare with QCDF S4.[]{data-label="tab:br12"}
--------------------------- ---------------- ---------------- ----------------------------------------------------- -------- ---------------------------------------- ----------------- -------------
Mode Data [@HFAG] QCDF S4[@QCDF] pQCD[@BsPV]
LO NLO+Vertex $a_{1,2}^{eff}$ $N_c^{eff}$
$B_s \to \pi^+ \pi^-$ $0.5\pm 0.5$ $0.155$ $0.57^{+0.16+0.09+0.01}_{-0.13-0.10-0.00}$ 0.19 $0.23^{+0.01+0.07}_{-0.01-0.05}$ 0.23 0.69
$B_s \to \pi^0 \pi^0$ **–** $0.078$ $0.28^{+0.08+0.04+0.01}_{-0.07-0.05-0.00}$ 0.10 $0.11^{+0.01+0.03}_{-0.01-0.02}$ 0.11 0.34
$B_s \to \pi^+ \bar{K}^-$ $5.0\pm1.25$ $8.3$ $7.6^{+3.2+0.7+0.5}_{-2.3-0.7-0.5}$ 6.96 $7.35^{+1.15+0.63}_{-0.94-0.51}$ 5.73 6.58
$B_s \to \pi^0 \bar{K}^0$ **–** $0.61$ $0.16^{+0.05+0.10+0.02}_{-0.04-0.05-0.01}$ 0.07 $0.17^{+0.04+0.04}_{-0.03-0.03}$ 0.69 0.60
$B_s\to K^+\bar{K}^- $ $24.4\pm4.8$ $36.1$ $13.6^{+4.2+7.5+0.7}_{-3.2-4.1-0.2}$ 13.26 $16.77^{+5.36+4.23}_{-3.69-3.17}$ 16.97 15.76
$B_s \to K^0\bar{K}^0 $ **–** $38.3$ $15.6^{+5.0+8.3+0.0}_{-3.8-4.7-0.0}$ 15.25 $18.94^{+5.72+4.34}_{-3.89-3.26}$ 18.94 16.63
$A_{CP}(\pi^+\pi^- )$ **–** **–** $-0.012^{+0.001+0.012+0.001}_{-0.004-0.012-0.001}$ 0.018 $0.015^{+0.028-0.003}_{-0.019+0.002}$ 0.015 0.016
$A_{CP}(\pi^0\pi^0 )$ **–** **–** $-0.012^{+0.001+0.012+0.001}_{-0.004-0.012-0.001}$ 0.018 $0.015^{+0.028-0.003}_{-0.019+0.002}$ 0.015 0.016
$A_{CP}(\pi^+\bar{K}^-)$ $0.39\pm 0.17$ $0.109$ $0.241^{+0.039+0.033+0.023}_{-0.036-0.030-0.012}$ 0.182 $0.182^{+0.009+0.015}_{-0.002-0.016}$ 0.207 0.171
$A_{CP}(\pi^0\bar{K}^0)$ **–** $0.046$ $0.594^{+0.018+0.074+0.022}_{-0.040-0.113-0.035}$ 0.128 $0.831^{+0.017+0.017}_{-0.011-0.006}$ -0.135 0.057
$A_{CP}(K^+\bar{K}^- )$ **–** $-0.047$ $-0.23.3^{+0.009+0.049+0.008}_{-0.002-0.044-0.011}$ -0.218 $-0.194^{+0.014+0.010}_{-0.011-0.010}$ -0.168 -0.191
$A_{CP}(K^0\bar{K}^0 )$ **–** $0.006$ $0$ 0.000 $0.000^{+0.000+0.000}_{-0.000-0.000}$ 0.000 0.000
--------------------------- ---------------- ---------------- ----------------------------------------------------- -------- ---------------------------------------- ----------------- -------------
: The same as Table III but including the vertex contributions and compare with QCDF S4.[]{data-label="tab:br13"}
$T$ $C$ $A$ $E$
---------- ------------ ------------------ -------------------- ------------------- ------------------- --
LO 81.48 $4.931e^{-0.75i}$ **–** $7.329e^{-3.07i}$
$\pi\pi$ NLO(+MP) 81.46 $6.952e^{-0.50i}$ **–** $7.321e^{-3.07i}$
NLO+Vertex $83.51e^{0.04i}$ $13.88e^{-1.60i}$ **–** $7.321e^{-3.07i}$
LO 100.0 $6.020e^{-0.75i}$ $39.86e^{-0.50i}$ **–**
$\pi K$ NLO(+MP) 100.0 $8.558e^{-0.488i}$ $39.58e^{-0.51i}$ **–**
NLO+Vertex $102.4e^{0.04i}$ $17.52e^{-1.59i}$ $39.58e^{-0.50i}$ **–**
LO **–** **–** $2.267e^{2.30i}$ $7.169e^{-0.85i}$
$K K$ NLO(+MP) **–** **–** $2.045e^{2.31i}$ $7.088e^{-0.85i}$
NLO+Vertex **–** **–** $2.045e^{2.31i}$ $7.088e^{-0.85i}$
: Diagrammatic amplitudes relating to the CKM matrix element $\lambda_u$ with $10^{-7}$ GeV. []{data-label="tab:Ttopo"}
$P_T$ $P_{EW}$ $P_{EW}^C$ $P_A$ $P_E$ $P_{EW}^A$ $P_{EW}^E$ $P=P_T+P_E$
---------- ------------ ------------------- ------------------- ------------------ ------------------ ------------------ ------------------- ------------------- ------------------ -- -- -- -- --
LO $6.555e^{-3.10i}$ $1.176e^{-3.13i}$ $0.101e^{0.50i}$ $2.210e^{0.06i}$ $3.137e^{1.51i}$ $0.021e^{-2.65i}$ $0.111e^{-0.62i}$ $6.987e^{2.72i}$
$\pi\pi$ NLO(+MP) $7.478e^{-3.11i}$ $1.175e^{-3.13i}$ $0.076e^{0.69i}$ $2.442e^{0.06i}$ $3.339e^{1.51i}$ $0.019e^{-2.59i}$ $0.112e^{-0.63i}$ $7.876e^{2.74i}$
NLO+Vertex $7.684e^{-3.07i}$ $1.201e^{-3.09i}$ $0.232e^{0.93i}$ $2.442e^{0.06i}$ $3.339e^{1.51i}$ $0.019e^{-2.59i}$ $0.112e^{-0.63i}$ $7.971e^{2.78i}$
LO $8.332e^{-3.10i}$ $1.490e^{-3.12i}$ $0.123e^{0.49i}$ **–** $4.604e^{1.88i}$ **–** $0.165e^{-0.58i}$ $10.54e^{2.74i}$
$\pi K$ NLO(+MP) $9.483e^{-3.11i}$ $1.489e^{-3.12i}$ $0.092e^{0.68i}$ **–** $5.069e^{1.91i}$ **–** $0.150e^{-0.60i}$ $12.00e^{2.76i}$
NLO+Vertex $9.731e^{-3.07i}$ $1.521e^{-3.07i}$ $0.287e^{0.96i}$ **–** $5.069e^{1.91i}$ **–** $0.150e^{-0.60i}$ $12.11e^{2.79i}$
LO $10.57e^{-3.08i}$ **–** $0.161e^{0.44i}$ $1.572e^{0.50i}$ $3.433e^{1.69i}$ $0.069e^{-1.26i}$ $0.053e^{-3.00i}$ $11.30e^{2.90i}$
$K K$ NLO(+MP) $12.03e^{-3.08i}$ **–** $0.120e^{0.59i}$ $1.667e^{0.54i}$ $3.663e^{1.70i}$ $0.063e^{-1.31i}$ $0.052e^{-2.94i}$ $12.79e^{2.92i}$
NLO+Vertex $12.35e^{-3.05i}$ **–** $0.362e^{0.95i}$ $1.667e^{0.54i}$ $3.663e^{1.70i}$ $0.063e^{-1.31i}$ $0.052e^{-2.94i}$ $12.99e^{2.95i}$
: Diagrammatic amplitudes relating to the CKM matrix element $\lambda_t$ with $10^{-7}$ GeV.[]{data-label="tab:Ptopo"}
$T_s$ $C_s$ $A_s$ $E_s$
---------- ------------ ------------------ ------------------- ------------------ ------------------- --
LO **–** **–** **–** $4.027e^{-2.28i}$
$\pi\pi$ NLO(+MP) **–** **–** **–** $3.980e^{-2.28i}$
NLO+Vertex **–** **–** **–** $3.980e^{-3.28i}$
LO $77.75e^{0.08i}$ $6.221e^{-0.84i}$ $53.96e^{2.58i}$ **–**
$\pi K$ NLO(+MP) $77.74e^{0.08i}$ $7.876e^{-0.60i}$ $53.76e^{2.59i}$ **–**
NLO+Vertex $79.63e^{0.08i}$ $14.96e^{-1.54i}$ $53.76e^{2.59i}$ **–**
LO $95.41e^{0.08i}$ $8.085e^{-0.84i}$ $2.705e^{1.77i}$ $7.313e^{-1.44i}$
$K K$ NLO(+MP) $95.41e^{0.08i}$ $10.10e^{-0.61i}$ $2.478e^{1.78i}$ $7.219e^{-1.43i}$
NLO+Vertex $97.73e^{0.08i}$ $19.36e^{-1.50i}$ $2.478e^{1.78i}$ $7.219e^{-1.43i}$
: Diagrammatic amplitudes relating to the CKM matrix element $\lambda_u$ in $B_s$ decays with $10^{-7}$ GeV. []{data-label="tab:Ttopos"}
$P_{T_s}$ $P_{sEW}$ $P_{sEW}^C$ $P_{sA}$ $P_{sE}$ $P_{sEW}^A$ $P_{sEW}^E$ $P_s=P_{sT}+P_{sE}$
---------- ------------ ------------------- ------------------- ------------------ ------------------ ------------------ ------------------- --------------------- ------------------ -- -- -- -- --
LO **–** **–** **–** $2.365e^{1.00i}$ **–** $0.026e^{-2.24i}$ **–** $2.265e^{1.00i}$
$\pi\pi$ NLO(+MP) **–** **–** **–** $2.597e^{0.99i}$ **–** $0.025e^{-2.23i}$ **–** $2.597e^{0.99i}$
NLO+Vertex **–** **–** **–** $2.597e^{0.99i}$ **–** $0.025e^{-2.23i}$ **–** $2.597e^{0.99i}$
LO $6.329e^{-3.01i}$ $1.122e^{-3.06i}$ $0.108e^{0.79i}$ **–** $4.105e^{1.12i}$ **–** $0.044e^{-2.78i}$ $5.315e^{2.58i}$
$\pi K$ NLO(+MP) $7.215e^{-3.01i}$ $1.121e^{-3.06i}$ $0.088e^{1.00i}$ **–** $4.362e^{1.05i}$ **–** $0.037e^{-2.25i}$ $5.745e^{2.62i}$
NLO+Vertex $7.410e^{-3.02i}$ $1.145e^{-3.05i}$ $0.243e^{1.07i}$ **–** $6.275e^{1.22i}$ **–** $0.093e^{-1.45i}$ $7.180e^{2.35i}$
LO $8.231e^{-3.00i}$ $0.918e^{+3.05i}$ $0.143e^{0.76i}$ $1.181e^{1.48i}$ $5.852e^{1.62i}$ $0.083e^{-1.47i}$ $0.043e^{-1.63i}$ $9.671e^{2.63i}$
$K K$ NLO(+MP) $9.348e^{-3.01i}$ $0.917e^{+3.05i}$ $0.118e^{0.95i}$ $1.320e^{1.50i}$ $6.238e^{1.62i}$ $0.078e^{-1.48i}$ $0.047e^{-1.61i}$ $10.80e^{2.66i}$
NLO+Vertex $9.597e^{-3.02i}$ $0.937e^{+3.04i}$ $0.307e^{1.04i}$ $1.320e^{1.50i}$ $6.238e^{1.62i}$ $0.078e^{-1.48i}$ $0.047e^{-1.61i}$ $10.80e^{2.66i}$
: Diagrammatic amplitudes relating to the CKM matrix element $\lambda_t$ in $B_s$ decays with $10^{-7}$ GeV.[]{data-label="tab:Ptopos"}
Conclusions {#sec:conc}
===========
Based on the approximate six quark operator effective Hamiltonian derived from perturbative QCD, the QCD factorization approach has naturally been applied to evaluate the hadronic matrix elements for charmless two body decays of bottom mesons. The resulting predictions for the decay amplitudes, branching ratios, and $CP$ asymmetries in $B^0,\ B^+,\ B_s\to \pi\pi,\ \pi K,\ KK$ decay channels have been found to be consistent with the current experimental measurements except for a few decay modes.
The puzzles for the observed large branching ratio in $B\to
\pi^0\pi^0$ decay and possible large positive CP violations in $B\to
\pi K^+$ decay need to be further investigated. As we have emphasized at the beginning that the six quark operator effective Hamiltonian considered in this paper is an approximate one, and a large number of six quark diagrams which suppressed at high energy scales have been ingored, but they may become sizable at low energy scales. Furthermore, when given the predictions, we have only considered the uncertainties caused by the choices of running scale $\mu$ and infrared energy scale $\mu_g$ as their effects are more significant than others. In general, the theoretical uncertainties could be much larger when the possible uncertainties for all the input parameters are included. The masses of light mesons are also neglected in comparison with the bottom meson masses, i.e, $m^2_{\pi}/m^2_B \sim 0$ and $m_K^2/m_B^2 \sim 0$.
Nevertheless, it is remarkable that such a simple theoretical framework based on the approximate six quark operator effective Hamiltonian from the perturbative QCD and the naive QCD factorization for the nonperturbative QCD effects can result in a satisfactory theoretical prediction for the charmless B meson decays $B,\ B_s \to \pi\pi, \pi K, KK$. It also shows that the singularity due to the on mass-shell fermion propagator can simply be treated with the principal integration method by apply the Cutkosky rules, and the one caused by the gluon exchanging interactions can well be regulated by the description used in the loop regularization method with the introduction of an intrinsic energy scale $\mu_g$. In particular, it is found that such a scale takes a typical value $\mu_g = (400\pm 50)$ MeV which is around the binding energy of hadron due to non-perturbative QCD effects.
We would like to point out that although the theoretical framework discussed above is a much simplified one, it turns out that as the first order approximation the six quark operator effective Hamiltonian considered in this paper can be taken as a good starting point. We have actually examined two interesting cases by considering teh effective Wilson coefficient functions $a_{1,2}^{eff}$ and the effective color number $N_c^{eff}$ in the color suppressed diagrams to bring the prediction for the branching ratio $B\to \pi^0\pi^0$ be consistent with the experimental data. It is of interest to calculate high order contributions though it is a challenging task. On the other hand, the precise measurements of charmless bottom meson decays, especially the measurements on CP-violations in $B\to KK$ and $B_s\to \pi\pi, \pi K, KK$ decays, will provide a useful test for various theoretical frameworks. It is expected that more and more precise experimental data in the future super B-factory and LHCb will guide us to arrive at a better understanding on perturbative and nonperturbative QCD.
The authors would like to thank I. Bigi, H.Y. Cheng, A. Khodjamirian, H.N. Li, G. Ricciardi, C. Sachrajda for useful discussions and conversations during the KITPC program on Flavor Physics at Beijing. The author (F.Su) is grateful to M. Beneke for his kind hospitality. This work was supported in part by the National Science Foundation of China (NSFC) under the grant 10475105, 10491306, and the key Project of Chinese Academy of Sciences (CAS).
Calculations of Hadronic Matrix Elements {#sec:CalcHME}
========================================
In this appendix, we are going to present the explicit expressions for all the hadronic matrix elements evaluated from the naive QCD factorization method based on effective six quark operators. To be specific, we shall first make the following convention for the momentums of quarks and mesons, which is explicitly shown in Fig. \[pic:definition\]
![Definition of momentum in $B\rightarrow M_1M_2$. The light-cone coordinate is adopted with $(n^+, n^-, k_{\bot})$[]{data-label="pic:definition"}](pic5.eps "fig:")\
Where we have ignored the light quark mass in external lines and light meson mass to simplify calculation.
Let us first give the factorizable emission contributions for the $(V-A)\times (V-A)$ and $(V-A)\times (V+A)$ effective four quark vertexes, they are simply denoted by $LL$ and $LR$ $$\begin{aligned}
\label{eq:ppf1}
T_{LL}^{FM_1M_2}(M)&=&T_{LLa}^{FM_1M_2}(M)+T_{LLb}^{FM_1M_2}(M),\nonumber\\
T_{LLa}^{FM_1M_2}(M)&=&i \frac{1}{4}\frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,m_B^2\phi_{M}(u)\nonumber\\&&
\big\{m_B (2 m_b-m_B x) \phi_{M_1}(x)+\mu_{M_1}(2 m_B x-m_b)[\phi^{p}_{M_1}(x)-\phi^{T}_{M_1}(x)]\big\}
\phi_{M_2}(y)h_{Ta}^F(u,x),\nonumber\\
T_{LLb}^{FM_1M_2}(M)&=&i \frac{1}{2} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\, m_B^3 \mu_{M_1} \phi_{M}(u)\phi_{M_2}(y) \phi^{p}_{M_1}(x)h_{Tb}^F(u,x),\nonumber\\
T_{LR}^{FM_1M_2}(M)&=&T_{LLa}^{FM_1M_2}(M)+T_{LLb}^{FM_1M_2}(M),\nonumber\\
T_{LRa}^{FM_1M_2}(M)&=&- T_{LLa}^{FM_1M_2}(M),\nonumber\\
T_{LRb}^{FM_1M_2}(M)&=&- T_{LLb}^{FM_1M_2}(M).\end{aligned}$$ The factorizable emission contributions for the $(S-P)\times (S+P)$ effective four quark vertex are found to be $$\begin{aligned}
\label{eq:ppf2}
T_{SP}^{FM_1M_2}(M)&=&T_{SPa}^{FM_1M_2}(M)+T_{SPb}^{FM_1M_2}(M),\nonumber\\
T_{SPa}^{FM_1M_2}(M)&=&i \frac{1}{2} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\, \ m_B\ \mu_{M_2}
\phi_{M}(u)\nonumber\\&&\big\{m_B(2 m_B-m_b)\phi_{M_1}(x)+\mu_{M_1}[4 m_b- (x+1)m_B] \phi^{p}_{M_1}(x)
+\mu_{M_1}m_B(1-x)\phi^{T}_{M_1}(x)\big\}\nonumber\\&& \phi^{p}_{M_2}(y)
h_{Ta}^F(u,x),\nonumber\\
T_{SPb}^{FM_1M_2}(M)&=&i \frac{1}{2} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\, \ m_B^2\ \mu_{M_2}\phi_{M}(u)
\nonumber\\&&[m_B u \phi_{M_1}(x)+2 (1-u)
\mu_{M_1}\phi^{p}_{M_1}(x)]\phi^{p}_{M_2}(y)h_{Tb}^F(u,x).\end{aligned}$$ Similarly, we obtain $$\begin{aligned}
\label{eq:ppnf1}
T_{LL}^{NM_1M_2}(M)&=&T_{LLa}^{NM_1M_2}(M)+T_{LLb}^{NM_1M_2}(M),\nonumber\\
T_{LLa}^{NM_1M_2}(M)&=&-i\frac{1}{4} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\ F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,
m_B^3\phi_{M}(u)\nonumber\\&&\big\{ (u-y) m_B\phi_{M_1}(x)
+ (1-x)\mu_{M_1} [\phi^{p}_{M_1}(x)+\phi^{T}_{M_1}(x)]\big\}\phi_{M_2}(y)
h_{Ta}^N(u,x,y),\nonumber\\
T_{LLb}^{NM_1M_2}(M)&=&i\frac{1}{4} \frac{C_{F}}{N_c}\ F_M\ F_1
\ F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\
m_B^3\phi_{M}(u)\nonumber\\&&\big\{(u+x+y-2) m_B \phi_{M_1}(x)
+(1-x)\mu_{M_1} [\phi^{p}_{M_1}(x)-\phi^{T}_{M_1}(x)]\big\}\phi_{M_2}(y)
h_{Tb}^N(u,x,y)\nonumber\\\end{aligned}$$ for non-factorizable emission contributions with the $(V-A)\times
(V-A)$ effective four quark vertex, and $$\begin{aligned}
\label{eq:ppnf3}
T_{LR}^{NM_1M_2}(M)&=&T_{LRa}^{NM_1M_2}(M)+T_{LRb}^{NM_1M_2}(M)\nonumber\\
T_{LRa}^{NM_1M_2}(M)&=&-i\frac{1}{4} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^2\phi_{M}(u)
\nonumber\\&&\Big\{\mu_{M_2}\ \mu_{M_1}\big\{[(u-x-y+1) \phi^{T}_{M_1}(x)+(u+x-y-1) \phi^{p}_{M_1}(x)]\phi^{p}_{M_2}(y)
\nonumber\\&&-[(u-x-y+1) \phi^{p}_{M_1}(x)+(u+x-y-1) \phi^{T}_{M_1}(x)]\phi^{T}_{M_2}(y)\}
\nonumber\\&&+(u-y) m_B\ \mu_{M_2} [\phi^{p}_{M_2}(y)-\phi^{T}_{M_2}(y)]\phi_{M_1}(x)\Big\}h_{Ta}^N(u,x,y)\nonumber\\
T_{LRb}^{NM_1M_2}(M)&=&i\frac{1}{4} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^2\phi_{M}(u)
\nonumber\\&&\Big\{\mu_{M_2}\ \mu_{M_1}\big\{[(u-x+y)\phi^{T}_{M_1}(x)
+(u+x+y-2) \phi^{p}_{M_1}(x)] \phi^{p}_{M_2}(y)
\nonumber\\&&+[(u-x+y)\phi^{p}_{M_1}(x)+(u+x+y-2)\phi^{T}_{M_1}(x)]\phi^{T}_{M_2}(y)\big\}
\nonumber\\&&+(u+y-1) m_B\ \mu_{M_2}[\phi^{p}_{M_2}(y)+\phi^{T}_{M_2}(y)]\phi_{M_1}(x)\Big\}h_{Tb}^N(u,x,y)\end{aligned}$$ for non-factorizable emission contributions with the $(V-A)\times
(V+A)$ effective four quark vertex, and $$\begin{aligned}
\label{eq:ppnf2}
T_{SP}^{NM_1M_2}(M)&=&T_{SPa}^{NM_1M_2}(M)+T_{SPb}^{NM_1M_2}(M)\nonumber\\
T_{SPa}^{NM_1M_2}(M)&=&i \frac{1}{4}\frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^3\phi_{M}(u)\nonumber\\&&
\big\{(u+x-y-1)m_B \phi_{M_1}(x)+ (1-x)\mu_{M_1} [\phi^{p}_{M_1}(x)-\phi^{T}_{M_1}(x)]\big\}
\phi_{M_2}(y)h_{Ta}^N(u,x,y)\nonumber\\
T_{SPb}^{NM_1M_2}(M)&=&-i \frac{1}{4}\frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^3\phi_{M}(u)\nonumber\\&&
\big\{(u+y-1)m_B \phi_{M_1}(x)+(1-x)\mu_{M_1} [\phi^{p}_{M_1}(x)+\phi^{T}_{M_1}(x)]\big\}\phi_{M_2}(y)
h_{Tb}^N(u,x,y)\end{aligned}$$ for non-factorizable emission contributions with the $(S-P)\times
(S+P)$ effective four quark vertex.
We now present the results from annihilation diagram contributions, $$\begin{aligned}
\label{eq:af1}
A_{LL}^{FM_1M_2}(M)&=&A_{LLa}^{FM_1M_2}(M)+A_{LLb}^{FM_1M_2}(M),\nonumber\\
A_{LLa}^{FM_1M_2}(M)&=&i \frac{1}{4} \frac{C_{F}}{N_c}\ F_M\
F_{M_1}\ F_{M_2} \int_0^1\int_0^1\int_0^1
\emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^2 \phi_{M}(u)
\nonumber\\&&\big\{(1-y)m_B^2 \phi_{M_2}(y) \phi_{M_1}(x) +2
\mu_{M_2}\ \mu_{M_1} [(2-y) \phi^{p}_{M_2}(y)+y\phi^{T}_{M_2}(y)]
\phi^{p}_{M_1}(x)\big\}h_{Aa}^F(x,y),\nonumber\\
A_{LLb}^{FM_1M_2}(M)&=&-i \frac{1}{4} \frac{C_{F}}{N_c}\ F_M\
F_{M_1} \ F_{M_2} \int_0^1\int_0^1\int_0^1
\emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^2 \phi_{M}(u)
\nonumber\\&&\big\{x m_B^2 \phi_{M_2}(y) \phi_{M_1}(x)+2 \mu_{M_2}\
\mu_{M_1} [(1+x)\phi^{p}_{M_1}(x)-
(1-x)\phi^{T}_{M_1}(x)]\phi^{p}_{M_2}(y)\}
h_{Ab}^F(x,y),\nonumber\\
A_{LR}^{FM_1M_2}(M)&=&A_{LRa}^{FM_1M_2}(M)+A_{LRb}^{FM_1M_2}(M),\nonumber\\
A_{LRa}^{FM_1M_2}(M)&=&A_{LLa}^{FM_1M_2}(M),\nonumber\\
A_{LRb}^{FM_1M_2}(M)&=&A_{LLb}^{FM_1M_2}(M)\end{aligned}$$ for the factorizable annihilation contributions with the $(V-A)\times (V-A)$ and $(V-A)\times (V+A)$ effective four quark vertexes, and $$\begin{aligned}
\label{eq:af3}
A_{SP}^{FM_1M_2}(M)&=&A_{SPa}^{FM_1M_2}(M)+A_{SPb}^{FM_1M_2}(M),\nonumber\\
A_{SPa}^{FM_1M_2}(M)&=&i\frac{1}{2} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\
m_B^3\phi_{M}(u)
\nonumber\\&&[(1-y)\mu_{M_2}
[\phi^{p}_{M_2}(y)+\phi^{T}_{M_2}(y)]\phi_{M_1}(x) +2 \mu_{M_1}
\phi_{M_2}(y) \phi^{p}_{M_1}(x)]h_{Aa}^F(x,y),\nonumber\\
A_{SPb}^{FM_1M_2}(M)&=&i\frac{1}{2} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\
m_B^3\phi_{M}(u)
\big\{2 \mu_{M_2}
\phi^{p}_{M_2}(y) \phi_{M_1}(x)\nonumber\\&&+ x\ \mu_{M_1}
\phi_{M_2}(y)[\phi^{p}_{M_1}(x)-\phi^{T}_{M_1}(x)]\big\}h_{Ab}^F(x,y)\end{aligned}$$ for the factorizable annihilation contributions with the $(S-P)\times (S+P)$ effective four quark vertex, and $$\begin{aligned}
\label{eq:anf1}
A_{LL}^{NM_1M_2}(M)&=&A_{LLa}^{NM_1M_2}(M)+A_{LLb}^{NM_1M_2}(M),\nonumber\\
A_{LLa}^{NM_1M_2}(M)&=&-i\frac{1}{4} \frac{C_{F}}{N_c}\ F_M\
F_{M_1}\ F_{M_2} \int_0^1\int_0^1\int_0^1
\emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^2 \phi_{M}(u) \Big\{[m_b+m_B
(u-y)] m_B^2 \phi_{M_2}(y)
\phi_{M_1}(x)\nonumber\\&&+\mu_{M_1}\mu_{M_2} \big\{[-(u-x-y+1)m_B
\phi^{p}_{M_1}(x)+ (-u-x+y+1)m_B\phi^{T}_{M_1}(x)]
\phi^{T}_{M_2}(y)\big\}\nonumber\\&&+[\big(4 m_b+(u+x-y-1)m_B \big)\phi^{p}_{M_1}(x)+(u-x-y+1)m_B \phi^{T}_{M_1}(x)]
\phi^{p}_{M_2}(y)\Big\}
h_{Aa}^N(u,x,y),\nonumber\\
A_{LLb}^{NM_1M_2}(M)&=&i\frac{1}{4} \frac{C_{F}}{N_c}\ F_M\ F_{M_1}\
F_{M_2} \int_0^1\int_0^1\int_0^1 \emph{d}u\,\emph{d}x\,\emph{d}y\,\
m_B^2 \phi_{M}(u) \Big\{x m_B^2 \phi_{M_2}(y)
\phi_{M_1}(x)\nonumber\\&&+\mu_{M_1}\mu_{M_2}\big\{
-[(u+x+y-1)\phi^{p}_{M_1}(x)+(-u+x-y+1)\phi^{T}_{M_1}(x)]
\phi^{T}_{M_2}(y)\nonumber\\&&+[(-u+x-y+1)\phi^{p}_{M_1}(x)
+(u+x+y-1)\phi^{T}_{M_1}(x)]
\phi^{p}_{M_2}(y)\big\}\Big\}h_{Ab}^N(u,x,y)\end{aligned}$$ for the non-factorizable annihilation contributions with the $(V-A)\times (V-A)$ effective four quark vertex, and $$\begin{aligned}
\label{eq:anf3}
A_{LR}^{NM_1M_2}(M)&=&A_{LRa}^{NM_1M_2}(M)+A_{LRb}^{NM_1M_2}(M),\nonumber\\
A_{LRa}^{NM_1M_2}(M)&=&i \frac{1}{4} \frac{C_{F}}{N_c}\ F_M\
F_{M_1}\ F_{M_2} \int_0^1\int_0^1\int_0^1
\emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^2\phi_{M}(u)
\nonumber\\&&\big\{\mu_{M_2} [m_b+(y-u)m_B
][\phi^{p}_{M_2}(y)-\phi^{T}_{M_2}(y)]
\phi_{M_1}(x)\nonumber\\&&-\mu_{M_1} [
(1-x)m_B+m_b][\phi^{p}_{M_1}(x)+\phi^{T}_{M_1}(x)]\phi^{p}_{M_2}(y)\big\}
h_{Aa}^N(u,x,y),\nonumber\\
A_{LRa}^{NM_1M_2}(M)&=&-i \frac{1}{4} \frac{C_{F}}{N_c}\ F_M\
F_{M_1}\ F_{M_2} \int_0^1\int_0^1\int_0^1
\emph{d}u\,\emph{d}x\,\emph{d}y\, \ m_B^3\phi_{M}(u)\big\{x\
\mu_{M_1} [\phi^{p}_{M_1}(x)+\phi^{T}_{M_1}(x)]
\phi^{p}_{M_2}(y)\nonumber\\&&-(1-u-y)\mu_{M_2}
[\phi^{p}_{M_2}(y)-\phi^{T}_{M_2}(y)] \phi_{M_1}(x)\}
h_{Ab}^N(u,x,y)\end{aligned}$$ for the non-factorizable annihilation contributions with the $(V-A)\times (V-A)$ and $(V-A)\times (V+A)$ effective four quark vertexes, and $$\begin{aligned}
\label{eq:anf2}
A_{SP}^{NM_1M_2}(M)&=&A_{SPa}^{NM_1M_2}(M)+A_{SPb}^{NM_1M_2}(M),\nonumber\\
A_{SPa}^{NM_1M_2}(M)&=&-i \frac{1}{4} \frac{C_{F}}{N_c}\ F_M\
F_{M_1}\ F_{M_2} \int_0^1\int_0^1\int_0^1
\emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B \phi_{M}(u)
\Big\{[m_b+(x-1)m_B ] m_B^2 \phi^{A}
_{M_2}(y)\nonumber\\&&+\mu_{M_1}\mu_{M_2} \big\{ \phi_{M_1}(x)[(u-x-y+1)m_B \phi^{p}_{M_1}(x)+
(-u-x+y+1)m_B\phi^{T}_{M_1}(x)]
\phi^{T}_{M_2}(y)\big\}\nonumber\\&&+\big\{[4 m_b-(-u-x+y+1)m_B ]\phi^{p}_{M_1}(x)-(u-x-y+1)m_B \phi^{T}_{M_1}(x)\big\}
\phi^{p}_{M_2}(y)\Big\}
h_{Aa}^N(u,x,y),\nonumber\\
A_{SPb}^{NM_1M_2}(M)&=&i \frac{1}{4} \frac{C_{F}}{N_c}\ F_M\
F_{M_1}\ F_{M_2} \int_0^1\int_0^1\int_0^1
\emph{d}u\,\emph{d}x\,\emph{d}y\,\ m_B^2 \phi_{M}(u) \Big\{(-u-y+1)
m_B^2 \phi_{M_2}(y) \phi_{M_1}(x)\nonumber\\&&+\mu_{M_1}\mu_{M_2}
\big\{[(u+x+y-1)\phi^{p}_{M_1}(x)-(-u+x-y+1)\phi^{T}_{M_1}(x)]
\phi^{T}_{M_2}(y)\big\}\nonumber\\&&+[(-u+x-y+1)\phi^{p}_{M_1}(x)-(u+x+y-1)\phi^{T}_{M_1}(x)]
\phi^{p}_{M_2}(y) \Big\} h_{Ab}^N(u,x,y)\end{aligned}$$ for the non-factorizable annihilation contributions with the $(S-P)\times (S+P)$ effective four quark vertex.
The functions $h^{Y}_{XA}$ with $(A=a,b)$ from Eqs. (\[eq:ppf1\]) to (\[eq:anf3\]) arise from propagators of gluon and quark, here $Y=F,N$ denote the factorizable and non-factorizable contributions respectively, and $X=T,A$ the emission and annihilation diagrams respectively. They have the following explicit forms: $$\begin{aligned}
&&h_{Ta}^F(u,x)=\frac{1}{(-u(1-x)m_B^2-\mu_g^2+i\epsilon)(x m_B^2-m_b^2+i\epsilon)},\nonumber\\
&&h_{Tb}^F(u,x)=\frac{1}{(-u(1-x)m_B^2-\mu_g^2+i\epsilon)(-u m_B^2-m_{q}^2+i\epsilon)},\nonumber\\
&&h_{Ta}^N(u,x,y)=\frac{1}{(-u(1-x)m_B^2-\mu_g^2+i\epsilon)((1-x)(1-u-y)m_B^2-m_{q}^2+i\epsilon)},\nonumber\\
&&h_{Tb}^N(u,x,y)=\frac{1}{(-u(1-x)m_B^2-\mu_g^2+i\epsilon)((1-x)(y-u)m_B^2-m_{q}^2+i\epsilon)},\nonumber\\
&&h_{Aa}^F(x,y)=\frac{1}{(x(1-y)m_B^2-\mu_g^2+i\epsilon)((1-y)m_B^2-m_{q}^2+i\epsilon)},\nonumber\\
&&h_{Ab}^F(x,y)=\frac{1}{(x(1-y)m_B^2-\mu_g^2+i\epsilon)(x\,m_B^2-m_{q}^2+i\epsilon)},\nonumber\\
&&h_{Aa}^N(u,x,y)=\frac{1}{(x(1-y)m_B^2-\mu_g^2+i\epsilon)((y-u)(1-x)m_B^2-m_b^2+i\epsilon)},\nonumber\\
&&h_{Ab}^N(u,x,y)=\frac{1}{(x(1-y)m_B^2-\mu_g^2+i\epsilon)((1-u-y)x\;m_B^2-m_{q}^2+i\epsilon)}.\label{eq:propN}\end{aligned}$$
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see, e.g., G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Review of Modern Physics, 68, 1125 (1996). M. Beneke and Th. Feldmann, Nucl. Phys. B[**592**]{}, 003(2000). R. E. Cutkosky, J. Math. Phys. 1, 429 (1960).
A. J. Buras, Les Houches 1997, Probing the standard model of particle interactions, Pt. 1\*281-539. e-Print: hep-ph/9806471. H. Y. Cheng and K. C. Yang, Phys. Lett. B [**511**]{}, 40 (2001); H. Y. Cheng and K. C. Yang, Phys. Rev. D [**64**]{}, 074004 (2001). H. N. Li, S. Mishima, A. I. Sanda, Phys.Rev. D[**72**]{}, 114005 (2005).e-Print: hep-ph/0508041. A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W, Wang and Y. M. Wang, Phys. Rev. D[**76**]{}, 074018(2007).e-Print: hep-ph/0703162 T. Kurimoto, Phys.Rev. D[**74**]{}, 014027 (2006).e-Print: hep-ph/0605112. M. Beneke, G.Buchalla, M. Neubert, C. T.Sachrajda, Nucl. Phys. B[**606**]{}, 245(2001). M. Beneke, M. Neubert, Nucl. Phys. B[**675**]{}, 333(2003). Y, L, Wu, M. Zhong and Y. B. Zuo, Int.J.Mod.Phys. A21:6125-6172,2006. e-Print: hep-ph/0604007. P. Ball, R. Zwicky, Phys. Rev. D[**71**]{}, 014015(2005); Phys.Rev. D71, 014029(2005). J. F. Hirschauer (BABAR Collaboration), talk presented at ICHEP08, the 34th Inter-national Conference on High Energy PhysicsPhiladelphia, Pennsylvania, July 30 - August5, 2008. J. P. Dalseno (Belle Collaboration), talk presented at ICHEP08, the 34th Inter-national Conference on High Energy PhysicsPhiladelphia, Pennsylvania, July 30 - August5, 2008. H. N. Li, S. Mishima, Phys. Rev. D[**71**]{}, 054025(2005).e-Print: hep-ph/0411146.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Alain Chenciner & Hugo Jiménez-Pérez\
Observatoire de Paris, IMCCE (UMR 8028), ASD\
77, avenue Denfert-Rochereau, 75014 Paris, France\
`chenciner@imcce.fr, jimenez@imcce.fr`
title: 'Angular momentum and Horn’s problem'
---
We prove a conjecture made in [@C1]: given an $n$-body central configuration $X_0$ in the euclidean space $E$ of dimension $2p$, let $Im{\cal F}$ be the set of ordered real $p$-tuples $\{\nu_1,\nu_2,\cdots,\nu_p\}$ such that $\{\pm i\nu_1,\pm i\nu_2,\cdots,\pm i\nu_p\}$ is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of $X_0$ in $E$. Then $Im {\cal F}$ is a convex polytope. The proof consists in showing that there exist two, generically $(p-1)$-dimensional, convex polytopes ${\cal P}_1$ and ${\cal P}_2$ in ${\ensuremath{\mathbb{R}}}^{p}$ such that ${\cal P}_1\subset Im{\cal F}\subset {\cal P}_2$ and that these two polytopes coincide.
${\cal P}_1$, introduced in [@C1], is the set of spectra corresponding to the hermitian structures $J$ on $E$ which are “adapted" to the symmetries of the inertia matrix $S_0$; it is associated with Horn’s problem for the sum of $p\times p$ real symmetric matrices with spectra $\sigma_-$ and $\sigma_+$ whose union is the spectrum of $S_0$.
${\cal P}_2$ is the orthogonal projection onto the set of “hermitian spectra" of the polytope ${\cal P}$ associated with Horn’s problem for the sum of $2p\times 2p$ real symmetric matrices having each the same spectrum as $S_0$.
The equality ${\cal P}_1={\cal P}_2$ follows directly from a deep combinatorial lemma, proved in [@FFLP], which characterizes those of the sums $C=A+B$ of two $2p\times 2p$ real symmetric matrices $A$ and $B$ with the same spectrum, which are hermitian for some hermitian structure.
Origin of the problem: $N$-body relative equilibria and their angular momenta
=============================================================================
We recall here the results of [@AC; @C1; @C2] which are needed in order to understand the mechanical origin of the purely algebraic conjecture solved in the present paper: given a configuration $x_0=(\vec r_1,\cdots,\vec r_N)\in E^N$ of $N$ punctual positive masses in the euclidean space $E$, a [*rigid motion*]{} of the configuration under Newton’s attraction is a motion in which the mutual distances $||\vec r_i-\vec r_j||$ between the bodies stay constant. It is proved in [@AC] (see also [@C2]) that such a motion is necessarily a [*relative equilibrium*]{}. This implies that the motion takes place in a space of even dimension $2p$, which can be supposed to coincide with $E$, and that, in a galilean frame fixing the center of mass at the origin, it is of the form $x(t)=(e^{\Omega t}\vec r_1,e^{\Omega t}\vec r_2,\cdots,e^{\Omega t}\vec r_N)$, where $\Omega$ is a $2p\times 2p$-antisymmetric endomorphism of the euclidean space $E$ which is non degenerate. Choosing an orthonormal basis where $\Omega$ is normalized, this amounts to saying that there exists a hermitian structure on the space $E$ of motion and an orthogonal decomposition $E\equiv{\ensuremath{\mathbb{C}}}^p={\ensuremath{\mathbb{C}}}^{k_1}\times\cdots\times{\ensuremath{\mathbb{C}}}^{k_r}$ such that $$x(t)=(x_1(t),\cdots,x_r(t))=(e^{i\omega_1t}x_1,\cdots,e^{i\omega_rt}x_r),$$ where $x_m$ is the orthogonal projection on ${\ensuremath{\mathbb{C}}}^{k_m}$ of the $N$-body configuration $x$ and the action of $e^{i\omega_mt}$ on $x_m$ is the diagonal action on each body of the projected configuration. Such quasi-periodic motions exist only for very special configurations, called [*balanced configurations*]{} (see [@AC; @C2] for their characterization). The most degenerate balanced configurations are the [*central configurations*]{} for which all the frequencies $\omega_i$ are the same; this means that $\Omega=\omega J$, with $J$ a hermitian structure on $E$, and the motion is $$x(t)=(\vec r_1(t),\cdots,\vec r_N(t))=e^{i\omega t}x_0=(e^{i\omega t}\vec r_1,\cdots,e^{i\omega t}\vec r_N)$$ in the hermitian space $E\equiv{\ensuremath{\mathbb{C}}}^{p}$; in particular, it is periodic. In a space of dimension at most 3, $E$ is necessarily of dimension 2 and the configuration of any relative equilibrium is central.
Given a configuration $x=(\vec r_1,\cdots,\vec r_N)$ and a configuration of velocities $y=\dot x=(\vec v_1,\cdots, \vec v_N)$, both with center of mass at the origin: $\sum_{k=1}^Nm_k\vec r_k=\sum_{k=1}^Nm_k\vec v_k=0$, the [*angular momentum*]{} of $(x,y)$ is the bivector ${\mathcal C}=\sum_{k=1}^Nm_k\vec r_k\wedge\vec v_k$. If we represent $x$ and $y$ by the $2p\times N$ matrices $X$ and $Y$ whose $i$th column are respectively made of the components $(r_{1i},\cdots,r_{2pi})$ and $(v_{1i},\cdots,v_{2pi})$ of $\vec r_i$ and $\vec v_i$ in an orthonormal basis of $E$ and if $M=\hbox{diag}(m_1,\cdots,m_N)$, this bivector is represented by the antisymmetric matrix [*(we use the french convention $^{t\!}X$ for the transposed of $X$)*]{} $$C=-XM^{t\!}Y+YM^{t\!\!}X\;\;\hbox{with coefficients}\;\; c_{ij}
=\sum_{k=1}^Nm_k(-r_{ik}v_{jk}+r_{jk}v_{ik}).$$ The dynamics of a solid body is determined by its [*inertia tensor*]{} (with respect to its center of mass), represented in the case of a point masses configuration $X$ by the symmetric matrix $$S=XM^{t\!\!}X\;\;\hbox{with coefficients}\;\;
s_{ij}=\sum_{k=1}^Nm_kr_{ik}r_{jk},$$ whose trace is the [*moment of inertia of the configuration $x$ with respect to its center of mass*]{}. In particular, the angular momentum of a relative equilibrium solution $X(t)=e^{t\Omega}X_0$ is represented by the antisymmetric matrix $C=S_0\Omega+\Omega S_0$, where $S_0=X_0M^{t\!\!}X_0$. Restricting to the case of central configurations, that is $\Omega=\omega J$, and making $\omega=1$, we consider in what follows the spectrum of $J$-skew-hermitian matrices of the form $S_0J+JS_0$ or, what amounts to the same, the spectrum of $J$-hermitian matrices[^1] of the form $J^{-1}S_0J+S_0$.
[*In the following, we identify $E$ with ${\ensuremath{\mathbb{R}}}^{2p}$ by the choice of some orthonormal basis. ${\ensuremath{\mathbb{R}}}^{2p}$ is endowed with its canonical basis $e_i=(0,\cdots,1,\cdots,0)$ and its canonical euclidean scalar product $x\cdot y=\sum_{i=1}^{2p}x_iy_i$; this allows identifying linear endomorphisms of $E={\ensuremath{\mathbb{R}}}^{2p}$ and $2p\times 2p$ matrices with real coefficients. When we say that $J$ is a hermitian structure, we mean that the standard euclidean structure is given and that $J$ is a complex structure which is orthogonal.*]{}
The frequency map
=================
We recall the definition, given in [@C1], of the [*frequency map*]{} ${\cal F}$ from the set of hermitian structures on ${\ensuremath{\mathbb{R}}}^{2p}$ to the positive Weyl chamber $W_p^+\subset {\ensuremath{\mathbb{R}}}^p$: given some $2p\times 2p$ real symmetric matrix $S_0$, we consider the mapping $J\mapsto J^{-1}S_0J+S_0$ from the space of hermitian structures on $E$ to the set of $2p\times 2p$ real symmetric matrices. We are only interested in the spectra of these matrices, hence choosing an orientation for $J$ is harmless and we shall consider only those of the form $J=R^{-1}J_0R$, where $J_0$ is the “standard" structure defined by $J_0=\begin{pmatrix}0&-Id\\ Id&0\end{pmatrix}$ and $R\in SO(2p)$. Two elements $R'$ and $R''$ of $SO(2p)$ defining the same $J$ if and only if there exists an element $U\in U(p)$ such that $R''=UR'$, it follows that the space of (oriented) hermitian structures is identified to the homogeneous space $U(p)\backslash SO(2p)$. The symmetric matrix $J^{-1}S_0J+S_0$ is $J$-hermitian, that is, it commutes with $J$. This implies that its spectrum is real, of the form $\{\nu_1,\nu_2,\cdots,\nu_p\}$ if considered as a $p\times p$ complex matrix (for the identification of ${\ensuremath{\mathbb{R}}}^{2p}$ to ${\ensuremath{\mathbb{C}}}^p$ defined by $J$) and of the form $\{(\nu_1, \nu_2, \cdots, \nu_p), (\nu_1, \nu_2, \cdots, \nu_p)\}$ if considered as a $2p\times 2p$ real matrix (see the next section for the trivial proof).
The [*frequency mapping*]{} $${\mathcal F}:U(p)\backslash SO(2p)\to W_p^+=\{(\nu_1,\cdots\nu_p)\in{\ensuremath{\mathbb{R}}}^p, \nu_1\ge\cdots\ge\nu_p\}$$ associates to each hermitian structure $J$ the ordered spectrum $(\nu_1,\cdots,\nu_p)$ of the $J$-hermitian matrix $J^{-1}S_0J+S_0$.
Hermitian spectra
=================
Let $C:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be a symmetric endomorphism. The following assertions are equivalent:
1\) There exists a hermitian structure $J=R^{-1}J_0R$ such that $C$ be $J$-hermitian (i.e. $JC=CJ$);
2\) The spectrum $\sigma(C)$ of $C$ is of the form $$\sigma(C)=\{(\nu_1, \nu_2, \cdots, \nu_p), (\nu_1, \nu_2, \cdots, \nu_p)\}.$$
Let $J=R^{-1}J_0R$; the mapping $C$ is $J$-hermitian if and only if $RCR^{-1}$ is $J_0$-hermitian. This is equivalent to the existence of an isomorphism $U\in U(p)\subset SO(2p)$ such that $$U^{-1}RCR^{-1}U=\hbox{diag}\bigl((\nu_1,\cdots,\nu_p),(\nu_1,\cdots,\nu_p)\bigr).$$ Conversely, the identity $$R^{-1}CR=\hbox{diag}\bigl((\nu_1,\cdots,\nu_p),(\nu_1,\cdots,\nu_p)\bigr)$$ implies the commutation of $R^{-1}CR$ with $J_0$ and hence the commutation of $C$ with $J=R^{-1}J_0R$.
[**Notations.**]{} We shall call [*hermitian*]{} the spectra of this form and [*the diagonal*]{} the linear subspace $\Delta$ of $W_{2p}^+$ defined by $$\Delta=\{(\mu_1\ge\mu_2\ge\cdots\ge\mu_{2p}),\; \mu_1=\mu_2,\, \mu_3=\mu_4,\, \, \cdots,\mu_{2p-1}=\mu_{2p}\}.$$ Hence the ordered hermitian spectra are the ones belonging to $\Delta$.
Two convex polytopes
====================
Let $S_0:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be a symmetric endomorphism with spectrum $$\sigma(S_0)=\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}.$$ To $S_0$ we associate the subsets ${\cal P}_1$ and ${\cal P}_2$ of ${\ensuremath{\mathbb{R}}}^p$ (in fact of the positive Weyl chamber $W_p^+$ of ordered real $p$-tuples), defined as follows:
1\) ${\cal P}_1$ is the set of ordered spectra $$\sigma(c)=\{\nu_1\ge\nu_2\ge\cdots\ge\nu_p\}$$ of symmetric endomorphisms $c$ of ${\ensuremath{\mathbb{R}}}^p$ of the form $c=a+b$, where $a:{\ensuremath{\mathbb{R}}}^p\to{\ensuremath{\mathbb{R}}}^p$ and $b:{\ensuremath{\mathbb{R}}}^p\to{\ensuremath{\mathbb{R}}}^p$ are arbitrary symmetric endomorphisms with respective spectra $$\sigma(a)=\sigma_-:= \{\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}\},\quad \sigma(b)=\sigma_+:=\{\sigma_2,\sigma_4,\cdots,\sigma_{2p}\} ;$$
2\) ${\cal P}_2$ is the set of $p$-tuples $\{\nu_1\ge\nu_2\ge\cdots\ge\nu_p\}$ such that $$\{(\nu_1,\nu_2\cdots,\nu_{p}),(\nu_1,\nu_2\cdots,\nu_{p})\}$$ is the spectrum of some symmetric endomorphism $C$ of ${\ensuremath{\mathbb{R}}}^{2p}$ of the form $C=A+B$, where $A:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ and $B:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ are arbitrary symmetric endomorphisms with the same spectrum $$\sigma(A)=\sigma(B)=\sigma(S_0).$$ In other words, identifying canonically the diagonal $\Delta$ with ${\ensuremath{\mathbb{R}}}^p$, one can write $${\cal P}_2={\cal P}\cap\Delta,$$ where ${\cal P}$ is the $(2p-1)$-dimensional Horn polytope which describes the ordered spectra of sums $C=A+B$ of $2p\times 2p$ real symmetric matrices $A,B$ with the same spectrum as $S_0$.
${\cal P}_1$ and ${\cal P}_2$ are both contained in the hyperplane of ${\ensuremath{\mathbb{R}}}^p$ with equation $$\sum_{i=1}^p\nu_i=\sum_{j=1}^{2p}\sigma_j.$$ Moreover, ${\cal P}_1$ and ${\cal P}_2$ are both $(p-1)$-dimensional convex polytopes and $${\cal P}_1\subset Im {\cal F} \subset{\cal P}_2.$$
The first identity comes from the additivity of the trace function. The fact that both ${\cal P}_1$, ${\cal P}$ and hence ${\cal P}_2={\cal P}\cap\Delta$, are convex polytopes is a general fact coming from the interpretation of the Horn problem as a moment map problem. Finally, the second inclusion comes from the very definition of ${\cal F}$ and the first comes from Lemma 1 and the following identity, where $\sigma_-$ and $\sigma_+$ are considered as $p\times p$ diagonal matrices and $\rho\in SO(p)$: $$\begin{cases}
&\begin{pmatrix}
\sigma_-&0\\
0&\sigma_+
\end{pmatrix}+
\begin{pmatrix}
0&-\rho^{-1}\\
\rho&0
\end{pmatrix}^{-1}
\begin{pmatrix}
\sigma_-&0\\
0&\sigma_+
\end{pmatrix}
\begin{pmatrix}
0&-\rho^{-1}\\
\rho&0
\end{pmatrix}\\
=&
\begin{pmatrix}
\sigma_-+\rho^{-1}\sigma_+\rho&0\\
0&\rho\sigma_-\rho^{-1}+\sigma_+
\end{pmatrix}\cdot
\end{cases}$$
[**Remark.**]{} The choice of the partition $\sigma=\sigma_-\cup\sigma_+$ of $\sigma$ is dictated by the following theorem, which proves that any other partition of $\sigma$ into two subsets with $p$ elements will lead to a smaller polytope ${\cal P}_1$:
Let $A$ and $B$ be $p\times p$ Hermitian matrices. Let $\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}$ be the eigenvalues of $A$ and $B$ arranged in descending order. Then there exist Hermitian matrices $\tilde A$ and $\tilde B$ with eigenvalues $\sigma_1\ge\sigma_3\ge\cdots\ge\sigma_{2p-1}$ and $\sigma_2\ge\sigma_4\ge\cdots\ge\sigma_{2p}$ respectively, such that $\tilde A+\tilde B=A+B$.
This was used in [@C1] to prove that ${\cal P}_1= Im {\cal A}$ is the image under the frequency map ${\cal F}$ of the [*adapted*]{} hermitian structures, i.e. those $J$ which send some $p$-dimensional subspace of ${\ensuremath{\mathbb{R}}}^{2p}$ generated by eigenvectors of $S_0$ onto its orthogonal.
The projection property
=======================
In this section, we prove the
The two polytopes ${\cal P}_1$ and ${\cal P}_2$ coincide.
${Im {\cal F}}={\cal P}_1= Im {\cal A}$. In other words, the image by the frequency map ${\cal F}$ of the adapted structures is already the full image $Im {\cal F}$.
We need recall the inductive definition of the Horn inequalities which define the Horn polytopes (see [@F]). For a sum $a+b=c$ of symmetric $p\times p$ matrices with respective (ordered in decreasing order) spectra $$\alpha=(\alpha_1,\cdots,\alpha_p),\; \beta=(\beta_1,\cdots,\beta_p),\; \gamma=(\gamma_1,\cdots,\gamma_p),$$ they read $$(^*IJK)\quad\quad\quad \forall r<p,\; \forall (I,J,K)\in T^p_r,\quad \sum_{k\in K}\gamma_k\le \sum_{i\in I}\alpha_i+\sum_{j\in J}\beta_j,$$ where $T^p_r$ (notation of [@F], noted $LR^p_r$ by reference to Littlewood-Richardson coefficients in [@FFLP]) is defined as follows: let $U^p_r$ be the set of triples $(I,J,K)$ of subsets of cardinal $r$ of $\{1,2,\cdots, p\}$ such that $$\sum_{i\in I}i+\sum_{j\in J}j=\sum_{k\in K}k+\frac{r(r+1)}{2}.$$ Then set $T^p_1=U^p_1$ and define recursively $T^p_r$ by $$T^p_r=
\begin{bmatrix}
(I,J,K)\in U^p_r, \forall s<r,\, \forall (F,G,H)\in T^r_s, \\
\sum_{f\in F}i_f+\sum_{g\in G}j_g\le \sum_{h\in H}k_h+\frac{s(s+1)}{2}
\end{bmatrix}$$
An immediate computation gives the following
Let $$\begin{cases}
I_2&=(2i_1-1,2i_2-1,\cdots,2i_r-1,2j_1,2j_2,\cdots,2j_r),\\
J_2&=(2i_1-1,2i_2-1,\cdots,2i_r-1,2j_1,2j_2,\cdots,2j_r),\\
K_2&=(2k_1-1,2k_1,2k_2-1,2k_2,\cdots,2k_r-1,2k_r),
\end{cases}$$
Then $(I_2,J_2,K_2)\in U^{2p}_{2r}$.
It suffices to check that $$2\left[\sum_{i\in I}(2i-1)+\sum_{j\in J}(2j)\right]=\sum_{k\in K}(2k-1)+\sum_{k\in K}2k+\frac{2r(2r+1)}{2}\cdot$$
Now, comes the key fact:
For any triple $(I,J,K)$ in $T^p_r$, the triple $(I_2,J_2,K_2)$ is in $T^{2p}_{2r}$
The proof of this theorem, which concerns the so-called “domino-decomposable Young diagrams", is based on a version of the Littlewood-Richardson rule due to Carré and Leclerc [@CL].
It implies that, for any a sum $A+B=C$ of real symmetric $2p\times 2p$ matrices with respective (ordered in decreasing order) spectra $$\hat\alpha=(\hat\alpha_1,\cdots,\hat\alpha_{2p}),\; \hat\beta=(\hat\beta_1,\cdots,\hat\beta_{2p}),\; \hat\gamma=(\hat\gamma_1,\cdots,\hat\gamma_{2p}),$$ $(^*I_2,J_2,K_2)$ holds, that is $$\sum_{k\in K}(\hat\gamma_{2k-1}+\hat\gamma_{2k})\le \sum_{i\in I}(\hat\alpha_{2i-1}+\hat\beta_{2i-1})+\sum_{j\in J}(\hat\alpha_{2j}+\hat\beta_{2j}).$$ In particular, if $$\hat\alpha=\hat\beta=\sigma=(\sigma_1,\sigma_2,\cdots,\sigma_{2p}),$$ we get that $$\sum_{k\in K}\frac{\hat\gamma_{2k-1}+\hat\gamma_{2k}}{2}\le \sum_{i\in I}\sigma_{2i-1}+\sum_{j\in J}\sigma_{2j}.$$ Note that the mapping $$(\hat\gamma_1,\hat\gamma_2,\cdots,\hat\gamma_{2p-1},\hat\gamma_{2p})\mapsto (\frac{\hat\gamma_1+\hat\gamma_2}{2},\frac{\hat\gamma_1+\hat\gamma_2}{2},\cdots, \frac{\hat\gamma_{2p-1}+\hat\gamma_{2p}}{2},\frac{\hat\gamma_{2p-1}+\hat\gamma_{2p}}{2})$$ is the orthogonal projection of the ordered set $(\hat\gamma_1,\hat\gamma_2,\cdots,\hat\gamma_{2p-1},\hat\gamma_{2p})$ on the [*diagonal*]{} $\Delta$ of ${\ensuremath{\mathbb{R}}}^{2p}$ defined by the equations $\hat\gamma_1=\hat\gamma_2,\cdots,\hat\gamma_{2p-1}=\hat\gamma_{2p}$, that is on the subset of “hermitian" spectra. Hence a paraphrase of the above theorem is
Let $C=A+B$ be the sum of two $2p\times 2p$ real symmetric matrices with the same spectrum $\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}$.
If $\{\nu_1\ge\cdots\ge\nu_p\}$ is the orthogonal projection on the diagonal $\Delta\equiv{\ensuremath{\mathbb{R}}}^p$ of the spectrum $\{\hat\gamma_1\ge\hat\gamma_2\ge\cdots\ge\hat\gamma_{2p}\}$ of $C$, that is if $\nu_k=\frac{\hat\gamma_{2k-1}+\hat\gamma_{2k}}{2}$, the triple of ordered spectra $$\alpha=(\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}),\; \beta=(\sigma_2,\sigma_4,\cdots,\sigma_{2p}),\; \gamma=(\nu_1,\nu_2,\cdots,\nu_p)$$ satisfies the Horn inequality $(^*I,J,K)$.
This implies the following extremal property of the subset of “hermitian" spectra:
The orthogonal projection on the diagonal $\Delta$ of the $(2p-1)$-dimensional Horn polytope ${\cal P}\subset {\ensuremath{\mathbb{R}}}^{2p}$ associated with the spectra $$\sigma(A)=\sigma(B)=\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}$$ coincides with the $(p-1)$-dimensional Horn polytope ${\cal P}_1\in{\ensuremath{\mathbb{R}}}^p$ associated with the spectra $$\sigma(a)=(\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}),\; \sigma(b)=(\sigma_2,\sigma_4,\cdots,\sigma_{2p}).$$
In particular, the intersection ${\cal P}_2={\cal P}\cap\Delta$ corresponding to the hermitian spectra, that is those such that $\hat\gamma_1=\hat\gamma_2,\cdots,\hat\gamma_{2p-1}=\hat\gamma_{2p}$, coincides with ${\cal P}_1$. Indeed, ${\cal P}_2$ coincides with the projection of ${\cal P}$, which itself coincides with ${\cal P}_1$.
[**Remark.**]{} The equality $Im {\cal F}={\cal P}_2$ implies the following
Let $C:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be the sum $C=A+B$ of two symmetric endomorphisms $A$ and $B$ with the same spectrum $\sigma(A)=\sigma(B)=\sigma(S_0)$. Then $C$ is $J$-hermitian for some hermitian structure $J$ on ${\ensuremath{\mathbb{R}}}^{2p}$ if and only if it is conjuguate by an element of $SO(2p)$ to a matrix of the form $S_0+\tilde J^{-1}S_0\tilde J$, where $\tilde J$ is a hermitian structure on $R^{2p}$.
[**Note.**]{} The proof of the above results has been written by the first author after he was convinced by the numerical experiments made by the second author that the equality ${\cal P}_1={\cal P}_2$ was plausible when $p=3$ and more precisely that not only the intersection ${\cal P}_2={\cal P}\cap\Delta$ but also the orthogonal projection of the Horn polytope ${\cal P}$ on $\Delta$ was contained in ${\cal P}_1$ after the canonical identification of $\Delta$ with ${\ensuremath{\mathbb{R}}}^p$. This led first to a proof when $p=2$ or $3$, obtained by coping directly with Horn’s inequalities and then to the discovery that the general case followed from a lemma which turned out to be exactly the lemma 1.18 of [@FFLP]. The numerical experiments are described in the next section.
Numerical experiments
=====================
The numerical checking of the conjecture that $Im {\cal F}={\cal P}_1$, was made on the matrix $S_0= \frac{1}{32}{\rm diag}\left\{ 13,8,5,3,2,1 \right\}$, whose spectrum satisfies the inequalities in [@C1] (section 8). We wrote a program in TRIP [@GL11] producing different rotation matrices $R\in SO(2p)$ in a random way. Starting from the canonical basis $\xi=\{\xi_1,\dots,\xi_m\},$ $m=p(2p-1)$, of $\mathfrak{so}(2p)$, we created a list containing the $m$ one-parameter subgroups $G_i(t)= e^{t\xi_i}\subset SO(2p)$. We created a second list of $m$ random values $[t_i]_{i=1}^m$ and a random permutation $[1,2,\dots,m]\to [i_1,i_2,\dots ,i_m]$. The random rotation matrix was defined as $$\begin{aligned}
R = \prod_{j=1}^m G_{i_j}(t_j),\quad 0\leq t_i \leq 2\pi.
\label{eqn:R}\end{aligned}$$ The program which plots $\mathcal P_1$ is very simple (the fact that we replaced the conjugation of $J_0$ by the conjugation of $S_0$ is immaterial and comes from the formulation of the conjecture in [@C1]):\
--------------------------------------------
`create` $S_0$ `and` $J_0$
`for` $i=1$ `to` $N_{max}$ `do`
$\quad$ `create` $R$ `and` $R^{-1}$;
$\quad$ $S = RS_0R^{-1}$;
$\quad$ $C = S - J_0 S J_0$;
$\quad$ `lst = eigenvalues(`$C$`)`;
$\quad$ `plot ( lst[5], lst[3], lst[1] )`;
`end for`.
--------------------------------------------
We have assigned the value $N_{max}=25000$ obtaining the results shown in Figure \[fig:AA\]. The figure shows also the simplex $\gamma_1+\gamma_2+\gamma_3=1$ and its intersection with $W_3^+$.
![${\cal P}_1=Im {\cal A}$: 25000 random adapted hermitian structures[]{data-label="fig:AA"}](fig1)
The modified algorithm to estimate the shape of $\mathcal P_2=\mathcal
P\cap\Delta$ in $W^+_3$ is similar. For a random $R$ in $SO(6)$, the ordered spectrum $spec(C)=\left( \gamma_1,\dots,\gamma_6 \right)$ of $C=S_0+R^{-1}S_0R$ is projected orthogonally onto the diagonal $\Delta$ by the map $\pi_\Delta:\mathbb R^6 \to \Delta$: $$\begin{aligned}
(\gamma_1,\gamma_2,\dots,\gamma_6)&\mapsto& \left( \frac{\gamma_1+\gamma_2}{2},
\frac{\gamma_3+\gamma_4}{2},
\frac{\gamma_5+\gamma_6}{2}
\right).\end{aligned}$$ At first, when $spec(C)$ was $\varepsilon$-close to $\Delta$ *i.e.*, if $\sum_{k=1}^3 |\gamma_{2k-1}-\gamma_{2k}|^2 < 2\varepsilon^2$ for $\varepsilon$ small, the projected point was plotted in green; otherwise it was plotted in red. No particular pattern was found for the green points meanwhile the red ones were all contained in $\mathcal P_1$. The algorithm to plot $\pi_\Delta(\mathcal P)$ is
------------------------------------------
`create` $S_0$
`for` $i=1$ `to` $N_{max}$ `do`
$\quad$ `create` $R$ `and` $R^{-1}$;
$\quad$ $C = S_0 + R^{-1} S_0 R$;
$\quad$ `lst = eigenvalues(`$C$`)`;
$\quad$ `sort( lst )`;
$\quad$ `plot` $\left(
\frac{lst[6]+lst[5]}{2},
\frac{lst[4]+lst[3]}{2},
\frac{lst[2]+lst[1]}{2}\right)$`;`
`end for`.
------------------------------------------
The results of the projection $\pi_\Delta(spec(C))$ for $50000$ random rotation matrices are shown in Figure \[fig:BB\].
![Projection of ${\cal P}$: 50000 random rotations[]{data-label="fig:BB"}](fig2)
The matrix $S_0$ and hence the polytope $\mathcal P_1$ are the same as in Figure \[fig:AA\] (the interior lines correspond to the polytopes associated to different partitions of the spectrum of $S_0$, as depicted in the corresponding figure in [@C1]). Recall that the polytope ${\cal P}$ has dimension 5; this explains that in order to get a better filling one should have taken many more points. This was not done because the evidence was sufficiently convincing to ask for a proof.
Acknowledgements
================
Warm thanks to Sun Shanzhong and Zhao Lei for their interest in this work and numerous discussions.
[99]{}
A. Albouy, A. Chenciner [*Le Problème des $N$ corps et les distances mutuelles*]{}, Inventiones mathematicae 131 (1998), 151-184.
A. Chenciner [*The angular momentum of a relative equilibrium*]{}, arXiv:1102.0025v1, final version to appear in D.C.D.S.
A. Chenciner [*The Lagrange reduction of the N-body problem: a survey*]{}, to appear in Acta Mathematica Vietnamica
C. Carré, B. Leclerc [*Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts*]{}, Journal of Algebraic Combinatorics 4 (1995), 201-231
W. Fulton [*Eigenvalues, invariant factors, highest weights, and Schubert calculus*]{}, Bull. Amer. Math. Soc. (N.S.) [**37**]{} no. 3, 209-249 (2000)
S. Fomin, W. Fulton, C.K. Li, Y.T. Poon, [*Eigenvalues, singular values, and Littlewood-Richardson coefficients*]{}, Amer. J. Math. [**127**]{}, no. 1, 101–127 (2005)
M. Gastineau and J. Laskar, 2011. [TRIP]{} 1.1.18, *TRIP Reference manual*. IMCCE, Paris Observatory. [http://www.imcce.fr/trip/]{}.
[^1]: Notice that this is the same as the spectrum of the $J_0$-hermitian matrix $\Sigma=J_0^{-1}SJ_0+S$, where $S=RS_0R^{-1}$, which was considered in [@C1].
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Jan-Peter Calliess$^{1}$ [^1] [^2]'
title: '**Lipschitz Optimisation for Lipschitz Interpolation$^*$** '
---
[^1]: \*This paper is an extended version of a conference paper that will appear in the Proceedings of the American Control Conference (ACC 2017).
[^2]: $^{1}$Jan-Peter Calliess is with the Engineering Department, University of Cambridge, UK. [jpc73@cam.ac.uk]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and $\mathbb{Z}_{h}$ be the residue class ring modulo $h$. Let $\mathbb{Z}_{h}^{n}$ be the $n$-dimensional row vector space over $\mathbb{Z}_{h}$. A generalized Grassmann graph for $\mathbb{Z}_{h}^n$, denoted by $G_r(m,n,\mathbb{Z}_{h})$ ($G_r$ for short), has all $m$-subspaces of $\mathbb{Z}_{h}^n$ as its vertices, and two distinct vertices are adjacent if their intersection is of dimension $>m-r$, where $2\leq r\leq m+1\leq n$. In this paper, we determine the clique number and geometric structures of maximum cliques of $G_r$. As a result, we obtain the Erdős-Ko-Rado theorem for $\mathbb{Z}_{h}^{n}$ and some bounds of the independence number of $G_r$.
[*AMS classification*]{}: 05C50, 05D05
[*Key words*]{}: Erdős-Ko-Rado theorem, Residue class ring, Grassmann graph, Clique number, Independence number
author:
- |
Jun Guo[^1]\
[Department of Mathematics, Langfang Normal University, Langfang 065000, China]{}
date:
title: '**Erdős-Ko-Rado theorem and generalized Grassmann graphs for vector spaces over residue class rings**'
---
Introduction
============
Let $\mathbb{Z}$ denote the integer ring. For $a,b,h\in \mathbb{Z}$, integers $a$ and $b$ are said to be [*congruent modulo*]{} $h$ if $h$ divides $a-b$, and denoted by $a\equiv b \mod h$. Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes. Let $\mathbb{Z}_{h}$ denote the residue class ring modulo $h$ and $\mathbb{Z}_{h}^\ast$ denote its unit group. Then $\mathbb{Z}_{h}$ is a principal ideal ring and $|\mathbb{Z}_{h}^\ast|=h\prod_{i=1}^t(1-p_i^{-1}).$ By [@Ireland], $\mathbb{Z}_{h}\cong\mathbb{Z}_{p_1^{s_1}}\oplus\mathbb{Z}_{p_2^{s_2}}\oplus\cdots\oplus\mathbb{Z}_{p_t^{s_t}}$ and $\mathbb{Z}_{h}^\ast\cong\mathbb{Z}_{p_1^{s_1}}^\ast\times\mathbb{Z}_{p_2^{s_2}}^\ast\times\cdots\times\mathbb{Z}_{p_t^{s_t}}^\ast$. Note that $(p_i),i=1,2,\ldots,t$, are all the maximal ideals of $\mathbb{Z}_{h}$. Write $J_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}=(\prod_{i=1}^{t}p_i^{\alpha_i})$, where $0\leq \alpha_i\leq s_i$ for $i=1,2,\ldots,t$. For brevity, we write $J_{(\alpha_1)}$ as $J_{\alpha_1}$ if $t=1$. For $a\in\mathbb{Z}$, we also denote by $a$ the congruence class of $a$ modulo $h$.
For a subset $S$ of $\mathbb{Z}_{h}$, let $S^{m\times n}$ be the set of all $m\times n$ matrices over $S$, and $S^{n}=S^{1\times n}$. A matrix in $S^n$ is also called an $n$-dimensional row vector over $S$. Let $I_r$ ($I$ for short) be the $r\times r$ identity matrix, and $0_{m,n}$ ($0$ for short) the $m\times n$ zero matrix. Let $\hbox{diag}(A_1,A_2,\ldots, A_k)$ denote the block diagonal matrix whose blocks along the main diagonal are matrices $A_1,A_2,\ldots, A_k$. The set of $n\times n$ invertible matrices forms a group under matrix multiplication, called the [*general linear group*]{} of degree $n$ over $\mathbb{Z}_{h}$ and denoted by $G\!L_n(\mathbb{Z}_{h})$. Let ${}^t\!A$ denote the transpose matrix of a matrix $A$ and $\det(X)$ the determinant of a square matrix $X$ over $\mathbb{Z}_{h}$. For $X\in \mathbb{Z}_{h}^{n\times n}$, by Corollary 2.21 in [@Brown], $X\in G\!L_n(\mathbb{Z}_{h})$ if and only if $\det(X)\in\mathbb{Z}_h^\ast$.
For $A\in\mathbb{Z}_{h}^{m\times n}$ and $B\in\mathbb{Z}_{h}^{n\times m}$, if $AB=I_m$, we say that $A$ has a [*right inverse*]{} and $B$ is a right inverse of $A$. Similarly, if $AB=I_m$, then $B$ has a [*left inverse*]{} and $A$ is a left inverse of $B$. $\mathbb{Z}_{h}^{n}$ is called the $n$-dimensional row vector space over $\mathbb{Z}_{h}$. Let $\alpha_i\in\mathbb{Z}_{h}^{n}$ for $i=1,2,\ldots,m$. The vector subset $\{\alpha_1,\alpha_2,\ldots,\alpha_m\}$ is called [*unimodular*]{} if the matrix ${}^t({}^t\alpha_1, {}^t\alpha_2, \ldots, {}^t\alpha_m)$ has a right inverse. By Lemma \[lem2.9\] below, a matrix $A\in\mathbb{Z}_{h}^{m\times n}$ has a right inverse if and only if all row vectors of $A$ are linearly independent in $\mathbb{Z}_{h}^{n}$.
Let $V\subseteq\mathbb{Z}_{h}^{n}$ be a [*linear subset*]{} (i.e., a $\mathbb{Z}_{h}$-module). A [*largest unimodular vector subset*]{} of $V$ is a unimodular vector subset of $V$ which has maximum number of vectors. The dimension of $V$, denoted by $\dim(V)$, is the number of vectors in a largest unimodular vector subset of $V$. Clearly, $\dim(V)=0$ if and only if $V$ does not contain a unimodular vector. If a linear subset $X$ of $\mathbb{Z}_{h}^{n}$ has a unimodular basis with $m$ vectors, then $X$ is called an $m$-[*dimensional vector subspace*]{} ($m$-[*subspace*]{} for short) of $\mathbb{Z}_{h}^{n}$. Every $m$-subspace of $\mathbb{Z}_{h}^{n}$ is isomorphic to $\mathbb{Z}_{h}^{m}$. Applying Lemma \[lem2.9\] below, it is easy to prove that every basis of a subspace of $\mathbb{Z}_{h}^{n}$ can be extended to a basis of $\mathbb{Z}_{h}^{n}$. We define the $0$-subspace to be $\{0\}$.
The Erdős-Ko-Rado theorem [@Erdos; @Wilson] is a classical result in extremal set theory which obtained an upper bound on the cardinality of a family of $m$-subsets of a set that every pairwise intersection has cardinality at least $r$ and describes exactly which families meet this bound. The results on Erdős-Ko-Rado theorem have inspired much research [@Frankl; @Godsi2; @Huang-T; @Tanaka; @Vanhove]. Let $0\leq r\leq m\leq n$ and ${\mathbb{Z}_{h}^{n}\brack m}$ be the set of all $m$-subspaces of $\mathbb{Z}_{h}^{n}$. A family ${\cal F}\subseteq{\mathbb{Z}_{h}^{n}\brack m}$ is called $r$-[*intersecting*]{} if $\dim(A\cap B)\geq r$ for all $A,B\in{\cal F}$. When $t=1$, Huang et al. [@Huang3] obtained an upper bound on the cardinality of an $r$-intersecting family in ${\mathbb{Z}_{h}^{n}\brack m}$ and described exactly which families meet this bound.
Let $0\leq2r\leq2m=n$ and $I\subseteq[t]:=\{1,2,\ldots,t\}$. Suppose that $\alpha_i=s_i,\pi_i(x)=1$ if $i\in I$, and $\alpha_i=0,\pi_i(x)=0$ if $i\in[t]\setminus I$. Define $$\label{equanew1}
{\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(r,m,n,I)}
=\left\{\left(\begin{array}{cc}
X & 0\\
Y & xI_{r}
\end{array}\right)\in{\mathbb{Z}_h^n\brack m} :
X\in\mathbb{Z}_h^{(m-r)\times(n-r)},Y\in J_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{r\times(n-r)} \right\}.$$ In this paper, we give the Erdőso-Rado theorem for $\mathbb{Z}_{h}^{n}$ as follows.
\[thm1.1\] Let $\lfloor n/2\rfloor\geq m\geq r\geq 0$ and ${\cal F}\subseteq{\mathbb{Z}_{h}^{n}\brack m}$ be an $r$-intersecting family. Then $$|{\cal F}|\leq \prod_{i=1}^tp_i^{(s_i-1)(n-m)(m-r)}{n-r\brack m-r}_{p_i},\;\hbox{where}\;
{n-r\brack m-r}_{p_i}=\prod_{j=0}^{m-r-1}\frac{(p_i^{n-r}-p_i^j)}{(p_i^{m-r}-p_i^j)},$$ and equality holds if and only if either [(a)]{} ${\cal F}$ consists of all $m$-subspaces of $\mathbb{Z}_{h}^{n}$ which contain a fixed $r$-subspace of $\mathbb{Z}_{h}^{n}$, [(b)]{} $n=2m$ and ${\cal F}$ is the set of all $m$-subspaces of $\mathbb{Z}_{h}^{n}$ contained in a fixed $(n-r)$-subspace of $\mathbb{Z}_{h}^{n}$, or [(c)]{} $n=2m$ and there exists some $T\in G\!L_n(\mathbb{Z}_h)$ such that ${\cal F}={\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(r,m,n,I)}T$ with $I\not=\emptyset,[t]$, where ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(r,m,n,I)}$ is given by (\[equanew1\]).
To prove Theorem \[thm1.1\], we need to discuss generalized Grassmann graphs over $\mathbb{Z}_{h}$. The Grassmann graph over a finite field plays an important role in geometry and combinatorics, see [@Brouwer; @Pankov; @Wan2]. As a natural extension, the [*generalized Grassmann graph*]{} over $\mathbb{Z}_{h}$, denoted by $G_r(m,n,\mathbb{Z}_{h})$, has the vertex set ${\mathbb{Z}_{h}^n\brack m}$, and two distinct vertices are adjacent if their intersection is of dimension $>m-r$, where $2\leq r\leq m+1\leq n$. Note that $G_2(m,n,\mathbb{Z}_{h})$ is the Grassmann graph $G(m,n,\mathbb{Z}_{h})$.
Let $V(\Gamma)$ denote the vertex set of a graph $\Gamma$. For $A,B\in V(\Gamma)$, we write $A\sim B$ if vertices $A$ and $B$ are adjacent. A [*clique*]{} of a graph $\Gamma$ is a complete subgraph of $\Gamma$. A clique ${\cal C}$ is [*maximal*]{} if there is no clique of $\Gamma$ which properly contains ${\cal C}$ as a subset. A [*maximum clique*]{} of $\Gamma$ is a clique of $\Gamma$ which has maximum cardinality. The [*clique number*]{} $\omega(\Gamma)$ of $\Gamma$ is the number of vertices in a maximum clique. An [*independent set*]{} of a graph $\Gamma$ is a subset of vertices such that no two vertices are adjacent. A [*largest independent set*]{} of $\Gamma$ is an independent set of maximum cardinality. The [*independence number*]{} $\alpha(\Gamma)$ is the number of vertices in a largest independent set of $\Gamma$.
The paper is organized as follows. In Section 2, we introduce the basic properties of matrices over $\mathbb{Z}_{h}$ for later reference. In Section 3, we obtain the dimension formula and the Anzahl theorem for subspaces of $\mathbb{Z}_{h}^n$. In Section 4, we calculate the clique number and determine geometric structures of maximum cliques of $G_r(m,n,\mathbb{Z}_{h})$. As a result, Theorem \[thm1.1\] is proved. In Section 5, we obtain some bounds of the independence number of $G_r(m,n,\mathbb{Z}_{h})$ by using method in [@Huang3].
Matrices over $\mathbb{Z}_h$
============================
In this section, we introduce some basic properties of matrices over $\mathbb{Z}_h$.
\[lem2.1\][(See Lemma 2.3 in [@Guo3].)]{} Every non-zero element $x$ in $\mathbb{Z}_{h}$ can be written as $x=u\prod_{i=1}^tp_i^{\alpha_i}$, where $u$ is a unit, and $0\leq\alpha_i\leq s_i$ for $i=1,2,\ldots,t$. Moreover, the vector $(\alpha_1,\alpha_2,\ldots,\alpha_t)$ is unique and $u$ is unique modulo the ideal $J_{(s_1-\alpha_1,s_2-\alpha_2,\ldots,s_t-\alpha_t)}$.
\[lem2.2\][(See Theorem 2.10 in [@Guo3].)]{} Let $m\leq n$ and $A\in\mathbb{Z}_{h}^{m\times n}$. Then there are $S\in G\!L_m(\mathbb{Z}_h)$ and $T\in G\!L_n(\mathbb{Z}_h)$ such that $$\label{equa1}
A=S{\rm diag}\,\left(\prod_{i=1}^tp_i^{\alpha_{i1}},\prod_{i=1}^tp_i^{\alpha_{i2}},\ldots,\prod_{i=1}^tp_i^{\alpha_{im}}\right)T,$$ where $0\leq\alpha_{i1}\leq\alpha_{i2}\leq\cdots\leq\alpha_{im}\leq s_i$ for $i=1,2,\ldots,t$. Moreover, the array $$\Omega=((\alpha_{11},\alpha_{12},\ldots,\alpha_{1m}),(\alpha_{21},\alpha_{22},\ldots,\alpha_{2m}),\ldots,(\alpha_{t1},\alpha_{t2},\ldots,\alpha_{tm}))$$ is uniquely determined by $A$.
Let $A\in \mathbb{Z}_{h}^{m\times n}$ be a non-zero matrix. By Cohn’s definition [@Cohn2], the [*inner rank*]{} of $A$, denoted by $\rho(A)$, is the least integer $r$ such that $$\label{equa2}
A = BC\quad \hbox{where}\; B \in \mathbb{Z}_{h}^{m\times r} \;\hbox{and}\; C \in \mathbb{Z}_{h}^{r\times n}.$$ Let $\rho(0) =0$. Any factorization as (\[equa2\]) with $r=\rho(A)$ is called a [*minimal factorization*]{} of $A$. For $A\in \mathbb{Z}_{h}^{m\times n}$, it is obvious that $\rho(A)\leq\min\{m,n\}$ and $\rho(A)=0$ if and only if $A=0$.
For matrices over $\mathbb{Z}_{h}$, from [@Cohn; @Cohn2], we deduce that the following hold: $$\label{equa3}
\rho(A)=\rho(SAT),\;\hbox{where}\;S\;\hbox {and}\;T\;\hbox{are invertible matrices}.$$ $$\label{equa4}
\rho(AB) \leq\min\{\rho(A),\rho(B)\}.$$ $$\label{equa5}
\rho\left(\begin{array}{cc}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{array}\right) \geq\max\{\rho(A_{ij}) :1\leq i,j\leq 2\}.$$
\[lem2.3\][(See Lemma 2.11 in [@Guo3].)]{} Let $A\in\mathbb{Z}_{h}^{m\times n}$ be as in (\[equa1\]). Then the inner rank of $A$ is $\max\{c: (\alpha_{1c},\alpha_{2c},\ldots,\alpha_{tc})\not=(s_1,s_2,\ldots,s_t)\}$.
For $i=1,2,\ldots,t$, let $\pi_i$ (resp. $\theta_i$) be the natural surjective homomorphism from $\mathbb{Z}_{h}$ to $\mathbb{Z}_{p_i^{s_i}}$ (resp. $\mathbb{Z}_{h/p_i^{s_i}}$), and $\pi_i(A)=(\pi_i(a_{uv}))$ (resp. $\theta_i(A)=(\theta_i(a_{uv}))$) for every $A=(a_{uv})\in\mathbb{Z}_{h}^{m\times n}$. For brevity, we write $\prod_{i=1}^{t}\mathbb{Z}_{p_i^{s_i}}^{m\times n}=\mathbb{Z}_{p_1^{s_1}}^{m\times n}\times\mathbb{Z}_{p_2^{s_2}}^{m\times n}\times\cdots\times\mathbb{Z}_{p_t^{s_t}}^{m\times n}$.
\[lem2.3.2\] For every $A\in\mathbb{Z}_{h}^{m\times n}$, let $\pi(A)=(\pi_1(A),\pi_2(A),\ldots,\pi_t(A))$ and $(\pi_i,\theta_i)(A)=(\pi_i(A),\theta_i(A))$ for $i=1,2,\ldots,t$. Then $\pi$ is a bijective map from $\mathbb{Z}_{h}^{m\times n}$ to $\prod_{i=1}^{t}\mathbb{Z}_{p_i^{s_i}}^{m\times n}$, and $(\pi_i,\theta_i)$ is a bijective map from $\mathbb{Z}_{h}^{m\times n}$ to $\mathbb{Z}_{p_i^{s_i}}^{m\times n}\times \mathbb{Z}_{h/p_i^{s_i}}^{m\times n}$ for $i=1,2,\ldots,t$.
[[*Proof.*]{}]{}For each $i$ with $1\leq i\leq t$ and $a,b\in\mathbb{Z}_h$, if $a=b$, then $(\pi_i,\theta_i)(a)=(\pi_i,\theta_i)(b)$. Suppose $(\pi_i,\theta_i)(a)=(\pi_i,\theta_i)(b)$. Then $\pi_i(a)=\pi_i(b)$ and $\theta_i(a)=\theta_i(b)$. Since $$a\equiv\pi_i(a)\mod p_i^{s_i}, b\equiv\pi_i(b)\mod p_i^{s_i},
a\equiv\theta_i(a) \mod h/p_i^{s_i},b\equiv\theta_i(b) \mod h/p_i^{s_i},$$ we obtain $p_i^{s_i}|(a-b)$ and $(h/p_i^{s_i})|(a-b)$, which imply that $a=b$. Therefore, $(\pi_i,\theta_i)$ is an injective map from $\mathbb{Z}_{h}$ to $\mathbb{Z}_{p_i^{s_i}}\times \mathbb{Z}_{h/p_i^{s_i}}$. By $|\mathbb{Z}_{h}|=|\mathbb{Z}_{p_i^{s_i}}\times \mathbb{Z}_{h/p_i^{s_i}}|$, $(\pi_i,\theta_i)$ is a bijective map from $\mathbb{Z}_{h}^{m\times n}$ to $\mathbb{Z}_{p_i^{s_i}}^{m\times n}\times \mathbb{Z}_{h/p_i^{s_i}}^{m\times n}$. Since $\mathbb{Z}_{h}\cong\mathbb{Z}_{p_1^{s_1}}\oplus\mathbb{Z}_{p_2^{s_2}}\oplus\cdots\oplus\mathbb{Z}_{p_t^{s_t}}$, $\pi$ is a bijective map from $\mathbb{Z}_{h}^{m\times n}$ to $\prod_{i=1}^{t}\mathbb{Z}_{p_i^{s_i}}^{m\times n}$. ${\hfill\Box\medskip}$
For matrices $A, B\in\mathbb{Z}_{h}^{m\times n}$ and $C\in\mathbb{Z}_{h}^{n\times k}$, it is easy to see that $$\label{equa6}
\pi_i(A+B)=\pi_i(A)+\pi_i(B)\quad\hbox{and}\quad\pi_i(AC)=\pi_i(A)\pi_i(C)\;\hbox{for}\;i=1,2,\ldots,t,$$ and $$\label{equa7}
\theta_i(A+B)=\theta_i(A)+\pi_i(B)\quad\hbox{and}\quad\theta_i(AC)=\theta_i(A)\pi_i(C)\;\hbox{for}\;i=1,2,\ldots,t.$$
\[lem2.4\][(See Lemma 2.12 in [@Guo3].)]{} Let $A\in\mathbb{Z}_{h}^{m\times n}$. Then $$\rho(A)=\max\{\rho(\pi_i(A)): i=1,2,\ldots,t\}=\max\{\rho(\theta_i(A)): i=1,2,\ldots,t\}.$$
Let $A\in\mathbb{Z}_{h}^{m\times n}$. Denote by $I_k(A)$ the ideal in $\mathbb{Z}_{h}$ generated by all $k\times k$ minors of $A$. Let $\hbox{Ann}_{\mathbb{Z}_{h}}(I_k(A)) =\{x\in \mathbb{Z}_{h}:xI_k(A)=0\}$ denote the annihilator of $I_k(A)$. The [*McCoy rank*]{} of $A$, denoted by $\hbox{rk}(A)$, is the following integer: $$\hbox{rk}(A) = \max\{k : \hbox{Ann}_{\mathbb{Z}_{h}}(I_k(A)) =(0)\} .$$ Note that $\hbox{rk}(A) =\hbox{rk}({}^t\!A)$; $\hbox{rk}(A) =\hbox{rk}(SAT)$ where $S$ and $T$ are invertible matrices; and $\hbox{rk}(A) =0$ if and only if $\hbox{Ann}_{\mathbb{Z}_{h}}(I_1(A)) \not=(0)$, see [@Brown].
\[lem2.5\] Let $A\in \mathbb{Z}_{h}^{m\times n}$ be as in (\[equa1\]). Then the McCoy rank of $A$ is $$\max\{c: (\alpha_{1c},\alpha_{2c},\ldots,\alpha_{tc})=0\}.$$
[[*Proof.*]{}]{}Let $\ell=\max\{c\in[m]:(\alpha_{1c},\alpha_{2c},\ldots,\alpha_{tc})=0\}$. Then $I_\ell(A)=\mathbb{Z}_{h}$ and $I_{\ell+1}(A)=(\prod_{i=1}^tp_i^{\alpha_{i,\ell+1}})$, which imply that $\hbox{Ann}_{\mathbb{Z}_{h}}(I_\ell(A))=(0)$ and $\hbox{Ann}_{\mathbb{Z}_{h}}(I_{\ell+1}(A))\not=(0)$. Similarly, $\hbox{Ann}_{\mathbb{Z}_{h}}(I_{k}(A))\not=(0)$ for $\ell+1 \leq k \leq \min\{m, n\}$. It follows that ${\rm rk}(A)=\ell$.${\hfill\Box\medskip}$
For matrices $A, B$ over $\mathbb{Z}_{h}$, by Lemmas \[lem2.2\], \[lem2.3\] and \[lem2.5\], if $\rho(A)={\rm rk}(A)$, then $$\label{equa8}
\rho\left(\begin{array}{cc}
A & 0\\ 0& B
\end{array}\right)=\rho(A)+\rho(B)\quad\hbox{and}\quad
\hbox{rk}\left(\begin{array}{cc}
A &0 \\ 0 & B
\end{array}\right)=\hbox{rk}(A)+\hbox{rk}(B).$$
\[lem2.6\] Let $A\in\mathbb{Z}_{h}^{m\times n}$. Then $${\rm rk}(A)=\min\{{\rm rk}(\pi_i(A)): i=1,2,\ldots,t\}=\min\{{\rm rk}(\theta_i(A)): i=1,2,\ldots,t\}.$$
[[*Proof.*]{}]{}By Lemma \[lem2.2\], we may assume that $A$ is as in (\[equa1\]). Let ${\rm rk}(A)=\ell$. For each $i$ with $1\leq i\leq t$, write $w_{ic}=\pi_i(\prod_{j=1}^tp_j^{\alpha_{jc}})$ for $c=\ell+1,\ldots,m$. By (\[equa6\]), $\pi_{i}(S)\in G\!L_m(\mathbb{Z}_{p_i^{s_i}}),\pi_{i}(T)\in G\!L_n(\mathbb{Z}_{p_i^{s_i}})$, and $$\pi_i(A)=\pi_{i}(S)\hbox{diag}(I_{\ell},w_{i,\ell+1},\dots,w_{im})\pi_{i}(T),$$ which imply that $${\rm rk}(\pi_i(A))={\rm rk}(\hbox{diag}(I_{\ell},w_{i,\ell+1},\dots,w_{im})\geq \ell={\rm rk}(A).$$ Therefore, ${\rm rk}(A)\leq\min\{{\rm rk}(\pi_i(A)): i=1,2,\ldots,t\}$. Since $(\alpha_{1,\ell+1},\alpha_{2,\ell+1},\ldots,\alpha_{t,\ell+1})\not=0$, there exists some $j$ such that $\alpha_{j,\ell+1}\not=0$, which implies that $w_{j,\ell+1}\not\in\mathbb{Z}_{p_j^{s_i}}^\ast$. It follows that ${\rm rk}(\pi_j(A))=\ell={\rm rk}(A)$.
For each $i$ with $1\leq i\leq t$, write $x_{ic}=\theta_i(\prod_{j=1}^tp_j^{\alpha_{jc}})$ for $c=\ell+1,\ldots,m$. By (\[equa7\]), $\theta_{i}(S)\in G\!L_m(\mathbb{Z}_{h/p_i^{s_i}}),\theta_{i}(T)\in G\!L_n(\mathbb{Z}_{h/p_i^{s_i}})$, and $$\theta_i(A)=\theta_{i}(S)\hbox{diag}(I_{\ell},x_{i,\ell+1},\dots,x_{im})\theta_{i}(T),$$ which imply that $${\rm rk}(\theta_i(A))={\rm rk}(\hbox{diag}(I_{\ell},x_{i,\ell+1},\dots,x_{im}))\geq \ell={\rm rk}(A).$$ Therefore, ${\rm rk}(A)\leq\min\{{\rm rk}(\theta_i(A)): i=1,2,\ldots,t\}$. Since $(\alpha_{1,\ell+1},\alpha_{2,\ell+1},\ldots,\alpha_{t,\ell+1})\not=0$, there exists some $j$ such that $\alpha_{j,\ell+1}\not=0$, which implies that $x_{i,\ell+1}\not\in\mathbb{Z}_{h/p_j^{s_j}}^\ast$ for $i\not=j$. It follows that ${\rm rk}(\theta_i(A))=\ell={\rm rk}(A)$ for $i\not=j$. Hence, the desired result follows. ${\hfill\Box\medskip}$
\[lem2.7\][(See [@Rosen].)]{} Let $\pi_i$ (resp. $\pi$) also denote the restriction of $\pi_i$ (resp. $\pi$) on $G\!L_n(\mathbb{Z}_{h})$ for $i=1,2,\ldots,t$. Then the following hold:
- $\pi_i$ is a natural surjective homomorphism from $G\!L_{n}(\mathbb{Z}_{h})$ to $G\!L_{n}(\mathbb{Z}_{p_i^{s_i}})$ for $i=1,2,\ldots,t$.
- $\pi$ is an isomorphism from $G\!L_{n}(\mathbb{Z}_{h})$ to $\prod_{i=1}^tG\!L_{n}(\mathbb{Z}_{p_i^{s_i}})$.
\[lem2.8\] Let $X\in\mathbb{Z}_{h}^{m\times n}$. Then the following hold:
- When $m\leq n$, $X$ has a right inverse if and only if ${\rm rk}(X)=m$.
- When $n=m$, $X$ is invertible if and only if $\pi_i(X)$ is invertible for $i=1,2,\ldots,t$. Moreover, if $X$ is invertible, then $\pi_i(X^{-1})=(\pi_i(X))^{-1}$ for $i=1,2,\ldots,t$.
[[*Proof.*]{}]{}(i). By Lemma 2.4 in [@Huang3], $\pi_i(X)$ has a right inverse if and only if ${\rm rk}(\pi_i(X))=m$ for $i=1,2,\ldots,t$. If $X$ has a right inverse, then (\[equa6\]) implies that $\pi_i(X)$ has a right inverse for $i=1,2,\ldots,t$. It follows that ${\rm rk}(\pi_i(X))=m$ for $i=1,2,\ldots,t$. By Lemma \[lem2.6\], ${\rm rk}(X)=m$.
Conversely, suppose ${\rm rk}(X)=m$. By Lemma \[lem2.2\], there are $S\in G\!L_m(\mathbb{Z}_h)$ and $T\in G\!L_n(\mathbb{Z}_h)$ such that $X=S(I_m,0_{m,n-m})T$. Pick $Y=T^{-1}\left(\begin{array}{c}
I_{m}\\
0_{n-m,m}
\end{array}\right)S^{-1}$. Then $XY=I_m$, which implies that $X$ has a right inverse.
(ii). By Lemma \[lem2.7\], $X$ is invertible if and only if $\pi_i(X)$ is invertible for $i=1,2,\ldots,t$. Suppose that $X\in\mathbb{Z}_{h}^{n\times n}$ is invertible. Then $XX^{-1}=I_n$, it follows from (\[equa6\]) that $\pi_i(XX^{-1})=\pi_i(X)\pi_i(X^{-1})=I_n$ for $i=1,2,\ldots,t$. Therefore, we have $\pi_i(X^{-1})=(\pi_i(X))^{-1}$ for $i=1,2,\ldots,t$. ${\hfill\Box\medskip}$
\[lem2.9\] Let $m\leq n$, and $\alpha_1,\alpha_2,\ldots,\alpha_m\in\mathbb{Z}_{h}^{n}$. Then the following hold:
- The matrix $A={}^t({}^t\alpha_1, {}^t\alpha_2, \ldots, {}^t\alpha_m)$ has a right inverse if and only if $\alpha_1,\alpha_2,\ldots,\alpha_m$ are linearly independent.
- If $\alpha_1,\alpha_2,\ldots,\alpha_m$ are linearly independent, then $\alpha_1,\alpha_2,\ldots,\alpha_m$ can be extended to a basis of $\mathbb{Z}_{h}^{n}$.
[[*Proof.*]{}]{}(i). If $A$ has a right inverse, then there exists some $B\in\mathbb{Z}_h^{n\times m}$ such that $AB=I_m$. From $\sum_{i=1}^mx_i\alpha_i=0$, we deduce that $(x_1,x_2,\ldots,x_m)A=0$, which implies that $(x_1,x_2,\ldots,x_m)=(x_1,x_2,\ldots,x_m)AB=0$, and therefore $\alpha_1,\alpha_2,\ldots,\alpha_m$ are linearly independent.
Conversely, suppose that $\alpha_1,\alpha_2,\ldots,\alpha_m$ are linearly independent. Without loss of generality, by Lemma \[lem2.2\], we may assume that $A$ is as in (\[equa1\]). By Lemma \[lem2.5\], ${\rm rk}(A)=m$ if and only if $(\alpha_{1m},\alpha_{2m},\ldots,\alpha_{tm})=0$, which implies that $A$ has a right inverse if and only if $(\alpha_{1m},\alpha_{2m},\ldots,\alpha_{tm})=0$ by Lemma \[lem2.8\]. Assume that $(\alpha_{1m},\alpha_{2m},\ldots,\alpha_{tm})\not=0$, then $x_m=\prod_{i=1}^tp_i^{s_i-\alpha_{im}}\not=0$ such that $(0,\ldots,0,x_m)S^{-1}\not=0$ and $(0,\ldots,0,x_m)S^{-1}A=0$, which implies that $\alpha_1,\alpha_2,\ldots,\alpha_m$ are not linearly independent, a contradiction.
(ii). Since $\alpha_1,\alpha_2,\ldots,\alpha_m$ are linearly independent, $A$ has a right inverse by (i), which implies that ${\rm rk}(A)=m$ by Lemma \[lem2.8\]. By Lemma \[lem2.2\], there are $S\in G\!L_m(\mathbb{Z}_h)$ and $T\in G\!L_n(\mathbb{Z}_h)$ such that $A=S(I_m,0_{m,n-m})T$. Then $$\widetilde{A}=\left(\begin{array}{c}
A\\ (0_{n-m,m},I_{n-m})T
\end{array}\right)
=\left(\begin{array}{cc}
S & 0\\ 0 & I_{n-m}
\end{array}\right)T$$ is invertible, which implies that all row vectors of $\widetilde{A}$ are a basis of $\mathbb{Z}_{h}^{n}$, and therefore $\alpha_1,\alpha_2,\ldots,\alpha_m$ can be extended to a basis of $\mathbb{Z}_{h}^{n}$. ${\hfill\Box\medskip}$
Subspaces of $\mathbb{Z}_{h}^{n}$
=================================
In this section, we study subspaces of $\mathbb{Z}_{h}^{n}$ and obtain some useful results. Notation and terminology will be adopted from [@Huang3].
Suppose that $\alpha_1,\alpha_2,\ldots,\alpha_m \;(m\leq n)$ are $m$ linearly independent row vectors in $\mathbb{Z}_{h}^{n}$. Let $[\alpha_1,\alpha_2,\ldots,\alpha_m]$ be the $\mathbb{Z}_{h}$-module generated by $\alpha_1,\alpha_2,\ldots,\alpha_m$. Then $X=[\alpha_1,\alpha_2,\ldots,\alpha_m]$ is an $m$-subspace of $\mathbb{Z}_{h}^{n}$, and the matrix ${}^t({}^t\alpha_1, {}^t\alpha_2, \ldots, {}^t\alpha_m)$ is called a [*matrix representation*]{} of $X$. We use an $m\times n$ matrix $A$ with ${\rm rk}(A)=m$ to represent an $m$-subspace of $\mathbb{Z}_{h}^{n}$. For $A,B\in\mathbb{Z}_{h}^{m\times n}$ with ${\rm rk}(A)={\rm rk}(B)=m$, both $A$ and $B$ represent the same $m$-subspace if and only if there exists an $S\in G\!L_m(\mathbb{Z}_h)$ such that $B=SA$. For convenience, if $A$ is a matrix representation of an $m$-subspace $X$, then we write $X=A$.
For an $m$-subspace $X$ of $\mathbb{Z}_{h}^{n}\;(m\leq n)$, by Lemma \[lem2.2\], there exists a $T\in G\!L_n(\mathbb{Z}_h)$ such that $X$ has matrix representations $$X=(0,I_m)T=S(0,I_m)T\; \hbox{for all}\; S\in G\!L_m(\mathbb{Z}_h).$$ By Lemma \[lem2.1\], every $m$-subspace $X$ of $\mathbb{Z}_{h}^{n}\;(m\leq n)$ has the unique matrix representation $(A_1,I_m)T$, which is a [*row-reduced echelon form*]{}, where $T$ is a permutation matrix and $A_1\in\mathbb{Z}_{h}^{m\times(n-m)}$.
Let $X$ and $Y$ be two subspaces of $\mathbb{Z}_{h}^{n}$. A [*join*]{} of $X$ and $Y$ is a minimum dimensional subspace containing $X$ and $Y$ in $\mathbb{Z}_{h}^{n}$. In general, the join of two subspaces in $\mathbb{Z}_{h}^{n}$ is not unique, see [@Huang3] for example. Denoted by $X\vee Y$ the set of all joins of subspaces $X$ and $Y$ with the same minimum dimension $\dim(X\vee Y )$. Then $X\vee Y=Y\vee X$, and $X\vee Y=\{Y\}$ if $X\subseteq Y$. Clearly, $X\cap Y$ contains a subspace of dimension $\dim(X\cap Y)$. In general, $X\cap Y$ is a linear subset but it may not be a subspace, see [@Huang3] for example.
\[lem3.1\] [(See Lemma 3.2 in [@Huang3].)]{} Let $1\leq m\leq k<n$. Suppose that $A$ is a $k$-subspace of $\mathbb{Z}_{p^s}^{n}$ and $B$ is an $m$-subspace of $\mathbb{Z}_{p^s}^{n}$ with $B\not\subseteq A$. Then there is a $T\in G\!L_n(\mathbb{Z}_{p^s})$ such that $$A=(0,I_k)T\quad\hbox{and}\quad B=(D,I_m)T,$$ where $D={\rm diag}\left(p^{\alpha_{1}},p^{\alpha_{2}},\ldots,p^{\alpha_{r}},0_{m-r,n-m-r}\right),1\leq r\leq \min\{m,n-k\}$ and $0\leq\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{r}\leq s-1$.
\[lem3.2\] Let $1\leq m\leq k<n$. Suppose that $A$ is a $k$-subspace of $\mathbb{Z}_{h}^{n}$ and $B$ is an $m$-subspace of $\mathbb{Z}_{h}^{n}$ with $B\not\subseteq A$. Then there is a $T\in G\!L_n(\mathbb{Z}_h)$ such that $$\label{equa10}
A=(0,I_k)T\quad\hbox{and}\quad B=(D,I_m)T,$$ where $D={\rm diag}\left(\prod_{i=1}^tp_i^{\alpha_{i1}},\prod_{i=1}^tp_i^{\alpha_{i2}},\ldots,\prod_{i=1}^tp_i^{\alpha_{ir}},0_{m-r,n-m-r}\right),1\leq r\leq \min\{m,n-k\},\rho(D)=r$, and $0\leq\alpha_{i1}\leq\alpha_{i2}\leq\cdots\leq\alpha_{ir}\leq s_i$ for $i=1,2,\ldots,t$.
[[*Proof.*]{}]{}By Lemmas \[lem2.6\] and \[lem2.8\], $\pi_i(A)$ is a $k$-subspace of $\mathbb{Z}_{p_i^{s_i}}^{n}$ and $\pi_i(B)$ is an $m$-subspace of $\mathbb{Z}_{p_i^{s_i}}^{n}$ for $i=1,2,\ldots,t$. For each $i$ with $1\leq i\leq t$, by Lemma \[lem3.1\], there is a $T_i\in G\!L_n(\mathbb{Z}_{p_i^{s_i}})$ such that $$\pi_i(A)=(0,I_k)T_i\quad\hbox{and}\quad \pi_i(B)=(D_i,I_m)T_i,$$ where $D_i={\rm diag}\,\left(p_i^{\alpha_{i1}},p_i^{\alpha_{i2}},\ldots,p_i^{\alpha_{ir_i}},0_{m-r_i,n-m-r_i}\right),1\leq r_i\leq \min\{m,n-k\}$ and $0\leq\alpha_{i1}\leq\alpha_{i2}\leq\cdots\leq\alpha_{ir_i}\leq s_i-1$.
Let $r=\max\{r_i:i=1,2,\ldots,t\}$ and $\alpha_{ij}=s_i$ for $i=1,2,\ldots,t$ and $j=r_i+1,\ldots,r$. Then $1\leq r\leq \min\{m,n-k\}$ and $0\leq\alpha_{i1}\leq\alpha_{i2}\leq\cdots\leq\alpha_{ir}\leq s_i$ for $i=1,2,\ldots,t$. Since $\mathbb{Z}_{h}\cong\mathbb{Z}_{p_1^{s_1}}\oplus\mathbb{Z}_{p_2^{s_2}}\oplus\cdots\oplus\mathbb{Z}_{p_t^{s_t}}$, there exist $d_c\in\mathbb{Z}_{h}$ such that $\pi_i(d_c)=p_i^{\alpha_{ic}}$ for $i=1,2,\ldots,t$ and $c=1,2,\ldots,r$. Let $\widetilde{D}={\rm diag}(d_1,d_2,\ldots,d_r,0_{m-r,n-m-r})$. By Lemma \[lem2.4\], $\rho(\widetilde{D})=r$ and $\pi_i(\widetilde{D})=D_i$ for $i=1,2,\ldots,t$.
Let $\pi_i(\prod_{j=1}^tp_j^{\alpha_{jc}})=d_{ic}p_i^{\alpha_{ic}}$, where $d_{ic}=\pi_i(\prod_{j\not=i}p_j^{\alpha_{jc}})\in\mathbb{Z}_{p_i^{s_i}}^\ast$, for $i=1,2,\ldots,t$ and $c=1,2,\ldots,r$. Since $\mathbb{Z}_{h}^\ast\cong\mathbb{Z}_{p_1^{s_1}}^\ast\times\mathbb{Z}_{p_2^{s_2}}^\ast\times\cdots\times\mathbb{Z}_{p_t^{s_t}}^\ast$, there exist $u_{c}\in\mathbb{Z}_h^\ast$ such that $\pi_i(u_c)=d_{ic}$ for $i=1,2,\ldots,t$ and $c=1,2,\ldots,r$. Since $\pi_i(u_cd_c)=\pi_i(\prod_{j=1}^tp_j^{\alpha_{jc}})$ for $i=1,2,\ldots,t$ and $c=1,2,\ldots,r$, we have $u_cd_c=\prod_{j=1}^tp_j^{\alpha_{jc}}$ for $c=1,2,\ldots,r$.
By Lemma \[lem2.7\], there exists the unique $\widetilde{T}\in G\!L_n(\mathbb{Z}_h)$ such that $\pi_i(\widetilde{T})=T_i$ for $i=1,2,\ldots,t$. By (\[equa6\]), $$\pi_i(A)=\pi_i((0,I_k)\widetilde{T})\quad\hbox{and}\quad \pi_i(B)=\pi_i((\widetilde{D},I_m)\widetilde{T})\;\hbox{for}\;i=1,2,\ldots,t,$$ which imply that $\pi(A)=\pi((0,I_k)\widetilde{T})$ and $\pi(B)=\pi((\widetilde{D},I_m)\widetilde{T}).$ Since $\pi$ is a bijective map, $A=(0,I_k)\widetilde{T}$ and $B=(\widetilde{D},I_m)\widetilde{T}.$ Let $D={\rm diag}(u_1d_1,u_2d_2,\ldots,u_rd_r,0_{m-r,n-m-r})$ and $T={\rm diag}\,(u_1^{-1},u_2^{-1},\ldots,u_r^{-1},I_{n-r})\widetilde{T}$. Since $r\leq n-k$, we have $$A=(0,I_k)T\quad\hbox{and}\quad B=(D,I_m)T.$$ Therefore, we complete the proof of this lemma. ${\hfill\Box\medskip}$
\[lem3.3\][(Dimensional formula.)]{} Let $A$ and $B$ be two subspaces of $\mathbb{Z}_{h}^n$. Then $$\label{equa11}
\dim(A\vee B)=\dim(A) + \dim(B) - \dim(A\cap B) = \rho\left(\begin{array}{c}A\\ B \end{array}\right).$$ Moreover, $\dim(A\cap B)=\min\{\dim(\pi_i(A)\cap\pi_i(B)):i=1,2,\ldots,t\}=\min\{\dim(\theta_i(A)\cap\theta_i(B)):i=1,2,\ldots,t\}.$
[[*Proof.*]{}]{}Let $k=\dim(A)$ and $m=\dim(B)$. Without loss of generality, we may assume that $1\leq m\leq k<n$ and $B\not\subseteq A$. By Lemma \[lem3.2\], there is a $T\in G\!L_n(\mathbb{Z}_h)$ such that $A$ and $B$ are as in (\[equa10\]). Let $D_1={\rm diag}\left(\prod_{i=1}^tp_i^{\alpha_{i1}},\prod_{i=1}^tp_i^{\alpha_{i2}},\ldots,\prod_{i=1}^tp_i^{\alpha_{ir}}\right)$. Without loss of generality, we may assume that $T=I_n$ and $m\geq2$. Hence we can assume further that $A$ and $B$ have matrix representations $$\label{equa12}
A=(0,I_k)\quad\hbox{and}\quad
B=\left(\begin{array}{cccc}
D_1 & 0_{r,n-m-r} & I_r & 0\\
0 & 0 & 0 & I_{m-r}
\end{array}\right),$$ respectively. Clearly, the $(m-r)$-subspace $(0_{m-r,n-m+r},I_{m-r})$ is contained in $A\cap B$, and the $(k+r)$-subspace $\left(\begin{array}{ccc}
I_r & 0_{r,n-k-r} & 0\\
0 & 0 & I_{k}
\end{array}\right)$ contains $A$ and $B$. It follows that $\dim(A\cap B)\geq m-r$ and $\dim(A\vee B)\leq k+r$.
Let $\alpha\in A\cap B$ be an $n$-dimensional row vector. Then there are matrices $C_1\in\mathbb{Z}_{h}^k$ and $C_2=(C_{21},C_{22})\in\mathbb{Z}_{h}^m$, where $C_{21}\in\mathbb{Z}_{h}^r$ and $C_{22}\in\mathbb{Z}_{h}^{m-r}$, such that $\alpha=C_1A=(0_{1,n-k},C_1)$ and $\alpha=C_2B=(C_{21}D_1,0,C_{21},C_{22})$. By $r\leq n-k$, we have that $C_{21}D_1=0$ and therefore $\alpha=(0_{1,n-m},C_{21},C_{22})$. From $C_{21}D_1=0$ and $\rho(D_1)=r$, we deduce that $C_{21}\in\bigcup_{i=1}^t(p_i)^{r}$, where $(p_i),i=1,2,\ldots,t,$ are the maximal ideals of $\mathbb{Z}_h$, which implies that $\dim(A\cap B)\leq m-r$. Hence, we have $\dim(A\cap B)=m-r$.
Let $d=\dim(A\vee B)$. Then there exists a $d$-subspace $W\in A\vee B$. Since $B\not\subseteq A$, one obtains $d>k$. By Lemma \[lem2.9\], $W$ has a matrix representation $W={\rm diag}(W_{11},I_k)$, where $W_{11}\in\mathbb{Z}_{h}^{(d-k)\times(n-k)}$ has a right inverse. By $B\subseteq W$, the $r$-subspace $(D_1,0_{r,n-m-r},I_r,0)\subseteq W$. Therefore, there is a matrix $C=(C_1,C_2)\in\mathbb{Z}_{h}^{r\times d}$, where $C_1\in\mathbb{Z}_{h}^{r\times(d-k)}$ and $C_2\in\mathbb{Z}_{h}^{r\times k}$, such that $(D_1,0_{r,n-m-r},I_r,0)=CW=(C_1W_{11},C_2)$. Hence $(D_1,0_{r,n-k-r})=C_1W_{11}$. It follows that $r=\rho(D_1)=\rho(C_1W_{11})\leq\rho(W_{11})\leq d-k$, which implies that $\dim(A\vee B)=k+r$.
From (\[equa8\]) and (\[equa12\]), we deduce that $$\rho\left(\begin{array}{c}A\\ B \end{array}\right)
=\rho\left(\begin{array}{ccc}
I_k&0&0\\
0&D_1 & 0\\
0 & 0 & 0
\end{array}\right)=k+r.$$ Therefore, we complete the proof of (\[equa11\]).
Let us continue to assume that $A$ and $B$ have matrix representations as in (\[equa12\]). From $r=\rho(D_1)$, we deduce that $\dim(A\cap B)=m-\rho(D_1)$. By Lemmas \[lem2.6\] and \[lem2.8\], $\pi_i(A)$ and $\pi_i(B)$ are two subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ for $i=1,2,\ldots,t$. Similarly, we can prove that $\dim(\pi_i(A)\cap \pi_i(B))=m-\rho(\pi_i(D_1))$ for $i=1,2,\ldots,t$. By Lemma \[lem2.4\], we have $$\begin{aligned}
\dim(A\cap B)&=&m-\rho(D_1)=m-\max\{\rho(\pi_i(D_1)): i=1,2,\ldots,t\}\\
&=&\min\{\dim(\pi_i(A)\cap\pi_i(B)):i=1,2,\ldots,t\}.\end{aligned}$$ By Lemmas \[lem2.4\], \[lem2.6\] and \[lem2.8\], we have similarly that $\dim(A\cap B)=\min\{\dim(\theta_i(A)\cap\theta_i(B)):i=1,2,\ldots,t\}$. ${\hfill\Box\medskip}$
\[lem3.4\][(See Theorem 3.5 in [@Huang3].)]{} Let $1\leq k<n$. Then the number of $k$-subspaces of $\mathbb{Z}_{p^s}^n$ is $p^{(s-1)k(n-k)}{n\brack k}_p$.
\[lem3.5\] Let $1\leq m\leq k<n$. Then the following hold:
- The number of $k$-subspaces of $\mathbb{Z}_{h}^n$ is $\prod_{i=1}^tp_i^{(s_i-1)k(n-k)}{n\brack k}_{p_i}.$
- In $\mathbb{Z}_{h}^n$, the number of $m$-subspaces in a given $k$-subspace is $\prod_{i=1}^tp_i^{(s_i-1)m(k-m)}{k\brack m}_{p_i}.$
- In $\mathbb{Z}_{h}^n$, the number of $k$-subspaces containing a fixed $m$-subspace is\
$\prod_{i=1}^tp_i^{(s_i-1)(k-m)(n-k)}{n-m\brack k-m}_{p_i}.$
[[*Proof.*]{}]{}(i). Let $\pi$ be the bijective map from $\mathbb{Z}_{h}^{k\times n}$ to $\prod_{i=1}^t\mathbb{Z}_{p_i^{s_i}}^{k\times n}$. By Lemma \[lem2.6\] and matrix representations of subspaces, $\pi$ induces a bijective map from ${\mathbb{Z}_{h}^{n}\brack k}$ to $\prod_{i=1}^t{\mathbb{Z}_{p_i^{s_i}}^{n}\brack k}$ such that $\pi(X)=(\pi_1(X),\pi_2(X),\ldots,\pi_t(X))$ for a matrix representation $X$ of every $k$-subspace of $\mathbb{Z}_h^n$. By Lemma \[lem3.4\], the number of $k$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ is $p_i^{(s_i-1)k(n-k)}{n\brack k}_{p_i}$ for $i=1,2,\ldots,t$, which imply that the number of $k$-subspaces of $\mathbb{Z}_{h}^n$ is $\prod_{i=1}^tp_i^{(s_i-1)k(n-k)}{n\brack k}_{p_i}.$
(ii). Since every $k$-subspace of $\mathbb{Z}_{h}^n$ is isomorphic to $\mathbb{Z}_{h}^k$, the desired result follows by (i).
(iii). Let $${\cal S}=\left\{(A,B):A\in{\mathbb{Z}_{h}^{n}\brack m},B\in{\mathbb{Z}_{h}^{n}\brack k},A\subseteq B\right\}.$$ We compute the cardinality of ${\cal S}$ in two ways. By (i) and (ii), the number of $k$-subspaces containing a fixed $m$-subspace is $$\frac{|{\cal S}|}{\left|{\mathbb{Z}_{h}^{n}\brack m}\right|}=\frac{\left|{\mathbb{Z}_{h}^{n}\brack k}\right|\cdot\left|{\mathbb{Z}_{h}^{k}\brack m}\right|}{\left|{\mathbb{Z}_{h}^{n}\brack m}\right|}
=\prod_{i=1}^tp_i^{(s_i-1)(k-m)(n-k)}{n-m\brack k-m}_{p_i},$$ as desired. ${\hfill\Box\medskip}$
Let $A$ be an $m$-subspace of $\mathbb{Z}_{h}^n$. Let $$A^\perp=\{y\in\mathbb{Z}_{h}^n: y{}^tx=0\;\hbox{for all}\;x\in A\}.$$ Then $A^\perp$ is a linear subset of $\mathbb{Z}_{h}^n$. By Lemma \[lem2.2\], $A$ has a matrix representation $A=(0,I_m)T$, where $T\in G\!L_n(\mathbb{Z}_h)$. It is easy to prove that $A^\perp$ has a matrix representation $$\label{equa13}
A^\perp=(I_{n-m},0){}^tT^{-1}\; \hbox{if}\; A=(0,I_m)T.$$ Hence $A^\perp$ is an $(n-m)$-subspace of $\mathbb{Z}_{h}^n$. The subspace $A^\perp$ is called the [*dual subspace*]{} of $A$. Note that $\dim(A)+\dim(A^\perp)=n$ and $(A^{\perp})^\perp=A.$ If $A_1$ and $A_2$ are two subspaces of $\mathbb{Z}_{h}^n$, then $A_1\subseteq A_2$ if and only if $A_2^\perp\subseteq A_1^\perp.$
\[lem3.6\] Let $m<n$ and let $A$ and $B$ be two $m$-subspaces of $\mathbb{Z}_{h}^n$. Then $$m-\dim(A\cap B)=n-m-\dim(A^\perp\cap B^\perp).$$
[[*Proof.*]{}]{}Without loss of generality, by Lemma \[lem3.2\], we may assume that $A=(0,I_m)$ and $B=(D,I_m)$, where $D={\rm diag}\left(\prod_{i=1}^tp_i^{\alpha_{i1}},\prod_{i=1}^tp_i^{\alpha_{i2}},\ldots,\prod_{i=1}^tp_i^{\alpha_{ir}},0_{m-r,n-m-r}\right),1\leq r\leq \min\{m,n-m\},\rho(D)=r$, and $0\leq\alpha_{i1}\leq\alpha_{i2}\leq\cdots\leq\alpha_{ir}\leq s_i$ for $i=1,2,\ldots,t$. Then $\dim(A\cap B)=m-r$.
Note that $A^\perp$ and $B^\perp$ are $(n-m)$-subspaces of $\mathbb{Z}_{h}^n$. By (\[equa13\]), $A^\perp$ has a matrix representation $A^\perp=(I_{n-m},0)$. Let $B^\perp=(B_1,B_2)$ be a matrix representation of $B^\perp$, where $B_1\in\mathbb{Z}_{h}^{(n-m)\times(n-m)}$ and $B_2\in\mathbb{Z}_{h}^{(n-m)\times m}$. Then $B_1{}^tD+B_2=0$, which implies that $\rho(B_2)=\rho(-B_1{}^tD)\leq r$. By Theorem \[lem3.3\], we have $$\begin{aligned}
n-m-\dim(A^\perp\cap B^\perp)&=&\rho\left(\begin{array}{c}
A^\perp\\ B^\perp
\end{array}\right)-(n-m)
=\rho\left(\begin{array}{cc}
I_{n-m} & 0\\ B_1 & B_2
\end{array}\right)-(n-m)\\
&=&\rho(B_2)\leq r=m-\dim(A\cap B).\end{aligned}$$ On the other hand, we have similarly $$m-\dim(A\cap B)=m-\dim((A^\perp)^\perp\cap(B^\perp)^\perp)
\leq n-m-\dim(A^\perp\cap B^\perp).$$ Therefore, we obtain $m-\dim(A\cap B)=n-m-\dim(A^\perp\cap B^\perp).$ ${\hfill\Box\medskip}$
Proof of Theorem \[thm1.1\]
===========================
In this section, we discuss the generalized Grassmann graph $G_r(m,n,\mathbb{Z}_{h})$ over $\mathbb{Z}_{h}$ and prove Theorem \[thm1.1\]. We always assume that $2\leq r\leq m+1\leq n$.
For convenience, if $X$ is a vertex of $G_r(m,n,\mathbb{Z}_{h})$, then the matrix representation of $X$ also is denoted by $X$. By Lemma \[lem2.3.2\], $\pi$ induces a bijective map from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))$, and $(\pi_i,\theta_i)$ induces a bijective map from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))\times V(G_r(m,n,\mathbb{Z}_{h/p_i^{s_i}}))$ for $i=1,2,\ldots,t$.
\[lem4.1\] Let $n\leq 2m$ and $2\leq r\leq \max\{m+1,n-m+1\}$. Then $G_r(m,n,\mathbb{Z}_{h})$ is a connected vertex transitive graph, and $G_r(m,n,\mathbb{Z}_{h})\cong G_r(n-m,n,\mathbb{Z}_{h})$.
[[*Proof.*]{}]{}If $X$ is a vertex of $G_r(m,n,\mathbb{Z}_{h})$, then there is a $T_X\in G\!L_n(\mathbb{Z}_h)$ such that $X=(0,I_m)T_X$ by Lemma \[lem2.2\]. Let $\varphi(X)=XT_X^{-1}$. Then $\varphi$ is an automorphism of $G_r(m,n,\mathbb{Z}_{h})$ and $\varphi(X)=(0,I_m)$. Hence $G_r(m,n,\mathbb{Z}_{h})$ is vertex transitive. By Lemma \[lem3.2\] and Theorem \[lem3.3\], it is easy to prove that $G_r(m,n,\mathbb{Z}_{h})$ is a connected graph.
By Theorem \[lem3.5\], we have $|V(G_r(m,n,\mathbb{Z}_{h}))|=|V(G_r(n-m,n,\mathbb{Z}_{h}))|.$ By Lemma \[lem3.6\], the map $X\mapsto X^\perp$ is an isomorphism from $G_r(m,n,\mathbb{Z}_{h})$ to $G_r(n-m,n,\mathbb{Z}_{h})$. Therefore, we obtain $G_r(m,n,\mathbb{Z}_{h})\cong G_r(n-m,n,\mathbb{Z}_{h})$. ${\hfill\Box\medskip}$
By Theorem \[lem4.1\], we may assume that $n\geq2m$ in our discussion on $G_r(m,n,\mathbb{Z}_{h})$.
\[lem4.2\][(See Lemma 4.7 and Theorem 4.9 in [@Huang3].)]{} Let $n\geq 2m$. Then the clique number of $G_r(m,n,\mathbb{Z}_{p^s})$ is $p^{(s-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p}.$ Moreover, ${\cal F}$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{p^s})$ if and only if either [(a)]{} ${\cal F}$ consists of all $m$-subspaces of $\mathbb{Z}_{p^s}^n$ which contain a fixed $(m-r+1)$-subspace of $\mathbb{Z}_{p^s}^n$, or [(b)]{} $n=2m$ and ${\cal F}$ is the set of all $m$-subspaces of $\mathbb{Z}_{p^s}^n$ contained in a fixed $(m+r-1)$-subspace of $\mathbb{Z}_{p^s}^n$.
\[lem4.3\] Let $n\geq 2m$, and $\pi_i$ be the surjective homomorphism from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))$ for $i=1,2\ldots,t$. Suppose that ${\cal F}$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{h})$. Then the following hold:
- The clique number of $G_r(m,n,\mathbb{Z}_{h})$ is $\prod_{i=1}^tp_i^{(s_i-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_i}.$
- $\pi_i({\cal F})$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{p_i^{s_i}})$ for $i=1,2,\ldots,t$.
[[*Proof.*]{}]{}Let ${\cal F}$ be a maximum clique of $G_r(m,n,\mathbb{Z}_{h})$. For each $i$ with $1\leq i\leq t$, let $\{\pi_i(A_{i1}),\pi_i(A_{i2}),\ldots,\pi_i(A_{ik_i})\}$ be the set of all different elements in $\pi_i({\cal F})$. Then $\pi_i({\cal F})=\{\pi_i(A_{i1}),\pi_i(A_{i2}),\ldots,\pi_i(A_{ik_i})\}$. By Lemma \[lem2.6\] and Theorem \[lem3.3\], $\pi_i({\cal F})$ is a clique of $G_r(m,n,\mathbb{Z}_{p_i^{s_i}})$. By Lemma \[lem4.2\], we have $$\label{equa14}
k_i\leq\omega(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))=p_i^{(s_i-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_i}.$$ Moreover, ${\cal F}$ has a partition into $k_i$ cliques: ${\cal F}=\bigcup_{j=1}^{k_i}{\cal C}_j$, where ${\cal C}_j$ is a clique with $\pi_i({\cal C}_j)=\{\pi_i(A_{ij})\}$ for $j=1,2,\ldots,k_i$, and ${\cal C}_u\cap{\cal C}_v=\emptyset$ for all $u\not=v$. It follows that $\omega(G_r(m,n,\mathbb{Z}_{h}))=\sum_{j=1}^{k_i}|{\cal C}_j|$.
(i). By Lemma \[lem4.2\], the result is trivial for $t=1$. Suppose that $t\geq2$ and the result is true for $t-1$. Let $n_j=|{\cal C}_j|$ for $j=1,2,\ldots,k_i$. Then there exists $\{B_{1j},B_{2j},\ldots,B_{n_jj}\}\subseteq (p_i^{s_i})^{m\times n}$ such that ${\cal C}_j=\{A_{ij}+B_{1j},A_{ij}+B_{2j},\ldots,A_{ij}+B_{n_jj}\}$. By Lemmas \[lem2.3.2\], \[lem2.6\] and Theorem \[lem3.3\], $|(\pi_i,\theta_i)({\cal C}_j)|=|\theta_i({\cal C}_j)|=|{\cal C}_j|$ and $\theta_i({\cal C}_j)$ is a clique of $G_r(m,n,\mathbb{Z}_{h/p_i^{s_i}})$. By induction, we have $$n_j\leq\omega(G_r(m,n,\mathbb{Z}_{h/p_i^{s_i}}))=\prod_{j\not=i}p_j^{(s_j-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_j}\;\hbox{for}\;j=1,2,\ldots,k_i.$$ Therefore, by (\[equa14\]), we have $$\label{equa15}
\omega(G_r(m,n,\mathbb{Z}_{h}))=\sum_{j=1}^{k_i}n_j\leq \prod_{i=1}^tp_i^{(s_i-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_i}.$$
On the other hand, $${\cal C}=\left\{\left(\begin{array}{cc}
X & 0\\
0 & I_{m-r+1}
\end{array}\right): X\in V(G_r(r-1,n-m+r-1,\mathbb{Z}_{h}))\right\}$$ is a clique of $G_r(m,n,\mathbb{Z}_{h})$ and $|{\cal C}|=\prod_{i=1}^tp_i^{(s_i-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_i}$. It follows that $\omega(G_r(m,n,\mathbb{Z}_{h}))\geq\prod_{i=1}^tp_i^{(s_i-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_i}$. Therefore, the clique number of $G_r(m,n,\mathbb{Z}_{h})$ is $\prod_{i=1}^tp_i^{(s_i-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_i}$.
(ii). By (\[equa14\]) and (\[equa15\]), $k_i=p_i^{(s_i-1)(n-m)(r-1)}{n-m+r-1\brack r-1}_{p_i}$ for $i=1,2,\ldots,t$. By Lemma \[lem4.2\], $\pi_i(\cal F)$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{p_i^{s_i}})$ for $i=1,2,\ldots,t$. ${\hfill\Box\medskip}$
\[lem4.4\] Let $4\leq2r\leq2m=n$ and $I\subseteq[t]$. Then ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{h})$, where ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}$ is given by (\[equanew1\]).
[[*Proof.*]{}]{}Since $\pi$ is a bijective map from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))$, $\pi$ induces a bijective map from ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}$ to $\pi\left({\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right)$. Therefore, we have $$\left|{\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right|=\left|\pi\left({\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right)\right|
=\prod_{i=1}^{t}\left|\pi_i\left({\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right)\right|.$$
For any $$U=\left(\begin{array}{cc}
X & 0\\
Y & xI_{m-r+1}
\end{array}\right)\in{\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)},$$ we have that $X$ is an $(r-1)$-subspace of $\mathbb{Z}_h^{m+r-1}$ since $U\in{\mathbb{Z}_h^n\brack m}$. By Theorem \[lem3.5\], one obtains $$\left|{\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right|
\geq\left|{\mathbb{Z}_h^{m+r-1}\brack r-1}\right|=\prod_{i=1}^tp_i^{(s_i-1)m(r-1)}{m+r-1\brack r-1}_{p_i}.$$
If $i\in I$, from Lemmas \[lem2.6\] and \[lem2.8\], we deduce that $\pi_i\left({\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right)\subseteq{\cal A}_i$, where ${\cal A}_i$ is the set of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ containing the fixed $(m-r+1)$-subspace $(0,I_{m-r+1})$. If $i\in[t]\setminus I$, from Lemmas \[lem2.6\] and \[lem2.8\], we deduce that $\pi_i\left({\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right)\subseteq{\cal A}_i$, where ${\cal A}_i$ is the set of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ contained in the fixed $(m+r-1)$-subspace $(I_{m+r-1},0)$. Therefore, by Theorem \[lem3.5\], we have $$\prod_{i=1}^{t}\left|\pi_i\left({\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right)\right|\leq\prod_{i=1}^t|{\cal A}_i|
=\prod_{i=1}^tp_i^{(s_i-1)m(r-1)}{m+r-1\brack r-1}_{p_i}.$$ Hence, we obtain $\left|{\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right|=\prod_{i=1}^tp_i^{(s_i-1)m(r-1)}{m+r-1\brack r-1}_{p_i}$.
Let $U$ and $W$ be two distinct subspaces in ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}$. If $i\in I$, then $\pi_i(U)\cap \pi_i(W)\supseteq(0,I_{m-r+1})$, which implies that $\dim(\pi_i(U)\cap \pi_i(W))\geq m-r+1$. If $i\in[t]\setminus I$, then $\pi_i(U),\pi_i(W)\subseteq(I_{m+r-1},0)$, which imply that $\dim(\pi_i(U)\cap \pi_i(W))\geq m-r+1$ by Theorem \[lem3.3\] and $\dim(\pi_i(U)\vee\pi_i(W))\leq m+r-1$. Therefore, we have $\dim(\pi_i(U)\cap \pi_i(W))\geq m-r+1$ for $i=1,2,\ldots,t$. By Theorem \[lem3.3\] again, we have $\dim(U\cap W)\geq m-r+1$, and therefore ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}$ is a clique of $G_r(m,n,\mathbb{Z}_{h})$. By Lemma \[lem4.3\] and $\left|{\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}\right|=\prod_{i=1}^tp_i^{(s_i-1)m(r-1)}{m+r-1\brack r-1}_{p_i}$, ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{h})$. ${\hfill\Box\medskip}$
\[lem4.5\] Let $n\geq 2m$. Then ${\cal F}$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{h})$ if and only if either [(a)]{} ${\cal F}$ consists of all $m$-subspaces of $\mathbb{Z}_{h}^n$ which contain a fixed $(m-r+1)$-subspace of $\mathbb{Z}_{h}^n$, [(b)]{} $n=2m$ and ${\cal F}$ is the set of all $m$-subspaces of $\mathbb{Z}_{h}^n$ contained in a fixed $(m+r-1)$-subspace of $\mathbb{Z}_{h}^n$, or [(c)]{} $n=2m$ and there exists some $T\in G\!L_n(\mathbb{Z}_h)$ such that ${\cal F}={\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}T$ with $I\not=\emptyset,[t]$, where ${\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}$ is given by (\[equanew1\]).
[[*Proof.*]{}]{}By the definition of $G_r(m,n,\mathbb{Z}_{h})$, we have $2\leq r\leq m+1$. When $r=m+1$ or $m=1$, $G_r(m,n,\mathbb{Z}_{h})$ is a clique, which implies that this theorem holds by Theorem \[lem3.5\]. From now on, we assume that $4\leq2r\leq2m\leq n$.
Suppose that either [(a)]{} ${\cal F}$ consists of all $m$-subspaces of $\mathbb{Z}_{h}^n$ which contain a fixed $(m-r+1)$-subspace of $\mathbb{Z}_{h}^n$, [(b)]{} $n=2m$ and ${\cal F}$ is the set of all $m$-subspaces of $\mathbb{Z}_{h}^n$ contained in a fixed $(m+r-1)$-subspace of $\mathbb{Z}_{h}^n$, or [(c)]{} $n=2m$ and there exists some $T\in G\!L_n(\mathbb{Z}_h)$ such that ${\cal F}={\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}T$ with $I\not=\emptyset,[t]$. By the definition of $G_r(m,n,\mathbb{Z}_{h})$, Theorem \[lem3.5\] and Lemma \[lem4.4\], ${\cal F}$ is a maximum clique of $G_r(m,n,\mathbb{Z}_{h})$.
Let ${\cal F}$ be a maximum clique of $G_r(m,n,\mathbb{Z}_{h})$. We will prove that either [(a)]{} ${\cal F}$ consists of all $m$-subspaces of $\mathbb{Z}_{h}^n$ which contain a fixed $(m-r+1)$-subspace of $\mathbb{Z}_{h}^n$, [(b)]{} $n=2m$ and ${\cal F}$ is the set of all $m$-subspaces of $\mathbb{Z}_{h}^n$ contained in a fixed $(m+r-1)$-subspace of $\mathbb{Z}_{h}^n$, or [(c)]{} $n=2m$ and there exists some $T\in G\!L_n(\mathbb{Z}_h)$ such that ${\cal F}={\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}T$ with $I\not=\emptyset,[t]$. There are the following two cases to be considered.
[**Case**]{} 1: $n>2m$. For each $i$ with $1\leq i\leq t$, by Lemmas \[lem4.2\] and \[lem4.3\], $\pi_i({\cal F})$ consists of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ which contain a fixed $(m-r+1)$-subspace $P_i$ of $\mathbb{Z}_{p_i^{s_i}}^n$. By Lemma \[lem2.2\], there is a $T_i\in G\!L_n(\mathbb{Z}_{p_i^{s_i}})$ such that $P_i=(0,I_{m-r+1})T_i$ for $i=1,2,\ldots,t$. So, we have $$\pi_i({\cal F})=\left\{\left(\begin{array}{cc}
X_i & 0\\
0 & I_{m-r+1}
\end{array}\right)T_i\in{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} : X_i\in{\mathbb{Z}_{p_i^{s_i}}^{n-m+r-1}\brack r-1}\right\}
\;\hbox{for}\;i=1,2,\ldots,t.$$ By Lemma \[lem2.7\], there is the unique $T\in G\!L_n(\mathbb{Z}_{h})$ such that $\pi_i(T)=T_i$ for $i=1,2,\ldots,t$. Since $\pi$ is a bijective map from $V(G_r(r-1,n-m+r-1,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(r-1,n-m+r-1,\mathbb{Z}_{p_i^{s_i}}))$, we have $$\pi_i({\cal F})=\left\{\pi_i\left(\left(\begin{array}{cc}
X & 0\\
0 & I_{m-r+1}
\end{array}\right)T\right)\in{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} : X\in{\mathbb{Z}_{h}^{n-m+r-1}\brack r-1}\right\}
\;\hbox{for}\;i=1,2,\ldots,t,$$ which imply that $$\pi({\cal F})=\left\{\pi\left(\left(\begin{array}{cc}
X & 0\\
0 & I_{m-r+1}
\end{array}\right)T\right)\in\prod_{i=1}^t{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} : X\in{\mathbb{Z}_{h}^{n-m+r-1}\brack r-1}\right\}.$$ Since $\pi$ is a bijective map from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))$, we have $${\cal F}=\left\{\left(\begin{array}{cc}
X & 0\\
0 & I_{m-r+1}
\end{array}\right)T\in{\mathbb{Z}_{h}^{n}\brack m} : X\in{\mathbb{Z}_{h}^{n-m+r-1}\brack r-1}\right\},$$ which implies that ${\cal F}$ consists of all $m$-subspaces of $\mathbb{Z}_{h}^n$ which contain a fixed $(m-r+1)$-subspace of $\mathbb{Z}_{h}^n$.
[**Case**]{} 2: $n=2m$. By Lemmas \[lem4.2\] and \[lem4.3\], there exists some subset $I$ of $[t]$ such that, $\pi_i({\cal F})$ consists of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ which contain a fixed $(m-r+1)$-subspace of $\mathbb{Z}_{p_i^{s_i}}^n$ if $i\in I$, and $\pi_i({\cal F})$ is the set of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ contained in a fixed $(m+r-1)$-subspace of $\mathbb{Z}_{p_i^{s_i}}^n$ if $i\in[t]\setminus I$. Then there are the following three cases to be considered.
[*Case*]{} 2.1: $I=[t]$. Similar to the proof of Case 1, we may obtain that ${\cal F}$ consists of all $m$-subspaces of $\mathbb{Z}_{h}^n$ which contain a fixed $(m-r+1)$-subspace of $\mathbb{Z}_{h}^n$.
[*Case*]{} 2.2: $I=\emptyset$. For each $i$ with $1\leq i\leq t$, $\pi_i({\cal F})$ is the set of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ contained in a fixed $(m+r-1)$-subspace $Q_i$ of $\mathbb{Z}_{p_i^{s_i}}^n$. By Lemma \[lem2.2\], there is a $T_i\in G\!L_n(\mathbb{Z}_{p_i^{s_i}})$ such that $Q_i=(I_{m+r-1},0)T_i$ for $i=1,2,\ldots,t$. So, we have $$\pi_i({\cal F})=\left\{(X_i, 0)T_i\in{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} : X_i\in{\mathbb{Z}_{p_i^{s_i}}^{m+r-1}\brack m}\right\}\;\hbox{for}\;i=1,2,\ldots,t.$$ By Lemma \[lem2.7\], there is the unique $T\in G\!L_n(\mathbb{Z}_{h})$ such that $\pi_i(T)=T_i$ for $i=1,2,\ldots,t$. Since $\pi$ is a bijective map from $V(G_r(m,m+r-1,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(m,m+r-1,\mathbb{Z}_{p_i^{s_i}}))$, we have $$\pi_i({\cal F})=\left\{\pi_i((X,0)T)\in{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} : X\in{\mathbb{Z}_{h}^{m+r-1}\brack m}\right\}\;\hbox{for}\;i=1,2,\ldots,t,$$ which imply that $$\pi({\cal F})=\left\{\pi((X,0)T)\in\prod_{i=1}^t{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} : X\in{\mathbb{Z}_{h}^{m+r-1}\brack m}\right\}.$$ Since $\pi$ is a bijective map from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))$, we have $${\cal F}=\left\{(X,0)T\in{\mathbb{Z}_{h}^{n}\brack m} : X\in{\mathbb{Z}_{h}^{m+r-1}\brack m}\right\},$$ which implies that ${\cal F}$ is the set of all $m$-subspaces of $\mathbb{Z}_{h}^n$ contained in a fixed $(m+r-1)$-subspace of $\mathbb{Z}_{h}^n$.
[*Case*]{} 2.3: $I\not=\emptyset,[t]$. If $i\in I$, $\pi_i({\cal F})$ consists of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ which contain a fixed $(m-r+1)$-subspace $P_i$ of $\mathbb{Z}_{p_i^{s_i}}^n$. By Lemma \[lem2.2\], there is a $T_i\in G\!L_n(\mathbb{Z}_{p_i^{s_i}})$ such that $P_i=(0,I_{m-r+1})T_i$ for each $i\in I$. So, we have $$\pi_i({\cal F})=\left\{\left(\begin{array}{cc}
X_i & 0\\
0 & I_{m-r+1}
\end{array}\right)T_i\in{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} : X_i\in{\mathbb{Z}_{p_i^{s_i}}^{m+r-1}\brack r-1}\right\}\;\hbox{for each}\;i\in I.$$
If $i\in[t]\setminus I$, $\pi_i({\cal F})$ is the set of all $m$-subspaces of $\mathbb{Z}_{p_i^{s_i}}^n$ contained in a fixed $(m+r-1)$-subspace $Q_i$ of $\mathbb{Z}_{p_i^{s_i}}^n$. By Lemma \[lem2.2\], there is a $T_i\in G\!L_n(\mathbb{Z}_{p_i^{s_i}})$ such that $Q_i=(I_{m+r-1},0)T_i$ for each $i\in [t]\setminus I$. So, we have $$\pi_i({\cal F})=\left\{\left(\begin{array}{cc}
X_i & 0\\
Y_i & 0
\end{array}\right)T_i\in{\mathbb{Z}_{p_i^{s_i}}^{n}\brack m} :
\left(\begin{array}{c}
X_i\\ Y_i
\end{array}\right)\in{\mathbb{Z}_{p_i^{s_i}}^{m+r-1}\brack m}\right\}\;\hbox{for each}\;i\in[t]\setminus I.$$
Let $T\in G\!L_n(\mathbb{Z}_{h})$ satisfy $\pi_i(T)=T_i$ for $i=1,2,\ldots,t$. Since $\pi$ is a bijective map from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))$ and $\pi_i(J_{(\alpha_1,\alpha_2,\ldots,\alpha_t)})=J_{\alpha_i}\subseteq\mathbb{Z}_{p_i^{s_i}}$ for $i=1,2,\ldots,t$, we have $$\begin{aligned}
\pi_i({\cal F})&\subseteq&\left\{\pi_i\left(\left(\begin{array}{cc}
X & 0\\
Y & xI_{m-r+1}
\end{array}\right)T\right)\in{\mathbb{Z}_{p_i^{s_i}}^n \brack m} :\right.\\
&&\left.X\in\mathbb{Z}_h^{(r-1)\times(m+r-1)},Y\in J_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1)\times(m+r-1)}\right\}\;\hbox{for}\;i=1,2,\ldots,t,\end{aligned}$$ where $\alpha_i=s_i,\pi_i(x)=1$ if $i\in I$, and $\alpha_i=0,\pi_i(x)=0$ if $i\in[t]\setminus I$, which imply that $\pi({\cal F})\subseteq\pi\left({\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}T\right)$. Since $\pi$ is a bijective map, one obtains ${\cal F}\subseteq{\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}T$. Since ${\cal F}$ is a maximum clique, ${\cal F}={\cal F}_{(\alpha_1,\alpha_2,\ldots,\alpha_t)}^{(m-r+1,m,n,I)}T$. ${\hfill\Box\medskip}$
[**Proof of Theorem \[thm1.1\].**]{} Let $\lfloor n/2\rfloor\geq m\geq r\geq 0$ and ${\cal F}\subseteq{\mathbb{Z}_{h}^{n}\brack m}$ be an $r$-intersecting family. Without loss of generality, we assume that $1\leq r\leq m-1$. By Theorem \[lem3.3\], ${\cal F}$ is a clique of the generalized Grassmann graph $G_{m-r+1}(m,n,\mathbb{Z}_{h})$. By Lemma \[lem4.3\], $|{\cal F}|\leq\prod_{i=1}^tp_i^{(s_i-1)(n-m)(m-r)}{n-r\brack m-r}_{p_i}$ and equality holds if and only if ${\cal F}$ is a maximum clique of $G_{m-r+1}(m,n,\mathbb{Z}_{h})$. By Theorem \[lem4.5\], this theorem holds. ${\hfill\Box\medskip}$
Independence numbers of $G_{r}(m,n,\mathbb{Z}_{h})$
===================================================
In this section, we discuss the independence number of $G_{r}(m,n,\mathbb{Z}_{h})$ and give some estimations of the independence number $\alpha(G_{r}(m,n,\mathbb{Z}_{h}))$.
If $\Gamma$ is a vertex transitive graph, by Lemma 2.7.2 in [@Roberson], we obtain the following inequality $$\label{equa17}
|V(\Gamma)|\geq\alpha(\Gamma)\omega(\Gamma).$$
\[lem5.1\][(See Theorem 5.1 in [@Huang3].)]{} If $n\geq 2m$, then $$p^{(s-1)(n-m)(m-r+1)}\alpha(G_{r}(m,n,\mathbb{Z}_{p}))\leq\alpha(G_{r}(m,n,\mathbb{Z}_{p^s}))\leq p^{(s-1)(n-m)(m-r+1)}\frac{{n\brack m}_p}{{n-m+r-1\brack r-1}_p}.$$
\[lem5.2\] If $n\geq 2m$, then $$\prod_{i=1}^tp_i^{\omega_i}\alpha(G_{r}(m,n,\mathbb{Z}_{p_i}))\leq\alpha(G_{r}(m,n,\mathbb{Z}_{h}))\leq
\prod_{i=1}^t\frac{p_i^{\omega_i}{n\brack m}_{p_i}}{{n-m+r-1\brack r-1}_{p_i}},$$ where $\omega_i=(s_i-1)(n-m)(m-r+1)$ for $i=1,2,\ldots,t$.
[[*Proof.*]{}]{}Let ${\cal S}_i=\{S_{i1},S_{i2},\ldots,S_{i\xi_i}\}$ be a largest independent set of $G_{r}(m,n,\mathbb{Z}_{p_i^{s_i}})$, where $S_{i\eta_i}\in {\mathbb{Z}_{p_i^{s_i}}^{n}\brack m}$ for $1\leq i\leq t$ and $1\leq \eta_i\leq \xi_i$. By the definition of $G_{r}(m,n,\mathbb{Z}_{p_i^{s_i}})$, we obtain $\dim(S_{ij}\cap S_{i\ell})\leq m-r$ for $1\leq i\leq t$ and $1\leq j,\ell\leq \xi_i$ with $j\not=\ell$. By Lemma \[lem5.1\], we have $$\xi_i=|{\cal S}_i|\geq p_i^{(s_i-1)(n-m)(m-r+1)}\alpha(G_{r}(m,n,\mathbb{Z}_{p_i}))\;\hbox{for}\;i=1,2,\ldots,t.$$ Let $${\cal S}=\{\pi^{-1}(S_{1\eta_1},S_{2\eta_2},\ldots,S_{t\eta_t}):1\leq \eta_i\leq \xi_i\;\hbox{for}\;i=1,2,\ldots,t\}.$$ Since $\pi$ is a bijective map from $V(G_r(m,n,\mathbb{Z}_{h}))$ to $\prod_{i=1}^{t}V(G_r(m,n,\mathbb{Z}_{p_i^{s_i}}))$, we have $$|{\cal S}|=\xi_1\xi_2\cdots\xi_t\geq \prod_{i=1}^tp_i^{(s_i-1)(n-m)(m-r+1)}\alpha(G_{r}(m,n,\mathbb{Z}_{p_i})).$$
For any two distinct vertices $A,B\in{\cal S}$, we have $\pi_i(A),\pi_i(B)\in{\cal S}_i$ for $i=1,2,\ldots,t$, which imply that $\dim(A\cap B)=\min\{\dim(\pi_i(A)\cap\pi_i(B)): i=1,2,\ldots,t\}\leq m-r$ by Theorem \[lem3.3\]. Therefore, ${\cal S}$ is a independent set of $G_{r}(m,n,\mathbb{Z}_{h})$, which implies that $\alpha(G_{r}(m,n,\mathbb{Z}_{h}))\geq |{\cal S}|\geq \prod_{i=1}^tp_i^{(s_i-1)(n-m)(m-r+1)}\alpha(G_{r}(m,n,\mathbb{Z}_{p_i}))$.
Since $G_{r}(m,n,\mathbb{Z}_{h})$ is vertex transitive, by (\[equa17\]) and Theorem \[lem3.5\], we have $$\alpha(G_{r}(m,n,\mathbb{Z}_{h}))\leq \prod_{i=1}^tp_i^{(s_i-1)(n-m)(m-r+1)}\frac{{n\brack m}_{p_i}}{{n-m+r-1\brack r-1}_{p_i}}.$$ Therefore, the desired result follows. ${\hfill\Box\medskip}$
For a vertex transitive graph $\Gamma$ and $\Delta\subseteq V(\Gamma)$, let $\alpha(\Delta)$ denote the independence number of the subgraph of $\Gamma$ induced by $\Delta$.
\[lem5.3\][(See Lemma 2.1 in [@Zhang].)]{} Let $\Gamma$ be a vertex transitive graph. Then $\frac{\alpha(\Gamma)}{|V(\Gamma)|}\leq\frac{\alpha(\Delta)}{|\Delta|}$ for any $\Delta\subseteq V(\Gamma)$, and equality implies that $|S\cap\Delta|=\alpha(\Delta)$ for every largest independent set $S$ of $\Gamma$.
\[lem5.4\] Let $n>m\geq2$. Then the following inequalities hold: $$\label{equa18}
\alpha(G_{r}(m,n,\mathbb{Z}_{h}))\leq\left\lfloor\beta_{0}\alpha(G_{r}(m-1,n-1,\mathbb{Z}_{h}))\right\rfloor,$$ $$\label{equa19}
\alpha(G_{r}(m,n,\mathbb{Z}_{h}))\leq\left\lfloor\beta_{0}
\left\lfloor\beta_{1}\cdots
\left\lfloor\beta_{m-r}\right\rfloor\cdots\right\rfloor\right\rfloor,$$ where $$\beta_{j}=\prod_{i=1}^tp_i^{(s_i-1)(n-m)}\frac{p_i^{n-j}-1}{p_i^{m-j}-1}\quad\hbox{for}\quad j=0,1,\ldots,m-r.$$
[[*Proof.*]{}]{}Let $$\Delta=\left(\begin{array}{cc}
1 & 0\\
0 & V(G_{r}(m-1,n-1,\mathbb{Z}_{h}))
\end{array}\right).$$ Then $\alpha(\Delta)=\alpha(G_{r}(m-1,n-1,\mathbb{Z}_{h}))$. By Lemma \[lem5.3\] and Theorem \[lem3.5\], we have $$\alpha(G_{r}(m,n,\mathbb{Z}_{h}))\leq\frac{|V(G_{r}(m,n,\mathbb{Z}_{h}))|}{|\Delta|}\alpha(\Delta)=\beta_{0}\alpha(G_{r}(m-1,n-1,\mathbb{Z}_{h})).$$ Since $\alpha(G_{r}(m,n,\mathbb{Z}_{h}))$ is a positive integer, we obtain (\[equa18\]). Since $G_{r}(r-1,n-m+r-1,\mathbb{Z}_{h})$ is a complete graph, $\alpha(G_{r}(r-1,n-m+r-1,\mathbb{Z}_{h}))=1$. By (\[equa18\]), we obtain (\[equa19\]). ${\hfill\Box\medskip}$
\[lem5.5\][(See Theorem 5.4 in [@Huang3].)]{} If $n>m\geq2$ and $m$ divides $n$, then $$\alpha(G_{m}(m,n,\mathbb{Z}_{p^s}))=p^{(s-1)(n-m)}\frac{p^{n}-1}{p^{m}-1}.$$
\[lem5.6\] If $n>m\geq2$ and $m$ divides $n$, then $$\alpha(G_{m}(m,n,\mathbb{Z}_{h}))=\prod_{i=1}^tp_i^{(s_i-1)(n-m)}\frac{p_i^{n}-1}{p_i^{m}-1}.$$
[[*Proof.*]{}]{}Since $G_{m}(m-1,n-1,\mathbb{Z}_{h})$ is a complete graph, $\alpha(G_{m}(m-1,n-1,\mathbb{Z}_{h}))=1$. By (\[equa18\]), when $n>m\geq 2$, we have $$\alpha(G_{m}(m,n,\mathbb{Z}_{h}))\leq\left\lfloor\prod_{i=1}^tp_i^{(s_i-1)(n-m)}\frac{p_i^{n}-1}{p_i^{m}-1}\right\rfloor.$$ If $n>m\geq2$ and $m$ divides $n$, by Lemma \[lem5.5\], $\alpha(G_{m}(m,n,\mathbb{Z}_{p_i}))=(p_i^n-1)/(p_i^m-1)$ for $i=1,2,\ldots,t$. By Theorem \[lem5.2\], we have $$\prod_{i=1}^tp_i^{(s_i-1)(n-m)}\frac{p_i^{n}-1}{p_i^{m}-1}\leq\alpha(G_{m}(m,n,\mathbb{Z}_{h})).$$ Therefore, the desired result follows. ${\hfill\Box\medskip}$
\[lem5.7\][(See Theorem 5.5 in [@Huang3].)]{} If $n>m\geq2$ and $n\equiv\; \ell\mod\; m$, then $$\alpha(G_{m}(m,n,\mathbb{Z}_{p^s}))\geq p^{(s-1)(n-m)}\frac{p^{n}-p^m(p^\ell-1)-1}{p^{m}-1}.$$
\[lem5.8\] If $n>m\geq2$ and $n\equiv\; \ell \mod\; m$, then $$\label{equa20}
\alpha(G_{m}(m,n,\mathbb{Z}_{h}))\geq\prod_{i=1}^tp_i^{(s_i-1)(n-m)}\frac{p_i^{n}-p_i^m(p_i^\ell-1)-1}{p_i^{m}-1}.$$
[[*Proof.*]{}]{}By Lemma \[lem5.7\], we have $$\alpha(G_{m}(m,n,\mathbb{Z}_{p_i}))\geq \frac{p_i^{n}-p_i^m(p_i^\ell-1)-1}{p_i^{m}-1}\;\hbox{for}\;i=1,2,\ldots,t.$$ By Theorem \[lem5.2\], we obtain (\[equa20\]). ${\hfill\Box\medskip}$
For two subspaces $A$ and $B$ of $\mathbb{Z}_h^n$, the [*subspace distance*]{} between $A$ and $B$ is $d(A,B):=\dim(A)+\dim(B)-2\dim(A\cap B)$. It is easy to prove that $d(A,B)\geq0,d(A,B)=0\Leftrightarrow A=B$, $d(A,B)=d(B,A)$ and $d(A,C)\leq d(A,B)+d(B,C)$ for subspaces $A,B$ and $C$ of $\mathbb{Z}_h^n$.
Huang et al. [@Huang3] discussed the Grassmannian code for $\mathbb{Z}_{p^s}^n$. As a natural extension, a [*Grassmannian code*]{} ${\cal C}$ for $\mathbb{Z}_h^n$ is a subset of ${\mathbb{Z}_h^n\brack m}$. An $(n,2r,m)_{\mathbb{Z}_h}$ [*code*]{} for $\mathbb{Z}_h^n$ is a Grassmannian code for $\mathbb{Z}_h^n$ with minimum subspace distance $2r$. An $(n,2r,m)_{\mathbb{Z}_h}$ code with the maximum cardinality is called an [*optimal*]{} $(n,2r,m)_{\mathbb{Z}_h}$ code for $\mathbb{Z}_h^n$.
Note that $d(A,B)\geq 2r$ if and only if $\dim(A\cap B)\leq m-r$ for subspaces $A,B$ in ${\mathbb{Z}_h^n\brack m}$. It follows that an $(n,2r,m)_{\mathbb{Z}_h}$ code for $\mathbb{Z}_h^n$ is an independent set of $G_r(m,n,\mathbb{Z}_h)$ and vice versa. Therefore, an optimal $(n,2r,m)_{\mathbb{Z}_h}$ code for $\mathbb{Z}_h^n$ is a largest independent set of $G_r(m,n,\mathbb{Z}_h)$ and vice versa.
Acknowledgment {#acknowledgment .unnumbered}
==============
This research is supported by National Natural Science Foundation of China (11971146).
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[^1]: Corresponding author. guojun$_-$lf@163.com
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
(BES Collaboration)\
\
M. Ablikim
- 'J. Z. Bai'
- 'Y. Ban'
- 'X. Cai'
- 'H. F. Chen'
- 'H. S. Chen'
- 'H. X. Chen'
- 'J. C. Chen'
- Jin Chen
- 'Y. B. Chen'
- 'Y. P. Chu'
- 'Y. S. Dai'
- 'L. Y. Diao'
- 'Z. Y. Deng'
- 'Q. F. Dong'
- 'S. X. Du'
- 'J. Fang'
- 'S. S. Fang[^1]'
- 'C. D. Fu'
- 'C. S. Gao'
- 'Y. N. Gao'
- 'S. D. Gu'
- 'Y. T. Gu'
- 'Y. N. Guo'
- 'Z. J. Guo[^2]'
- 'F. A. Harris'
- 'K. L. He'
- 'M. He'
- 'Y. K. Heng'
- 'J. Hou'
- 'H. M. Hu'
- 'J. H. Hu'
- 'T. Hu'
- 'G. S. Huang[^3]'
- 'X. T. Huang'
- 'X. B. Ji'
- 'X. S. Jiang'
- 'X. Y. Jiang'
- 'J. B. Jiao'
- 'D. P. Jin'
- 'S. Jin'
- 'Y. F. Lai'
- 'G. Li, [^4]'
- 'H. B. Li'
- 'J. Li'
- 'R. Y. Li'
- 'S. M. Li'
- 'W. D. Li'
- 'W. G. Li'
- 'X. L. Li'
- 'X. N. Li'
- 'X. Q. Li'
- 'Y. F. Liang'
- 'H. B. Liao'
- 'B. J. Liu'
- 'C. X. Liu'
- 'F. Liu'
- Fang Liu
- 'H. H. Liu'
- 'H. M. Liu'
- 'J. Liu[^5]'
- 'J. B. Liu'
- 'J. P. Liu'
- Jian Liu
- 'Q. Liu'
- 'R. G. Liu'
- 'Z. A. Liu'
- 'Y. C. Lou'
- 'F. Lu'
- 'G. R. Lu'
- 'J. G. Lu'
- 'C. L. Luo'
- 'F. C. Ma'
- 'H. L. Ma'
- 'L. L. Ma[^6]'
- 'Q. M. Ma'
- 'Z. P. Mao'
- 'X. H. Mo'
- 'J. Nie'
- 'S. L. Olsen'
- 'R. G. Ping'
- 'N. D. Qi'
- 'H. Qin'
- 'J. F. Qiu'
- 'Z. Y. Ren'
- 'G. Rong'
- 'X. D. Ruan'
- 'L. Y. Shan'
- 'L. Shang'
- 'C. P. Shen'
- 'D. L. Shen'
- 'X. Y. Shen'
- 'H. Y. Sheng'
- 'H. S. Sun'
- 'S. S. Sun'
- 'Y. Z. Sun'
- 'Z. J. Sun'
- 'X. Tang'
- 'G. L. Tong'
- 'G. S. Varner'
- 'D. Y. Wang[^7]'
- 'L. Wang'
- 'L. L. Wang'
- 'L. S. Wang'
- 'M. Wang'
- 'P. Wang'
- 'P. L. Wang'
- 'W. F. Wang[^8]'
- 'Y. F. Wang'
- 'Z. Wang'
- 'Z. Y. Wang'
- Zheng Wang
- 'C. L. Wei'
- 'D. H. Wei'
- 'Y. Weng'
- 'N. Wu'
- 'X. M. Xia'
- 'X. X. Xie'
- 'G. F. Xu'
- 'X. P. Xu'
- 'Y. Xu'
- 'M. L. Yan'
- 'H. X. Yang'
- 'Y. X. Yang'
- 'M. H. Ye'
- 'Y. X. Ye'
- 'G. W. Yu'
- 'C. Z. Yuan'
- 'Y. Yuan'
- 'S. L. Zang'
- 'Y. Zeng'
- 'B. X. Zhang'
- 'B. Y. Zhang'
- 'C. C. Zhang'
- 'D. H. Zhang'
- 'H. Q. Zhang'
- 'H. Y. Zhang'
- 'J. W. Zhang'
- 'J. Y. Zhang'
- 'S. H. Zhang'
- 'X. Y. Zhang'
- Yiyun Zhang
- 'Z. X. Zhang'
- 'Z. P. Zhang'
- 'D. X. Zhao'
- 'J. W. Zhao'
- 'M. G. Zhao'
- 'P. P. Zhao'
- 'W. R. Zhao'
- 'Z. G. Zhao[^9]'
- 'H. Q. Zheng'
- 'J. P. Zheng'
- 'Z. P. Zheng'
- 'L. Zhou'
- 'K. J. Zhu'
- 'Q. M. Zhu'
- 'Y. C. Zhu'
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- 'Z. A. Zhu'
- 'B. A. Zhuang'
- 'X. A. Zhuang'
- 'B. S. Zou'
date: 'Received: date / Revised version: date'
title: '**Study of ${J/\psi}$ decaying into $\omega{p\bar{p}}$**'
---
Introduction
============
Decays of the $J/\psi$ meson are regarded as being well suited for searches for new types of hadrons and for systematic studies of light hadron spectroscopy. Recently, a number of new structures have been observed in $J/\psi$ decays. These include strong near-threshold mass enhancements in the $p{\bar{p}}$ invariant mass spectrum from ${J/\psi}\rightarrow\gamma p{\bar{p}}$ decays [@bes1860], the $p \bar \Lambda$ and $K^-\bar \Lambda$ threshold enhancements in the $p \bar \Lambda$ and $K^-\bar \Lambda$ mass spectra in $J/\psi \rightarrow p K^- \bar \Lambda$ decays [@pkl], the $\omega\phi$ resonance in the $\omega\phi$ mass spectrum in the double-OZI suppressed decay $J/\psi\to\gamma \omega\phi$ [@goph], and a new resonance, the $X(1835)$, in $J/\psi\to\gamma \pi^+\pi^-\eta'$ decays [@x1835].
The enhancement $X(1860)$ in ${J/\psi}\rightarrow\gamma p{\bar{p}}$ can be fitted with an $S$- or $P$-wave Breit-Wigner (BW) resonance function. In the case of the $S$-wave fit, the mass is $1859^{+3}_{-10}$$^{+5}_{-25}$ MeV/$c^2$ and the width is smaller than 30 MeV/$c^2$ at the 90$\%$ confidence level (C.L.). It is of interest to note that a corresponding mass threshold enhancement is not observed in either $p \bar p$ cross section measurements or in $B$-meson decays [@B-ppbar].
This surprising experimental observation has stimulated a number of theoretical interpretations. Some have suggested that it is a $p \bar p$ bound state ([*baryonium*]{}) [@ppbar; @theory; @gao; @yan; @baryonium]. Others suggest that the enhancement is primarily due to final state interactions (FSI) between the proton and antiproton [@fsi1; @fsi2].
The CLEO Collaboration published results on the radiative decay of the $\Upsilon (1S)$ to the $p\bar p$ system [@cleoc], where no $p \bar p$ threshold enhancement is observed and the upper limit of the branching fraction is set at $B(\Upsilon(1S)\rightarrow\gamma
X(1860))B(X(1860)\rightarrow p\overline{p})<5\times10^{-7}$ at 90$\%$ C.L.. This enhancement is not observed in BES2 $\psi(2S) \to
\gamma p \bar p$ data either [@bes2psip] and the upper limit is set at $B(\psi(2S)\rightarrow\gamma X(1860))B(X(1860)\rightarrow
p\overline{p})<5.4\times10^{-6}$ at 90$\%$ C.L..
The investigation of the near-threshold ${p\bar{p}}$ invariant mass spectrum in other $J/\psi$ decay modes will be helpful in understanding the nature of the observed new structures and in clarifying the role of ${p\bar{p}}$ FSI effects. If the enhancement seen in ${J/\psi}{\rightarrow}{\gamma}{p\bar{p}}$ is from FSI, it should also be observed in other decays, such as ${J/\psi}{\rightarrow}\omega{p\bar{p}}$, which motivated our study of this channel. In this paper, we present results from an analysis of ${J/\psi}{\rightarrow}{\pi^+}{\pi^-}{\pi^0}{p\bar{p}}$ using a sample of $5.8 \times 10^7 J/\psi$ decays recorded by the BESII detector at the Beijing Electron-Positron Collider (BEPC). BES is a conventional solenoidal magnetic detector that is described in detail in Ref. [@bes]. BESII is the upgraded version of the BES detector [@besii]. A twelve-layer Vertex Chamber (VC) surrounds a beryllium beam pipe and provides track and trigger information. A forty-layer main drift chamber (MDC) located just outside the VC provides measurements of charged particle trajectories over $85\%$ of the total solid angle; it also provides ionization energy loss ($dE/dx$) measurements that are used for particle identification (PID). A momentum resolution of $\sigma
_p/p =1.78\%\sqrt{1+p^2}$ ($p$ in GeV/$c$) and a $dE/dx$ resolution of $\sim$8% are obtained. An array of 48 scintillation counters surrounding the MDC measures the time of flight (TOF) of charged particles with a resolution of about 200 ps for hadrons. Outside of the TOF counters is a 12 radiation length, lead-gas barrel shower counter (BSC), that operates in self quenching streamer mode and measures the energies and positions of electrons and photons over $80\%$ of the total solid angle with resolutions of $\sigma_{E}/E=0.21/\sqrt{E}$ ($E$ in GeV/$c^{2}$), $\sigma_{\phi}=7.9$ mrad, and $\sigma_{z}=2.3$ cm. External to a solenoidal coil, which provides a 0.4 T magnetic field over the tracking volume, is an iron flux return that is instrumented with three double-layer muon counters that identify muons with momentum greater than 500 MeV$/c$.
Monte-Carlo simulation is used to determine the mass resolution and detection efficiency, as well as to estimate the contributions from background processes. In this analysis, a GEANT3-based Monte-Carlo program (SIMBES), with a detailed simulation of the detector performance, is used. As described in detail in Ref. [@SIMBES], the consistency between data and Monte-Carlo has been validated using many physics channels from both $J/\psi$ and $\psi(2S)$ decays.
Analysis of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$, $\omega \to {\pi^+}{\pi^-}{\pi^0}$ {#analysis}
=======================================================================================
For candidate ${J/\psi}{\rightarrow}{\pi^+}{\pi^-}{\pi^0}{p\bar{p}}$ events, we require four well reconstructed charged tracks with net charge zero in the MDC and at least two isolated photons in the BSC. Each charged track is required to be well fitted to a helix, be within the polar angle region $|\cos\theta| <
0.8$, have a transverse momentum larger than 70 MeV/$c$, and have a point of closest approach of the track to the beam axis that is within 2 cm of the beam axis and within 20 cm from the center of the interaction region along the beam line. For each track, the TOF and $dE/dx$ information is combined to form a particle identification confidence level for the $\pi, K$ and $p$ hypotheses; the particle type with the highest confidence level is assigned to each track. The four charged tracks are required to consist of an unambiguously identified $p$, $\bar{p}$, ${\pi^+}$ and ${\pi^-}$ combination. An isolated neutral cluster is considered as a photon candidate when the angle between the nearest charged track and the cluster is greater than 5$^{\circ}$, the angle between the $\bar{p}$ track and the cluster is greater than 25$^{\circ}$ [@angpb], the first hit is in the beginning of six radiation lengths of the BSC, the difference between the angle of the cluster development direction in the BSC and the photon emission direction is less than 30$^{\circ}$, and the energy deposited in the shower counter is greater than 50 MeV. A four-constraint kinematic fit is performed to the hypothesis $J/\psi \to
{p\bar{p}}{\pi^+}{\pi^-}{\gamma}{\gamma}$, and, in the cases where the number of photon candidates exceeds two, the combination with the smallest $\chi^{2}_{{p\bar{p}}{\pi^+}{\pi^-}{\gamma}{\gamma}}$ value is selected. We further require that $\chi^{2}_{{p\bar{p}}{\pi^+}{\pi^-}{\gamma}{\gamma}} < 20$.
Figure \[ompi0\] shows the $\gamma \gamma$ invariant mass of the events which survive the above-listed criteria, where a distinct ${\pi^0}\to {\gamma}{\gamma}$ signal is evident. Candidate ${\pi^0}$ mesons are selected by requiring $|M_{{\gamma}{\gamma}}-m_{{\pi^0}}|<0.04$ GeV/$c^2$. After this selection, a total of 15260 events is retained. The ${{\pi^+}{\pi^-}{\pi^0}}$ invariant mass spectrum for these events is shown as data points with error bars in Fig. \[momegafit\], where prominent $\omega$ and $\eta$ signals are observed.
The backgrounds in the selected event sample are studied with Monte-Carlo simulations. We generated ${J/\psi}\rightarrow$ ${p\bar{p}}{\pi^+}{\pi^-}{\pi^0}$ decays as well as a variety of processes that are potential sources of background: ${J/\psi}\to {p\bar{p}}{\eta^{\prime}}({\eta^{\prime}}\to{\pi^+}{\pi^-}\eta)$; ${p\bar{p}}{\eta^{\prime}}({\eta^{\prime}}\to\rho^{0}{\gamma})$; ${p\bar{p}}{\pi^+}{\pi^-}$; ${\Lambda\bar{\Lambda}}{\pi^0}$; $\Sigma^0\bar{\Sigma}^0$; $\Sigma(1385)^-\bar{\Sigma}^+$; $\eta_c{\gamma}$; $\Delta^{++}\Delta^{--}$; ${\gamma}{p\bar{p}}{\pi^+}{\pi^-}$; $\Delta^{++}\bar{p}{\pi^-}$; $\Lambda\bar{\Sigma}^-\pi^+$ (+ [*c.c.*]{}); $\Sigma^0\pi^0\bar{\Lambda}$; $\Sigma(1385)^0\bar{\Sigma}^0$; $\Delta^{++}\Delta^{--}{\pi^0}$; and $\Xi^0\bar{\Xi}^0$, in proportion to the branching fractions listed in the Particle Data Group (PDG) Tables [@pdg2004]. The main background sources are found to be the decays $J/\psi \to \Lambda\bar{\Sigma}^-\pi^+$ (+ [*c.c.*]{}) and $\Delta^{++}\Delta^{--}\pi^0$. The ${\pi^+}{\pi^-}{\pi^0}$ invariant mass spectrum for background events that survive the selection criteria is shown as a solid histogram in Fig. \[momegafit\]; here no signal for $\omega{p\bar{p}}$ is evident.
The branching fraction for $J/\psi \to \omega p \bar p$ is computed using the relation $$B({J/\psi}{\rightarrow}\omega{p\bar{p}}) = \frac{N_{obs}}{N_{{J/\psi}}\cdot {\varepsilon}\cdot
B(\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0})\cdot B({\pi^0}{\rightarrow}{\gamma}{\gamma})}.$$
Here, $N_{obs}$ is the number of observed events; $N_{{J/\psi}}$ is the number of ${J/\psi}$ events, $(57.7\pm 2.6)\times 10^6$ [@jpsinum]; ${\varepsilon}$ is the Monte-Carlo determined detection efficiency; and $B(\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0})$ and $B({\pi^0}{\rightarrow}{\gamma}{\gamma})$ are the $\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$ and ${\pi^0}{\rightarrow}{\gamma}{\gamma}$ branching fractions.
The ${{\pi^+}{\pi^-}{\pi^0}}$ invariant mass spectrum shown in Fig. \[momegafit\] is fitted using an unbinned maximum likelihood fit with resolution broadened BW functions to represent the $\omega$ and $\eta$ signal peaks. The mass resolutions are obtained from Monte-Carlo simulation to be 12 MeV$/c^{2}$ for the $\omega$ and and 14 MeV$/c^{2}$ for the $\eta$. The masses and widths of the $\omega$ and $\eta$ are fixed at their PDG values [@pdg2004]. A 4th-order Chebychev polynomial is used to describe the background. The fit gives an $\omega$ signal yield of 2449$\pm$69 events. The detection efficiency from a uniform-phase-space Monte-Carlo simulation of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ ($\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$, ${\pi^0}{\rightarrow}{\gamma}{\gamma}$) is $4.9 \pm 0.1)$%. The branching fraction is determined to be: $$B({J/\psi}{\rightarrow}\omega{p\bar{p}})=(9.8\pm0.3)\times 10^{-4},$$ where the error is statistical only.
We use this sample with $|M_{{\pi^+}{\pi^-}{\pi^0}}-0.783|<0.03$ GeV/$c^2$ to study the near-threshold region of the ${p\bar{p}}$ invariant mass spectrum. Figure \[dalitz\] shows a Dalitz plot for the selected ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ candidates, where no obvious structure is observed although it is not a uniform distribution. Figure \[mppbfit\] shows the threshold behavior of the $p \bar p$ invariant mass distribution. The dotted curve in the figure indicates how the acceptance varies with invariant mass.
The backgrounds in the $p \bar p$ threshold region mainly come from the decays of $J/\psi \to \Lambda\bar{\Sigma}^-\pi^+$ (+ [*c.c.*]{}) and $\Delta^{++}\Delta^{--}\pi^0$. The $M({p\bar{p}})$ dependence of this background can be modeled by appropriately scaled data from the $\omega$ sidebands (0.663 GeV/c$^2$ $<M_{{{\pi^+}{\pi^-}{\pi^0}}}<$0.723 GeV/c$^2$ and 0.843 GeV/c$^2$ $<M_{{{\pi^+}{\pi^-}{\pi^0}}}<$0.903 GeV/c$^2$). The contributions of sideband and non-resonant $\omega{p\bar{p}}$ events can be well described by a function of the form $$f(\delta) =
N(\delta^{\frac{1}{2}}+a_1\delta^{\frac{3}{2}}+a_2\delta^{\frac{5}{2}})$$ with $\delta\equiv M_{{p\bar{p}}} - 2m_p $.
In Fig. \[mppbfit\], no significant excess over the background plus non-resonant terms is evident. A Bayesian approach [@pdg2004] is employed to extract the upper limit on the branching fraction of ${J/\psi}{\rightarrow}\omega X(1860)$. An acceptance-weighted $S$-wave BW function $$BW(M) \propto \frac{q^{(2l+1)}k^3}{(M^2-M_0^2)^2-M_0^2\Gamma^2}\cdot\varepsilon(M)$$ is used to represent the low-mass enhancement. Here, $\Gamma$ is a constant width, $q$ is the momentum of proton in the ${p\bar{p}}$ rest frame, $l$ is the relative orbital angular momentum of $p$ and $\bar
p$, $k$ is the momentum of $\omega$, and $\varepsilon(M)$ is the detection efficiency obtained from Monte-Carlo simulation. The mass and width of the BW signal function are fixed to 1860 MeV/c$^2$ and 30 MeV/c$^2$, respectively. The contributions of background and non-resonant $\omega{p\bar{p}}$ events are presented by the function form $f(\delta)$, where the parameters $a_1$ and $a_2$ are allowed to float. As shown in Fig. \[mppbfit\], the solid curve is the fit of the $M_{{p\bar{p}}}$ - 2$m_p$ with the BW signal function and $f(\delta)$ function described above. Using the Bayesian method, the 95% C.L. upper limit on the number of observed signal events is 29. Since the $J^{PC}$ of X(1860) is unknown, we use simulated events distributed uniformly in phase space to determine a detection efficiency of ${J/\psi}{\rightarrow}\omega
X(1860)$ ($X(1860){\rightarrow}{p\bar{p}}$, $\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$, ${\pi^0}{\rightarrow}{\gamma}{\gamma}$) of $(4.7 \pm 0.1)$%. The upper limit of the branching fraction, without considering the systematic errors, is then: $$B({J/\psi}{\rightarrow}\omega X(1860))\cdot B( X(1860){\rightarrow}{p\bar{p}}))$$ $$< \frac{N_{obs}^{UL}}{N_{{J/\psi}}\cdot {\varepsilon}\cdot B(\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0})\cdot B({\pi^0}{\rightarrow}{\gamma}{\gamma})} = 1.2\times 10^{-5}.$$
Systematic errors {#syserrs}
=================
The systematic errors on the branching fractions are mainly due to uncertainties in the MDC tracking, kinematic fitting, particle identification (PID), photon detection, background estimation, the model used to describe hadronic interactions in the material of the detector, and the uncertainty of the total number of $J/\psi$ decays in the data sample.
The systematic error associated with the tracking efficiency has been carefully studied [@SIMBES]. The difference of the tracking efficiencies between data and Monte-Carlo is 2% per charged track; an 8% contribution to the systematic error associated with the efficiency for detecting the four-track final state is assigned. In Ref. [@SIMBES; @pnpi], the efficiencies for charged particle identification and photon detection are analyzed in detail. The systematic errors from PID and photon detection are 2% per proton (antiproton), 1% per pion and 2% per photon. In this analysis, with four charged tracks and two isolated photons; 6% is taken as the systematic error due to PID and 4% due to photon detection. The uncertainty due to kinematic fitting is studied using a number of exclusive $J/\psi$ and $\psi(2S)$ decay channels that are cleanly isolated without a kinematic fit [@rhopi; @etanpi]. It is found that the Monte-Carlo simulates the kinematic fit efficiency at the 5% or less level of uncertainty for almost all channels tested. Therefore, we take 5% as the systematic error due to the kinematic fit.
The background uncertainties come from the uncertainty of the background shape. For the branching fraction measurement of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$, changing the order of the polynomial background causes an uncertainty in the number of background events. For the upper limit determination of ${J/\psi}{\rightarrow}\omega X(1860)$, the uncertainty of background shape can be determined by the fitting results with the background shape fixed to the function form $f(\delta)$, derived from fitting the scaled $\omega$ sideband data plus phase-space generated $\omega{p\bar{p}}$ MC events. Respectively, 5% and 10% are taken as the systematic errors due to the background uncertainties in the branching fraction measurement of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ and the upper limit determination of ${J/\psi}{\rightarrow}\omega X(1860)$.
Different simulation models for the hadronic interactions in the material of the detector (GCALOR/FLUKA) [@gcalor; @fluka] give different efficiencies. Respectively, 4.8% and 11.4% are taken as the systemic errors due to the different hadronic models in the branching fraction measurement of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ and the upper limit determination of ${J/\psi}{\rightarrow}\omega X(1860)$. In addition, if the $J^{P}$ of X(1860) is $0^{-}$ , the angular distribution of the $\omega$ would be $1+cos^{ 2}\theta$. A Monte-Carlo sample generated with the $\omega$ produced with a $1+cos^{2}\theta$ distribution and a uniform distribution for the X(1860) decay into ${p\bar{p}}$ results in an 8.5% reduction in detection efficiency. This difference is taken as the systematic error associated with the production model.
The branching fractions of $\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$ and ${\pi^0}{\rightarrow}{\gamma}{\gamma}$ are taken from the PDG tables. The errors of the intermediate decay branching fractions, as well as the uncertainty of the number of $J/\psi$ events [@jpsinum] also result in the systematic errors in the measurements.
The systematic errors from the different sources are listed in Table \[syserr\]. The total systematic errors for the branching fractions are obtained by adding up all the systematic sources in quadrature.
B(${J/\psi}{\rightarrow}\omega{p\bar{p}}$) Upper Limit
------------------------- -------------------------------------------- ------------- --
Tracking efficiency 8 8
Photon efficiency 4 4
Particle ID 6 6
Kinematic fit 5 5
Background uncertainty 5 10
Hadronic model 4.8 11.4
Production model - 8.5
Intermediate decays 0.8 0.8
Total ${J/\psi}$ events 4.7 4.7
Total systematic error 14.6 21.6
: Systematic error sources and contributions (%).
\[syserr\]
Summary
=======
With a $5.8 \times 10^7 {J/\psi}$ event sample in the BESII detector, the branching fraction ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ is measured as: $$B({J/\psi}{\rightarrow}\omega{p\bar{p}})=(9.8\pm 0.3\pm 1.4)\times 10^{-4}.$$
No obvious near-threshold ${p\bar{p}}$ mass enhancement in ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ is observed, and the FSI interpretation of the $p \bar p$ enhancement in $J/\psi \to \gamma p \bar p$ is disfavored. A conservative estimate of the upper limit is determined by lowering the efficiency by one standard deviation. In this way, a 95% confidence level upper limit on the branching fraction $$B({J/\psi}{\rightarrow}\omega X(1860))\cdot B( X(1860){\rightarrow}{p\bar{p}}))< 1.5 \times 10^{-5}$$ is determined. The absence of the enhancement $X(1860)$ in $J/\psi \to \omega p \bar p$, $\Upsilon(1S) \to \gamma p \bar p$ and $\psi(2S) \to \gamma p \bar p$ also indicates its similar production property to that of $\eta'$ [@ichep06; @klempt], [*i.e.*]{}, $X(1860)$ is only largely produced in $J/\psi$ radiative decays.
Acknowledgments
===============
The BES collaboration thanks the staff of BEPC and computing center for their hard efforts. This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10491300, 10225524, 10225525, 10425523, 10625524, 10521003, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos. U-11, U-24, U-25, and the Knowledge Innovation Project of CAS under Contract Nos. U-602, U-34 (IHEP), the National Natural Science Foundation of China under Contract No. 10225522 (Tsinghua University), and the Department of Energy under Contract No. DE-FG02-04ER41291 (U. Hawaii).
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[^1]: Current address: DESY, D-22607, Hamburg, Germany
[^2]: Current address: Johns Hopkins University, Baltimore, MD 21218, USA
[^3]: Current address: University of Oklahoma, Norman, OK 73019, USA
[^4]: Current address: Universite Paris XI, LAL-Bat. 208–BP34, 91898 ORSAY Cedex, France
[^5]: Current address: Max-Plank-Institut fuer Physik, Foehringer Ring 6, 80805 Munich, Germany
[^6]: Current address: University of Toronto, Toronto M5S 1A7, Canada
[^7]: Current address: CERN, CH-1211 Geneva 23, Switzerland
[^8]: Current address: Laboratoire de l’Acc[é]{}l[é]{}rateur Lin[é]{}aire, Orsay, F-91898, France
[^9]: Current address: University of Michigan, Ann Arbor, MI 48109, USA
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Matthew Buican$^{\diamondsuit, 1}$ and Takahiro Nishinaka$^{\clubsuit, 2,3}$'
bibliography:
- 'chetdocbib.bib'
date: May 2017
title: |
On Irregular Singularity Wave Functions\
and Superconformal Indices
---
Introduction
============
Inspired by constructions of certain four-dimensional (4D) superconformal indices as correlators in 2D topological field theory (TFT) on a (punctured) Riemann surface ${\mathcal{C}}$ [@Gadde:2011ik], we proposed a generalization in [@Buican:2015ina] that leads to closed-form expressions for the Schur limit of the superconformal indices of two infinite sets of Argyres-Douglas (AD) theories that arise from twisted compactifications of the 6D $A_1$ $(2,0)$ theory on ${\mathcal{C}}$—the so-called $(A_1, A_{2n-3})$ and $(A_1, D_{2n})$ superconformal field theories (SCFTs).[^1] In addition to giving exact information about non-trivial sectors of these theories (the so-called Schur operators [@Gadde:2011uv; @Beem:2013sza]) and characterizing new states in 2D $SU(2)$ $q$-deformed Yang-Mills (see [@Cordes:1994fc] for a review and, e.g., [@deHaro:2006uvl; @Kimura:2008gs; @Szabo:2013vva] for other recent developments), these indices contain a surprise: they encode information about the ${\mathcal{N}}=2$ chiral operators[^2] parameterizing the Coulomb branch even though ${\mathcal{N}}=2$ chiral operators do not contribute directly to the Schur index [@Buican:2015hsa]. These results may point to the existence of a deeper structure at work in 4D ${\mathcal{N}}=2$ SCFTs (see [@Fredrickson:2017yka] for interesting recent progress in this direction). In fact, Coulomb branch physics is at the heart of a complementary approach to computing these indices via BPS state counting [@Cordova:2015nma] (building on results in [@Iqbal:2012xm]).
More recently, many papers have appeared that include generalizations to other classes of Argyres-Douglas theories[^3] and other limits of the superconformal index [@Buican:2015tda; @Song:2015wta; @Cecotti:2015lab; @Xie:2016evu] as well as to the full superconformal index [@Maruyoshi:2016tqk; @Maruyoshi:2016aim; @Agarwal:2016pjo] (and also to minimal interacting deformations of AD theories [@Xie:2016hny; @Buican:2016hnq]). However, many interesting Argyres-Douglas theories remain to be explored, and various aspects of the structure underlying these theories remain to be uncovered (see [@Cordova:2016uwk; @Cordova:2017ohl; @Cordova:2017mhb] for interesting recent progress).
In this paper, we generalize our discussion in [@Buican:2015ina] and propose the following simple wave functions for certain irregular punctures in $SU(N)$ $q$-deformed Yang-Mills theory (thus generalizing our earlier results from $N=2$ to all $N\ge2$) $$\begin{aligned}
{\widetilde}{f}_{R}^{(n)}(q;{\bf x}) &= \prod_{k=1}^\infty\left(\frac{1}{1-q^k}\right)^{N-1}q^{nC_2(R)}\text{Tr}_R\left[q^{-\frac{n}{2}F^{ij}h_ih_j}\prod_{i=1}^{N-1}(x_1\cdots x_i)^{h_i}\right]~,
\label{eq:wf1}\end{aligned}$$ where $n\ge2$ is an integer, $q$ is a fugacity, $R$ is an irreducible representation of $A_{N-1}$ with quadratic Casimir $C(R)$ and Cartans $h_i$ (in the Chevalley basis[^4]), ${\bf x}=(x_1,\cdots,x_{N-1})$ are flavor fugacities, and the factor $F^{ij}$ is the quadratic form matrix, i.e., the inverse of the Cartan matrix $$\begin{aligned}
\label{Cartinv}
F
= \frac{1}{N}\left[
\begin{array}{ccccc}
N-1 & N-2 & N-3 & \cdots & 1\\
N-2 & 2(N-2)& 2(N-3)& \cdots & 2\\
N-3 & 2(N-3) & 3(N-3) & \cdots & 3\\
\vdots &\vdots& \vdots& \ddots & \vdots \\
1 & 2& 3&\cdots& N-1\\
\end{array}
\right]~.\end{aligned}$$ In particular, this wave function can be used to construct Schur indices for the $(A_{N-1}, A_{N(n-1)-1})$ Argyres-Douglas theories $$\begin{aligned}
\mathcal{I}_{(A_{N-1},A_{N(n-1)-1})}(q;{\bf x}) = \sum_{R}C_R(q){\widetilde}{f}^{(n)}_R(q;{\bf x})~,
\label{eq:general} \end{aligned}$$ where the sum is taken over all irreducible representations of $A_{N-1}$, and the coefficients, $C_R$, take the form $$\begin{aligned}
\label{topological}
C_{R}(q) = \frac{\prod_{k=1}^{N-1}(1-q^k)^{N-k}}{(q;q)_\infty^{N-1}}\text{dim}_q R = \frac{\prod_{k=1}^{N-1}(1-q^k)^{N-k}}{(q;q)_\infty^{N-1}}\chi_R^{su(N)}(q^{-\frac{N-1}{2}},q^{-\frac{N-3}{2}},\cdots,q^{\frac{N-1}{2}}) ~,\end{aligned}$$ as conjectured in [@Gadde:2011ik].[^5] Here, our convention for the character is such that $\chi^{su(N)}_R(x_1,\cdots,x_{N}) \equiv \text{Tr}_{R} \left[\prod_{i=1}^{N-1}(x_1\cdots x_i)^{h_i}\right]$ with $x_{N} \equiv (x_1\cdots x_{N-1})^{-1}$.[^6]
Therefore, in addition to providing a description of new states in $SU(N)$ $q$-deformed Yang-Mills theory, our expression in can be used to construct closed-form expressions for Schur indices of a doubly infinite set of strongly interacting SCFTs. This proposal completes the construction of all such indices for Argyres-Douglas theories of type $(A_N, A_M)$. Indeed, these indices have not been previously constructed for theories of type $(A_{N-1}, A_{N(n-1)-1})$ with $N>2$ (with the exception of the $(A_3, A_3)$ and $(A_2, A_5)$ cases for which expressions involving integrals over gauge groups exist [@Buican:2015ina; @Buican:2015tda], but no simple sum of the type in has been found[^7]).
One interesting aspect of the $(A_{N-1}, A_{N(n-1)-1})$ theories with $N>2$ is that they typically have exactly marginal deformations (if $n=2$, there are $N-3$ such deformations, and, if $n>2$, there are $N-2$ exactly marginal deformations). While the index is an invariant of the resulting conformal manifolds, the $S$-duality groups (see the interesting recent discussion in [@Caorsi:2016ebt]) act on the index through discrete symmetries. Our compact expressions for the Schur indices make it possible to explore the discrete symmetries of the index efficiently.
Moreover, as we will see in detail, our formulae encode a highly non-trivial set of renormalization group (RG) flows that typically take conformal manifolds in the ultraviolet (UV) and often map them to products of conformal manifolds in the IR along with various isolated factors. While we leave a deeper exploration of such RG flows and the laws they obey to future work, we develop a simple monopole vev RG flow" formalism to study these flows in the theories related by mirror symmetry to the $S^1$ reductions of our AD theories of interest (we explain why the reduction along the circle commutes with the RG flow).
Another aspect of our proposal is that it immediately gives us an infinite set of new superconformal indices for free. Indeed, simply by including already-existing expressions for wave functions corresponding to an additional regular puncture in the $SU(N)$ $q$-deformed Yang-Mills theory, we generate Schur indices for infinitely many co-called type IV" Argyres-Douglas theories [@Xie:2012hs; @Xie:2013jc] $$\begin{aligned}
\label{2puncture}
\mathcal{I}_{(I_{N,N(n-1)},R_Y)}(q;{\bf x};{\bf y}) = \sum_{R}{\widetilde}{f}^{(n)}_R(q;{\bf x})f_R^{Y}(q;{\bf y})~, \end{aligned}$$ where $$\begin{aligned}
\label{Regular}
f_{R}^{Y}(q;{\bf y}) &= P.E.\left[\frac{q}{1-q}\chi^{\rho(Y)}(q,{\bf y})\right]\chi^{\text{SU}(N)}_R\left(\Lambda^Y({\bf y})\right)~,\end{aligned}$$ ${\bf y}$ is the flavor fugacity associated with the regular puncture, $\Lambda^Y({\bf y})$ is the set of $N-1$ fugacities corresponding to the Young diagram $Y$ (see [@Gadde:2011ik]), $\chi^{\rho(Y)}(q,{\bf y})$ is a polynomial in $q$ determined by $Y$, and $$P.E.\left[G(a_1,\cdots,a_p)\right]\equiv\exp\left[\sum_{n=1}^{\infty}{1\over n}G(a_1^n,\cdots, a_p^n)\right]~,$$ for any function of the fugacities, $G$.
Finally, the fact that our expressions are written in terms of simple Lie algebra data makes it tempting to speculate about possible generalizations of our expressions beyond $A_{N-1}$ (and perhaps beyond the theories one can engineer from compactifications of the $(2,0)$ theory on a Riemann surface, ${\mathcal{C}}$). In section \[speculation\], we explore these ideas further.
The plan of this paper is as follows. In the next section, we briefly review basic aspects of the $(A_{N-1}, A_{N(n-1)-1})$ theories including their Seiberg-Witten curves and exactly marginal deformations. We then move on to discuss some basic theory-specific consistency checks of our proposal that allow us to make contact with previous results in the literature. Afterwards, we describe our monopole vev RG flow formalism and explain how our formulae for the $(A_{N-1}, A_{N(n-1)-1})$ indices encode 4D ancestors of these flows. We then discuss the discrete symmetries of the index and the implications for the corresponding chiral algebras (in the sense of [@Beem:2013sza]). Before concluding, we speculate about the hypothetical exotic ${\mathcal{N}}=2$ SCFTs alluded to above.
AD theories from 6d (2,0) $A_{N-1}$ theories {#sec:Hitchin}
============================================
All of the $(A_{N-1}, A_{N(n-1)-1})$ and $(I_{N,N(n-1)}, R_Y)$ SCFTs discussed above are in class ${\mathcal{S}}$. A theory, ${\mathcal{T}}_{{\mathcal{C}}}$, of class ${\mathcal{S}}$ can be engineered by taking the six-dimensional $(2,0)$ theory with any ADE Lie algebra, $\mathfrak{g}$, and compactifying it on a (punctured) Riemann surface, ${\mathcal{C}}$. In order to obtain a 4D theory with ${\mathcal{N}}=2$ supersymmetry (SUSY), one performs a partial topological twist on ${\mathcal{C}}$ (thereby breaking $SO(5)_R\to SO(2)_R\times SO(3)_R$) and imposes a BPS boundary condition at each puncture [@Gaiotto:2009hg; @Gaiotto:2009we; @Witten:1997sc]. One can understand these boundary conditions as corresponding to half-BPS co-dimension two defects that fill the four-dimensional spacetime and endow ${\mathcal{T}}_{{\mathcal{C}}}$ with flavor symmetries.
In our earlier paper [@Buican:2015ina] we focused on the case $\mathfrak{g}=A_1$, while in this paper we generalize to the case $\mathfrak{g}=A_{N-1}$ for $N\ge2$. This theory supports a plethora of somewhat simpler to understand regular“ defects [@Gaiotto:2009hg; @Gaiotto:2009we; @Gukov:2006jk; @Chacaltana:2012zy] in addition to a bewildering array of so-called irregular” defects [@Witten:2007td; @Gaiotto:2009hg; @Bonelli:2011aa; @Xie:2012hs; @Wang:2015mra]. In all cases, we can describe the Coulomb branch physics and aspects of the flavor symmetries by considering a Higgs field, $\varphi$, of an $A_{N-1}$ Hitchin system living on ${\mathcal{C}}$ (this is an ${(\rm End}V)$-valued meromorphic $(1,0)$ form on ${\mathcal{C}}$, where $V$ is an $SU(N)$ bundle). Casimirs of $\varphi$ then correspond to vevs of ${\mathcal{N}}=2$ chiral operators, $\langle{\mathcal{O}}\rangle$, that parameterize the Coulomb branch of the 4D theory. In the case of a regular defect, $\varphi$ has a simple pole at the insertion point (which we refer to as a regular puncture"), and the corresponding flavor symmetry is a subgroup $G\subseteq SU(N)$ depending on the nature of the regular puncture.[^8] If all the punctures are regular, the theory is typically superconformal.
On the other hand, in the case of an irregular defect, the $\varphi$ field has a pole of order $n+1$ (with $n>0$ referred to as the rank“ of the defect) at the insertion point (which we refer to as an irregular puncture” on ${\mathcal{C}}$). In the theories we study below, $n\in\mathbb{Z}_{\ge0}$, but there are more general possibilities. The flavor symmetry group induced by such a singularity will generically be $U(1)^{N-1}$ in the cases we consider below (but, again, there are more general possibilities). According to the discussion in [@Bonelli:2011aa; @Xie:2012hs], ${\mathcal{T}}_{{\mathcal{C}}}$ is an SCFT only if ${\mathcal{C}}$ is a $\mathbb{CP}^1$ with either one irregular puncture and no regular punctures or with one irregular puncture and one regular puncture. In either of these two cases, ${\mathcal{T}}_{{\mathcal{C}}}$ will typically have non-integer scaling dimension ${\mathcal{N}}=2$ chiral primaries and will therefore be an AD SCFT. We will encounter both situations below.
Irregular singularities of type I
---------------------------------
In this section, we briefly describe the form of the Higgs field $\varphi$ near the irregular punctures we introduced above. The particular irregular singularities we are interested in are called type I" in the nomenclature of [@Xie:2012hs]. Placing such a singularity at the point $z=\infty$ on ${\mathcal{C}}$ yields an expansion for $\varphi$ of the form $$\label{irregsing}
\varphi(z)=dz\left[M_1z^{n-1}+M_2z^{n-2}+\cdots+M_n+{M_{n+1}\over z}+{\mathcal{O}}(z^{-2})\right]~,$$ where the $M_i$ are arbitrary traceless $N\times N$ matrices. The singular terms above (i.e., those up to ${\mathcal{O}}(z^{-2})$) encode the various relevant and exactly marginal deformations of the ${\mathcal{T}}_{{\mathcal{C}}}$ SCFT. In the case of the $(A_{N-1}, A_{N(n-1)-1})$ theories, $\varphi$ is regular for $z\ne\infty$. On the other hand, for the theories of type $(I_n, R_Y)$, $\varphi$ has an additional regular singularity at $z=0$. The corresponding simple pole encodes the additional flavor symmetry arising from the puncture at $z=0$.
Seiberg-Witten curves of the AD theories and exactly marginal couplings
-----------------------------------------------------------------------
Given the expansion for the Higgs field in , we can obtain the Seiberg-Witten curve for the $(A_{N-1},A_{N(n-1)-1})$ SCFT by considering the spectral curve (i.e., $\det\left(xdz-\varphi(z)\right)=0$) and using reparametrizations that do not affect the physics $$\begin{aligned}
\label{curve}
0 = x^{N} + a_2 x^{N-2}z^{2(n-1)} + a_3 x^{N-3}z^{3(n-1)} + \cdots + a_{N-1}xz^{(N-1)(n-1)} + z^{N(n-1)} + \cdots~,\end{aligned}$$ where the final ellipses contain lower-dimensional terms.[^9] The $a_i$ coefficients we have written explicitly in are dimensionless and give the $N-2$ exactly marginal couplings of the theory for $n>2$. Note that for $n=2$, there are only $N-3$ such couplings because one of the $a_i$ can be eliminated by a change of coordinates. There are also $N-1$ mass parameters corresponding to the deformations of the curve by $x^k z^{(n-1)(N-k)-n}$ for $k=0,\cdots,N-2$.
Hence, to summarize, for sufficiently large $N$ we have conformal manifolds of the following dimensions $$\begin{aligned}
\label{confmfld}
&&\dim_{\mathbb{C}}\left({\mathcal{M}}^{\rm conf}_{(A_{N-1},A_{N(n-1)-1})}\right)=N-2~, \ \ \ (n>2) ~,\cr&&\dim_{\mathbb{C}}\left({\mathcal{M}}^{\rm conf}_{(A_{N-1},A_{N-1})}\right)=N-3~.\end{aligned}$$ Moreover, using methods described in [@Xie:2012hs], it is straightforward to compute the corresponding conformal anomalies $$\begin{aligned}
\label{acAD}
a&=&{(N-1)(2N^4(n-1)^2+2N^3(n-1)^2-5N(n-1)-5)\over24(N(n-1)+1)}~, \cr c&=&{(N-1)(N(n-1)(N^2(1+N)(n-1)-2)-2)\over12(N(n-1)+1)}~.\end{aligned}$$
The curves, spaces of marginal deformations, and anomalies for the theories with an additional regular singularity are more complicated, but can be found using similar methods. We will not discuss their detailed form here but instead refer the reader to [@Xie:2012hs; @Xie:2013jc] for details.
Reduction to three dimensions
-----------------------------
As we will see below, our formulas in and contain a wealth of information about RG flows between different theories of type $(A_{N-1}, A_{N(n-1)-1})$ via the residues of various poles in the Schur index. To check these predictions, we will find it useful to compare our results with RG flows for these same theories compactified on a circle (where the $(A_{N-1}, A_{N(n-1)-1})$ SCFTs flow to 3D ${\mathcal{N}}=4$ SCFTs that are believed to be described by a 3D ${\mathcal{N}}=4$ Lagrangian). The fundamental reason we can learn about 4D RG flows from 3D ones is that our procedure manifestly preserves eight Poincaré supercharges throughout and so non-perturbative superpotentials cannot be generated.
In fact, it will be simpler to study the 3D mirrors of the direct $S^1$ reductions of these theories [@Xie:2012hs; @Xie:2013jc]. For the $(A_{N-1}, A_{N(n-1)-1})$ SCFT, the 3D mirror is given by a $U(1)^{N-1}$ quiver gauge theory with each $U(1)$ factor having $n-1$ fundamental hypermultiplets ($Q_{i,k}, {\widetilde}Q_{i,k}^{\dagger}$ with $i\ne j$ labeling the $U(1)$ nodes) and $n-1$ bifundamental hypermultiplets ($Q_{i,j,k}, {\widetilde}Q_{i,j,k}^{\dagger}$) between each pair of nodes (where we define $Q_{i,j,k}=Q_{j,i,k}, {\widetilde}Q_{i,j,k}^{\dagger}={\widetilde}Q_{j,i,k}^{\dagger}$). The resulting ${\mathcal{N}}=4$ superpotential reads $$\label{N4super}
W=\sum_{i=1}^{N-1}\sum_{k=1}^{n-1}\Phi_i\left(Q_{i,k}{\widetilde}Q_{i,k}+q_{i,j}\sum_{j\ne i}Q_{i,j,k}{\widetilde}Q_{i,j,k}\right)~,$$ where $i,j=1,\cdots, N-1$, $k=1,\cdots,n-1$, and $q_{i,j}=-q_{j,i}=1$ for $i<j$.
The theories with an additional regular puncture have closely related 3D mirrors that have been proposed in [@Xie:2012hs; @Xie:2013jc]. We will not discuss these latter dimensionally reduced theories further below, but our discussion of RG flows can be extended to these cases as well.
The Schur index {#sec:formula}
===============
Before discussing our formulas in greater detail, we would like to briefly review the construction of the superconformal index. This quantity is a refined Witten index that counts operators in short representations of the superconformal group weighted by three superconformal fugacities ($p,q,t$) and arbitrarily many flavor fugacities ($x_i$). The counting is modulo short multiplets that can pair up to form long multiplets, thereby guaranteeing the invariance of the index under exactly marginal deformations (as long as the spectrum is discrete). The index is written with respect to some supercharge ${\mathcal{Q}}$ (with $\left\{{\mathcal{Q}},{\mathcal{Q}}^{\dagger}\right\}=\Delta$) and a mutually commuting set of charges as $${\mathcal{I}}(p,q,t,x_i)={\rm Tr}_{{\mathcal{H}}}(-1)^Fe^{-\beta\Delta}p^{j_1+j_2-r}q^{j_1-j_2-r}t^{R+r}\prod_i x_i^{f_i}~,$$ where the trace is taken over the Hilbert space of local operators, ${\mathcal{H}}$, $j_{1,2}$ are the $SO(4)$ spins, $R$ is the $SU(2)_R$ Cartan, $r$ is the $U(1)_R\subset U(1)_R\times SU(2)_R$ superconformal $R$-charge, and the $f_i$ are flavor charges. By standard arguments, only operators annihilated by ${\mathcal{Q}}$ and ${\mathcal{Q}}^{\dagger}$ contribute to the index (these states have $\Delta=E-2j_1-2R+r=0$ with $E$ the scaling dimension).
A simpler but nonetheless physically rich limit of the index, called the Schur index, is obtained by taking $t\to q$. It is straightforward to check that the $p$ dependence drops out and we are left with $$\label{Schurlim}
{\mathcal{I}}(q,x_i)={\rm Tr}_{{\mathcal{H}}}(-1)^Fe^{-\beta\Delta}q^{E-R}\prod_ix_i^{f_i}~.$$ This limit of the index is intimately connected with 2D chiral algebras via the correspondence in [@Beem:2013sza], with Coulomb branch physics [@Cecotti:2010fi; @Buican:2015hsa; @Cordova:2015nma; @Fredrickson:2017yka] (even though the Coulomb branch operators themselves do not contribute to ), with Higgs branch physics (such operators, of type $\hat{\mathcal{B}}_R$ in the nomenclature of [@Dolan:2002zh] (see also [@Dobrev:1985qv]), contribute directly to ),[^10], $S^3$ partition functions for dimensionally reduced theories [@Buican:2015hsa; @Gadde:2011ia; @Nishioka:2011dq], and, crucially for us below, with $q$-deformed Yang-Mills theory in the class ${\mathcal{S}}$ context [@Gadde:2011ik]. In the next subsection, we will motivate our proposals and for this limit of the index.
Motivating our generalization {#subsec:motivation}
-----------------------------
To explain our proposal in , it is useful to recall the corresponding result in the case of $SU(N)$ $q$-deformed Yang-Mills theory on a Riemann surface, ${\mathcal{C}}$, of genus $g$ with $m$ regular punctures (defined by Young diagrams, $Y_i$, with $i=1,\cdots,m$) [@Gadde:2011ik]. Indeed, the authors of [@Gadde:2011ik] argued that the Schur index in this case can be computed (up to an overall prefactor) as an $m$-point correlator in the zero-area limit of $q$-deformed Yang-Mills theory $$\label{regex}
{\mathcal{I}}_{{\mathcal{T}}_{{\mathcal{C}}}}(q;{\bf x})=\sum_RC_R(q)^{2-2g-m}\prod_{k=1}^mf_R^{Y_k}(q;{\bf x}_k)~,$$ where $C_R(q)$ is defined as in , and ${\bf x}_k\equiv(x_{1,k},\cdots, x_{N-1,k})$ are $SU(N)$ fugacities. The $R$ dummy variables in correspond to irreducible representations of $SU(N)$ labeling intermediate states in the TFT correlator, and $f_R^{Y_k}$ is the inner product of the state $|{\bf x}_k,Y_k\rangle$ corresponding to the holonomy around the $k$-th regular puncture with the state corresponding to the representation $R$, i.e. $$\langle R|{\bf x}, Y_k\rangle=f_R^{Y_k}(q;{\bf x}_k)~.$$ The topological nature of the index in is reflected in the fact that it is independent of the ordering of the punctures.
Therefore, in order to compute the index for the $(A_{N-1}, A_{N(n-1)-1})$ theory from this perspective, the main question is to fix the irregular wave function, ${\widetilde}f^{(n)}_R(q;{\bf x})$. Indeed, the factor of $C_R(q)$ in follows from the discussion in the regular puncture case by noting that ${\mathcal{C}}=\mathbb{CP}^1$ with a single puncture (and so $2-2g-m=1$).[^11]
In our earlier paper [@Buican:2015ina] we provided strong evidence (confirmed using other techniques [@Cordova:2015nma; @Maruyoshi:2016aim]) that, in the $SU(2)$ case, the index is given by with $$\label{su2wfn}
{\widetilde}f^{(n)}_R(q;x)=\prod_{k=1}^{\infty}\left({1\over1-q^k}\right)q^{nC_2(R)}{\rm Tr}_R\left[x^{2J_3}q^{-n(J_3)^2}\right]~,$$ where $J_3=\pm{1\over2}$ for the fundamental representation. One important aspect of this formula is that it is manifestly invariant under the $SU(2)$ Weyl group (in this case $S_2\simeq Z_2$). This invariance is natural since the Hitchin system description of our theory has $S_N$ invariance.
To emphasize this invariance and also to see how to generalize in an $SU(N)$ Weyl-invariant way, we can re-write our expression as $$\begin{aligned}
\label{rewrite}
{\widetilde}{f}^{(n)}_R(q;x)&=&\prod_{k=1}^\infty \left(\frac{1}{1-q^k}\right)q^{nC_2(R)}\text{Tr}_R\left[q^{-\frac{n}{2}h_1\left(\frac{1}{2}\right)h_1}x^{h_1}\right]=\cr&=& \prod_{k=1}^\infty\left(\frac{1}{1-q^k}\right)q^{nC_2(R)}\text{Tr}_R\left[q^{-\frac{n}{2}F^{11}h_1h_1}x_1^{h_1}\right]~,\end{aligned}$$ where the various quantities appearing are written in terms of the Chevalley basis $e_1,\, f_1,\,h_1$ of $SU(2),$[^12] and $F^{11}={1\over2}$ is the (in this case $1\times 1$) inverse Cartan matrix appearing in (with $N=2$). Weyl invariance follows from the fact that any highest weight representation, $R$, is spanned by states $|\lambda\rangle$ such that $h_i|\lambda\rangle = \lambda_i |\lambda\rangle$, and $F^{ij}\lambda_i\lambda_j = (\lambda,\lambda)$ is Weyl invariant.
It is now clear how to generalize the $R$-dependent part of in for general $N$ in a minimal way that respects the $SU(N)$ Weyl invariance (recall that ${\rm Weyl}[SU(N)]\simeq S_N)$ of $q$-deformed Yang-Mills theory $$\label{weyltr}
q^{nC_2(R)}\text{Tr}_R\left[q^{-\frac{n}{2}F^{11}h_1h_1}x_1^{h_1}\right]\to q^{nC_2(R)}\text{Tr}_R\left[q^{-\frac{n}{2}F^{ij}h_ih_j}\prod_{i=1}^{N-1}(x_1\cdots x_i)^{h_i}\right]~.$$
Generalizing the first factor in is also straightforward. In [@Buican:2015ina], we argued that, for $N=2$, was the natural generalization of the plethystic exponential in the regular puncture wavefunction to the irregular wave function case, where the global symmetry is generically reduced to $U(1)$. For $N\ge2$, the global symmetry is generically $U(1)^{N-1}$, and so a straightforward generalization gives $$\label{cartans}\prod_{k=1}^{\infty}\left({1\over1-q^k}\right)\to\prod_{k=1}^{\infty}\left({1\over1-q^k}\right)^{N-1}~.$$ This latter expression has the added benefit that, when combined with $C_R(q)$, it gives the expected asymptotic Cardy-like behavior in the limit $q\to1$ (we will make this statement more precise below).
Finally, our expression has the important property that $$\begin{aligned}
\label{bound}
{\widetilde}{f}_R^{(n)}(q; {\bf x}) = O\left( q^{n(\lambda, \rho)} \right)~, \ \ \ {\rm dim}_q R = O\left( q^{-(\lambda, \rho)} \right)~,\end{aligned}$$ where $\lambda$ is the Dynkin label of the highest weight of the representation, $R$, of $SU(N)$, $\rho$ is the Weyl vector of $SU(N)$, and the inner product is the standard one $$(\lambda,\rho)\equiv\sum_{i,j}\lambda_iF^{ij}\rho_j=\sum_{i,j}\lambda_iF^{ij}~,$$ where we have used the fact that $\rho=(1,\cdots,1)$. One immediate consequence of is that our expressions for the indices have only positive powers of $q$ (as required by unitarity). Combined with the fact that $C_R(q)$ has only positive contributions in $q$ (in the sense of the coefficients), we have that flavor contributions to the index obey $$\label{flavor}
{\rm Flavor} = {\mathcal{O}}\left(q^{(n-1)(\lambda,\rho)}\right)~.$$ We will be able to use to argue that our indices capture the correct symmetry structure of the $(A_{N-1}, A_{N(n-1)-1})$ SCFTs and to explain how baryon contributions to the index arise from $SU(N)$ representations of $q$-deformed Yang-Mills theory.
Some general properties
-----------------------
On general grounds, the $O(q)$ terms in the Schur index can only come from flavor symmetry moment maps.[^13] In the $(A_{N-1}, A_{N(n-1)-1})$ theories, we have a flavor symmetry of rank $N-1$. In the case $n>3$, this symmetry is simply $U(1)^{N-1}$. For $n=3$ and $N=2$, the symmetry is enhanced to $SU(2)$, while it remains $U(1)^{N-1}$ for $N>2$. Finally, for $n=2$ and $N=2$ we have $SU(2)$ flavor symmetry, for $N=3$ we have $SU(3)$ flavor symmetry, and for $N>3$ we again have $U(1)^{N-1}$ flavor symmetry.
Using the result in , we can easily argue for this flavor structure. First, suppose $n>3$. In this case, we expect $U(1)^{N-1}$ flavor symmetry and hence $$\label{abflavor}
{\mathcal{I}}_{A_{(N-1}, A_{N(n-1)-1})}=1+(N-1)q+\cdots~, \ \ \ n>3~,$$ since the adjoint of $U(1)^{N-1}$ is $N-1$ singlet representations. Indeed, from , we see that the power of $q$ at which flavor comes in is $$\label{flbound}
(n-1)(\lambda,\rho)\ge{(n-1)(N-1)\over2}~.$$ For $n>3$, we see that the flavor contributions come in at order $q^{3\over2}$ and higher. Therefore, we can only have singlets at $O(q)$. In this case, it is easy to see that the only contributions at $O(q)$ come from the $(1-q)^{-(N-1)}=1+(N-1)q+\cdots$ factor in .
Next, consider the case $n=3$. Clearly, if $N>2$, then by flavor contributions come in at order $q^{2}$ and higher, and we can again argue that the index takes the form in . On the other hand, if $N=2$, then there will be contributions at $O(q)$. These contributions are from $SU(2)$ moment maps corresponding to raising and lowering operators of $SU(2)$ and were described in our paper [@Buican:2015ina].
Finally, consider setting $n=2$. In this case, if $N>3$, then flavor contributions come in at order $q^{3\over2}$, and we are back to the analysis presented in the $n>3$ case. If $N=3$, then there are flavor contributions at order $q$ corresponding to the flavor symmetry enhancement $U(1)^2\to SU(3)$ of the $(A_2, A_2)\simeq(A_1, D_4)$ theory. The $(A_1, D_4)$ index was described in [@Buican:2015ina] and corresponds to the vacuum partition of $\widehat{su(3)}_{-{3\over2}}$. Below we will describe the same theory from the dual $(A_2, A_2)$ perspective and explicitly check this flavor symmetry enhancement. If $N=2$, then we are in the case of the free hypermultiplet, $(A_1, A_1)$. This theory again has flavor contributions at order $q$ reflecting the $U(1)\to SU(2)$ flavor symmetry enhancement (this case was covered in [@Buican:2015ina]).
The result in also allows us to describe how baryons arise in the index of the $(A_{N-1}, A_{N(n-1)-1})$ SCFT. Indeed, these theories have baryons (and also monopoles in the mirrors of the $S^1$ reductions that we will study further below) $$\label{baryoncont}
(E-R)|_{\rm Baryons} =E^{3d}_{\rm Monopoles}= {(n-1)\nu(N-\nu)\over2}~, \ \ \ \nu=1,\cdots,N-1~.$$ Using our formula in , we can reproduce these baryonic contributions from $SU(N)$ representations with Dynkin labels $$\label{baryonreps}
\lambda_i = \delta_{i,\nu}~,$$ thus furnishing an interesting algebraic description of these operators and the corresponding moduli (sub)-spaces they parameterize.
Let us also note that if we have a conformal manifold with $n=2$ and complex dimension $N-3>0$ as in , then the flavor symmetry is $U(1)^{N-1}$ and there are at least three moment maps at $O(q)$. These moment maps are related to three AKM currents via the correspondence in [@Beem:2013sza] and hence we find that the resulting chiral algebra has at least three generators in accord with the general bounds of [@Buican:2016arp]. If we have a conformal manifold with $n>2$ and complex dimension $N-2>0$, then the flavor symmetry is at least $U(1)^2$. However, we must also have baryons from . These are generators of the Hall-Littlewood ring [@Gadde:2011uv] and hence are also generators of the corresponding chiral algebra [@Beem:2013sza] and so the bounds of [@Buican:2016arp] are again obeyed.
Finally, we close by noting that this discussion on the irregular singularities and the flavor symmetry guarantees that the theories with an additional regular singularity as in have the correct flavor symmetry dependence at $O(q)$ as well.
Consistency checks {#sec:consistency}
==================
In the first part of this section, we perform various non-trivial consistency checks of our generalization by matching it onto previously known indices that can be constructed by duality with $A_1$ Hitchin systems or by conformally gauging isolated theories constructed from $A_1$ Hitchin systems for small $N$ and $n$.
We then describe the proposed Lagrangian mirrors of the $S^1$ reductions of the $(A_{N-1}, A_{N(n-1)-1})$ SCFTs and certain RG flows that interpolate between such theories. These flows involve giving vevs to monopole operators, and we will present a simple formalism to capture the resulting physics. We then compare our results with the parent RG flows in 4D encoded in our indices via the procedure described in [@Gaiotto:2012xa] (and already exploited in the $A_1$ irregular singularity case in [@Buican:2015ina]). This gives a strong check of our proposal for all $N, n\ge2$ and also of the Lagrangians in [@Xie:2012hs; @Xie:2013jc]. Finally, we conclude with a non-perturbative-in-$q$ check of the expected Cardy-like behavior of these indices [@DiPietro:2014bca; @Ardehali:2015bla; @Buican:2015ina; @DiPietro:2016ond].
Lower rank checks {#subsec:lower}
-----------------
The checks that follow, while highly non-trivial, are only for small $N, n$. In the next subsection, we will give checks for all $N, n\ge2$. Since we have already seen that our formula for general $N$ is consistent with our earlier work on $N=2$ in [@Buican:2015ina], here we focus on the case of $N>2$.
### The $(A_2,A_2)$ index
The $(A_2, A_2)$ theory is dual to the $(A_1, D_4)$ theory [@Cecotti:2010fi]. In [@Buican:2015ina] we gave a closed-form expression for the Schur index of this latter theory (it is just the vacuum character of $\widehat{su(3)}_{-{3\over2}}$).
Given our generalization, we see that the $(A_2, A_2)$ index (and hence the vacuum character of $\widehat{su(3)}_{-{3\over2}}$) must also be expressible as $$\begin{aligned}
\mathcal{I}_{(A_2,A_2)}(q;x,y) &= \sum_{R}C_R\, {\widetilde}{f}_R^{(2)}(q;x,y)~,\end{aligned}$$ where $R$ runs over irreducible $SU(3)$ representations, and $C_R$ is given by (with $N=3$) $$\begin{aligned}
C_R = \frac{(1-q)^2(1-q^2)}{(q;q)_\infty^2}\chi_R^{su(3)}(q,1,q^{-1})~.
\label{eq:A2-C_R}\end{aligned}$$ In this case, our proposed wave function is $$\begin{aligned}
{\widetilde}{f}_{R}^{(2)}(q;x,y) = \left(\prod_{k=1}^\infty \frac{1}{1-q^k}\right)^2 q^{2C_2(R)} \text{Tr}_{R}\left[q^{-F^{ij}h_ih_j}x_1^{h_1}(x_1x_2)^{h_2}\right]~.\end{aligned}$$ We have checked that the above formula reproduces the $(A_1,D_4)\simeq (A_2,A_2)$ index that we described in [@Buican:2015ina] correctly up to high order in $q$.
### $(A_3,A_3)$ index
For the $(A_3,A_3)$ theory, our above discussion implies that $$\begin{aligned}
\mathcal{I}_{(A_3,A_3)}(q;x_1,x_2,x_3) = \sum_{R}C_R\, {\widetilde}{f}_R^{(2)}(q;x_1,x_2,x_3)~,
\label{eq:A3A3}\end{aligned}$$ where $$\begin{aligned}
C_R
= \frac{(1-q)^3(1-q^2)^2(1-q^3)}{(q;q)_\infty^3}\chi_{R}^{su(4)}(q^{\frac{3}{2}},q^{\frac{1}{2}},q^{-\frac{1}{2}},q^{-\frac{3}{2}})~.\end{aligned}$$ The wave function corresponding to the irregular puncture is given by $$\begin{aligned}
{\widetilde}{f}^{(2)}_R(q;x_1,x_2,x_3) &= \left(\prod_{k=1}^\infty\frac{1}{1-q^k}\right)^3 q^{2C_2(R)}\text{Tr}_R\left[q^{-F^{ij}h_ih_j}x_1^{h_1} (x_1x_2)^{h_2} (x_1x_2x_3)^{h_3}\right]~,\end{aligned}$$ where $R$ runs over irreducible representations of $SU(4)$.
Our expression for the index can be expanded in powers of $q$ as follows $$\begin{aligned}
\mathcal{I}_{(A_3,A_3)}(q;x_1,x_2,x_3) =& 1 + 3q + (a+1/a)(b+1/b+c+1/c)q^{\frac{3}{2}}
\nonumber\\
& + \left[10 + a^2+1/a^2+(b+1/b)(c+1/c)\right]q^2
\nonumber\\
& + 4(a+1/a)(b+1/b+c+1/c)q^{\frac{5}{2}} + \cdots~,\end{aligned}$$ where we reparameterize $x_{1,2,3}$ by $x_1 \equiv a b,\, x_2 = c/a$, and $x_3=a/b$. We have checked that the above expression coincides with the $(A_3,A_3)$ index evaluated in [@Buican:2015ina] by gauging a diagonal $SU(2)$ subgroup of the flavor symmetry of two $(A_1, D_4)$ theories and two $(A_1, A_1)$ theories (our expression here is considerably simpler since there is no integration over a gauge group). In terms of the quiver description in figure \[fig:A3A3\], the fugacity $a$ corresponds to the $U(1)$ flavor symmetry acting on the hyper multiplets while $b$ and $c$ are fugacities associated with the two $(A_1,D_4)$ theories.
.4cm
\(1) at (-2,0) [$(A_1,D_4)$]{}; (2) at (0,0) \[shape=circle\] [$2$]{} edge \[-\] node\[auto\] (1); (3) at (2,0) [$(A_1,D_4)$]{} edge \[-\] node\[auto\] (2); (9) at (0,-1.4) [$\;1\;$]{} edge\[-\] (2);
The S-duality transformation (see [@Buican:2014hfa; @DelZotto:2015rca; @Cecotti:2015hca] for a discussion of this duality) $(a,b,c)\to (\sqrt{b/c},\,a\sqrt{bc},\,\sqrt{bc}/a)$ corresponds to $$\begin{aligned}
x_3 \to \frac{1}{x_1x_2x_3}~,\ \ \ \text{with}\ \ x_1~,x_2 \ \ \text{fixed}~.
\label{eq:S-dual}\end{aligned}$$ This transformation is identical to the Weyl reflection associated with $\alpha_3$.[^14] Since is invariant under the action of the Weyl group of $A_3$, our conjecture is manifestly invariant under the S-duality transformation.[^15] Note also that the other generators of the Weyl group exchange $x_1$ and $x_2$, or $x_2$ and $x_3$. In terms of the quiver description shown in figure \[fig:A3A3\], these two exchanges correspond to the combination of and a symmetry manifest in the weak coupling description shown in Fig .\[fig:A3A3\].
### $(A_2,A_5)$ theory
A simple $S$-duality for the $(A_2, A_5)$ theory was worked out in [@DelZotto:2015rca] and was studied at the level of the Macdonald index in [@Buican:2015tda]. This theory has a quiver description given in Fig. \[fig:A2A5\] which enabled us to compute its Macdonald index in [@Buican:2015tda] as an integral over the diagonal $SU(2)$ gauge group. On the other hand, from our expressions above, we see that the Schur index of this theory should be expressible in terms of an expansion over $SU(3)$ characters $$\begin{aligned}
\mathcal{I}_{(A_2,A_5)} (q;x_1,x_2) = \sum_{R}C_R\, \widetilde{f}_{R}^{(3)}(q;x_1,x_2)~,\end{aligned}$$ where $C_R$ is given by , and the irregular wave function is $$\begin{aligned}
\widetilde{f}_{R}^{(3)} =& \left(\prod_{k=1}^\infty\frac{1}{1-q^k}\right)^2 q^{3C_2(R)}\text{Tr}_R\left[q^{-\frac{3}{2}F^{ij}h_ih_j} x_1^{h_1}(x_1x_2)^{h_2}\right]~.\end{aligned}$$ By an explicit computation, we obtain $$\begin{aligned}
\mathcal{I}_{(A_2,A_5)}(q;x_1,x_2) =& 1 + 2q + \left(6+c^2 + bc + \frac{b}{c} + \frac{c}{b} + \frac{1}{bc} + \frac{1}{c^2}\right)q^2\nonumber\\
& \qquad + \left(14 + 3c^2 + 3bc + \frac{3c}{b} + \frac{3b}{c} + \frac{3}{bc} + \frac{3}{c^2}\right)q^3 + \cdots~,
\label{eq:A2A5}\end{aligned}$$ where the flavor fugacities $x_{1,2}$ are rewritten in terms of $b \equiv x_1\sqrt{x_2}$ and $c\equiv 1/\sqrt{x_2}$. The above expression coincides with the index of the quiver theory shown in figure \[fig:A2A5\], where $b$ is the flavor fugacity for the $U(1)\supset U(1)\times SU(2)$ flavor symmetry of the $(A_1,D_6)$ theory while $c$ is the fugacity for $U(1)$ flavor symmetry acting on the hyper multiplets.
.4cm
\(1) at (-2,0) [$(A_1,A_3)$]{}; (2) at (0,0) \[shape=circle\] [$2$]{} edge \[-\] node\[auto\] (1); (3) at (2,0) [$(A_1,D_6)$]{} edge \[-\] node\[auto\] (2); (9) at (0,-1.4) [$\;1\;$]{} edge\[-\] (2);
The S-duality transformation is, as found in [@Buican:2015tda], given by $(b,c)\to (\sqrt{c^3/b},\, 1/\sqrt{bc})$. Indeed, the index is invariant under this transformation. Up to the symmetry $b\to b^{-1}$, which is manifest in the weak coupling description shown in Fig. \[fig:A2A5\], this S-dual transformation is equivalent to $$\begin{aligned}
x_2 \to \frac{1}{x_1x_2}~,\ \ \ x_1 \text{ : fixed}~,\end{aligned}$$ which corresponds to the Weyl transformation associated with $\alpha_2$.
RG flows
--------
In this section, we give a general non-perturbative check of our proposal in for all $N, n\ge2$. The RG flows we will discuss can be studied both in 4D and 3D and are interpolations between the following SCFTs in our class of theories $$\label{genflow}
(A_{N-1}, A_{N(n-1)-1})\to (A_{\nu-1},A_{\nu(n-1)-1})\oplus(A_{N-\nu-1},A_{(N-\nu)(n-1)-1})\oplus (A_1, A_1)~, \ \ \ 1<\nu< N-1~,$$ as well as $$\label{origflow}
(A_{N-1}, A_{N(n-1)-1})\to(A_{N-2}, A_{(N-1)(n-1)-1})\oplus(A_{1}, A_{1})~.$$ In the 4D description, these RG flows are triggered by turning on baryonic vevs.[^16] On general grounds [@Intriligator:1996ex], in the mirror description of the $S^1$ reductions of these theories, the RG flow should be triggered by turning on vevs for monopole operators.
The consistency of our RG flow picture in both 4D and 3D is a highly non-trivial check of our proposal in as well as for the 3D mirrors of the dimensional reductions of the $(A_{N-1}, A_{N(n-1)-1})$ SCFTs in [@Xie:2012hs; @Xie:2013jc].[^17]
### The 3D picture and the mirror RG flow {#3DRG}
Around , we briefly described the mirrors of the $S^1$ reductions of the $(A_{N-1}, A_{N(n-1)-1})$ SCFTs [@Xie:2013jc]: they are 3D ${\mathcal{N}}=4$ SCFTs that arise from gauge-coupling RG flows of Lagrangian $U(1)^{N-1}$ quiver gauge theories with each $U(1)$ factor having $n-1$ fundamental hypermultiplets ($Q_{i,k}, {\widetilde}Q_{i,k}^{\dagger}$ with $i\ne j$ labeling the $U(1)$ nodes) and $n-1$ bifundamental hypermultiplets ($Q_{i,j,k}, {\widetilde}Q_{i,j,k}^{\dagger}$) between each pair of nodes (where we define $Q_{i,j,k}=Q_{j,i,k}, {\widetilde}Q_{i,j,k}^{\dagger}={\widetilde}Q_{j,i,k}^{\dagger}$). The resulting ${\mathcal{N}}=4$ superpotential was described in and is reproduced below for ease of reference $$\label{N4super2}
W=\sum_{i=1}^{N-1}\sum_{k=1}^{n-1}\Phi_i\left(Q_{i,k}{\widetilde}Q_{i,k}+q_{i,j}\sum_{j\ne i}Q_{i,j,k}{\widetilde}Q_{i,j,k}\right)~,$$ where $i,j=1,\cdots, N-1$, $k=1,\cdots,n-1$, and $q_{i,j}=-q_{j,i}=1$ for $i<j$.
The particular 3D flows we will study are triggered by turning on vevs for monopole operators in the above theories (so that the 4D flows correspond to flows induced by turning on vevs for baryons). These operators are superconformal primaries (at the IR endpoint of the flows described by the Lagrangians in ) that have scaling dimension [@Gaiotto:2008ak] $$E^{3d}_{\rm Monopoles} = j_L ={n-1\over2}\left(\sum_{i=1}^{N-1}|a_i|+\sum_{i<j}|a_i-a_j|\right) ~,$$ where $j_L$ is the spin under the $SU(2)_L\subset SO(4)_R$ $R$-symmetry that acts on the (mirror) Coulomb branch, and $a_i\in\mathbb{Z}$ is the charge of the monopole operator under the $i$th $U(1)$ topological symmetry (there are $N-1$ such global symmetries corresponding to each $U(1)\subset U(1)^{N-1}$ gauge factor). By mirror symmetry, these operators map to baryons that have $E^{3d}_{\rm Baryons}=j_R$, where $j_R$ is the spin under the $SU(2)_R\subset SO(4)_R$ $R$-symmetry factor that acts on the Higgs branch.[^18] The topological charges map to charges under the baryonic symmetries of the $(A_{N-1}, A_{N(n-1)-1})$ SCFTs in 4D.
Since monopoles do not appear in the superpotential , describing the flows resulting from turning on their vevs is non-trivial.[^19] Let us first study the somewhat simpler flow arising from the mirror of the $S^1$ reduction of $$\label{origflow2}
(A_{N-1}, A_{N(n-1)-1})\to(A_{N-2}, A_{(N-1)(n-1)-1})\oplus(A_{1}, A_{1})~.$$ Note that the $N=2$ case was already studied in [@Buican:2015ina] (the first factor on the RHS disappears in this case) where the flow was analyzed in the direct $S^1$ reduction. Here we give a simple description of this flow in the mirror theory for $N\ge2$. In this case, we claim that the flow in can be triggered by turning on the following vev $$\label{monovevsymm}
\langle\mathcal{O}_{1,\cdots,1}\rangle\ne0~,$$ where ${\mathcal{O}}_{1,\cdots,1}$ is the highest $SU(2)_L$ weight component of the monopole primary with $a_1=a_2=\cdots=a_{N-1}=1$ and $$E^{3d}({\mathcal{O}}_{1,\cdots,1})={(N-1)(n-1)\over2}=I_3^L({\mathcal{O}}_{1,\cdots,1})~,$$ where the RHS denotes the $SU(2)_L$ weight of the operator.[^20]
To analyze the flow[^21] resulting from , we note that, by Goldstone’s theorem, [**(i)**]{} there will be a Goldstone Boson for the broken overall topological symmetry in the IR, and [**(ii)**]{} this particle must be coupled irrelevantly to the rest of the theory. Moreover, at long distances, the corresponding topological current must have the form $j_{\mu}=\partial_{\mu}\phi$, where $\phi$ is the Goldstone Boson. In the case of the topological symmetry, we know $j_{\mu}=\left(\star F\right)_{\mu}$ and so the Goldstone boson is just the scalar dual to the photon (see [@Gaiotto:2014kfa] for related discussions).
Therefore, we see that the overall $U(1)$ vector multiplet must decouple in the IR. In particular, it cannot couple to anything charged under the corresponding gauge symmetry. This fact implies that, for each $U(1)_i$ node ($i=1,\cdots, N-1$) all $n-1$ fundamentals get a mass. On the other hand, the bi-fundamentals are all neutral under the overall $U(1)$ gauge symmetry and are therefore unaffected by the VEV in . Once we remove the overall $U(1)$, the remaining theory is precisely the $U(1)^{N-2}$ gauge theory that describes the mirror of the $S^1$ reduction of the $(A_{N-2}, A_{(N-1)(n-1)-1})$ theory. Combined with the decoupled free vector (which in 4D becomes a decoupled $(A_1, A_1)$ theory), we find the RG flow in .
We claim that the more general RG flows in can be analyzed in a similar spirit by turning on a VEV for any one of the monopoles with charges $$a_{i_1}=a_{i_2}=\cdots=a_{i_{\nu}}=1~, \ \ \ i_{a}\ne i_b\ \forall a\ne b~,$$ and all other topological quantum numbers vanishing.[^22] This monopole operator has scaling dimension $$E^{3d}({\mathcal{O}}_{1,\cdots,1})={(N-\nu)\nu(n-1)\over2}=I_3^L({\mathcal{O}}_{1,\cdots,1})~.$$ Moreover, reasoning similar to the one above with $\nu=N-1$ (now we take the overall $U(1)$ to be $U(1)_{a_{i_1}}+\cdots+ U(1)_{a_{i_{\nu}}}$) suggests that we have the following RG flow when turning on a VEV for this operator $$(A_{N-1}, A_{N(n-1)-1})\to (A_{\nu-1},A_{\nu(n-1)-1})\oplus(A_{N-\nu-1},A_{(N-\nu)(n-1)-1})\oplus (A_1, A_1)~,$$ where $1<\nu<N-1$.
Next we will describe the 3D Coulomb branch flows of this sub-section as Higgs branch flows in the parent 4D theory. Our approach will be to study poles and residues corresponding to the baryonic vevs of the monopole ancestors.
### The 4D RG flow and poles in the Index
In this sub-section, we study the behavior of our formula for the Schur index under RG-flows triggered by turning on vevs for the 4D baryonic ancestors of the monopole operators discussed above. We expect that these 4D RG flows are in one-to-one correspondence with the flows discussed above in 3D and that the RG flows in 4D commute with the reduction along the circle.[^23]
In what follows, we give a vacuum expectation value (vev) to a Higgs branch operator, $\mathcal{O}$. The authors of [@Gaiotto:2012xa] argued that the superconformal indices of the UV and IR SCFTs of such an RG flow are related to each other by $$\begin{aligned}
\mathcal{I}_\text{vect}(q)^{-1}\cdot \mathcal{I}_\text{IR}(q;{\bf y}) = -f_{i,\mathcal{O}}\cdot\text{Res}_{x_i=x_i^*}\left(\frac{1}{x_i}\mathcal{I}_\text{UV}(q;{\bf x})\right)~,
\label{eq:index-relation}\end{aligned}$$ where $f_{k,\mathcal{O}}$ is the flavor charge of $\mathcal{O}$ under the $k$-th flavor symmetry of the UV SCFT, and $x_i$ is a flavor fugacity associated with a flavor symmetry with non-vanishing $f_{i,\mathcal{O}}$. The contribution $\mathcal{I}_\text{vect}(q) \equiv [(q;q)_\infty]^2$ is the Schur index of a free vector multiplet. In , $x_i^*$ is the value of $x_i$ such that $$\begin{aligned}
q^{R_{\mathcal{O}}}\prod_{k=1}^{\text{rank}\, G_F}(x_k)^{f_{k,\mathcal{O}}} = 1~,\end{aligned}$$ where $R_{\mathcal{O}}$ is the $SU(2)_R$ weight of $\mathcal{O}$, and $G_F$ is the flavor symmetry of the UV SCFT. We will see below that our formula for the Schur index of the $(A_{N-1},A_{N(n-1)-1})$ theory is perfectly consistent with the above index relation for various non-trivial RG flows.
### $(A_{N-1},A_{N(n-1)-1}) \to (A_{N-2},A_{(N-1)(n-1)-1})\oplus (A_1,A_1)$ flow
Let us first study the RG-flow of the form $(A_{N-1},A_{N(n-1)-1}) \to (A_{N-2},A_{(N-1)(n-1)-1})\oplus (A_1,A_1)$. For generic $N$ and $n$ this is a flow between conformal manifolds in the UV and IR (with a decoupled axion-dilaton multiplet in the IR). To study this flow from the 4D index perspective, we first rewrite our formula for the index so that it is suitable for the above residue computation.
To that end, the expression for the $(A_{N-1},A_{N(n-1)-1})$ index that we proposed in can be rewritten as $$\begin{aligned}
\mathcal{I}_{(A_{N-1},A_{N(n-1)-1})}(q;{{\bf x}}) &= \frac{\prod_{k=1}^{N-1}(1-q^k)^{N-k}}{[(q;q)_\infty]^{2N-2}}\sum_{\lambda}q^{\frac{n}{2}(\lambda,\lambda+2\rho)}(\text{dim}_q\,R_\lambda)\,
\text{Tr}_{R_\lambda}\left[q^{-\frac{n}{2}F^{ij}h_ih_j}\prod_{i=1}^{N-1}(x_1\cdots x_i)^{h_i}\right]~,
\label{eq:index1}\end{aligned}$$ where $\lambda$ runs over the non-negative weights, and $\rho$ is the Weyl vector of $A_{N-1}$.[^24] Here $R_\lambda$ stands for the highest weight representation whose highest weight is given by $\lambda$.
We first note that the quantum dimension of the representation $R_\lambda$ can be split into a $\lambda_{N-1}$-independent part and a $\lambda_{N-1}$-dependent one as follows[^25] $$\begin{aligned}
\text{dim}_q\,R_\lambda
&= \text{dim}_q^{A_{N-2}} R_{(\lambda_1,\cdots,\lambda_{N-2})}^{A_{N-2}} \times \prod_{i=1}^{N-1}\frac{[\ell_i+N-i]_q}{[N-i]_q}~,
\label{eq:q-dim0}\end{aligned}$$ where $R^{A_{N-2}}_{(\lambda_1,\cdots,\lambda_{N-2})}$ is the representation of $A_{N-2}$ corresponding to the dynkin label $(\lambda_1,\cdots,\lambda_{N-2})$ and the $q$-number is defined as $$\label{qnum}
\left[x\right]_q\equiv{q^{-{x\over2}}-q^{x\over2}\over q^{-{1\over2}}-q^{1\over2}}~.$$ Now, note that the last factor on the RHS of can be expanded as follows $$\begin{aligned}
\prod_{i=1}^{N-1}\frac{[\ell_i+N-i]_q}{[N-i]_q} = \frac{(-1)^{N-1}q^{-\frac{N-1}{2}\lambda_{N-1}-\frac{1}{2}\sum_{k=1}^{N-2}k\lambda_k-\frac{N(N-1)}{4}} + (\text{higher powers of }q^{\lambda_{N-1}} )}{\prod_{i=1}[N-i]_q}~.
\label{eq:q-dim}\end{aligned}$$ Furthermore, since $F^{ij}h_ih_j$ is Weyl invariant, we have[^26] $$\begin{aligned}
\text{Tr}_{R_\lambda}\left[q^{-\frac{n}{2}F^{ij}h_ih_j}\prod_{i=1}^{N-1}(x_1\cdots x_i)^{h_i}\right] &= \sum_{\vec{n}\in M_\lambda}q^{-\frac{n}{2}|\lambda-\vec{n}\cdot\vec{\alpha}|^2}\sum_{\mu\in W_{\lambda-\vec{n}\cdot\vec{\alpha}}}\prod_{i=1}^{N-1}(x_1\cdots x_i)^{\mu_i}~,
\label{eq:trace1}\end{aligned}$$ where $M_\lambda \subset\mathbb{N}^{N-1}$ is the set of non-negative integers, $\vec{n}$, such that $(\lambda-\vec{n}\cdot \vec{\alpha})_i$ is non-negative for all $i$, and $W_{\lambda-\vec{n}\cdot\vec{\alpha}}$ is the Weyl orbit including the weight $\lambda-\vec{n}\cdot\vec{\alpha}$.[^27] Combining the expressions –, we obtain $$\begin{aligned}
&\mathcal{I}_{(A_{N-1},A_{N(n-1)-1})}(q;{\bf x}) = \frac{\prod_{k=1}^{N-1}(1-q^k)^{N-k}}{[(q;q)_\infty]^{2N-2}}\frac{(-1)^{N-1}}{\prod_{i=1}^{N-1}[N-i]_q}\sum_{\lambda}\sum_{\vec{n}\in M_\lambda} \text{dim}_q^{A_{N-2}}R_{(\lambda_1,\cdots,\lambda_{N-2})}^{A_{N-2}}
\nonumber\\
& \times \left(q^{\left(\frac{(n-1)(N-1)}{2}+nn_{N-1}\right)\lambda_{N-1}-\frac{1}{2}\sum_{k=1}^{N-2}k\lambda_k-\frac{N(N-1)}{4}} + (\text{higher powers of }q^{\lambda_{N-1}} )\right)
\nonumber\\
&\times q^{\frac{n}{2}\sum_{i=1}^{N-2}i(N-i)\lambda_i+n\sum_{i=1}^{N-2}\lambda_i n_i- \frac{n}{2}\left|\vec{n}\cdot\vec{\alpha}\right|^2}\left( \prod_{k=1}^{N-1}(x_1\cdots x_{k})^{\lambda_k +n_{k-1}-2n_{k}+n_{k+1}} + \text{Weyl conjugates}\right)~,
\label{eq:residue1}\end{aligned}$$ where we defined $n_{0},\,n_{N}\equiv 0$. Note here that $$\begin{aligned}
\sum_{\lambda}\sum_{\vec{n}\in M_\lambda} &= \sum_{n_1,\cdots,n_{N-1}=0}^\infty \sum_{\lambda \in {\widetilde}{M}_{\vec{n}}}~,\end{aligned}$$ where ${\widetilde}{M}_{\vec{n}}$ is the set of weights $\lambda$ such that $(\lambda - \vec{n}\cdot \vec{\alpha})_i\geq 0$ for all $i=1,\cdots,N-1$. For a given $\vec{n}$, while there is no maximum value of $\lambda_{N-1}$ so that $\lambda\in {\widetilde}{M}_{\vec{n}}$, there exists a minimum value, say $\lambda_{N-1}^*$. Then we see that contains the following sum $$\begin{aligned}
&\sum_{\lambda_{N-1}=\lambda_{N-1}^*}^\infty \bigg(q^{\left(\frac{(n-1)(N-1)}{2}+nn_{N-1}\right)\lambda_{N-1} -\frac{1}{2} \sum_{k=1}^{N-2}k\lambda_k -\frac{N(N-1)}{4}} + (\text{higher powers of }q^{\lambda_{N-1}} )\bigg)
\nonumber\\
& \qquad \times \left(\prod_{k=1}^{N-1}(x_1\cdots x_{k})^{\lambda_k +n_{k-1}-2n_{k}+n_{k+1}} + \text{Weyl conjugates}\right)~.\end{aligned}$$ Since the sum over $\lambda_{N-1}$ gives a geometric series in $x_1x_2\cdots x_{N-1}q^{\frac{(N-1)(n-1)}{2}+nn_{N-1}}$, the index has a pole at $x_1x_2\cdots x_{N-1}q^{\frac{(N-1)(n-1)}{2}+nn_{N-1}}=1$. In particular, the index has a pole at $$\begin{aligned}
x_1x_2\cdots x_{N-1}q^{\frac{(N-1)(n-1)}{2}}=1~,
\label{eq:pole}\end{aligned}$$ which arises from the contributions with $n_{N-1}=0$. The corresponding residue is evaluated as $$\begin{aligned}
-\text{Res}\;\mathcal{I}_{(A_{N-1},A_{N(n-1)-1})}(q;{\bf x})
&= \left[\mathcal{I}_\text{vect}(q)\right]^{-1}\cdot \mathcal{I}_{(A_{N-2},A_{(N-1)(n-1)-1})}(q;{\bf y})~,
\label{eq:res1}\end{aligned}$$ where $y_i\equiv x_iq^{\frac{n-1}{2}}$.[^28] Note that the condition now reduces to $$\begin{aligned}
y_1\cdots y_{N-1}=1~,\end{aligned}$$ which is required for the Weyl symmetry of $A_{N-2}$ that corresponds to the permutations of $y_1,\cdots, y_{N-1}$. The result agrees with the expected index relation for the RG-flow $$(A_{N-1},A_{N(n-1)-1}) \to (A_{N-2},A_{(N-1)(n-1)-1})\oplus (A_1,A_1)~,$$ and is compatible with the discussion in Sec. \[3DRG\] from the perspective of the dimensionally reduced mirror. This is a highly non-trivial check of our formula for the Schur index of the $(A_{N-1},A_{N(n-1)-1})$ theory and also of the proposed mirror in [@Xie:2012hs; @Xie:2013jc].
### The $(A_{N-1},A_{N(n-1)-1}) \to (A_{\nu-1},A_{\nu(N-1)-1})\oplus(A_{N-\nu-1},A_{(N-\nu)(n-1)-1})\oplus (A_1,A_1)$ flow
It is straightforward to generalize the above discussion to the more general RG flow $$(A_{N-1},A_{N(n-1)-1}) \to (A_{\nu-1},A_{\nu(N-1)-1})\oplus(A_{N-\nu-1},A_{(N-\nu)(n-1)-1})\oplus (A_1,A_1)~, \ \ \ 1<\nu<N-1~.$$ For generic values of $N$ and $n$, this is an RG flow from a UV conformal manifold to an IR theory that is a direct product of two smaller conformal manifolds and a decoupled theory (comprising the axion-dilaton multiplet). The 3D descendant of this RG flow was described above.
To reproduce this flow from the index, we first note that the quantum dimension can also be rewritten as[^29] $$\begin{aligned}
\text{dim}_q\,R_\lambda &= \text{dim}_q\,R_{(\lambda_1,\cdots,\lambda_{\nu-1})}^{A_{\nu-1}}\times \text{dim}_q\,R_{(\lambda_{\nu+1},\cdots,\lambda_{N-1})}^{A_{N-\nu-1}} \times \prod_{1\leq i\leq \nu<j\leq N} \frac{[\ell_i-\ell_j+j-i]_q}{[j-i]_q}~.\end{aligned}$$ Moreover, the last factor on the RHS can be expanded as follows $$\begin{aligned}
&
(-1)^{\nu(N-\nu)}\frac{q^{-\frac{1}{2}\left(\nu(N-\nu)\lambda_\nu + (N-\nu)\sum_{k=1}^{\nu-1}k{\lambda_k} +\nu\sum_{k=\nu+1}^{N-1}(N-k)\lambda_{k} + \frac{1}{2}N\nu(N-\nu)\right)}}{\prod_{1\leq i\leq \nu<j\leq N}[j-i]_q} +\; \left(\text{higher powers of }q^{\lambda_{\nu}}\right)~.\end{aligned}$$ Combining these expressions with , we will see that the sum over $\lambda_\nu$ gives a geometric series in $x_1\cdots x_{\nu}q^{\frac{(n-1)\nu(N-\nu)}{2}+n n_\nu}$ and therefore the index $\mathcal{I}_{(A_{N-1},A_{N(n-1)-1})}(q;{\bf x})$ has a pole at $x_1\cdots x_\nu q^{\frac{(n-1)\nu(N-\nu)}{2}+n n_\nu}=1$ for all $n_\nu = 0,1,2,3,\cdots$. In particular, the index has a pole at $$\begin{aligned}
x_1\cdots x_{\nu}\, q^{\frac{(n-1)\nu(N-\nu)}{2}}=1~,
\label{eq:cond}\end{aligned}$$ with the residue given by $$\begin{aligned}
-\text{Res}\;\mathcal{I}_{(A_{N-1},A_{N(n-1)-1})}(q;{\bf x})
&= [\mathcal{I}_\text{vect}(q)]^{-1}\cdot\mathcal{I}_{(A_{\nu-1},A_{\nu(n-1)-1})}(q;{\bf y})\cdot \mathcal{I}_{(A_{N-\nu-1},A_{(N-\nu)(n-1)-1})}(q;{\bf z})~.
\label{eq:RG2}\end{aligned}$$ Here, the flavor fugacities on the RHS are defined as $y_i \equiv x_i q^{\frac{(n-1)(N-\nu)}{2}}$ for $i=1,\cdots,\nu$ and $z_i\equiv x_{\nu+i} q^{-\frac{(n-1)\nu}{2}}$ for $i=1,\cdots,N-\nu$. Note that and $x_1\cdots x_{N}=1$ imply $$\begin{aligned}
\prod_{i=1}^{\nu} y_i = 1~,\qquad \prod_{i=1}^{N-\nu}z_i = 1~.\end{aligned}$$ The relation agrees with the expected index relation for the RG-flow, $(A_{N-1},A_{N(n-1)-1}) \to (A_{\nu-1},A_{\nu(n-1)-1})\oplus (A_{N-\nu-1},A_{(N-\nu)(n-1)-1})\oplus (A_1,A_1)$. This result is also compatible with the 3D discussion in Sec. \[3DRG\].
Cardy-like behavior
-------------------
In this section, we would like to study the essential singularity that arises in the index when we take $q\to1$. On general grounds, we expect that the Schur index behaves as follows in this limit [@Buican:2015ina; @Ardehali:2015bla; @DiPietro:2016ond; @DiPietro:2014bca] $$\label{Schurasy}
\lim_{\beta\to0}\log\mathcal{I}_{\rm Schur}=-{8\pi^2\over\beta}(a-c)+\cdots~,$$ where $\beta$ is defined as $q=e^{-\beta}$ and is proportional to the $S^1$ radius (when thinking of the index as being related to the twisted partition function of the theory on $S^3\times S^1$).[^30]
Furthermore, if the theory in question has a genuine Higgs branch (i.e., a branch on which there are, at generic points, just free hypermultiplets), then $U(1)_R$ ’t Hooft anomaly matching guarantees that $$\label{higgs}
\lim_{\beta\to0}\log\mathcal{I}_{\rm Schur}={\pi^2\over3\beta}\dim_{\mathbb{Q}}\mathcal{M}_{H}~,$$ where $\dim_{\mathbb{Q}}\mathcal{M}_H$ is the quaternionic dimension of the Higgs branch, ${\mathcal{M}}_H$.
Now, our conjecture in is a sum over products of terms whose power is fixed by the topology of ${\mathcal{C}}$ (which we will refer to as the topological" terms) and components of irregular wavefunctions . It is clear that the only terms that contribute to the pole in are those involving factors of $(q;q)^{-1}$. The topological terms supply $N-1$ such factors, and the irregular wavefunction supplies an additional $N-1$ factors. Each such factor is just the $q\to1$ contribution of a half-hypermultiplet. Therefore, we have $$\label{cardyI}
\lim_{\beta\to0}\log\mathcal{I}_{(A_{N-1},A_{N(n-1)-1})}={\pi^2\over3\beta}(N-1)+\cdots~,$$ which reproduces the known $a-c$ for these theories given by taking the difference of the expressions in .
As a simple application to see how easily our results extend to the case of type IV AD theories [@Xie:2013jc] (recall from the introduction that these theories are characterized by an additional regular singularity), let us turn our attention to the $\mathcal{I}_{(I_{N,N(n-1)}, F)}$ SCFT (here $F$“ refers to the fact that the regular puncture is full”). The irregular wave function is the same as in the $(A_{N-1}, A_{N(n-1)-1})$ case and therefore again contributes a factor of ${\pi^2(N-1)\over6\beta}$ to . On the other hand, the full regular puncture corresponds to the following Young diagram: $[1,\cdots ,1]$. The resulting flavor symmetry is $SU(N)$. The adjoint representation has dimension $N^2-1$, and so we find that $$\label{cardyI3}
\lim_{\beta\to0}\log\mathcal{I}_{(I_{N,N(n-1)},F)}={\pi^2\over6\beta}(N^2+N-2)+\cdots={\pi^2\over6\beta}(N+2)(N-1)+\cdots~,$$ which matches the known results in (2.43) of [@Xie:2013jc].
Discrete Symmetries of the Index and $S$-duality
================================================
As we discussed in Sec. \[subsec:motivation\], our wave function $$\begin{aligned}
{\widetilde}{f}_{R}^{(n)}(q;{\bf x}) &= \prod_{k=1}^\infty\left(\frac{1}{1-q^k}\right)^{N-1}q^{nC_2(R)}\text{Tr}_R\left[q^{-\frac{n}{2}F^{ij}h_ih_j}\prod_{i=1}^{N-1}(x_1\cdots x_i)^{h_i}\right]~,\end{aligned}$$ for the irregular singularity is invariant under the action of Weyl group of $SU(N)$ on the flavor fugacities, $x_i$, even though the flavor symmetry associated with the irregular puncture is not $SU(N)$ but $U(1)^{N-1}$. More concretely, the above wave function is invariant under $$\begin{aligned}
s_i: x_i\leftrightarrow x_{i+1}
\label{eq:Weyl}\end{aligned}$$ for $i=1,\cdots, N-1$, which generate $S_N$ acting on the flavor fugacities, ${\bf x}$.[^31] This invariance implies that the Schur indices of the $(A_{N-1},A_{N(n-1)-1})$ and $(I_n, R_{Y})$ Arygres-Douglas theories are invariant under the $S_{N}$ action on the flavor fugacities.
This $S_N$ symmetry of the index is consistent with the fact that our irregular puncture is type I and therefore corresponds to the boundary condition of the Hitchin system. To see this, let us collect the singular terms in as $$\begin{aligned}
M(z) \equiv M_1z^{n-1} + M_2z^{n-2} + \cdots M_n + \frac{M_{n+1}}{z}~.
\label{eq:boundary}\end{aligned}$$ Since we can diagonalize this matrix order by order without changing the spectral curve, $\det (xdz - \varphi(z))=0$, we assume the $M_i$ are all diagonal. Then the (diagonal) elements of the $M_i$ are the coupling constants and mass parameters of the correspoinding 4d $\mathcal{N}=2$ theory. In particular, the parameters in $M_1$ correspond to exactly marginal couplings and those in $M_{n+1}$ are mass parameters, while those in $M_i$ for $2\leq i\leq n$ correspond to relevant couplings. Note here that the only constraint on $M_i$ is that the sum of its eigenvalues is vanishing. Therefore there is an action of $S_N$ that permutes the $N$ eigenvalues of $M(z)$. Since such an $S_N$ action can be realized by a gauge transformation in the Hitchin system,[^32] it preserves the Seiberg-Witten curve, $\det(xdz -\varphi(z))=0$, of the 4d $\mathcal{N}=2$ theory up to re-labeling the Coulomb branch parameters.[^33] In the same spirit as [@Seiberg:1994aj; @Gaiotto:2009we; @Buican:2014hfa], this discussion suggests that the $(A_{N-1},A_{N(n-1)-1})$ and $(I_{N,N(n-1)},R_Y)$ Argyres-Douglas theories are invariant under this $S_N$. Note here that, since it permutes the diagonal elements of $M_1$, this $S_N$ symmetry generically relates the theory at different points on the conformal manifold. Therefore this $S_N$ contains an S-duality group of the theory as a sub-group.[^34]
This $S_N$ action on the boundary condition can naturally be identified with the $S_N$ action on the flavor fugacities, since a permutation of the mass parameters encoded in $M_{n+1}$ corresponds to a change of the basis of flavor charges. Then, the $S_N$ invariance of the index is consistent with the $S_N$ symmetry of the Seiberg-Witten curve. Indeed, in Sec. \[subsec:lower\], we have seen that the $S_4$ invariance of the $(A_3,A_3)$ index and the $S_3$ invariance of the $(A_2,A_5)$ index are related to the S-duality discussed in [@Buican:2014hfa; @Buican:2015tda]. This identification of the two $S_N$ will tell us how the Schur operators of general $(A_{N-1},A_{N(n-1)-1})$ and $(I_{N,N},R_{Y})$ theories are mapped by the S-duality. This will be useful to identify the 2d chiral algebras associated, in the sense of [@Beem:2013sza], with these Argyres-Douglas theories.
As a final point, we note that the index is also invariant under a $\mathbb{Z}_2$ charge conjugation $$\begin{aligned}
x_1\to (x_1)^{-1}~, \quad x_2 \to (x_2)^{-1}~,\quad x_3\to (x_3)^{-1}~,\quad \cdots~,\quad x_{N-1}\to (x_{N-1})^{-1}~.\end{aligned}$$ Combined with the above $S_N$ symmetry, we see that, for generic $N$, the discrete symmetry of the index includes $S_N\times\mathbb{Z}_2$ (for $N=2$, this group is reduced to $\mathbb{Z}_2$). However, the discrete symmetry group acting on the index may be larger (as it will generally be when there is also a regular puncture). To study such a possibility, it is useful to recall . A more general discrete symmetry acting only on ${\bf x}$ and preserving $C_R(q)$ can interchange different components of the wavefunctions, ${\widetilde}f^{(n)}_{R_{\lambda_i}}(q;{\bf x})$ ($i=1,\cdots,N'$), only if $(\lambda_1,\rho)=(\lambda_2,\rho)=\cdots=(\lambda_{N'},\rho)$. In the case of conjugate representations, this latter quantity is the same. However, it would be interesting to see if there are more general group actions that are consistent with this constraint and are indeed symmetries of the index.
Beyond $A_{N-1}$? {#speculation}
=================
One amusing aspect of our wave function formula in is that it can, in principle, be defined for any Lie algebra, $\mathfrak{g}$, by a mild re-writing $$\label{moregen}
{\widetilde}{f}_{R}^{(n)}(q;{\bf x}) =\prod_{k=1}^\infty\left(\frac{1}{1-q^k}\right)^{r_{\mathfrak{g}}}q^{nC_2(R)}\text{Tr}_R\left[q^{-\frac{n}{2}F^{ij}h_ih_j}\prod_{i=1}^{N-1}(x_1\cdots x_i)^{h_i}\right]~,$$ where $r_{\mathfrak{g}}$ is the rank of $\mathfrak{g}$, $F^{ij}$ is the quadratic form for $\mathfrak{g}$, and the $h_i$ are the Cartans for $\mathfrak{g}$.[^35] We can also try to generalize as follows $$\label{rewriteCr}
C_R(q)=\prod_{k=1}^{r_{\mathfrak{g}}}\prod_{i=1}^{\infty}{1\over1-q^{d_k-1+i}}{\rm dim}_qR~,$$ where, in the regular puncture case with simply laced $\mathfrak{g}$, the authors of [@Mekareeya:2012tn; @Lemos:2012ph] argued that the ${\bf d}_{\mathfrak{g}}=\left\{d_1,\cdots,d_{r_{\mathfrak{g}}}\right\}$ are the degrees of invariants of $\mathfrak{g}$,[^36] and ${\rm dim}_qR$ is defined as $$\label{genqdim}
{\rm dim}_qR\equiv\prod_{\alpha>0}{\left[(\lambda+\rho,{\widetilde}\alpha)\right]_q\over\left[(\rho,{\widetilde}\alpha)\right]_q}~, \ \ \ {\widetilde}\alpha\equiv{2\alpha\over(\theta,\theta)}~.$$ Here $\lambda$ are the Dynkin labels, $\alpha$ is any positive root, $\rho$ is the Weyl vector, $\theta$ is the highest root, and $[x]_q$ was defined in .
Given these expressions, we can attempt to construct a putative Schur index $$\label{genind}
\mathcal{I}_{(\mathfrak{g},n)}(q;{\bf x}) = \sum_{R}C_R(q){\widetilde}{f}^{(n)}_R(q;{\bf x})~,$$ where, for $\mathfrak{g}=A_{N-1}$, the hypothetical $(\mathfrak{g},n)$ SCFT is the actual $(A_{N-1},A_{N(n-1)-1})$ theory. If such expressions are meaningful for more general $\mathfrak{g}$, they may not come from a $(2,0)$ SCFT compactified on ${\mathcal{C}}$.[^37] Therefore, it is not completely clear what the rules are (or even if they exist!) for writing an index given the data in and . However, we can simply press on and ask if satisfies the basic rules for a Schur index.
The most obvious generalizations to check are those involving $\mathfrak{g}=D_N, E_{6,7,8}$ (where there might be some hope that, at least for some paris $(\mathfrak{g},n)$, a theory of class ${\mathcal{S}}$ can be associated with —see the discussion in the regular puncture case given in [@Mekareeya:2012tn; @Lemos:2012ph]). We defer such a discussion to future work and instead focus on an even more speculative avenue with non-simply laced $\mathfrak{g}$.
In this spirit, one can check that for $\mathfrak{g}=G_2$ and $n=2$, the expression in cannot correspond to a Schur index. Indeed, plugging the following group theory data $$F_{G_2}=\left[\begin{array}{cc}
{2\over3} & 1 \\
1 & 2\\
\end{array}\right]~, \ \ \ {\bf d}_{G_2}=\left\{2, 6\right\}~,$$ into results in powers of $q$ that are not integer or half-integer. Clearly, we must, at the very least, specialize to a subset of $\mathfrak{g}$ and $n$.
With these facts in mind, consider for example the case $\mathfrak{g}=F_4$ and $n=2$ with $$F_{F_4}=\left[\begin{array}{cccc}
2 & 3 & 2 & 1 \\
3 & 6 & 4 & 2 \\
2 & 4 & 3 & {3\over2} \\
1 & 2 & {3\over2} & 1\\
\end{array}\right]~, \ \ \ {\bf d}_{F_4}=\left\{2, 6, 8, 12\right\}~.$$ It is straightforward to verify that all powers appearing in are integer or half-integer and that the expansion in $q$ of this index reads $$\label{F4index}
{\mathcal{I}}_{(F_4,2)}(q;{\bf x})=1+4q+15q^2+45q^3+125q^4+316q^5+{\mathcal{O}}(q^{11\over2})~,$$ where the first flavor dependence comes in at ${\mathcal{O}}(q^{11\over2})$. If the corresponding theory exists, it has flavor symmetry $U(1)^4$ and has a candidate stress tensor contribution at ${\mathcal{O}}(q^2)$. Note that this theory cannot be the putative $F_4$ SCFT discussed in [@Beem:2013sza; @Shimizu:2017kzs] since the flavor symmetry here is not $F_4$ but rather is its Cartan subalgebra (note however, that the order 1152 Weyl group of $F_4$ must still be a symmetry of ).[^38] It would be interesting to see if one can bootstrap a chiral algebra (with a finite number of generators) whose vacuum character satisfies and gives the Schur sector of a genuine 4D ${\mathcal{N}}=2$ SCFT.[^39]
Discussion and Conclusion
=========================
From our simple proposal for the wave function for an irregular singularity of type I in $SU(N)$ $q$-deformed Yang-Mills theory , we extracted a great deal of physics concerning new superconformal indices, conformal manifolds, $S$-duality, RG flows, and $S^1$ reductions. Based on this discussion, we suggest the following open problems:
- It would be interesting to find the chiral algebras associated with the $(A_{N-1}, A_{N(n-1)-1})$ and $(I_{N,N(n-1)}, R_Y)$ SCFTs (see [@Creutzig:2017qyf; @BR] for a beautiful recent discussion of the $(A_1, A_{2n-3})$ and $(I_{2,2(n-1)}, F_{A_1})$ cases). Our discussion implies there is a non-trivial $S_N\times\mathbb{Z}_2$ action on the associated chiral algebra.
- It would be worthwhile to understand the full symmetry group that acts on the index and which subgroup corresponds to the action of the $S$-duality group. In so doing, perhaps we can make closer contact with the results in [@Caorsi:2016ebt].
- Since we can compute the $(A_2, A_2)\sim (A_1, D_4)$ index either as a correlator in $SU(3)$ $q$-deformed Yang-Mills on a surface with a single irregular puncture or as a correlator in $SU(2)$ $q$-deformed Yang-Mills theory with both a regular and an irregular puncture, we see that certain observables in different $q$-deformed Yang-Mills theories (on different Riemann surfaces) must be related. Can this result be promoted to a duality between different subsectors of these theories?
- Can our expressions for wavefunctions in the $A_{N-1}$ case be extended to other Lie algebras and give information about new SCFTs (perhaps even making contact with some of the results in [@Argyres:2016xua] or their generalizations to higher ranks)? We made a speculative proposal in this regard and briefly discussed the hypothetical case of an SCFT associated with $F_4$ (and $n=2$) (note again that this theory is not the hypothetical $F_4$ SCFT discussed in [@Beem:2013sza; @Shimizu:2017kzs] since it only has $U(1)^4\subset F_4$ flavor symmetry, although it does have the full discrete Weyl symmetry, $W(F_4)$). The $D_N$ and $E_{6,7,8}$ cases may be even more promising avenues for study. Can one find an associated chiral algebra and 4D ${\mathcal{N}}=2$ theory using certain bootstrap techniques? Even though our expressions typically don’t manifest the full $\mathfrak{g}$ flavor symmetry, they are still invariant under the (sometimes very large) Weyl groups, $W(\mathfrak{g})$, which may furnish strong constraints on the resulting algebras.
- We described some intricate RG flows between conformal manifolds in the UV and IR. It would be interesting to understand if these flows obey any new constraints beyond the $a$-theorem.
[^1]: See [@Cecotti:2010fi; @Xie:2012hs] for nomenclature and the classic references [@Argyres:1995jj; @Argyres:1995xn; @Eguchi:1996vu] for the original constructions of these theories as endpoints of renormalization group flows from ${\mathcal{N}}=2$ gauge theories.
[^2]: By ${\mathcal{N}}=2$ chiral operators, we mean operators annihilated by all the anti-chiral ${\mathcal{N}}=2$ Poincaré supercharges.
[^3]: We define an Argyres-Douglas theory to be any ${\mathcal{N}}=2$ SCFT with at least one ${\mathcal{N}}=2$ chiral operator of non-integer scaling dimension.
[^4]: Namely, the “rasing” and “lowering” generators, $e_i$ and $f_i$, satisfy $[h_i,\, e_j] = A_{ji} e_j,\, [h_i, f_j] = -A_{ji}f_j$ and $[e_i,f_j] = \delta_{ij}h_j$ with the Cartan matrix, $A_{ij}$.
[^5]: These coefficients are fixed by the rank of the $q$-deformed Yang-Mills theory and the topology of the surface on which it lives. As we will review below, both these properties are in turn fixed—up to duality—by the particular AD theory we consider.
[^6]: \[foot:ell\]An equivalent expression is $$\begin{aligned}
\chi^{su(N)}_R(x_1,\cdots,x_{N}) = \frac{\det (x_j^{\ell_i+N-i})}{\det (x_j^{N-i})}~,\end{aligned}$$ where $\ell_1,\,\cdots,\ell_{N}$ are given by $\ell_i = \sum_{j=i}^{N-1}\lambda_j$, in terms of the Dynkin labels, $(\lambda_1,\cdots\lambda_{N-1})$, of the representaiton $R$.
[^7]: One may also try to infer additional integral expressions for some of these theories using the methods described in [@Xie:2016uqq; @Xie:2017vaf] and results in the existing literature.
[^8]: Although it will not be particularly important in our discussion below, we note that it is possible for there to be additional flavor symmetries that are not manifest in this way of describing the physics.
[^9]: Here the dimensions of the coordinates are $[x]={n-1\over n}$ and $[z]={1\over n}$. This statement follows from the fact that $\lambda=xdz$ is the 1-form and that therefore $[x]+[z]=1$.
[^10]: We can think of Higgs branch physics as contributing perturbatively in $q$ to while Coulomb branch physics contributes only non-perturbatively in $q$ [@Buican:2015hsa].
[^11]: Similarly, there is no factor of $C_R(q)$ in since $2-2g-m=0$.
[^12]: Recall again that the Chevalley basis satisfies $[h_i,\,e_j] = A_{ji}e_j,\, [h_i,\,f_j] = -A_{ji}f_j$ and $[e_i,\,f_j] = \delta_{ij}h_j$, where $A_{ij}$ is the Cartan matrix. In the case of $SU(2)$, we can identify $J_3 = \frac{1}{2}h_1$.
[^13]: Via the correspondence of [@Beem:2013sza], which maps Schur indices in 4D to vacuum characters of chiral algebras in 2D and 4D flavor symmetry moment maps to 2D affine Kac-Moody (AKM) currents, this statement corresponds to the fact that any $O(q)$ term in the vacuum character of a chiral algebra is an AKM current (here we normalize the character so that its expansion starts with a 1").
[^14]: Indeed, the above transformation implies $\lambda_k \to \lambda_k- \langle \alpha_k,\alpha_3\rangle\lambda_3$, or, equivalently, $\lambda \to \lambda - \langle \lambda,\alpha_3\rangle \alpha_3$.
[^15]: Recall from the discussion above that the Weyl invariance follows from the fact that any highest weight representation $R$ is spanned by states $|\lambda\rangle$ such that $h_i|\lambda\rangle = \lambda_i |\lambda\rangle$, and $F^{ij}\lambda_i\lambda_j = (\lambda,\lambda)$ is Weyl invariant.
[^16]: The decoupled $(A_1, A_1)$ factors on the right-hand side (RHS) of and contain the axion-dilaton of spontaneous conformal breaking and their ${\mathcal{N}}=2$ partners.
[^17]: In the case of the $(A_1, A_{2n-3})$ and $(A_1, D_{2n})$ theories, we have additional checks of the 3D mirrors arising from the $S^3$ partition function computations in [@Buican:2015hsa] and the Higgs branch Hilbert series analysis in [@DelZotto:2014kka] (the Hilbert series analysis counts a proper subset of the Schur operators but also applies to the other $(A_N, A_M)$ theories).
[^18]: Therefore, these baryons descend from 4D baryons of $SU(2)_R$ spin $j_R$ and scaling dimension $E_{\rm Baryons}=2j_R$.
[^19]: Although such an analysis can in principle be carried using standard matter operators built out of fields appearing in the superpotential of the mirror theory to (using the techniques in [@deBoer:1996ck]), this avenue becomes very tedious for the theories we study once $N$ and $n$ become sufficiently large.
[^20]: This flow can also be triggered by turning on a similar vev for the $j$th monopole with $a_i=\delta_{i,j}$ and $j=1,\cdots,N-1$. Together with the monopole we explicitly consider in the text and the same monopoles but with $a_i\to-a_i$, the 4D baryonic ancestors of these operators contribute to the Schur index as a fundamental plus an anti-fundamental of $SU(N)$ $q$-deformed Yang-Mills theory.
[^21]: Although it will not be important for us below, we note that the necessary RG analysis can be more subtle in theories with accidental IR superconformal $R$ symmetries (i.e., theories that are often referred to as bad“ and ugly”).
[^22]: There are ${(N-1)!\over(N-\nu-1)!\nu!}$ such operators. Combined with another set of ${(N-1)!\over(\nu-1)!(N-\nu)!}$ operators with $a_{j_1}=a_{j_2}=\cdots=a_{j_{N-\nu}}=1$ and $j_{a}\ne j_b$ $\forall a\ne b$ (and all other topological quantum numbers vanishing) and similar operators with $a_i\to-a_i$, we get precisely the number of operators in a $\nu$-index antisymmetric representation of $SU(N)$ and its conjugate, i.e., an $N-\nu$-index antisymmetric representation (unless $N$ is even and $\nu={N\over2}$; in this case we simply get a single ${N\over2}$-index antisymmetric combination). All in all, we see that the monopole operators are in one-to-one correspondence with the $\nu$-index anti symmetric representations of $SU(N)$ for $\nu=1,\cdots,N-1$ (i.e., the representations discussed around ).
[^23]: This expectation is based on the fact that a non-perturbative superpotential is not compatible with eight supercharges.
[^24]: We use the fact that $C_2(R) = \frac{1}{2}(\lambda,\lambda + 2\rho)$.
[^25]: Here, $\ell_i$ is defined by $\ell_i = \sum_{j=i}^{N-1}\lambda_j$ as in footnote \[foot:ell\].
[^26]: As usual, $|\eta|^2 \equiv (\eta,\eta)$ and $\vec{n}\cdot \vec{\alpha} \equiv \sum_{i=1}^{N-1}n_i\alpha_i$.
[^27]: Here we used the fact that the multiplicity of $\lambda-\vec{n}\cdot\vec{\alpha}$ in the representation $R_\lambda$ is the number of ways of writing $\lambda-\vec{n}\cdot\vec{\alpha}$ as a linear combination of positive roots. If $(\lambda-\vec{n}\cdot\vec{\alpha})_i\neq 0$ for all $i$, then the Weyl orbit, $W_{\lambda-\vec{n}\cdot\vec{\alpha}}$, contains $N!$ weights, while, if $(\lambda-\vec{n}\cdot\vec{\alpha})_i=0$ for some $i$, then there are fewer weights.
[^28]: Here we used the fact that $\lambda_{N-1}^*=0$ for $n_{N-1}=0$.
[^29]: Recall here that $\ell_i \equiv \sum_{k=i}^{N-1}\lambda_k$.
[^30]: A more rigorous statement is that if the $S^3$ partition function of the theory reduced on $S^1$ is finite, then holds [@Ardehali:2015bla; @DiPietro:2016ond] (although we are not aware of any ${\mathcal{N}}=2$ SCFT counterexamples to the behavior in ). The $(A_{N-1}, A_{N(n-1)-1})$ theories we are considering in this note satisfy the above criteria: their mirror duals consist of $N-1$ $U(1)$ gauge groups with $n-1$ fundamentals for each $U(1)$ and $n-1$ bifundamentals between each pair of abelian nodes.
[^31]: Recall here that $x_{N}\equiv (x_1\cdots x_{N-1})^{-1}$. With this definition, $s_i$ corresponds to the Weyl reflection $\lambda_k \to \lambda_k - (\alpha_i,\alpha_k)\lambda_i$.
[^32]: We can make this gauge transformation consistent with a possible regular puncture at $z=0$.
[^33]: The Coulomb branch parameters of the 4d theory correspond to the Hitchin moduli that are not fixed by the boundary condition at punctures. Since they correspond to the vacuum expectation values of Coulomb branch operators, this re-labeling of the Coulomb branch parameters corresponds to a change of a basis of the Coulomb branch chiral ring.
[^34]: In the case of an additional regular singularity, we may also have additional discrete symmetry (and part of this discrete symmetry may also be part of the $S$-duality group).
[^35]: Of course, the above generalization is not unique. For example, we could take $r_{\mathfrak{g}}\to h_{\mathfrak{g}}^{\vee}-1$ in and below (where $h_{\mathfrak{g}}^{\vee}$ is the dual Coxeter number). Note that for generic $\mathfrak{g}$, $h^{\vee}_{\mathfrak{g}}-1\ne r_{\mathfrak{g}}$. The choice we make is somewhat more natural: it guarantees that the rank of the flavor symmetry (as defined by the number of flavor-singlet terms in the index at ${\mathcal{O}}(q)$) is the same as the number of fugacities in .
[^36]: Since this is a topological factor, it is likely to apply to the irregular puncture case as well.
[^37]: If $\mathfrak{g}\ne A,D,E$, there are various obstructions to a $(2,0)$ origin of the theory (e.g., see [@Cordova:2015vwa] for an interesting recent discussion). Moreover, even if $\mathfrak{g}=D,E$, we are not sure which SCFTs—if any—, , and correspond to.
[^38]: Moreover, one would guess from our formula that the hypothetical Higgs branch has quaternionic dimension 4 instead of 8.
[^39]: Note that the chiral algebra for the hypothetical $(F_4,2)$ SCFT would need to include new AKM primaries at ${\mathcal{O}}(q^{11\over2})$.
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Introduction
============
The Standard Model of particle physics accounts successfully for all subatomic observational data. The gauge charges of the Standard Model matter states suggest its embedding in $SO(10)$ Grand Unified Theory, which is broken to the Standard Model at the GUT or string scale. The $SO(10)$ unification picture is further supported by: the logarithmic evolution of the Standard Model gauge parameters; the proton longevity; and the suppression of left–handed neutrino masses. The heterotic–string [@gross] produces chiral $SO(10)$ representations in its perturbative spectrum, and is therefore the one suited to explore the $SO(10)$ GUTs structure underlying the Standard Model. Phenomenological studies of the heterotic–string have been pursued since the mid–eighties [@candelas], using a variety of world–sheet [@fff; @gepner; @bert] and target space techniques [@cy; @orbifolds].
The free fermionic construction of the heterotic–string in four dimensions produced a rich space of phenomenological three generation models. These models admit the underlying $SO(10)$ GUT embedding of the Standard Model spectrum. However, the $SO(10)$ symmetry is broken directly at the string level. The early studies of these models consisted of isolated examples that shared an underlying NAHE–base structure [@nahe]. Examples in which the $SO(10)$ symmetry is broken to the: flipped $SU(5)$ (FSU5) [@revamp]; $SO(6)\times SO(4)$ Heterotic String Pati–Salam Models (HSPSM) [@alr]; $SU(3)\times SU(2)\times U(1)^2$ Standard–like Models (SLM) [@slm]; $SU(3)\times SU(2)^2\times U(1)$ left–right symmetric (LRS) [@lrs]; and $SU(4)\times SU(2)\times U(1)$ (SU421) [@su421]; subgroups were studied. Among those the FSU5; SLM; HSPSM; LRS cases produced quasi–realistic three generation models, whereas the SU421 case did not produce any viable three generation model. The advantage of the SU421 models compared to the FSU5 and HSPSM is that they admit both the doublet–triplet, as well as the doublet–doublet spitting mechanism [@su421]. We also note the recent interest in SU421 models from purely phenomenological considerations [@wise].
The phenomenological free fermionic heterotic–string models are ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ orbifolds that are constructed at enhanced symmetry points in the moduli space [@Z2Z2Faraggi1994; @Z2Z2Kounnas1997]. Many of the phenomenological properties of the models are rooted in their underlying ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ structure [@recentreview]. In recent years systematic methods for the classification of symmetric ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ free fermionic orbifolds were developed in [@typeIIclassi] for type II superstrings and in refs. [@fknr; @fkr] for symmetric ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ heterotic–string orbifolds with $SO(10)$ GUT symmetry. The classification was extended in refs. [@acfkr; @cfr; @rizos] and [@frs] to string vacua in which the $SO(10)$ symmetry is broken to the $SO(6)\times SO(4)$ Pati–Salam and to the flipped $SU(5)$ subgroups, respectively. The Pati–Salam class of free fermionic vacua produced examples of three generation exophobic models in which exotic fractionally charged states only appear in the massive string spectrum [@acfkr; @cfr], whereas the flipped $SU(5)$ class of models did not produce exophobic models with an odd number of generations [@frs].
In this paper we discuss the classification for the class of SU421 heterotic–string models. We provide a general argument that breaking the $SO(10)$ symmetry to this subgroup cannot produce three chiral generations in the prevalent free fermionic construction which is based on symmetric ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ toroidal compactification with a ${\mathbb{Z}}_2 \times {\mathbb{Z}}_4$ fermionic boundary conditions that break the $SO(10)$ symmetry to $SU(4)\times SU(2)\times U(1)$.
$SU(4) \times SU(2) \times U(1)$ Phenomenology {#analysis}
==============================================
The field theory content of the $N=1$ supersymmetric $SU(4)_C\times SU(2)_L\times U(1)_R$ model[^1] was discussed in ref. [@su421]. The SU421 class of heterotic–string models differs from the HSPSM models in the breaking of $SU(2)_R\rightarrow U(1)_R$ directly at the string level. Similar to the HSPSM, the SU421 heterotic–string models admit the $SO(10)$ embedding and the chiral states are obtained from the spinorial [**16**]{} representations of $SO(10)$ which decomposes in the following way: F\_L\^[i]{} &=& ( 4 ,2, 0) = (3 ,2, [13]{}, 0) + (1,2, -[1]{}, 0) = [ud]{}\^i+[e]{}\^i,\[SU421fl\]\
U\_R\^[i]{} &=& ([4]{},1,-[12]{}) = ([3]{},1,-[13]{},-[12]{}) + (1,1,+[1]{},-[12]{}) = [u\^[c]{}]{}\^i+[N\^[c]{}]{}\^i,\[SU421ur\]\
D\_R\^[i]{} &=& ([4]{},1,+[12]{}) = ([3]{},1,-[13]{},+[12]{}) + (1,2,+[1]{},+[12]{}) = [d\^[c]{}]{}\^i+[e\^[c]{}]{}\^i. \[SU421dr\] The first and second equalities show the decomposition under $SU(4)_C\times SU(2)_L\times U(1)_R$ and $SU(3)_C\times SU(2)_L\times U(1)_{B-L}\times U(1)_R$, respectively. The electroweak $U(1)_Y$ current is given by U(1)\_Y=[12]{}U(1)\_[B-L]{}+U(1)\_R. \[ewu1current\] From eq. (\[SU421fl\]) we note that $F_L$ produces the quarks and leptons weak doublets, and that $U_R$ and $D_R$ produces the right–handed weak singlets. The two Higgs multiplets of the Minimal Supersymmetric Standard Model, $h^u$ and $h^d$, are given by, h\^d &=& ( 1 ,2,-[12]{}),\
h\^u &=& ( 1 ,2,+[12]{}). \[SU421mssmhigss\] The heavy Higgs states that are responsible for breaking $SU(4)_C\times U(1)_{R}$ gauge symmetry to the Standard Model groups $SU(3)\times U(1)_Y$ are given by the fields &=& ([4]{},1,-[12]{})\
[H]{} &=& ([ 4]{},1,+[12]{}) \[SU421Higgs\] The SU421 heterotic–string models may also contain states that transform as $$(6,1,0)= ({3},1,{1\over3},0)+ ({\overline 3},1,-{1\over3},0).$$ These multiplets arise from the vectorial [**10**]{} representation of $SO(10)$. These coloured states generate proton decay from dimension five operators, and therefore must be sufficiently heavy to be in agreement with the proton lifetime limits. An important benefit of the SU421 symmetry breaking pattern is that these colour triplets may be projected out by the Generalised GSO (GGSO) projections [@ps], and need not be present in the low energy spectrum. The string doublet–triplet mechanism works in all models that include the symmetry breaking pattern $SO(10)\rightarrow SO(6)\times SO(4)$. The HSPSM heavy Higgs states, which break $SU(4)\times SU(2)_R\rightarrow SU(3)_C\times U(1)_{Y}$, contain colour triplets with the charges of the states in (\[SU421dr\]) that may give rise to dimension five proton decay mediating operators. In the HSPSM the superpotential terms $\lambda_2HHD+\lambda_3 {\bar H}{\bar H}{\bar D}$ couples the colour triplets from the vectorial representation $(6,1,1)$ to the colour triplets arising from the heavy Higgs field. The GUT scale VEVs of the heavy Higgs fields $H$ and ${\bar H}$ are used to give heavy mass to the Higgs colour triplets. However, the heavy Higgs representations in the SU421 heterotic–string models, eq. (\[SU421Higgs\]), do not contain the states with the charges of eq. (\[SU421dr\]). Consequently, the stringy doublet–triplet splitting mechanism works only in models in which the $SO(10)$ symmetry is broken to $SU(3)_C\times SU(2)_L\times U(1)^2$, $SU(4)_C\times SU(2)_L\times U(1)_R$, or $SU(3)_C\times SU(2)_L\times SU(2)_R \times U(1)_{B-L}$.
Another important advantage of the SU421 class of models versus the PS and LRS models is with respect to the light Higgs representations. In the LRS and PS models, the light Higgs states exist in bi–doublet representations and couple simultaneously to the up– and down–type quarks, which may give rise to Flavor Changing Neutral Currents (FCNC) at an unacceptable rate [@lrsfcnc]. This introduces a bi–doublet splitting problem. The solutions that have been proposed in the literature [@bidoub] use a $SU(2)_L$ triplet representation that are not present in string models in which the gauge symmetry is realised as a level one Kac–Moody algebra. On the other hand, in SU421 models $SU(2)_R$ is broken at the string level and consequently the Higgs bi–doublet is split at the string level.
The solutions to the doublet–doublet as well as the doublet–triplet splitting problems are the two appealing properties offered by the SU421 free fermionic heterotic–string models. However, as we argue in the next section the free fermionic ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ orbifold models, with additional ${\mathbb{Z}}_2 \times {\mathbb{Z}}_4$ basis vectors that are used to break the $SO(10)$ symmetry to $SU(4)_C\times SU(2)_L\times U(1)_R$, cannot in fact produce three complete chiral generations and therefore, like the NAHE–based free fermionic models [@su421], these models do not produce viable SU421 string models.
The $SU(4) \times SU(2) \times U(1)$ Free Fermionic Construction
----------------------------------------------------------------
The string vacuum in the free fermionic formulation [@fff] is defined in terms of a set of boundary condition basis vectors and the Generalised GSO projection coefficients, which span the one–loop partition function. The basis vectors generate a finite additive group $\Xi=\sum_k{{n_k}{b_k}}$ where $n_k=0,\cdots,{{N_{z_k}}-1}$. The physical states in the Hilbert space of a sector $\alpha\in{\Xi}$ are obtained by acting on the vacuum with fermionic and bosonic oscillators and by applying the GGSO projections. Each fermionic complex oscillator acting on the vacuum is counted by a fermion number operator as $F_\alpha(f)=1$ and $\alpha(f^*)=-1$. For periodic complex fermions with $\alpha(f)=1$, the vacuum is in a doubly degenerate spinorial representation ${\vert \pm\rangle}$, annihilated by the zero modes $f_0$ and ${{f_0}^*}$ and with fermion numbers $F(f)=0,-1$, respectively. The $U(1)$ charges $Q(f)$ of the unbroken Cartan generators of the right–moving gauge group are given in terms of the boundary conditions and fermion numbers of the complex right–moving world–sheet fermions by $${Q(f) = {1\over 2}\alpha(f) + F(f)}.
\label{u1charges}$$
In the light–cone gauge, the free fermionic heterotic–string models in four dimensions require $20$ and $44$, left–moving and right–moving real world–sheet fermions respectively, to cancel the conformal anomaly. In the usual notation these are denoted as: $\psi^\mu, \chi^{1,\dots,6},y^{1,\dots,6}, \omega^{1,\dots,6}$ and $\overline{y}^{1,\dots,6},\overline{\omega}^{1,\dots,6}$, $\overline{\psi}^{1,\dots,5}$, $\overline{\eta}^{1,2,3}$, $\overline{\phi}^{1,\dots,8}$.
The $SU(4) \times SU(2) \times U(1)$ Gauge Group
------------------------------------------------
In the following we set up the necessary ingredients for the classification of the SU421 free fermionic heterotic–string models. The analysis is along similar lines to the one performed in the classification of the $SO(10)$ [@fknr]; heterotic–string Pati–Salam models [@acfkr]; and flipped $SU(5)$ models [@frs]. The novelty compared to these cases is that the SU421 models employ two basis vectors that break the $SO(10)$ symmetry, whereas the HSPSM and FSU5 models use only one. However, we argue below that this class of heterotic–string vacua cannot in fact produce phenomenologically viable models. The basis vectors that generate our $SU(4) \times SU(2) \times U(1)$ heterotic–string models are given by the following 14 basis vectors $$\begin{aligned}
\label{421}
v_1={\bf1}&=&\{\psi^\mu,\
\chi^{1,\dots,6},y^{1,\dots,6}, \omega^{1,\dots,6}|\overline{y}^{1,\dots,6},
\overline{\omega}^{1,\dots,6},
\overline{\eta}^{1,2,3},
\overline{\psi}^{1,\dots,5},\overline{\phi}^{1,\dots,8}\},\nonumber\\
v_2=S&=&\{{\psi^\mu},\chi^{1,\dots,6}\},\nonumber\\
v_{2+i}={e_i}&=&\{y^{i},\omega^{i}|\overline{y}^i,\overline{\omega}^i\}, \
i=1,\dots,6,\nonumber\\
v_{9}={b_1}&=&\{\chi^{34},\chi^{56},y^{34},y^{56}|\overline{y}^{34},
\overline{y}^{56},\overline{\eta}^1,\overline{\psi}^{1,\dots,5}\},\label{basis}\\
v_{10}={b_2}&=&\{\chi^{12},\chi^{56},y^{12},y^{56}|\overline{y}^{12},
\overline{y}^{56},\overline{\eta}^2,\overline{\psi}^{1,\dots,5}\},\nonumber\\
v_{11}=z_1&=&\{\overline{\phi}^{1,\dots,4}\},\nonumber\\
v_{12}=z_2&=&\{\overline{\phi}^{5,\dots,8}\},\nonumber\\
v_{13}=\alpha&=&\{\overline{\psi}^{4,5},\overline{\phi}^{1,2}\},\nonumber\\
v_{14}=\beta&=&\{\overline{\psi}^{4,5}=\textstyle\frac{1}{2},
\overline{\phi}^{1,...,6}=\textstyle\frac{1}{2}\}.
\nonumber\end{aligned}$$ The basis vector [**1**]{} generates models with $SO(44)$ gauge group from the Neveu–Schwarz sector. The vector $S$ produces ${N} = 4$ space–time supersymmetry. The vectors $e_{1}$,$\dots$,$e_{6}$ break the $SO(44)$ gauge group to $SO(32) \times U(1)^6$ and preserve the ${N = 4}$ space–time supersymmetry. The $e_i$ basis vectors correspond to all the possible symmetric shifts of the six internal bosonic coordinates. The basis vectors $b_1$ and $b_2$ correspond to ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ orbifold twists and break ${N} = 4$ space–time supersymmetry to $N=1$. Additionally, they reduce the rank of gauge group by breaking the $U(1)^6$ symmetry. Combined with the projections of the basis vectors $z_{1}$ and $z_{2}$ the $SO(32)$ gauge group is reduced to $SO(10) \times U(1)^3 \times SO(8)_1\times SO(8)_2$, where $SO(10) \times U(1)^3$ and $SO(8)_1\times SO(8)_2$ correspond to the observable and hidden gauge groups, respectively. The combined projection of the basis vectors $\alpha$ and $\beta$ breaks the $SO(10)$ GUT symmetry to $SU(4) \times SU(2) \times U(1)$, where $\alpha$ is identical to the basis vector used in the classification of the Pati–Salam models, and hence breaks the $SO(10)$ symmetry to $SO(6)\times SO(4)$ and finally using the $\beta$ basis vector with fractional boundary conditions reduces the $SO(10)$ gauge symmetry to $SU(4) \times SU(2) \times U(1)$.
The String Spectrum {#analysisspec}
-------------------
The space–time vector bosons that are obtained from the Neveu–Schwarz (NS) sector and that survive the GGSO projections, defined by the basis vectors in (\[basis\]) generate the observable and hidden gauge groups given by: $$\begin{aligned}
{\rm Observable} &: &~~~~SU(4) \times SU(2)_L
\times U(1)_R \times{U(1)}^3 \nonumber\\
{\rm Hidden} &: &~~~~SU(2)_A \times U(1)_A \times SU(2)_B
\times U(1)_B \times SU(2)_C \times U(1)_C
\times SO(4)_2 \nonumber\end{aligned}$$ The string states arising in other sectors transform under these gauge group factors. Additional space–time vector bosons that enhance the NS observable and/or hidden gauge groups may arise from additional sectors. In order to preserve the above gauge groups, all these additional space–time vector bosons need to be projected out. These additional space–time vector bosons arise from the following 36 sectors $$\mathbf{G}_{Enh} =
\left\{ \begin{array}{ccccc}
\,\,\,\, z_1 ,&
\,\,\,\, z_1 + \beta ,&
\,\,\,\, z_1 + 2\beta ,\\
\,\,\,\,\, z_1 + \alpha ,&
\,\,\,\,\, z_1 + \alpha + \beta ,&
\,\,\, z_1 + \alpha + 2\beta ,\\
\,\, z_2 ,&
\,\,\,\,\,\,\,\, z_2 + \beta ,&
\,\,\,\,\,\, z_2 + 2\beta ,\\
\,\,\,\,\, z_2 + \alpha ,&
\,\,\,\,\, z_2 + \alpha + \beta ,&
\,\,\, z_2 + \alpha + 2\beta ,\\
\,\, z_1 + z_2 ,&
\,\,\,\,\,\,\,\, z_1 + z_2 + \beta ,&
\,\,\,\,\,\, z_1 + z_2 + 2\beta ,\\
\,\,\,\,\, z_1 + z_2 + \alpha ,&
\,\,\,\,\, z_1 + z_2 + \alpha + \beta ,&
\,\,\, z_1 + z_2 + \alpha + 2\beta ,\\
\,\, \beta ,&
\,\,\,\,\,\,\,\, 2\beta ,&
\,\,\, \alpha,\\
\,\,\,\,\, \alpha + \beta ,&
\,\,\,\,\, \alpha + 2\beta ,&
\,\, x,\\
\,\,\,\,\, z_1 + x + \beta ,&
\,\,\,\,\, z_1 + x + 2\beta ,&
\,\, z_1 + x + \alpha,\\
\,\,\,\,\, z_1 + x + \alpha + \beta ,&
\,\,\, z_2 + x + \beta ,&
\,\,\,\,\, z_2 + x + \alpha + \beta ,\\
\,\,\, z_1 + z_2 + x + \beta ,&
\,\, z_1 + z_2 + x + 2\beta,&
\,\,\,\,\, z_1 + z_2 + x + \alpha + \beta ,\\
\,\, x + \beta,&
\,\,\,\,\, x + \alpha ,&
\,\,\, x + \alpha + \beta ,\\
\end{array} \right\}, \label{ggsectors1}$$\
where $x = 1 + S + \textstyle\sum_{i = 1}^{6} e_i + z_1 + z_2$.
The Matter Content {#analysis2}
------------------
The observable matter states in heterotic–string vacuum with $(2,2)$ world–sheet supersymmetry is embedded in the $\bf{27}$ representation of $E_6$. In the free fermionic construction that we adopt here, and using the basis vectors in (\[421\]), the $E_6$ is first broken to the $SO(10)\times U(1)$ symmetry. Therefore, the $\bf{27}$ of $E_6$ decomposes in the following way $$\begin{aligned}
\textbf{27} &= & \textbf{16} + \textbf{10} + \textbf{1}.\end{aligned}$$ Where the $\textbf{16}$ transforms under the spinorial representation of $SO(10)$ and **10** transforms in the vectorial representation of the $SO(10)$, and similarly for $\bf{\overline{27}}$. The following 48 sectors produce states that give the spinorial $\bf{16}$ or $\bf{\overline{16}}$ of $SO(10)$ $$\begin{aligned}
\label{obspin}
B_{pqrs}^{(1)}&=& S + {b_1 + p e_3+ q e_4 + r e_5 + s e_6} \nonumber\\
&=&\{\psi^\mu,\chi^{12},(1-p)y^{3}\overline{y}^3,p\omega^{3}\overline{\omega}^3,
(1-q)y^{4}\overline{y}^4,q\omega^{4}\overline{\omega}^4, \nonumber\\
& & ~~~(1-r)y^{5}\overline{y}^5,r\omega^{5}\overline{\omega}^5,
(1-s)y^{6}\overline{y}^6,s\omega^{6}\overline{\omega}^6,
\overline{\eta}^1,\overline{\psi}^{1,...,5}\},
\\
B_{pqrs}^{(2)}&=& S + {b_2 + p e_1+ q e_2 + r e_5 + s e_6},
\label{twochiralspinorials}
\nonumber\\
B_{pqrs}^{(3)}&=& S + {b_3 + p e_1+ q e_2 + r e_3 + s e_4}, \nonumber\end{aligned}$$ where $p,q,r,s=0,1$ and $b_3=b_1+b_2+x$. In order to distinguish between the spinorial $\bf{16}$ and $\bf{\overline{16}}$ in the states given above, the following chirality operators are used
$$\begin{aligned}
\label{so10operators}
X_{pqrs}^{(1)_{SO(10)}} & = &
C\binom{B^{(1)}_{pqrs}}{b_{2} + (1-r)e_{5} + (1-s)e_{6}},\nonumber\\
X_{pqrs}^{(2)_{SO(10)}} & = &
C\binom{B^{(2)}_{pqrs}}{b_{1} + (1-r)e_{5} + (1-s)e_{6}},\\
X_{pqrs}^{(3)_{SO(10)}} & = &
C\binom{B^{(3)}_{pqrs}}{b_{1} + (1-r)e_{3} + (1-s)e_{4}}.\nonumber\end{aligned}$$
Where $X_{pqrs}^{(1,2,3)_{SO(10)}} = 1$ implies the states corresponds to the $\bf{16}$ of $SO(10)$ and $X_{pqrs}^{(i)_{SO(10)}} = -1$ to the $\bf{\overline{16}}$ of $SO(10)$. Moreover, we note that the states here can be projected in or out depending on the GGSO projections of the basis vectors $e_1,....,e_6$, $z_1$ and $z_2$. Therefore, we define below a projector $P$, such that $P=1$ implies the state is projected in and $P=0$ implies the state is projected out. The projector $P$ is given by
$$\begin{aligned}
\label{matrixequations}
P_{pqrs}^{(1)} &= \frac{1}{16}
\left( 1-C \binom {e_1} {B_{pqrs}^{(1)}}\right) .
\left( 1-C \binom {e_2} {B_{pqrs}^{(1)}}\right) .
\left( 1-C \binom {z_1} {B_{pqrs}^{(1)}}\right) .
\left( 1-C \binom {z_2} {B_{pqrs}^{(1)}}\right),\nonumber\\
P_{pqrs}^{(2)} &= \frac{1}{16}
\left( 1-C \binom {e_3} {B_{pqrs}^{(2)}}\right) .
\left( 1-C \binom {e_4} {B_{pqrs}^{(2)}}\right) .
\left( 1-C \binom {z_1} {B_{pqrs}^{(2)}}\right) .
\left( 1-C \binom {z_2} {B_{pqrs}^{(2)}}\right),\\
P_{pqrs}^{(3)} &= \frac{1}{16}
\left( 1-C \binom {e_5} {B_{pqrs}^{(3)}}\right) .
\left( 1-C \binom {e_6} {B_{pqrs}^{(3)}}\right) .
\left( 1-C \binom {z_1} {B_{pqrs}^{(3)}}\right) .
\left( 1-C \binom {z_2} {B_{pqrs}^{(3)}}\right).\nonumber\end{aligned}$$ These projectors above can in fact be expressed as matrix equations given by
$$\begin{aligned}
\begin{pmatrix} (e_1|e_3)&(e_1|e_4)&(e_1|e_5)&(e_1|e_6)\\
(e_2|e_3)&(e_2|e_4)&(e_2|e_5)&(e_2|e_6)\\
(z_1|e_3)&(z_1|e_4)&(z_1|e_5)&(z_1|e_6)\\
(z_2|e_3)&(z_2|e_4)&(z_2|e_5)&(z_2|e_6) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_1|b_1)\\
(e_2|b_1)\\
(z_1|b_1)\\
(z_2|b_1)
\end{pmatrix},\nonumber
\\[0.3cm]
\begin{pmatrix} (e_3|e_1)&(e_3|e_2)&(e_3|e_5)&(e_3|e_6)\\
(e_4|e_1)&(e_4|e_2)&(e_4|e_5)&(e_4|e_6)\\
(z_1|e_1)&(z_1|e_2)&(z_1|e_5)&(z_1|e_6)\\
(z_2|e_1)&(z_2|e_2)&(z_2|e_5)&(z_2|e_6) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_3|b_2)\\
(e_4|b_2)\\
(z_1|b_2)\\
(z_2|b_2)
\end{pmatrix},
\\[0.3cm]
\begin{pmatrix} (e_5|e_1)&(e_5|e_2)&(e_5|e_3)&(e_5|e_4)\\
(e_6|e_1)&(e_6|e_2)&(e_6|e_3)&(e_6|e_4)\\
(z_1|e_1)&(z_1|e_2)&(z_1|e_3)&(z_1|e_4)\\
(z_2|e_1)&(z_2|e_2)&(z_2|e_3)&(z_2|e_4) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_5|b_3)\\
(e_6|b_3)\\
(z_1|b_3)\\
(z_2|b_3)
\end{pmatrix}.\nonumber\end{aligned}$$
Writing the projectors as matrix equations given above entails solving systems of linear equations. These algebraic equations can be solved using a computerised code, which can be used to scan a vast space of models.
Similar to the spinorial representations singlet and vectorial $\bf{10}$ representations of $SO(10)$ are obtained from the following 48 sectors $$\begin{aligned}
\label{lighthiggssectors}
B_{pqrs}^{(4)}&=& B_{pqrs}^{(1)} + x
\nonumber\\
&=&\{\psi^\mu,\chi^{12},(1-p)y^{3}\overline{y}^3,p\omega^{3}\overline{\omega}^3,
(1-q)y^{4}\overline{y}^4,q\omega^{4}\overline{\omega}^4, \nonumber\\
& & ~~~~~~~~~(1-r)y^{5}\overline{y}^5,r\omega^{5}\overline{\omega}^5,
(1-s)y^{6}\overline{y}^6,s\omega^{6}\overline{\omega}^6,\overline{\eta}^{2,3} \},
\label{nonchiralvectorials}\\
B_{pqrs}^{(5,6)}&=& B_{pqrs}^{(2,3)} + x. \nonumber\end{aligned}$$ Massless states that arise in these sectors are obtained by acting on the vacuum with a NS oscillator. The type of states therefore depend on the type of oscillator, and may correspond to $SO(10)$ singlets or vectorial $\bf{10}$ representation of $SO(10)$, which is needed for electroweak symmetry breaking. The different type of $SO(10)$ singlets arising from eq. (\[nonchiralvectorials\]) are
- $\{\overline\eta^{i}\}|R \rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\eta^{*i}\}|R\rangle_{pqrs}^{(4,5,6)}$, $i = 1,2,3$, where $|R\rangle_{pqrs}^{(4,5,6)}$ is the degenerated Ramond vacuum of the $B_{pqrs}^{(4,5,6)}$ sector. These states transform as a vector–like representations under the $U(1)_i$’s.
- $\{\overline\phi^{1,2}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\phi^{*1,2}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SU(2)_A \times U(1)_A$.
- $\{\overline\phi^{3,4}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\phi^{*3,4}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SU(2)_B \times U(1)_B$.
- $\{\overline\phi^{5,6}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\phi^{*5,6}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SU(2)_C \times U(1)_C$.
- $\{\overline\phi^{7,8}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\phi^{*7,8}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SO(4)$.
Similarly, for the matrix equations given above in eq. (\[matrixequations\]), we can write algebraic equations for the sectors in eq. (\[lighthiggssectors\]) given as follows:
$$\begin{aligned}
\begin{pmatrix} (e_1|e_3)&(e_1|e_4)&(e_1|e_5)&(e_1|e_6)\\
(e_2|e_3)&(e_2|e_4)&(e_2|e_5)&(e_2|e_6)\\
(z_1|e_3)&(z_1|e_4)&(z_1|e_5)&(z_1|e_6)\\
(z_2|e_3)&(z_2|e_4)&(z_2|e_5)&(z_2|e_6) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_1|b_1 + x)\\
(e_2|b_1 + x)\\
(z_1|b_1 + x)\\
(z_2|b_1 + x)
\end{pmatrix},\nonumber
\\[0.3cm]
\begin{pmatrix} (e_3|e_1)&(e_3|e_2)&(e_3|e_5)&(e_3|e_6)\\
(e_4|e_1)&(e_4|e_2)&(e_4|e_5)&(e_4|e_6)\\
(z_1|e_1)&(z_1|e_2)&(z_1|e_5)&(z_1|e_6)\\
(z_2|e_1)&(z_2|e_2)&(z_2|e_5)&(z_2|e_6) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_3|b_2 + x)\\
(e_4|b_2 + x)\\
(z_1|b_2 + x)\\
(z_2|b_2 + x)
\end{pmatrix},
\\[0.3cm]
\begin{pmatrix} (e_5|e_1)&(e_5|e_2)&(e_5|e_3)&(e_5|e_4)\\
(e_6|e_1)&(e_6|e_2)&(e_6|e_3)&(e_6|e_4)\\
(z_1|e_1)&(z_1|e_2)&(z_1|e_3)&(z_1|e_4)\\
(z_2|e_1)&(z_2|e_2)&(z_2|e_3)&(z_2|e_4) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (es_5|b_3 + x)\\
(e_6|b_3 + x)\\
(z_1|b_3 + x)\\
(z_2|b_3 + x)
\end{pmatrix}.\nonumber\end{aligned}$$
The Observable Matter Spectrum {#observable}
==============================
The basis vectors $\alpha$ and $\beta$ given in eq. (\[421\]) break the $SO(10)$ symmetry to $SU(4) \times SU(2)_L \times U(1)_R$. Following the $\alpha$ and $\beta$ GGSO projections, the decomposition of the spinorial $\bf{16}$ and $\bf{\overline{16}}$ representations of $SO(10)$, under the $SU(4) \times SU(2)_L \times U(1)_L$ gauge group is given as follows: $$\begin{aligned}
\textbf{16} &= &\left({\textbf{4}},{\textbf{2}},
0\right) + \left(\overline{{\textbf{4}}},{\textbf{1}},
-1\right) + \left(\overline{{\textbf{4}}},{\textbf{1}},
+1\right),\nonumber\\
\overline{\textbf{16}} &= &\left(\overline{{\textbf{4}}},{\textbf{2}},
0\right) + \left({\textbf{4}},{\textbf{1}},
-1\right) + \left({\textbf{4}},{\textbf{1}},
+1\right).\nonumber\end{aligned}$$ Here to break the $SU(4) \times SU(2)_L \times U(1)_L$ gauge group to the standard model group, we require the heavy higgs pair. This pair is given by $$\begin{aligned}
\left(\overline{{\textbf{4}}},{\textbf{1}},
-1\right) + \left({\textbf{4}},{\textbf{1}},
-1\right).\nonumber\end{aligned}$$ Similarly, the vectorial representation $\bf{10}$ of $SO(10)$ decomposed under the $SU(4) \times SU(2)_L \times U(1)_L$ gauge group is given as follows $$\begin{aligned}
\textbf{10} &= &\left({\textbf{6}},{\textbf{1}},
0\right) + \left({\textbf{1}},{\textbf{2}},
-1\right) + \left({\textbf{1}},{\textbf{2}},
+1\right)\nonumber,\end{aligned}$$ Furthermore, we take the following normalizations of the hypercharge and electromagnetic charge $$\begin{aligned}
Y &=& \frac{1}{3} (Q_1 + Q_2 + Q_3) + \frac{1}{2} (Q_4 + Q_5), \nonumber\\
Q_{em} &=& Y + \frac{1}{2} (Q_4 - Q_5). \nonumber\end{aligned}$$ where the $Q_{i}$ charges of a state arise due to $\psi^{i}$ for $i =1,...,5$. The following table summaries the charges of the colour $SU(3)$ and electroweak $SU(2) \times U(1)$ Cartan generators of the states which form the $SU(4) \times SU(2)_L \times U(1)_L$ matter representations:
Representation $\overline{\psi}^{1,2,3}$ $\overline{\psi}^{4,5}$ $Y$ $Q_{em}$
---------------------------------------------------------------------- --------------------------- ------------------------- ------ ----------
($+,+,-$) ($+,-$) 1/6 2/3
($+,+,-$) ($-,+$) 1/6 -1/3
$\left( \, \textbf{4} \, , \textbf{2} , \, 0 \, \right)$ $(-,-,-)$ ($+,-$) -1/2 0
$(-,-,-)$ ($-,+$) -1/2 -1
($+,-,-$) $(-,-)$ -2/3 -2/3
$\left( \, \overline{\textbf{4}} \, , \textbf{1} , \, -1 \, \right)$ $(+,+,+)$ $(-,-)$ 0 0
($+,-,-$) $(+,+)$ 1/3 1/3
$\left( \, \overline{\textbf{4}} \, , \textbf{1} , \, +1 \, \right)$ $(+,+,+)$ $(+,+)$ 1 1
($+,-,-$) ($+,-$) -1/6 -2/3
($+,-,-$) ($-,+$) -1/6 1/3
$\left( \, \overline{\textbf{4}} \, , \textbf{2} , \, 0 \, \right)$ $(+,+,+)$ ($+,-$) 1/2 0
$(+,+,+)$ ($-,+$) 1/2 1
($+,+,-$) $(+,+)$ 2/3 2/3
$\left( \, \textbf{4} \, , \textbf{1} , \, -1 \, \right)$ $(-,-,-)$ $(+,+)$ 0 0
($+,+,-$) $(-,-)$ -1/3 -1/3
$\left( \, \textbf{4} \, , \textbf{1} , \, +1 \, \right)$ $(-,-,-)$ $(-,-)$ -1 -1
Here $``+"$ and $``-"$, label the contribution of an oscillator with fermion number $F = 0$ or $F = -1$, to the degenerate vacuum. These states correspond to particles of the Standard Model. More precisely we can decompose these representations under $SU(3) \times SU(2) \times U(1)$ as $$\begin{aligned}
\label{16decomposition}
\left( \textbf{4} , \textbf{2} , 0 \right)&
= \left(\textbf{3},\textbf{2},+\frac{1}{6}\right)_{Q} +
\left(\textbf{1},\textbf{2},-\frac{1}{2}\right)_{L}, \nonumber \\
\left( \overline{\textbf{4}} , \textbf{1} , -1 \right) &=
\left(\overline{\textbf{3}},\textbf{1},-\frac{2}{3}\right)_{u^c}+
\left(\textbf{1},\textbf{1},0\right)_{\nu^c},\nonumber\\
\left( \overline{\textbf{4}} , \textbf{1} , +1 \right)&=
\left(\overline{\textbf{3}},\textbf{1},+\frac{1}{3}\right)_{d^c}+
\left(\textbf{1},\textbf{1},+1 \, \right)_{e^c}. \nonumber\end{aligned}$$ Where $L$ is the lepton–doublet; $Q$ is the quark–doublet; $d^c,~u^c,~e^c$ and $\nu^c$ are the quark and lepton singlets. Because of the $\alpha$- and $\beta$-projections, which projects on incomplete $\textbf{16}$ and $\overline{\textbf{16}}$ representations, complete families and anti–families are formed by combining states from different sectors.
Nonviability of the $SU(4) \times SU(2) \times U(1)$ model {#nonv}
==========================================================
We now discuss why in our free fermionic construction, the $SU(4) \times SU(2) \times U(1)$ GUT models are not viable. As mentioned in the previous section, the matter content comes from the $\bf{16}$ of $SO(10)$. However, with the addition of the $\alpha$ and $\beta$ basis vectors from eq. (\[421\]), the $\bf{16}$ representation is broken by the GGSO projections that are in general given by $$\label{gso}
e^{i\pi v_i\cdot F_{\xi}} |S_{\xi}> =
\delta_{{\xi}}\ C\binom {\xi} {v_i}^* |S_{\xi}>.$$ Here $\delta_{{\xi}}=\pm1$ is a spacetime spin statistics index and $F_{\xi}$ is the fermion number operator. In the SU421 models spanned by eq. (\[basis\]) the GGSO projection coefficients $C \binom {\xi} {v_i}$ can take the values $\pm1; \pm i$. Therefore, firstly considering the $\alpha$ GGSO projection, we decompose the $\bf{16}$ into the Pati-Salam group representation. Moreover, using the following chirality operators $$\begin{aligned}
\label{patioperators}
X_{pqrs}^{(1)_{SO(6)}} & = &
C\binom{B^{(1)}_{pqrs}}{\alpha},\nonumber\\
X_{pqrs}^{(2)_{SO(6)}} & = &
C\binom{B^{(2)}_{pqrs}}{\alpha},\\
X_{pqrs}^{(3)_{SO(6)}} & = &
C\binom{B^{(3)}_{pqrs}}{\alpha},\nonumber\end{aligned}$$ we deduce that for $X_{pqrs}^{(i)_{SO(6)}} = 1$ we get the $Q_R\equiv(\bf{\overline{4}},\bf{1},\bf{2})$ states under $SU(4) \times SU(2)_L \times SU(2)_R$, whereas the $Q_L\equiv(\bf{4},\bf{2},\bf{1})$ states correspond to $X_{pqrs}^{(i)_{SO(6)}} = -1$. Next, considering the $\beta$ GGSO projection, the operators $$\begin{aligned}
\label{su4operators}
X_{pqrs}^{(1)_{421}} & = &
C\binom{B^{(1)}_{pqrs}}{\beta},\nonumber\\
X_{pqrs}^{(2)_{421}} & = &
C\binom{B^{(2)}_{pqrs}}{\beta},\\
X_{pqrs}^{(3)_{421}} & = &
C\binom{B^{(3)}_{pqrs}}{\beta}.\nonumber\end{aligned}$$ determine the decomposition of the $Q_L$ and $Q_R$ states under $SU(4)\times SU(2)\times U(1)$. Here, the product $\beta\cdot B_j^{pqrs}= -1$ with $(j=1,2,3)$, and the modular invariance constraints, impose that $X_{pqrs}^{(1,2,3)_{421}} = \pm \, i$. Therefore, this implies the states cannot be completed to form a family. Thus, to complete the $\bf{16}$ the states: $(\textbf{4} , \textbf{2} , 0)$, $(\overline{\textbf{4}} , \textbf{1} , -1)$ and $(\overline{\textbf{4}} , \textbf{1} , +1)$ under the $SU(4) \times SU(2)_L \times U(1)_R$ group all need to survive the GGSO projections, but in order for the $(\overline{\textbf{4}} , \textbf{1} , -1)$ and $(\overline{\textbf{4}} , \textbf{1} , +1)$ states to survive, we need $X_{pqrs}^{(1,2,3)_{421}} = \pm \, 1$, which is forbidden in this case by modular invariance. To see more clearly why this is the case we consider the decomposition of the $\bf{16}$ representation in the combinatorial notation of ref. [@xmap] $$\begin{aligned}
{\bf 16}
& \equiv &
\left[ \binom{5}{0} + \binom{5}{2} + \binom{5}{4} \right] \label{so1016}\\
& \equiv &
\left[ \binom{3}{0} + \binom{3}{2} \right]
\left[ \binom{2}{0} + \binom{2}{2} \right]
~+~
\left[ \binom{3}{1} \right]
\left[ \binom{2}{1} \right] \label{so64}\\
& \equiv &
\left[ \binom{3}{0} + \binom{3}{2} \right]
\left[ \binom{2}{0} \right]
~+~
\left[ \binom{3}{0} + \binom{3}{2} \right]
\left[ \binom{2}{2} \right]
~+~
\left[ \binom{3}{1} \right]
\left[ \binom{2}{1} \right]~~~~~
\label{su421decomposition}\end{aligned}$$ where the combinatorial factor counts the number of periodic fermions in the $\vert -\rangle$ state. The second line in eq. (\[so64\]) corresponds to the decomposition of the ${\bf16}$ under the Pati–Salam subgroup, whereas eq. (\[su421decomposition\]) shows its decomposition under the SU421 subgroup. The key point here, as seen from eq. (\[su421decomposition\]), is the even number of fermions in the $\vert -\rangle$ vacuum of the $Q_R$ states, resulting in $\pm1$ projections on the left–hand side of eq. (\[gso\]), whereas the right–hand side is fixed by the product $\beta\cdot B_j^{pqrs}= -1$ to be $\pm i$. Thus, the exclusion arises because the $\beta$ projection fixes the chirality of the vacuum of the world–sheet fermions ${\overline\psi}^{4,5}$ that generate the $SU(2)_L\times U(1)_R$ symmetry. We note that the situation here is different from the case of the SU421 models of ref. [@su421]. The reason is that our classification method only allows for symmetric boundary conditions for the set of internal fermions $\{y,\omega\vert{\overline y},{\overline\omega}\}^{1,\cdots,6}$, whereas the models of ref. [@su421] introduce additional freedom by allowing asymmetric boundary conditions. Thus, while the NAHE–based models of ref. [@su421] did not yield any model with three complete generations they contain both the $Q_L$ and $Q_R$ states in their spectra, whereas vacua with only symmetric boundary conditions with respect to the set $\{y,\omega\vert{\overline y},{\overline\omega}\}^{1,\cdots,6}$ do not contain $Q_R$ states and are therefore categorically excluded. It is of further interest to note that in the case of the LRS models the chirality of the $Q_L+L_L$ and $Q_R+L_R$ is similarly affected [@lrs]. However, there it is compensated by the chirality of the ${\overline\eta}^j$ worldsheet fermions leading to opposite charges under the $U(1)_j$ gauge symmetries. The SLM models [@slm] are obtained by combining the PS and FSU5 breaking vectors. Therefore, the SLM models produce complete ${\bf16}$ multiplets decomposed under the SLM group and with equal $U(1)_j$ charges. The SU421 class of models is the only case that is excluded in vacua with symmetric internal boundary conditions.
Conclusion
==========
In this paper we discussed the classification of the SU421 models with symmetric internal boundary conditions. This continues the development of the classification program initiated in ref. [@fknr], which led to the discovery of spinor–vector duality [@spinvecdual] and exophobic string vacua [@acfkr; @cfr; @SU6SU2]. The novel feature in the classification of the SU421 models compared to the PS and FSU5 vacua is the introduction of two basis vectors that break the $SO(10)$ symmetry. An appealing feature of the SU421 models is the admission of both the triplet–doublet as well as the doublet–doublet splitting mechanism, which is shared only with the standard–like models. However, as we showed in section \[nonv\] these models cannot accommodate the weak $SU(2)$ singlet states of the Standard Model and are therefore excluded. The next step in our classification program is the classification of standard–like models that will be reported in a future publication.
Acknowledgements
================
We are grateful to John Rizos for fruitful discussions. The work of A.F. is partially supported by the UK Science and Technology Facilities Council (STFC) under grant number ST/G00062X/1.
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[^1]: we note that $U(1)_R$ as defined here is equal to $1/2\, U(1)_L$ as defined in ref. [@slm].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'An efficient first principles method was developed to calculate spin transfer torques in layered system with noncollinear magnetization. The complete scattering wave function is determined by matching the wave function in the scattering region with the Bloch states in the leads. The spin transfer torques are obtained with aid of the scattering wave function. We applied our method to the ferromagnetic spin valve and found that the material (Co, Ni and Ni$_{80}$Fe$_{20}$) dependence of the spin transfer torques could be well understood by the Fermi surface. Ni has much longer spin injection penetration length than Co. Interfacial disorder is also considered. It is found that the spin transfer torques could be enhanced by the interfacial disorder in some system.'
author:
- Shuai Wang
- Yuan Xu
- Ke Xia
title: First principles study on the spin transfer torques
---
Introduction
============
Spin angular momentum can be transferred by the flowing electrons from one ferromagnetic (FM) material to another FM material, which is so-called spin transfer torques (STT) introduced by Slonczewski[@J.Slonc96] and Berger[@Berger96]. Those two seminal studies have shown that the dynamics of magnetization in FM material could be dominated by the spin torques carried by electric current. The excitation of coherent precession of magnetization and spin wave were predicted. The STT was soon identified in the experiments[@experiment] by clear observation of the magnetization switching in FM spin valve, which excites great interests in experiment and theory[sun00,zhangsc98,Waintal,Brataas\_circuit,MDstiles02,stiles02,Edwards05,PMHaney]{}.
The theories[@Waintal; @Brataas_circuit; @MDstiles02; @stiles02] combining the quantum treatment of the interface scattering and the Boltzmann-like treatment of the bulk scattering work reasonable well with the experiments of metallic system. However, recent experiments on the tunnelling system[Fuchs]{} and magnetic domain wall[@exp_domainwall] call for a full quantum treatment of the whole system. Edwards * et.al.,*[Edwards05]{} obtained the torques of spin valve in the empirical tight-binding frame and Haney *et.al.,*[@PMHaney] calculated the torques in the similar structure with nonequilibrium Green’s function (NEGF) based on LCAO basis.
Both semiclassical and quantum mechanical study show that the STT is most significant near the nonmagnet(NM)$|$FM interfaces in the spin valve. Up to now, only a few studies have addressed the material dependence of spin torque, which could be an important issue as the spin dependent transport is greatly affected by the electronic structure in FM[Maciej\_decay,mixing\_G\_Turek]{}. Furthermore, previous studies focused on ideal structure without considering the disorder at the FM$|$NM interface, which should exist in the realistic spin valve[@flip06].
The main aim of this paper is to formulate a method to calculate STT of a noncollinear magnetized system within the first principles frame. Differing from the previous Green function based work[@PMHaney], we obtained the complete scattering wave functions of the whole system[@Xia06]. The STT[@MDstiles02] is formulated in the tight-binding representation. Large system such as domain wall can be well treated in this framework[@tang06]. We apply our formulism to the Co$|$Cu$|$FM$|$Cu spin valve system with impurity scattering at the FM$|$NM interface. Our study shows that the STT can penetrate deep into the ferromagnetic materials for Ni, which is quite different from Co. It is also found that average torques are enhanced in the presence of interfacial disorder.
This paper is organized as following. In Sec. II, we present the details of the formalism for constructing the eigenmodes of the lead and computing the STT in spin valve. Note that not only the transmission and reflection coefficient are obtained but also the wave function in the scattering regime is obtained explicitly. In Section III, the method is used to calculate the conductance and STT in the systems of Co$|$Cu$|$FM$|$Cu(111), with FM is Co, Ni and Ni$_{80}$Fe$_{20}$, respectively. The effect of interfacial disorder is discussed. In Sec. IV, we summarize our results.
Theoretical model
=================
Let us focus on the spin transport and STT in the layered systems sketched in Fig.\[config\]. The scattering region $\mathbf{S}$, which is denoted by the layer index $1\leq I\leq N$, is sandwiched by left$\left( \mathbf{L}%
\right) $ and right$\left( \mathbf{R} \right) $ leads. For this device, there exists perfect lattice periodicity in the $X$-$Z$ plane. Particle current flows along $Y$ axis. In scattering region no periodicity is assumed along current direction. Here the atomic potentials were determined by the tight-binding linearized muffin-tin-orbital (TB-LMTO) surface Green’s function (SGF) method[@I.Turek97book]. When combined with the coherent potential approximation (CPA), this method allows the electronic structure, charge, and spin densities of layered materials with substitutional disorder to be calculated self-consistently with high efficiency. To model the noncollinear system in the spin valve, the rigid potential approximation is used. In this approximation, we rotate the potential of fixed magnet in spin space to construct the relative angle between the polarization directions of fixed magnet and free magnet, which is a good approximation as the two magnets are spaced far enough by a Cu layer.
![(color online) Sketch of the configuration used for current-induced switching. A scattering region is sandwiched by left-($\mathbf{L}$) and right-hand($\mathbf{R}$)leads which have translational symmetry and are partitioned into principle layers perpendicular to the transport direction. The scattering region contains $N$ principle layers but the structure and chemical composition are in principle arbitrary. The switching layer FM can be Co, Ni, Ni$_{80}$Fe$_{20}$.[]{data-label="config"}](config_1){width="8.6cm"}
Following previous work[@Xia06], we describe the theoretical frame developed with wave-function matching (WFM) based on TB-LMTO basis for studying the STT. In Sec. II A, we review the Hamiltonian and KKR equation for a device with noncollinear magnetization. The equation of motion (EOM) for layered system is extracted from KKR equation. In the Sec.II B, the boundary conditions of the EOM are formulated in terms of the Bloch states in the leads. In the Sec. II C, by solving the EOM in the scattering region with embedding potentials of the two leads, we obtain the complete scattering wave function of the scattering region. In the Sec. II D and E, the particle current and spin current are formulated with those obtained scattering wave function expanded in TB-LMTO basis.
Hamiltonian and KKR equation
----------------------------
For layered systems, atoms can always be grouped into principle layers defined as to be so thick that the interactions between layers $I$ and $I\pm
2$ are negligible as shown in Fig.\[config\].
The EOM for $I$th principal layer can be written as $$\mathbf{H}_{I,I-1}\mathbf{a}_{I-1}+\left( \mathbf{H}-E\right) _{II}\mathbf{a}%
_{I}+\mathbf{H}_{I,I+1}\mathbf{a}_{I+1}=0, \label{eom}$$where $E$ is always set to the Fermi energy $E_{F}$ for the transport problem. Here, $\mathbf{a}_{I}$ is the a vector describing the amplitudes of the $I$th layer in terms of the localized orbital basis $\left\vert \mathbf{R%
}L\zeta \right\rangle $, where $\mathbf{R}$ is the site index and $L$ can be defined by $L\equiv (l,m)$. $l$ and $m$ are the azimuthal and magnetic quantum numbers respectively. $\zeta =\uparrow \left( \downarrow \right) $ denotes that the basis is eigenstate in spin space, which is parallel (antiparallel) to spin quantization axis.
To the first order approximation of the full LMTO Hamiltonian, a short-range TB-LMTO Hamiltonian in the $\alpha $ representation[J.Kudron00,Andersen85book]{} in the global coordinate system can be written as $$\begin{aligned}
\mathbf{H}_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\alpha } &=&U_{%
\mathbf{R}}\mathcal{\overline{C}}_{\mathbf{R}L}^{\alpha }U_{\mathbf{R}%
^{\prime }}^{\dagger }\delta _{\mathbf{R}^{\prime }L^{\prime }\mathbf{R}L}
\notag \label{hamiltonian} \\
&&+[U_{\mathbf{R}}\left( \overline{\Delta }_{\mathbf{R}L}^{\alpha }\right) ^{%
\frac{1}{2}}U_{\mathbf{R}}^{\dag }S_{\mathbf{R}L,\mathbf{R}^{\prime
}L^{\prime }}^{\alpha } \notag \\
&&\times U_{\mathbf{R}^{\prime }}\left( \overline{\Delta }_{\mathbf{R}%
^{\prime }L^{\prime }}^{\alpha }\right) ^{\frac{1}{2}}U_{\mathbf{R}^{\prime
}}^{\dagger }],\end{aligned}$$where $\mathcal{\overline{C}}_{\mathbf{R}L}^{\alpha }$ and $\overline{\Delta
}_{\mathbf{R}L}^{\alpha }$ are $2\times 2$ potential parameter matrices expanded in spin space and diagonal in the local coordinate system. The unitary rotation matrix at site $\mathbf{R}$ can be defined by $$U_{\mathbf{R}}\left( \theta _{\mathbf{R}},\varphi _{\mathbf{R}}\right) =%
\left[
\begin{array}{cc}
\cos \frac{\theta _{\mathbf{R}}}{2}e^{-i\frac{\varphi _{\mathbf{R}}}{2}} &
-\sin \frac{\theta _{\mathbf{R}}}{2}e^{-i\frac{\varphi _{\mathbf{R}}}{2}} \\
\sin \frac{\theta _{\mathbf{R}}}{2}e^{i\frac{\varphi _{\mathbf{R}}}{2}} &
\cos \frac{\theta _{\mathbf{R}}}{2}e^{i\frac{\varphi _{\mathbf{R}}}{2}}%
\end{array}%
\right] , \label{u}$$where $\theta _{\mathbf{R}},\varphi _{\mathbf{R}}$ are the azimuth angles of the local quantization axis. Screened structure constants $S_{\mathbf{R}L,%
\mathbf{R}^{\prime }L^{\prime }}^{\alpha }$ contain all information about the structure, which are block diagonal in the spin space, $$S_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\alpha }=\left[
\begin{array}{cc}
s_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\alpha } & 0 \\
0 & s_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\alpha }%
\end{array}
\right] . \label{s}$$ Note that $s_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\alpha }$ is spin independent. The Hamiltonian of Eq.(\[hamiltonian\]) yields eigenvalues corrected to first order in $\left( E-E_{F}\right) $ and is exact when we set $E=E_{F}$.
For a noncollinear magnetized system, the “tail-cancellation” condition yields the KKR equation[@Andersen85book], $$\underset{\mathbf{R}^{\prime },L^{\prime }}{\sum }\left( -U_{\mathbf{R}}%
\overline{P}_{\mathbf{R}L}^{\alpha }\left( E\right) U_{\mathbf{R}}^{\dag
}\delta _{\mathbf{RR}^{\prime }}\delta _{LL^{\prime }}-S_{\mathbf{R}L,%
\mathbf{R}^{\prime }L^{\prime }}^{\mathbf{\alpha }}\right) C_{\mathbf{R}%
^{\prime }L^{\prime }}=0, \label{tail_cancellation}$$where $\mathbf{C}_{\mathbf{R}L}=\left( \mathbf{C}_{\mathbf{R}L\uparrow },%
\mathbf{C}_{\mathbf{R}L\downarrow }\right) ^{T}$ has the relation to the wave amplitude $\mathbf{a}_{\mathbf{R}L}$ of $L$ orbital at site $\mathbf{R}$ as $\mathbf{C}_{\mathbf{R}L}=U_{\mathbf{R}}\left( \overline{\Delta }_{%
\mathbf{R}L}^{\alpha }\right) ^{\frac{1}{2}}U_{\mathbf{R}}^{\dagger }\mathbf{%
a}_{\mathbf{R}L}$. $\overline{P}_{\mathbf{R}L}^{\alpha }\left( E\right) $ is the screened potential function matrix and contains all information about the atomic species at site $\mathbf{R}$ for calculating the electronic structure. It is diagonal in the local coordinate system, $$\overline{P}_{\mathbf{R}L}^{\alpha }\left( E\right) \equiv \left[
\begin{array}{cc}
\overline{p}_{\mathbf{R}L}^{\alpha ,\uparrow } & 0 \\
0 & \overline{p}_{\mathbf{R}L}^{\alpha ,\downarrow }%
\end{array}%
\right] , \label{p_func}$$where $\overline{p}_{\mathbf{R}L}^{\alpha ,\uparrow \left( \downarrow
\right) }\equiv \left( E-\mathcal{\overline{C}}_{\mathbf{R}L}^{\alpha
,\uparrow \left( \downarrow \right) }\right) \left( \overline{\Delta }_{%
\mathbf{R}L}^{\alpha ,\uparrow \left( \downarrow \right) }\right) ^{-1}$ and $E$ is set to $E_{F}$ for the transport problem we considered.
As there exists two-dimensional translational symmetry in the lateral plane, the states along the transport direction can be characterized by a lateral wave vector $\mathbf{k}_{\parallel }$ in the corresponding 2-dimensional Brillouin zone (2D BZ). The screened KKR equation in the mixed representation of $\mathbf{k}_{\parallel }$ can be expressed in terms of principal layers as,
$$-S_{I,I-1}^{\mathbf{k}_{\parallel }}\mathbf{C}_{I-1}\left(
\mathbf{k}_{\parallel}\right) +\left(
U_{I}\overline{P}_{I,I}\left( E_{F}\right) U_{I}^{\dag
}-S_{I,I}^{\mathbf{k}_{\parallel }}\right) \mathbf{C}_{I}\left(
\mathbf{k}_{\parallel}\right) -S_{I,I+1}^{\mathbf{k}_{\parallel
}}\mathbf{C}_{I+1}\left( \mathbf{k}_{\parallel}\right) =0,
\label{kkr}$$
where $\mathbf{C}_{I}\left( \mathbf{k}_{\parallel }\right) $ is the wave vector describing the wave function amplitudes of the $I$th principal layer consisting of $h$ atom sites and has the dimension of $%
2\left( l_{\max }+1\right) ^{2}h=2M$. $\overline{P}_{I,I}$ is $2M\times 2M$ diagonal matrix. and $S_{I,I}^{\mathbf{k}_{\parallel }}$ is also $2M\times
2M $ matrices with its sub matrix $S_{\mathbf{R}L,\mathbf{R}^{\prime
}L^{\prime }}^{\mathbf{k}_{\Vert }}$ defined by $$\begin{aligned}
&&S_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\mathbf{k}_{\Vert }}
\notag \\
&=&\underset{\mathbf{T}_{\Vert }}{\sum }\exp \left[ i\mathbf{k}_{\Vert
}\cdot \mathbf{T}_{\parallel }\right] S_{\mathbf{R}L,\left( \mathbf{R}%
^{\prime }+\mathbf{T}_{\parallel }\right) L^{\prime }}^{\alpha }\text{ \ }%
\begin{array}{c}
\mathbf{R}\in I \\
\mathbf{R}^{\prime }\mathbf{+T}_{\parallel }\in I^{\prime }%
\end{array}%
, \label{structure_k}\end{aligned}$$where $I$ and $I^{\prime }$ are layer index and $\mathbf{T}_{\parallel }$ is 2-dimensional translational vector in the plane of principal layer.
Note that Eq.(\[kkr\]) is the EOM by analogy with Eq.(\[eom\]). We will solve it for a given energy $E_{F}$ of electrons to obtain the wave function of the scattering state. The reference to $\mathbf{k}_{\parallel}$ and $%
E_{F} $ in the formulism will be suppressed in the following two parts Sec. II B and Sec. II C.
Eigenmodes of the leads
-----------------------
For the scattering problem, far enough away from the scattering region the wave function can be expressed rigorously with asymptotic forms in terms of reflection and transmission coefficients and Bloch states in the leads. As the wave function should satisfy Bloch’s theorem in a periodic potential, we set $\mathbf{\overline{C}}_{n}\mathbf{=}\lambda ^{n}\mathbf{\overline{C}}%
_{0} $. In local coordinate system, the EOM in lead becomes $$\begin{aligned}
&&\left(
\begin{array}{cc}
S_{0,1}^{-1}\left( \overline{P}_{00}-S_{0,0}\right) & -S_{0,1}^{-1}S_{1,0}
\\
1 & 0%
\end{array}%
\right) \left(
\begin{array}{c}
\mathbf{\overline{C}}_{0} \\
\mathbf{\overline{C}}_{-1}%
\end{array}%
\right) \notag \\
&=&\lambda \left(
\begin{array}{c}
\mathbf{\overline{C}}_{0} \\
\mathbf{\overline{C}}_{-1}%
\end{array}%
\right). \label{lead_u}\end{aligned}$$Details for solving Bloch states $\mathbf{\overline{C}}_{0}$ can be found in Ref.\[\]. To overcome the numerical difficult of the spin degeneracy in NM lead and reduce the calculation efforts, we solve the EOM in the leads in local coordinate system for each spin separately. In global coordinate system, Bloch states can be obtained after an unitary transformation. For the amplitude of 0th layer, we have $\mathbf{C}_{0}=U_{0}%
\mathbf{\overline{C}} _{0}$.
The propagating states and evanescent states can be identified and sorted into right-going$\left( +\right) $ or left-going $\left( -\right) $. Letting $\mathbf{\overline{w}}_{\mu }^{\uparrow \left( \downarrow \right) }\left(
\pm\right) $ denotes the solutions of $\mathbf{\overline{C}}$ corresponding to eigenvalue $\lambda _{\mu }\left( \pm\right) $, where $\uparrow $($%
\downarrow $) denotes the eigenstate parallel (antiparallel) to the local spin quantization direction. Constructing the matrix $\mathbf{W}\left( \pm
\right) $ as
$$\begin{aligned}
\mathbf{W}\left( \pm \right) &=&U_{0}\mathbf{\overline{W}}\left( \pm \right)
\notag \\
&\equiv & U_{0}[\overline{\mathbf{w}}_{1}^{\uparrow }\left( \pm \right)
,\cdots ,\overline{\mathbf{w}}_{M}^{\uparrow }\left( \pm \right) , \notag \\
&&\text{ \ \ \ \ \ }\overline{\mathbf{w}}_{1}^{\downarrow }\left( \pm
\right) ,\cdots ,\overline{\mathbf{w}}_{M}^{\downarrow }\left( \pm \right) ].
\label{u_b}\end{aligned}$$
Following Ando[@T.Ando91], define the Bloch factor as
$$F\left( \pm \right) \equiv \mathbf{W}\left( \pm \right) \mathbf{\Lambda }%
\left( \pm \right) \mathbf{W}^{-1}\left( \pm \right) . \label{bloch_ma}$$
where $\mathbf{\Lambda }\left( \pm \right) $ is a diagonal matrix with the diagonal elements given by $\left[ \lambda _{1}^{\uparrow }\left( \pm
\right) ,\cdots ,\lambda _{M}^{\uparrow }\left( \pm \right) ,\lambda
_{1}^{\downarrow }\left( \pm \right) ,\cdots ,\lambda _{M}^{\downarrow
}\left( \pm \right) \right] $. In local coordinate system, we have the relation $\mathbf{\overline{C}}_{I}\left( \pm \right) =\overline{F}%
^{I-J}\left( \pm \right) \mathbf{\overline{C}}_{J}\left( \pm \right) $[Xia06]{}.It is easy to proof that the Bloch factor defined above satisfies the Bloch relation in global coordinate system $$\mathbf{C}_{I}\left( \pm \right) =F\left( \pm \right) ^{I-J}\mathbf{C}%
_{J}\left( \pm \right) . \label{c_relation}$$ Bloch factors matrix $F\left( \pm \right) $ relates the wave amplitude in the $I$th layer to that in the $J$th layer for a state in the lead.
Scattering problem
------------------
The equations of motion with open boundary conditions for a device usually contain infinite number of equations. By incorporating the boundary conditions in the leads, the scattering problem can be reduced to a set of coupled linear equations with finite number of equations[@Xia06].
For an electron coming from the left lead, Eq.(\[kkr\]) for $I=0$ can be rewritten as
$$\begin{aligned}
&&\left( U_{0}\overline{P}_{0,0}U_{0}^{\dag }-\tilde{S}_{0,0}\right) \mathbf{%
C}_{0}-S_{0,1}\mathbf{C}_{1} \notag \\
&=&S_{0,-1}\left[ F_{L}^{-1}\left( +\right) -F_{L}^{-1}\left( -\right) %
\right] \mathbf{C}_{0}\left( +\right) ,\end{aligned}$$
where $L$ denotes the left lead and with $\tilde{S}_{0,0}\equiv
S_{0,0}+S_{0,-1}F_{L}^{-1}\left( -\right) $. The $S_{0,-1}F_{L}^{-1}\left(
-\right) $ is the embedding potential for the left lead.
In the right lead, only right-going waves exist in the $\left( N+1\right)$th layer. The EOM for $I=N+1$ is $$\left( U_{N+1}\overline{P}_{N+1,N+1}U_{N+1}^{\dag }-\tilde{S}%
_{N+1,N+1}\right) \mathbf{C}_{N+1}-S_{N+1,N}\mathbf{C}_{N}=0,$$where $\tilde{S}_{N+1,N+1}=S_{N+1,N+1}+S_{N+1,N+2}F_{R}\left( +\right) $ and $S_{N+1,N+2}F_{R}\left( +\right) $ is the embedding potential for the right lead.
Making use of the lead boundary conditions for $0$th and $\left( N+1\right) $ layer, the scattering wave function can be found as
$\left(
\begin{array}{c}
\mathbf{C}_{0} \\
\mathbf{C}_{1} \\
\mathbf{C}_{2} \\
\mathbf{\vdots } \\
\mathbf{C}_{N} \\
\mathbf{C}_{N+1}%
\end{array}%
\right) =\left( U\mathbf{\overline{P}}U^{\dag }\mathbf{-\tilde{S}}\right)
^{-1}$
$$\text{ \ \ \ \ \ \ \ \ }\times \left(
\begin{array}{c}
S_{1,-1}\left[ F_{L}^{-1}\left( +\right) -F_{L}^{-1}\left( -\right) \right]
\mathbf{C}_{0}\left( +\right) \\
0 \\
\vdots \\
0 \\
0%
\end{array}%
\right), \label{coeff}$$
where $\mathbf{\tilde{S}}$ is of block tridiagonal matrix containing $%
S_{I,J} $ except the $\tilde{S}_{0,0}$ and $\tilde{S}_{N+1,N+1}$ are defined as above. The spin polarization direction at different sites can be incorporated by the unitary rotation $U$ at corresponding site.
To obtain the scattering state, we need to specify an incoming state $%
\mathbf{C}_{0}\left( +\right) $ at the right side of Eq.(\[coeff\]). This can be achieved by introducing the right going eigenmodes of left lead as the incoming states by setting $\mathbf{C}_{0}\left( +\right)$ to be $%
\mathbf{w}_{\lambda }\left( +\right) $, where $\mathbf{w}_{\lambda }\left(
+\right) $ should be renormalized so as to carry an unit flux. Each $\mathbf{%
w}_{\lambda }\left( +\right) $ corresponds to a scattering state in device.
The amplitude of layers from $0$ to $N+1$ solved from Eq.(\[coeff\]) serves for computing the particle current and spin current. Also, the scattering matrix can be obtained[@Xia06].
Particle Current
----------------
Let us consider the particle current operator of a quasi one-dimensional TB model for a special $\mathbf{k}_{\parallel }$ vector at $E=E_{F}$. The MTO-basis functions $\left\vert \mathbf{R}L\zeta ^{\mathbf{k}_{\parallel
}}\right\rangle $ are obtained from the Bloch sum of the particle waves:$$\left\vert \mathbf{R}L\zeta ^{\mathbf{k}_{\parallel }}\right\rangle =%
\underset{T_{\parallel }}{\sum }e^{i\mathbf{k}_{\parallel }\cdot \mathbf{T}%
_{\parallel }}\left\vert \mathbf{R+T}_{\parallel },L\zeta ^{\alpha
}\right\rangle . \label{orbital_tran}$$So the density operator at $\mathbf{R}$ site in the mixed representation for a special $\mathbf{k}_{\parallel }$ vectors can be defined by $$\mathbf{\hat{\rho}}_{\mathbf{R}}^{\mathbf{k}_{\parallel }}\equiv \underset{%
L\zeta }{\sum }\left\vert \mathbf{R}L\zeta ^{\mathbf{k}_{\parallel
}}\right\rangle \left\langle \mathbf{R}L\zeta ^{\mathbf{k}_{\parallel
}}\right\vert . \label{density_k}$$
Neglecting the electron motion inside the atomic cells, the velocity operators can be expressed by the intersite hopping[@Turek02] and will give the total current for subspace. The velocity (current) operator can be defined by$$\mathbf{\hat{V}}=\frac{1}{i\hbar }\left[ \mathbf{\hat{X},\hat{H}}\right] ,
\label{v_d}$$where $\mathbf{\hat{X}}$ is the coordinate operator, which can be represented in TB model by a diagonal matrix $\mathbf{\hat{X}}_{\mathbf{R}L,%
\mathbf{R}^{\prime }L^{\prime }}=\mathbf{X}_{\mathbf{R}}\delta _{\mathbf{RR}%
^{\prime }}\delta _{LL^{\prime }}$ [@Turek02].
With aid of Eq.(\[v\_d\]), the current operator $\mathbf{\hat{J}}_{\mathbf{R%
}^{\prime }\mathbf{R}}^{\mathbf{k}_{\parallel }}$ from $\mathbf{R}^{\prime }$th to $\mathbf{R}$th site ($\mathbf{R}\neq \mathbf{R}^{\prime }$) can be written as $$\mathbf{\hat{J}}_{\mathbf{R}^{\prime }\mathbf{R}}\left( \mathbf{k}%
_{\parallel }\right) =\underset{LL^{\prime }}{\sum }\frac{1}{i\hslash }\left[
\mathbf{\hat{H}}_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\mathbf{k}%
_{\parallel }}-h.c.\right] . \label{j_operator1}$$where $\mathbf{\hat{H}}_{\mathbf{R}L\zeta ,\mathbf{R}^{\prime }L^{\prime
}\zeta ^{\prime }}^{\mathbf{k}_{\parallel }}=\left\vert \mathbf{R}L\zeta ^{%
\mathbf{k}_{\parallel }}\right\rangle \mathbf{H}_{\mathbf{R}L\zeta ,\mathbf{R%
}^{\prime }L^{\prime }\zeta ^{\prime }}^{\mathbf{k}_{\parallel
}}\left\langle \mathbf{R}^{\prime }L^{\prime }\zeta ^{\prime \mathbf{k}%
_{\parallel }}\right\vert $ and $\mathbf{H}_{\mathbf{R}L,\mathbf{R}^{\prime
}L^{\prime }}^{\mathbf{k}_{\parallel }}$ is the Hamiltonian matrix in spin space, which has relation with Eq.(\[hamiltonian\]) as$$\mathbf{H}_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\mathbf{k}%
_{\parallel }}=\underset{\mathbf{T}_{\parallel }}{\sum }\exp \left[ i\mathbf{%
k}_{\parallel }\cdot \mathbf{T}_{\parallel }\right] \mathbf{H}_{\mathbf{R}L%
\mathbf{,}\left( \mathbf{R}^{\prime }+\mathbf{T}_{\parallel }\right)
L^{\prime }}^{\alpha }. \label{hami_k}$$
The expectation value of operator $\mathbf{\hat{A}}$ is $\left\langle
\mathbf{\hat{A}}\right\rangle \equiv \left\langle \Psi \left\vert \mathbf{%
\hat{A}}\right\vert \Psi \right\rangle $. The particle current can be expressed as $$\left\langle \mathbf{\hat{J}}_{\mathbf{R}^{\prime }\mathbf{R}}\left( \mathbf{%
k}_{\parallel }\right) \right\rangle =\underset{LL^{\prime }}{\sum }\frac{1}{%
i\hslash }[\mathbf{a}_{\mathbf{R}L}^{\dagger }\left( \mathbf{k}_{\parallel
}\right) \mathbf{H}_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\mathbf{k}%
_{\parallel }}\mathbf{a}_{\mathbf{R}^{\prime }L^{\prime }}\left( \mathbf{k}%
_{\parallel }\right) -h.c.], \label{j_k_a}$$where where $\mathbf{a}_{\mathbf{R}L}\left( \mathbf{k}_{\parallel }\right)
=\left( \mathbf{a}_{\mathbf{R}L\uparrow }\left( \mathbf{k}_{\parallel
}\right) ,\mathbf{a}_{\mathbf{R}L\downarrow }\left( \mathbf{k}_{\parallel
}\right) \right) ^{T}$, and $\mathbf{a}_{\mathbf{R}L\zeta }\left( \mathbf{k}%
_{\parallel }\right) =\left\langle \mathbf{R}L\zeta ^{\mathbf{k}_{\parallel
}}|\Psi \right\rangle $. $\mathbf{a}_{\mathbf{R}L}\left( \mathbf{k}%
_{\parallel }\right) $ has the relation with $\mathbf{C}_{\mathbf{R}L}\left(
\mathbf{k}_{\parallel }\right) $ as follow$$\mathbf{a}_{\mathbf{R}L}\left( \mathbf{k}_{\parallel }\right) =U_{\mathbf{R}%
}\left( \overline{\Delta }_{\mathbf{R}L}^{\alpha }\right) ^{-\frac{1}{2}}U_{%
\mathbf{R}}^{\dagger }\mathbf{C}_{\mathbf{R}L}\left( \mathbf{k}_{\parallel
}\right) . \label{a_c}$$
The $\mathbf{C}_{\mathbf{R}L}\left( \mathbf{k}_\parallel\right)$ can be obtained by Eq.(\[coeff\]) for a given $k_\parallel$. Within the MTO formulism, the current can also be expressed with structure constants matrix as in Ref. $$\left\langle \mathbf{\hat{J}}_{\mathbf{R}^{\prime }\mathbf{R}}\left( \mathbf{%
k}_{\parallel}\right) \right\rangle =\underset{LL^{\prime }}{\sum }\frac{1}{%
i\hslash }[ \mathbf{C}_{\mathbf{R}L}^{\dagger }\left( \mathbf{k}%
_{\parallel}\right) S_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{\mathbf{%
k}_{\parallel}}\mathbf{C}_{\mathbf{R}^{\prime }L^{\prime }}\left( \mathbf{k}%
_{\parallel}\right)-h.c.]. \label{j_lmto}$$
![Illustration of incoming current and outcoming current for the $%
\mathbf{R}$th site. Assuming the particle current comes from $I-1$th layer to $I+1$th layer. Arrow lines denote the current related to $\mathbf{R}$th site and dot lines denote the coupling between sites irrelevant to $\mathbf{R%
}$th site. []{data-label="c2"}](cc_2){width="6cm"}
The continuity equation of particle current at $\mathbf{R}$ site in the $I $th principle layer reads $$\begin{aligned}
&&\underset{\mathbf{R}^{\prime }\in I-1,I}{\sum }\left\langle \mathbf{\hat{J}%
}_{\mathbf{R}^{\prime },\mathbf{R}}\left( \mathbf{k}_{\parallel}\right)
\right\rangle -\underset{\mathbf{R}^{\prime }\in I,I+1}{\sum }\left\langle
\mathbf{\hat{J}}_{\mathbf{R},\mathbf{R}^{\prime }}\left( \mathbf{k}%
_{\parallel}\right) \right\rangle \notag \\
&=&\frac{\mathbf{d}\left\langle \mathbf{\hat{\rho}}_{\mathbf{R}}^{\mathbf{k}%
_{\parallel}}\right\rangle }{\mathbf{d}t}\text{ \ }, \label{j_conti}\end{aligned}$$ where the first term at the left side of Eq.(\[j\_conti\]) is the incoming current to the $\mathbf{R}$ site and the second term is outgoing current from this site.
As shown in Fig.(\[c2\]), the current is assumed to flow from $I-1$th layer to $I+1$th layer. Considering the $\mathbf{R}$ site, the incoming current is composed of current from the sites in the $I-1$th layer and the sites ahead $\mathbf{R}$ site (relative to transport direction) in $I$th layer. If there exist other atoms in the same plane of $\mathbf{R}$(see Fig.(\[c2\])), the current from those atoms also are considered as the component of incoming current to $\mathbf{R}$ site. The outgoing current is composed of current to the sites in $I+1$th layer and those sites behinds $%
\mathbf{R}$ site in $I$th layer. Note that treating the current between atoms in the same plane as the incoming current or outgoing current will not result in any physical consequence. Careful check has been carried out that the particle current conservation law can be satisfied atom by atom and layer by layer. For the scattering states we calculated, the right side of the Eq.(\[j\_conti\] ) is zero.
In the linear response regime, the particle current under a small bias $%
V_{b} $ at zero temperature can be expressed as[@Datta95], $$\mathbf{J}_{\mathbf{RR}^{\prime }}=\frac{e}{h}\frac{1}{N_{\parallel }}%
\underset{\mathbf{k}_{\parallel }}{\sum }\left\langle \mathbf{\hat{J}}_{%
\mathbf{RR}^{\prime }}\left( \mathbf{k}_{\parallel }\right) \right\rangle
V_{b}, \label{j_linear}$$where the bias is given by the difference between the electrochemical potentials of the two leads as $eV_{b}=\mu _{L}-\mu _{R}$, and $N_{\parallel
}$ is the number of $\mathbf{k}_{\parallel }$ points in 2D BZ.
Spin current and spin torque
----------------------------
The spin current is defined similar to the particle current in Section II.D. Considering a quasi one-dimensional TB mode for a special $\mathbf{k}%
_{\parallel }$ vector, the spin density operator at site $\mathbf{R}$ can be defined as $$\mathbf{\hat{S}}_{\mathbf{R}}^{\mathbf{k}_{\parallel }}\equiv \underset{%
L\zeta }{\sum }\left\vert \mathbf{R}L\zeta ^{\mathbf{k}_{\parallel
}}\right\rangle \mathbf{\hat{\sigma}}\left\langle \mathbf{R}L\zeta ^{\mathbf{%
k}_{\parallel }}\right\vert , \label{s_density}$$where $\mathbf{\hat{\sigma}}$ is $2\times 2$ Pauli spin matrix. The spin current operator generally can be defined as $$\hat{\mathcal{J}}\equiv \frac{1}{2}\left[ \mathbf{\hat{\sigma}}\otimes
\mathbf{\hat{V}}+\mathbf{\hat{V}}\otimes \mathbf{\hat{\sigma}}\right] .
\label{js_d}$$ note that $\hat{\mathcal{J}}$ is a tensor. For spin current between $\mathbf{%
R}$th and $\mathbf{R}^{\prime }$th site ($\mathbf{R}\neq \mathbf{R}^{\prime
} $), we could project $\hat{\mathcal{J}}$ along the direction vector $%
\mathbf{x}_{\mathbf{R},\mathbf{R}^{\prime }}$ in real space as $\hat{%
\mathcal{J}}\cdot \mathbf{x}_{\mathbf{R},\mathbf{R}^{\prime }}$. Then the spin current operator $\hat{\mathcal{J}}_{\mathbf{R}^{\prime },\mathbf{R}%
}\left( \mathbf{k}_{\parallel }\right) $ from $\mathbf{R}^{\prime }$th to $%
\mathbf{R}$th site ($\mathbf{R}\neq \mathbf{R}^{\prime }$) can be written as
$$\hat{\mathcal{J}}_{\mathbf{R}^{\prime },\mathbf{R}}\left( \mathbf{k}%
_{\parallel }\right) =\underset{LL^{\prime }}{\sum }\frac{1}{2i\hslash }[%
\mathbf{\hat{\sigma}\hat{H}}_{\mathbf{R}L,\mathbf{R}^{\prime }L^{\prime }}^{%
\mathbf{k}_{\parallel }}+\mathbf{\hat{H}}_{\mathbf{R}L,\mathbf{R}^{\prime
}L^{\prime }}^{\mathbf{k}_{\parallel }}\mathbf{\hat{\sigma}}-h.c.].
\label{js_operator}$$
where $\hat{\mathcal{J}}_{\mathbf{R}^{\prime },\mathbf{R}}\left( \mathbf{k}%
_{\parallel }\right) $ is a vector only in spin space.
For a specific state $|\Psi \rangle $, the expectation value is
$$\left\langle \hat{\mathcal{J}}_{\mathbf{R}^{\prime
},\mathbf{R}}\left(
\mathbf{k}_{\parallel }\right) \right\rangle =\underset{LL^{\prime }}{\sum }%
\frac{1}{2i\hslash }\left[ \mathbf{a}_{\mathbf{R}L}^{\dag }\left( \mathbf{k}%
_{\parallel }\right) \mathbf{\hat{\sigma}H_{\mathbf{R}L,\mathbf{R}^{\prime
}L^{\prime }}^{\mathbf{k}_{\parallel }}a}_{\mathbf{R}^{\prime }L^{\prime
}}\left( \mathbf{k}_{\parallel }\right) +\mathbf{a}_{\mathbf{R}L}^{\dag
}\left( \mathbf{k}_{\parallel }\right) \mathbf{H}_{\mathbf{R}L,\mathbf{R}%
^{\prime }L^{\prime }}^{\mathbf{k}_{\parallel }}\mathbf{\hat{\sigma}a}_{%
\mathbf{R}^{\prime }L^{\prime }}\left( \mathbf{k}_{\parallel }\right) -h.c.%
\right] . \label{js_average}$$
The STT $\left\langle \mathbf{\hat{T}}_{\mathbf{R}}^{s}\left( \mathbf{k}%
_{\parallel }\right) \right\rangle $ can be defined as the difference of the incoming spin current and outgoing spin current of $\mathbf{R}$ site in the $%
I$th principal layer: $$\begin{aligned}
&&\left\langle \mathbf{\hat{T}}_{\mathbf{R}}^{s}\left( \mathbf{k}_{\parallel
}\right) \right\rangle \notag \label{t_average} \\
&=&\underset{\mathbf{R}^{\prime }\in I-1,I}{\sum }\left\langle \hat{\mathcal{%
J}}_{\mathbf{R}^{\prime },\mathbf{R}}^{s}\left( \mathbf{k}_{\parallel
}\right) \right\rangle -\underset{\mathbf{R}^{\prime }\in I,I+1}{\sum }%
\left\langle \hat{\mathcal{J}}_{\mathbf{R},\mathbf{R}^{\prime }}^{s}\left(
\mathbf{k}_{\parallel }\right) \right\rangle . \notag \\
&&\end{aligned}$$ where the superscript $s$ is used to denote the incoming state is parallel or antiparallel to the local spin quantization axis of injection lead, which is very helpful, e.g. in ferromagnet we can distinguish the contribution to the total torques from the majority spin or minority spin. Such definition consists with those in Ref.\[\], where analytic analysis shows that for STT defined in this way equals to the exchange torques acted on the injected spin defined in Eq.(\[s\_density\]) with only a sign difference. After summation over 2D BZ, spin torque acted on $\mathbf{R}$ th atom can be expressed as $$\mathbf{T}_{\mathbf{R}}=\left( \frac{\hbar }{2}\right) \frac{e}{h}\frac{1}{%
N_{\parallel }}\underset{s,\mathbf{k}_{\parallel }}{\sum }\left\langle
\mathbf{\hat{T}}_{\mathbf{R}}^{s}\left( \mathbf{k}_{\parallel }\right)
\right\rangle V_{b}, \label{t_linear}$$where the bias is given by the difference between the electrochemical potentials of the two leads as $eV_{b}=\mu _{L}-\mu _{R}$.
Spin transfer torques in Co/Cu/FM/Cu (111) Spin valves
======================================================
Ordered interfaces
------------------
A spin valve of Co$|$Cu$|$FM$|$Cu as shown in Fig.\[config\] is used as an example to illustrate our method. The left lead consists of semi-infinite Co with the polarization direction $\theta $ (see Fig.\[config\]). Cu spacer of $9$ monolayer (ML) is located between fixed magnet Co and free magnet FM. The free magnet contains $d$ ML, which could be Co, Ni, or Ni$_{80}$Fe$_{20}$ in this study. The lattice constants is assumed to be uniform in the whole spin valve, that is, $a_{Cu}=a_{FM}=3.54\mathring{A}$ and the transport is along *fcc*\[111\]. With $spd$ -basis, exchang-correlation potential is calculated and parameterized by Vosko-Wilk-Nusair [@vosko]. Our calculation gives the magnetic moments as $1.64\mu _{B}/$Co atom, $2.60\mu
_{B}/$Fe atom and $0.62\mu _{B}/$Ni atom. For the calculation of transport, total 90000 $\mathbf{k}_{\parallel}$ points in 2D BZ are summed.
![(color online) (a)The total conductance of Co($\protect\theta $)$|$Cu(9ML)$|$Co(15ML)$|$Cu versus polarization direction $\protect\theta $ of fixed magnet Co. (b) The angular dependence of total spin torques on free magnet Co, where the electron current flow from the fixed magnet to the free magnet. []{data-label="cocucond"}](cu_co_G_T_theta_3){width="8cm"}
Firstly, we present the angular dependence of total conductance $G(\theta)$ of the spin valve with the free magnet to be 15ML Co in Fig.\[cocucond\] (a). The monotonic decrease with increase of $\theta$ is consistent with the previous *ab.initio* results[@PMHaney]. Giant magnetoresistance (GMR) can be defined as $GMR\equiv \frac{G(0^{o})-G(180^{o})}{G(180^{o})}%
100\%$, which is $24\%$ in this case. With electron current flowing from the fixed magnet to the free magnet, Fig.\[cocucond\] (b) gives the angular dependence of total spin torques on the free magnet Co, which restore the line shape of spin torques obtained in previous work[@PMHaney]. With the drive of the in-plane torque, magnetization of free layer is going to parallel to that of fixed layer. Due to the breakdown of time reverse symmetry for spin current, if the direction of electron current is reversed, the in-plane torques is going to drive free layer to antiparallel to fixed layer. Such phenomenon is exactly the current induced switching of magnetization observed in spin valve.
![(color online) The layer dependence of STT on interfacial unit cell in the spin valve Co($\protect\theta =90$)$|$Cu$|$FM$|$Cu , where the free magnet FM are Co, Ni, Ni$_{80}$Fe$_{20}$ respectively.[]{data-label="torque3"}](torque3_4){width="8cm"}
*Layer resolved Spin Torque:* The layer resolved STT contains the information about whether the spin angle moment is absorbed near the interface or not. Fig.\[torque3\] gives the comparison of the layer dependence of STT in the spin valves with three different free magnet. Here the polarization of the fixed magnet Co is set to $\theta =90^{o}$ without lost of generality. In our frame, $\mathbf{T}_{x}$ corresponds to the in-plane torques and $\mathbf{T}_{y}$ corresponds to the out-of-plane torque. The decay and oscillation of the STT are greatly different among those materials we studied. When free magnet is Co as shown in Fig.[torque3]{}(a), our result almost reproduces the previous result[Edwards05,PMHaney]{}. The fast decay of the STT indicated the surface atoms absorbed most of the spin angle moment as the current passes by.
However, when Ni serves as the free magnet as shown in Fig\[torque3\](b), the maximum torques is not on the surface atom and the decay is very slow with much longer oscillation. This observation is quite different with our previous knowledge[@Maciej_decay]. The similar behavior is also found in Fig\[torque3\](c) as Ni$_{80}$Fe$_{20}$ is free magnet, the oscillation looks like that in Ni, but decaying faster. Due to the lack of obvious oscillation, the total in-plane torques on Ni$_{80}$Fe$_{20}$ ($7.7\times
10^{-3}eV_{b}$) is greater than that on Co ($5.0\times 10^{-3}eV_{b}$) and that on Ni ($6.3\times 10^{-3}eV_{b}$).
The layer resolved STT shown in Fig.\[torque3\] could be affected by the multiple scattering between the two interfaces with Cu. To remove multiple scattering effect on the torque, we perform the calculation for single interfaces of Cu($90^{o}$)$|$Co and Cu($90^{o}$)$|$Ni, with 100% polarized electrons injected from Cu side. Here Cu($90^{o}$) indicates the polarization direction of the injected electrons. The results are shown in Fig.\[coni\]. For both interfaces, the maximum torques exists on the surface atoms and the oscillation spectra in ferromagnet become smoother and clear. Still the oscillation behavior strongly depends on the materials. The STT exists only near the Cu$|$Co interface, while the STT penetrates deep into the Ni for Cu$|$Ni interface. Due to the long penetration length, the multiple scattering between two Cu$|$Ni interface does appear in Fig.[torque3]{}(b).
![(color online) The layer dependence of STT on interfacial unit cell for the spin injection setup of single interface. (a) Cu$|$Co (b)Cu$|$Ni.[]{data-label="coni"}](cu_co_cu_ni_5){width="8.6cm"}
*Simple Model for Spin Torques in FM:*For the layered system such as spin valve, the incoming state of the injection lead can be labelled by $%
\mathbf{k}_{\parallel}$ in 2D BZ. Generally, these states will be coupled to the propagating states and evanescent states of another side of the injection interface. The STT can be expressed as
![(color online) (a) and (b) The layer dependence of STT on interfacial unit cell when the spin injected through a single Cu$|$Co interface for different $\mathbf{k}_{\parallel}$ points in 2D BZ. []{data-label="decay_bz"}](decay_bz_6){width="8.6cm"}
$$\label{torque_pression}
\Gamma\varpropto\sum_{\mu,\nu } C_{\mu\nu} e^{i[(k_{\mu }^{\downarrow
}-k_{\nu }^{\uparrow })x+\varphi_{\mu\nu}]}+\Im_{decay}(x),$$
where first term denotes the contribution from the propagating states[MDstiles02]{} and $k_{\mu }^{\downarrow }-k_{\nu}^{\uparrow }$ gives the spatial precession frequency $\bigtriangleup k_{\mu \nu }$. The contribution from the propagating states should oscillate as function of position and will not decay as shown in Fig.\[decay\_bz\](b). However, the frequency could be quite different as $\mathbf{k}_{\parallel}$ runs over the 2DBZ, so the final contribution will decay after summation over 2DBZ. The second term of Eq. (\[torque\_pression\]), $\Im_{decay}(x)$, is the contribution from the evanescent states. As we have known that no particle current can be carried by evanescent state, however, such states do give effect on the spin current and also on the STT. Evanescent states do contribute to spin torques and should responds for the initio decay of the STT in the system as Co$|$Cu$%
|$Co$|$Cu, as shown in Fig.\[decay\_bz\](a) where the evanescent state dominates.
Two reasons could account for the decay of the STT away from the interface. (i) Vanishing of the evanescent states’ contribution. For Cu$|$Co, our calculation shows that this part of contribution is about 10% of the total torques on the first layer close to injection interface. (ii) Cancellation effect among different $\mathbf{k}_{\parallel}$ in 2D BZ [@MDstiles02].
The materials dependency of the STT could be understood based on the Fermi surface of Ni and Co. The wave vector $k_{\mu(\nu)}^{\downarrow(\uparrow)}$ can be found by the projection of minority spin (majority spin) Fermi surface of ferromagnet along the current direction, where $\mu$ ($\nu$) denotes the different sheets of Fermi surface for minority spin (majority spin). In Fig.\[fs\], the Fermi surface for Co and Ni viewed along the (111) direction for majority and minority spins is shown. As the shape of Fermi surface for majority spin and minority spin in Co are greatly different to each other, the precession frequencies $\bigtriangleup k_{\mu
\nu }$ of injected spin are varied rapidly as $\mathbf{k}_{\parallel}$ running over the 2DBZ. After summation of 2D BZ, the strong cancellation is expected as shown in Fig.\[torque3\](a) and Fig.\[coni\](a). However, for Cu$|$Ni, due to the similar symmetry between the wave function of the sheet $\mu =6$ of minority spin and that of Cu,the electrons pass Cu$|$Ni interface mainly through this channel($\mu =6$). While for majority spin, the Fermi surface contains only one sheet. The precession frequency is dominated by $\bigtriangleup k_{6 6 }$. Similar shape of sheet $\mu =6$ of minority spin and sheet $\nu =6$ of majority spin will result in amount of propagating states with similar precession frequency. After summation of those states, the cancellation could be week and collective oscillation must be of long period.
![(color online) The First row: Fermi surface projection of the Co bulk fcc Brillouin zone on to a plane perpendicular the (111) direction. Left-hand panel is for majority electron and right-hand panel is for minority electron with band 3,4,5 FS. The Second row is Ni bulk. []{data-label="fs"}](fs_7){width="8.6cm"}
The above physical picture about spin torques in FM should be qualitatively applicable to the system with FM to be Ni$_{80}$Fe$_{20}$. For Ni$_{80}$Fe$%
_{20}$, the overall band structure resembles that of Ni, however, due to the scattering of Fe impurity atoms, the fine structure at Fermi surface could be much more complicated than that of Ni. The dispersion of precession frequency $\bigtriangleup k_{\mu \nu }$ is large and the cancellation should be strong. As a result, the decay of spin torque is much faster than in the conventional FM, Ni and Co. Besides, spin-orbit coupling is not included in our calculation yet, which could introduce new mechanism of decay in Ni$%
_{80} $Fe$_{20}$.
Interfacial disorder
--------------------
![(color online) The concentration $x$ dependence of the conductance (a) and total spin torques on the free magnet (b) of spin valve Co($\protect%
\theta =90$)$|$Cu$|$FM$|$Cu with interfacial alloy Cu$_{x}$FM$_{1-x}$, where the solid symbol for FM to be Co and open symbol for FM to be Ni. In (b), the black symbol for in-plane torques and red symbol for out-of-plane torque. []{data-label="codisordercondx"}](co_cu_co_cu_disor_cond_x_8){width="8cm"}
![(color online) The layer dependence of STT on interfacial unit cell in the spin valve Co($\protect\theta =90$)$|$Cu$|$FM$|$Cu with interfacial disorder, (a) FM is Co, (b) FM is Ni. The zone labelled by $\Pi$ is the layers with substitutional alloy Cu$_{50}$FM$_{50}$.[]{data-label="codisorder"}](conidisorder_9){width="9cm"}
Interfacial disorder is likely to exist in the metallic system. Previous studies[@Xia06] showed that the interfacial alloy could change the polarization of the interface resistance. How the interfacial disorder affects the STT is question we would like to answer in this section. The interfacial disorder is introduced by two layers of substitutional alloy Cu$%
_{x}$FM$_{1-x}$ and Cu$_{1-x}$FM$_{x}$at interface between Cu and FM. In present study, alloy is modelled by a $8\times 8$ lateral supercell, which was shown to be a good modelling of the interfacial alloy[@Xia06].
![(color online) The thickness $d$ dependence of the averaged spin torques on the atom in free magnet and the total conductance of the spin valve.[]{data-label="cuco_disord_layer"}](cuco_disord_layer_10){width="9cm"}
In Fig.\[codisordercondx\](a)&(b), the concentration $x$ dependence of total conductance and total torque(in-plane and out-of-plane) on free magnet are given for the realistic spin valve of Co($\theta=90^{o}$)$|$Cu$|$Co(Ni)$%
| $Cu. The total conductance decreases with increase of the concentration $x$, which means the interfacial disorder will suppress electronic transport in this system. However, as shown in Fig.\[codisordercondx\](b), the total in-plane torques acted on free magnet increase when the disorder is enhanced in spite of the almost constant out-of-plane spin torque. The increase is around 50% for Co and 30% for Ni.
In Fig.\[codisorder\], the layer dependence of spin torques is shown. Comparing with Fig.\[torque3\](a), it is found that the torques on atoms near the interface with Cu spacer have been enhanced. This result means that the interfacial roughness will not kill the interfacial spin torques, on the contrary, the dirty interface may be helpful to enhance the torques transfer.
For the spin valve with FM is Co, the free magnet thickness $d$ dependence of the total conductance and STT are shown in Fig.\[cuco\_disord\_layer\], where the spin torques is obtained by average of total torques on free magnet over all atoms in free magnet. Due to the quantum size effect, the conductance decays with a small oscillation and tend to be constant with increase of free magnet thickness. The in-plane torques dominates over the out-of-plane torques for all thickness.
Summary
=======
Based on the first principles frame, a method was developed to calculate the transport and spin torques of the layered system with noncollinear magnetization in linear response regime. STT in the ferromagnetic (FM) spin valves are calculated. We found that the behavior of spin torques in the free layer are greatly dependent on the materials. The cancellation effect of the STT due to the different precessional frequency is sensitive to the band structure of material. The contribution of evanescent states to the STT is found to be nontrivial at the NM$|$FM interface. The effect due to interfacial disorder is also considered, it is found that average torques are enhanced in the presence of disorder.
This work is supported by NSF(10634070) and MOST(2006CB933000, 2006AA03Z402) of China.We are grateful to: Paul Kelly for useful discussion; Ilja Turek for his TB-LMTO-SGF layer code which we used to generate self-consistent potentials; Anton Starikov for permission to use his version of the TB-MTO code based upon sparse matrix techniques with which we can solve Eq.([coeff]{}) in an efficient way.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The coherence of electron spin qubits in semiconductor quantum dots suffers mostly from low-frequency noise. During the last decade, efforts have been devoted to mitigate such noise by material engineering, leading to substantial enhancement of the spin dephasing time for an idling qubit. However, the role of the environmental noise during spin manipulation, which determines the control fidelity, is less understood. We demonstrate an electron spin qubit whose coherence in the driven evolution is limited by high-frequency charge noise rather than the quasi-static noise inherent to any semiconductor device. We employed a feedback control technique to actively suppress the latter, demonstrating a $\pi$-flip gate fidelity as high as $99.04\pm 0.23\,\%$ in a gallium arsenide quantum dot. We show that the driven-evolution coherence is limited by the longitudinal noise at the Rabi frequency, whose spectrum resembles the $1/f$ noise observed in isotopically purified silicon qubits.'
author:
- Takashi Nakajima
- Akito Noiri
- Kento Kawasaki
- Jun Yoneda
- Peter Stano
- Shinichi Amaha
- Tomohiro Otsuka
- Kenta Takeda
- 'Matthieu R. Delbecq'
- Giles Allison
- Arne Ludwig
- 'Andreas D. Wieck'
- Daniel Loss
- Seigo Tarucha
title: 'Coherence of a driven electron spin qubit actively decoupled from quasi-static noise'
---
[^1]
[^2]
Introduction: Noise in Spin Qubits
==================================
Since electrical manipulation of a single spin was demonstrated in semiconductor quantum dots[@Koppens2006], enormous efforts have been devoted to improve spin coherence by controlling[@Foletti2009; @Bluhm2010] or eliminating[@Veldhorst2014; @Eng2015; @Yoneda2017] nuclear spins, a magnetic noise source inherent to the host material[@Coish2004; @Merkulov:2002ft; @Khaetskii:2002jw]. The progress is impressive: for example, dephasing times of $120\,\mu\text{s}$ in $^{28}$Si and $2\,\mu\text{s}$ in GaAs have been demonstrated[@Veldhorst2014; @Shulman2014]. It is natural to expect that prolonging the spin coherence also improves the qubit control fidelity. However, while the spin coherence is dominated by low-frequency (quasi-static) noise, control fidelity of a qubit is often impeded by noise at higher frequencies[@Martinis2003; @Ithier2005; @Lisenfeld2010; @Yoshihara2014]. The underlying relationship between the control fidelity and spin coherence remains elusive because there are different noise sources that could dominate in different frequency ranges, such as nuclear spin diffusion and charge fluctuators (see Fig. \[fig:noise\]). The former shows a $1/f^{\beta}$ spectrum with $3 > \beta > 1$ in GaAs[@Medford2012; @Malinowski2017a] and possibly in natural Si devices[@Kawakami2016], while the latter with $\beta \sim 1$ can dominate in $^{28}$Si devices[@Yoneda2017]. In general, the dominant noise source depends on the material and structure of the quantum dot device as well as the frequency range of interest.
![Example of noise power spectra for spin qubits with and without feedback. A typical noise spectrum composed of $1/f^2$ and $1/f$ noise is shown in a log-log plot (black). The feedback control acts like an active filter suppressing the low-frequency noise (red). Shown on the bottom are relevant frequencies with $\Delta t$ the feedback latency, $t$ the qubit evolution time at which the coherence is evaluated, and $f_\text{rabi}$ the Rabi frequency. \[fig:noise\]](Fig0){width="40.00000%"}
To understand the limits on the qubit control fidelity imposed by those different mechanisms, we build a feedback-controlled circuit which implements realtime Hamiltonian estimation[@Shulman2014]. It allows us to suppress the low-frequency noise[@Yang2019] and resolve the $1/f$ charge and nuclear spin noise at high frequencies. We analyze how the low-frequency and high-frequency parts of the noise compete with each other and discuss the limitations of the high-fidelity control.
Device and experimental setup
=============================
We use a triple quantum dot (TQD) device fabricated on a GaAs/AlGaAs heterostructure wafer. An electron is confined in each quantum dot (QD) by the electrostatic potentials induced by Ti/Au gate electrodes. A Co micromagnet is placed on the surface and magnetized by a magnetic field of $B_\text{ext}=1.01\,\text{T}$ applied in the $z$-direction (see Fig. \[fig:ramsey\]a), creating inhomogeneous magnetic field over the QD array. The single electron spin qubit reported in this work is located in the middle QD and manipulated by the electric-dipole spin resonance (EDSR)[@Pioro-Ladriere:2008kx; @Tokura:2006ir; @Yoneda2014]. It is initialized and measured using the ancilla electron spin in the right QD[@Noiri2016], see Fig. \[fig:ramsey\]b. An up-spin state of the qubit is prepared by initializing a doubly-occupied singlet ground state in the right QD and loading one of the electrons to the middle QD. The voltage ramp is chosen to be adiabatic with respect to the inter-dot tunnel gap and the local magnetic field difference between the two dots but non-adiabatic with respect to the hyperfine gap. The final state is read out by unloading an up-spin state to the right QD in the reverse process, while leaving a down-spin state blocked in the middle QD. The experiment is conducted in a dilution refrigerator with an electron temperature of $120\,\text{mK}$.
![Ramsey meausurement and feedback-control scheme of an electron spin qubit. (a) False-colored scanning electron micrograph image of the TQD device. An electron spin qubit in the middle QD (red arrow with a circle) is controlled by the EDSR where the spin is coupled to a microwave (MW) electric field via a stray magnetic field of the micromagnet deposited on the wafer surface[@Pioro-Ladriere:2008kx]. The right QD hosts an electron spin (blue arrow with a circle) used as a readout ancilla while the left QD hosts another electron which is unused and decoupled from the two spins. The energy detuning between the middle and the right QDs ($\varepsilon$) is gate-tunable and the QD electron occupancies are probed by a proximal single-electron transistor (SET)[@Barthel:2010fk]. (b) Schematic of the Ramsey measurement. Two electrons (qubit and ancilla) are initialized to a doubly-occupied singlet state in the right QD and an up-spin qubit is prepared by adiabatically loading one of the electrons to the middle QD[@Noiri2016]. Two $\pi/2$ microwave bursts, separated by time $t_\text{R}$, are applied (before and during these, off-resonant microwave bursts are optionally applied). The ancilla-spin state is not affected by the microwave bursts. The final state is read out by unloading an up-spin (anti-parallel to the ancilla) state from the middle QD while a down-spin (parallel to the ancilla) state remains blocked. (c) Up-spin probability $P_\uparrow$ as a function of $t_\text{R}$. The lower panel shows the Ramsey oscillations whose frequency varies with the laboratory time due to Overhauser field fluctuations. Each data point of $P_\uparrow$ is calculated from one hundred single-shot readout outcomes. The upper panel shows the trace obtained by averaging all the oscillations in the lower panel. The decay envelope gives the dephasing time of $T_{2}^{*}=28.4\,\text{ns}$, a value typical for electron spins in GaAs heterostructures. (d) Schematic of the feedback control loop for a spin qubit. Data of a Ramsey oscillation as shown in (c) are processed in a digital signal processing (DSP) hardware with programmable logic (FPGA) to estimate the frequency detuning $\delta f = f_\text{qubit} - {f^\text{est}_\text{qubit}}$ between the current qubit frequency $f_\text{qubit}$ and its previous estimate ${f^\text{est}_\text{qubit}}$ (“probe” step). The value of ${f^\text{est}_\text{qubit}}$ is updated to ${f^\text{est}_\text{qubit}}\mapsto {f^\text{est}_\text{qubit}}+ \delta f$ (“update” step), after which the target experiment follows (“target” step). In the ideal case, the subsequent qubit algorithms can be executed with a microwave frequency $f_\text{MW}$ matching $f_\text{qubit}$ exactly (by choosing $\Delta=0$). \[fig:ramsey\]](Fig1){width="80.00000%"}
We first perform a standard Rabi measurement[@Noiri2016] to roughly identify the Rabi frequency $f_\text{rabi}$ and the qubit resonance frequency $f_\text{qubit}=|g\mu_\text{B}B_\text{total}|/h$. Here $g$ is the electron $g$-factor, $\mu_\text{B}$ is the Bohr magneton, and $B_\text{total}$ is the sum of $B_\text{ext}$ and the $z$ components of the Overhauser field $B^\text{nuc}_{z}$ and the micromagnet stray field $B^\text{MM}_{z}$. After that, we measure Ramsey oscillations using two $\pi/2$ microwave bursts of duration $(4f_\text{rabi})^{-1}$ separated by a time interval $t_\text{R}$. The lower panel of Fig. \[fig:ramsey\]c shows the data gathered over $1200\,\text{s}$ with a fixed microwave frequency of $f_\text{MW}=5.55\,\text{GHz}$. The frequency of the measured oscillations fluctuates around a mean value $f_\text{MW}-f_\text{qubit} \approx 55\,\text{MHz}$. The fluctuations arise from changes of $B^\text{nuc}_z$ due to nuclear spin diffusion[@Reilly2008], leading to inhomogeneous broadening of $f_\text{qubit}$. Averaging all the measured data results in damped oscillations shown in the upper panel of Fig. \[fig:ramsey\]c. Fitting it with a Gaussian envelope gives a spin dephasing time of $T_\text{2}^{*}=28.4\,\text{ns}$.
The Feedback protocol
=====================
To suppress this dephasing, rooted in slow fluctuations[@Coish2004] (quasi-static noise) of $f_\text{qubit}$, we employ a feedback-control scheme based on the realtime Hamiltonian estimation[@Giedke2006; @Klauser2006](see Fig. \[fig:ramsey\]d). A similar technique was previously adopted for singlet-triplet qubits[@Shulman2014] to evaluate the stability of the idle qubit frequency, expressed by the dephasing time $T_\text{2}^{*}$, and its improvement upon noise estimation. Here we apply this technique to a single spin. The difference from a singlet-triplet qubit first of all requires changes in the protocol details, as given below. Second, the noise field couples to the spin differently, through its local value rather than its spatial gradient. However, the most important difference is that we focus here on the stability of the qubit being driven, rather than sitting idle. The rest of this section describes the details of the feedback protocol and its benchmarking, by examining how it boosts the dephasing time $T_\text{2}^{*}$. Readers interested solely on its benefits for the driven qubit stability can proceed directly to the next section.
The feedback scheme alternates the “probe” and “target” steps. In the former, the qubit frequency is probed by sampling $150$ up or down-spin outcomes of a Ramsey oscillation with $t_\text{R}=2,4,\cdots 300\,\text{ns}$ using $f_\text{MW}={f^\text{est}_\text{qubit}}+ \Delta_\text{p}$. Here, ${f^\text{est}_\text{qubit}}$ is the result of the qubit frequency estimation in the preceding probe step and $\Delta_\text{p}=50\,\text{MHz}$ is a fixed frequency offset inserted to ensure $f_\text{MW}>f_\text{qubit}$. With these settings, we use a Bayesian algorithm to estimate the instantaneous frequency detuning, $\delta f = f_\text{qubit} - {f^\text{est}_\text{qubit}}$. At the end of the probe step, the value of ${f^\text{est}_\text{qubit}}$ is updated to ${f^\text{est}_\text{qubit}}\mapsto {f^\text{est}_\text{qubit}}+ \delta f$ and the microwave frequency is set to $f_\text{MW}={f^\text{est}_\text{qubit}}+\Delta$. The subsequent “target” step begins after a short delay ($\sim$ms) to stabilize the signal output. The variable $\Delta$ is a controllable offset such that $\Delta = 0$ corresponds to the target algorithm executed with $f_\text{MW}$ equal to $f_\text{qubit}$. By continuously looping the probe-target sequence, we can compensate low-frequency fluctuations of $f_\text{qubit}$ and remove their contribution to various qubit errors. For example, the dephasing time $T_\text{2}^{*}$ is expected to be boosted by employing such compensation protocol.
![Suppressed dephasing of an electron spin qubit in a feedback-controlled rotating frame. (a) Time dependence of the frequency detuning $\delta f = f_\text{qubit} - {f^\text{est}_\text{qubit}}$ extracted from Ramsey measurements. The blue trace is taken with a fixed $f_\text{MW}$ and the red trace is taken with a feedback-controlled $f_\text{MW}$. The inset shows the histograms of $\delta f$ in the two cases. The histogram with the feedback control exhibits a normal distribution with a variance $\sigma^{2}=\left<(\delta f)^{2}\right>=(0.294\,\text{MHz})^2$ (black dashed curve). (b) Ramsey oscillations as in Fig. \[fig:ramsey\]c but obtained with the feedback control ($\Delta = 50\,\text{MHz}$). The envelope of the oscillation in the upper panel is a Gaussian decay function drawn using dephasing time $T_{2}^{*}=1/(\pi\sqrt{2}\sigma)=766.7\,\text{ns}$. (c) Variance of $\delta f$, $\sigma^{2} = \left<(\delta f)^{2}\right>$, as a function of the latency $\Delta t$ between the probe and target steps (orange circles) and that of the frequency correlator $\sigma_\text{B}^{2}$ in the laboratory frame (blue circles) as a function of the time difference $\Delta t$. A blue line is a fit to $\sigma_\text{B}^{2} = D\,\Delta t^\alpha$ and shows subdiffusive behaviour with the exponent $\alpha=0.84$ similar to a value found for singlet-triplet oscillations[@Delbecq2015]. The orange curve is a fit to $\sigma^{2} = D\,\Delta t^\alpha + (0.288\,\text{MHz})^{2}$. \[fig:feedback\]](Fig2){width="46.00000%"}
We now evaluate the performance of the feedback control by executing in the target step a Ramsey measurement similar to the one in the probe step ($\Delta = \Delta_\text{p} = 50\,\text{MHz}$). Figure \[fig:feedback\]a shows the values of $\delta f$ obtained from Ramsey measurements with feedback off ($f_\text{MW} = 5.55\,\text{GHz}$) and feedback on ($f_\text{MW}$ adjusted to ${f^\text{est}_\text{qubit}}$), respectively. The fluctuation of $\delta f$ is significantly suppressed by the feedback, exhibiting a Gaussian distribution with a variance $\sigma^{2}=(0.294\,\text{MHz})^{2}$ as shown in the inset. As a result, the Ramsey oscillation is substantially prolonged, as shown in Fig. \[fig:feedback\]b. Averaging the data with the overall acquisition time of $1200\,\text{s}$ gives a decay envelope well in line with the dephasing time expected from $\sigma$, being $T_{2}^{*}=1/(\pi\sqrt{2}\sigma)=766.7\,\text{ns}$.
The gain of $T_{2}^{*}$ is therefore directly associated with the achievable value of $\sigma$, which in turn is limited by the resolution of the Bayesian estimation. To demonstrate this, we plot $\sigma^{2}$ in Fig. \[fig:feedback\]c as a function of the feedback latency $\Delta t$, defined as the time interval between the probe and target steps plus the average time spent in each step (see Appendix A). For large latency, $\sigma^{2}$ approaches the variance of the qubit frequency correlator in the laboratory frame, $\sigma_\text{B}^{2}(\Delta t) \equiv \left<(f_\text{qubit}(t+\Delta t) - f_\text{qubit}(t))^{2}\right>$. It implies that in this regime $\sigma^{2}$ is dominated[@Delbecq2015] by the Overhauser field fluctuations during $\Delta t$. For small latency, $\sigma^2$ converges to $(0.288\,\text{MHz})^2$. This value is comparable to the bin width of the frequency discretization ($0.25\,\text{MHz}$) used in the Bayesian estimation algorithm performed by the feedback hardware. We believe that the variance $\sigma^{2}$ could be further decreased by using a smaller bin width as $\sigma_\text{B}^{2}$ continues to decrease with $\Delta t$ within the measured range, although we were not able to do so due to hardware limitations. The value of $\sigma^{2}$, and thereby that of $T_{2}^{*}$, can be controlled by $\Delta t$, allowing for studying the effects of noise in different frequency ranges.
![Coherent control and benchmarking of a single-electron spin with the feedback. (a) Typical Rabi oscillations versus the offset $\Delta$ obtained in the feedback-controlled rotating frame. A horizontal white dashed line indicates the microwave-induced shift of the qubit frequency, $\Delta f_\text{qubit}$. (b) Microwave-amplitude dependence of the qubit frequency shift. A blue line is a fit to the data showing that the amplitude dependence is quadratic ($\propto E^{1.95\pm 0.18}$). (c) Rabi oscillations obtained at zero detuning upon compensating for the induced shift $\Delta f_\text{qubit}$. Namely, an off-resonant microwave burst with the same amplitude as the one used for the Rabi measurement is applied before and during the Ramsey measurement (see Fig. \[fig:ramsey\]c). A similar off-resonant burst is also applied for $200\,\text{ns}$ before the Rabi drive to stabilize the value of the frequency shift. (d) Microwave amplitude dependence of the Rabi frequency (blue circles) and the Rabi decay time (red squares with lines). A blue line is a linear fit to the low-amplitude data of the Rabi frequency. (e) Normalized sequence fidelities for standard (top) and interleaved (others) randomized benchmarking. Traces are offset by $0.4$ for clarity. The standard sequence shows an exponential decay with an average single-gate fidelity of $97.50\pm 0.05\%$. Interleaved sequences are annotated with corresponding single-qubit gates and extracted fidelities. \[fig:control\]](Fig3){width="70.00000%"}
Improvements of the qubit control
=================================
We now turn to benchmarking of the qubit-control fidelity with the boosted dephasing time. Figure \[fig:control\]a shows Rabi oscillations of the single-spin qubit measured with a varied frequency offset $\Delta$. We observe a clear chevron pattern with the symmetry axis offset from $\Delta=0$. This implies that the qubit frequency $f_\text{qubit}$ is shifted by a finite ac electric field $E$ of the driving microwave burst, while the frequency probed in a Ramsey measurement corresponds to that of $E=0$. The magnitude of this shift, $\Delta f_\text{qubit}(E,t) = f_\text{qubit}(E,t) - f_\text{qubit}(0,t)$, increases quadratically with $E$, as shown in Fig. \[fig:control\]b. Similar frequency shifts are observed in silicon devices with micromagnets, attributed to a spatial displacement of the electron wavefunction[@Watson2017; @Takeda2018]. This effect is detrimental to our feedback protocol because $\Delta f_\text{qubit}(E,t)$ may vary with time $t$ due to the spatial dependence of the Overhauser field. In order to obtain ${f^\text{est}_\text{qubit}}=f_\text{qubit}(E,t)$ directly in the probe step, we apply an off-resonant microwave burst at $f_\text{off}=5.4\,\text{GHz}$ \[$f_\text{qubit}-f_\text{off}>200\,\text{MHz}$\] during the interval of $t_\text{R}$ which induces nominally the same displacement and the same $\Delta f_\text{qubit}(E,t)$ as for the target step. In addition, a $200\,\text{ns}$-long off-resonant pre-burst is applied to stabilize a transient component of the microwave-induced frequency shift[@Takeda2018]. The frequency is switched between $f_\text{off}$ and $f_\text{MW}$ within $1\,\text{ns}$ using a high-speed microwave switch (see Supplementary Information). We use this modified Ramsey sequence in the probe step and focus on the target step performed at zero detuning \[$\Delta = 0,\, {f^\text{est}_\text{qubit}}= f_\text{qubit}(E,t)$\] in all measurements described below.
Figure \[fig:control\]c shows a typical Rabi oscillation at zero detuning. It shows an exponential decay, which is common for silicon QDs[@Takeda2016; @Yoneda2017] but atypical for GaAs[@Yoneda2014; @Nichol2016]. We extract the Rabi frequency $f_\text{rabi}$ and the decay time of the driven oscillation $T_{2}^\text{rabi}$ from fitting and plot their dependence on the driving field amplitude in Fig. \[fig:control\]d. For a given field amplitude, we find that both $f_\text{rabi}$ and $T_{2}^\text{rabi}$ are influenced by inter-dot detuning energy that modulates spin-electric-coupling (SEC) strength (see Supplementary Information). We therefore optimize the detuning energy for the highest quality factor, defined as the number of typical qubit operations available within the Rabi decay time. We reach a quality factor of $Q=2f_\text{rabi}T_{2}^\text{rabi}=85\pm 8$ (see Fig. S4b of Supplementary Information), comparable to natural silicon quantum dots[@Kawakami2016; @Takeda2016]. The value predicts the fidelity of a $\pi$-gate of $e^{-1/Q}=98.8\pm 0.1\,\%$. We tested this prediction using randomized benchmarking[@Muhonen2015] and find an $\text{X}_\pi$ gate fidelity of $99.04\pm 0.23\,\%$ (see Fig. \[fig:control\]e), close to the $Q$-factor limited value. This is the highest fidelity for single-spin qubits in GaAs reported so far. We notice however that the average single-gate fidelity is $97.50\pm 0.05\,\%$, most likely limited by systematic errors in the other gates (unitary errors) due to the microwave setup in the present study (see Supplementary Information). This issue would be readily resolved by integrating an established technique of IQ modulation[@Takeda2016; @Yang2019] with the FPGA in the microwave generation setup. The $\text{X}_\pi$ gate is least affected by microwave imperfections as we calibrate the control pulse line primarily for this gate.
Discussion: limits on the spin-qubit control fidelity
=====================================================
What is the physical mechanism limiting the Rabi decay time and the ultimate control fidelity of a single-spin qubit in this system? One obvious candidate is the residual inhomogeneity of $f_\text{qubit}$ or, in other words, the quasi-static noise $\delta f$. However, this contribution should lead to a power-law envelope[@Koppens2007; @Cucchietti2005] of the Rabi decay $[1+(2\pi\sigma^2 t/f_\text{rabi})^2]^{-1/4}$, as opposed to the exponential one seen in Fig. \[fig:control\]c. For $f_\text{rabi}\gg \sigma$, the initial decay could be approximated by a Gaussian envelope with[@Yoneda2014] $T_{2}^{\text{rabi}^\prime}=f_\text{rabi}/(\pi\sigma^{2})$, leading to $T_{2}^{\text{rabi}^\prime}=74\,\mu\text{s}$ with $f_\text{rabi}=20\,\text{MHz}$. This value is an order of magnitude larger than the measured decay time. Also, assuming that unitary errors are removed[@Yang2019], the qubit control fidelity as high as $\exp\left[-(2f_\text{rabi} T_{2}^{\text{rabi}^\prime})^{-2}\right]>99.9999\,\%$ could be reached. We conclude that such quasi-static noise is therefore not the main limiting factor of the ultimate qubit control fidelity.
We consider three other noise sources as possibly relevant to the Rabi decay[@Yan2013; @Yoshihara2014]: the quasi-static noise in $f_\text{rabi}$, the transverse noise at the electron Larmor frequency leading to spin relaxation, and the longitudinal noise in $f_\text{qubit}$ at the Rabi frequency. The quasi-static noise in $f_\text{rabi}$ could be caused by fluctuations of the microwave driving amplitude or SEC. However, this mechanism would also lead to a Gaussian decay envelope with $T_{2}^\text{rabi}\propto f_\text{rabi}^{-1}$, inconsistent with Fig. \[fig:control\]c. The spin relaxation is also unlikely, at least in the range of $f_\text{rabi}<20\,\text{MHz}$, because it cannot explain the increase of $T_{2}^\text{rabi}$ with $f_\text{rabi}$. Therefore, we conclude that $T_{2}^\text{rabi}$ is most likely dominated by the high-frequency (on the order of $f_\text{rabi}$) longitudinal noise in $f_\text{qubit}$, which inherently leads to an exponential Rabi decay.
When the high-frequency noise in $f_\text{qubit}$ dominates, one can relate[@Yan2013; @Yoshihara2014] the noise power spectral density $S(f)$ at the Rabi frequency to the exponential-decay rate of the Rabi oscillations \[see Supplementary Information\]. The power spectral density $S(f)$ extracted in this way is plotted in Fig. \[fig:spectroscopy\]a. For $f>20\,\text{MHz}$, it grows rapidly with $f$. It could be due to thermal noise caused by microwave-induced heating[@Takeda2018], with the consistent scaling $P\propto E^{2}\propto f_\text{rabi}^{2}$. On the other hand, we cannot exclude a Rabi decay through spin relaxation in this range, possibly caused by, for example, electron exchange with reservoirs due to photon-assisted tunneling[@Fujisawa1997; @Oosterkamp:1998tm]. Since we could not extract more information of the intrinsic noise density, we do not discuss this range further.
We turn to the other frequency range, $f<20\,\text{MHz}$. Here, $S(f)$ shows three prominent peaks at nuclear Larmor precession frequencies of $^{75}$As, $^{69}$Ga and $^{71}$Ga. It clearly suggests that such high-frequency noise sources indeed influence the Rabi decay of the spin qubit. The hyperfine coupling between a single electron spin $\hat{\bm{S}}$ and nuclear spins $\hat{\bm{I}}_k$ is given by $H_\text{hf}=\sum_k A_k \hat{\bm{S}}\cdot\hat{\bm{I}}_k$, where $A_k$ is a coupling constant dependent on each nuclear site $\bm{r}_k$ indexed by $k$. This results in an Overhauser field component parallel to the external field, being $B^\text{nuc}_{z}\propto \sum_k A_k \hat{I}_k^z$. When nuclear spins precess around the $z$ axis, $B^\text{nuc}_{z}$ is constant and there is no noise at the Larmor frequencies. In the present device, however, the stray magnetic field from the micromagnet induces field inhomogeneity, making each nuclear spin at $\bm{r}_k$ precess around a local magnetic field vector $\bm{B}(\bm{r}_k)$ slightly off the $z$ axis (see the inset of Fig. \[fig:spectroscopy\]a). The inhomogeneity of the nuclear spin polarization leads to small but finite residual oscillations of $B^\text{nuc}_{z}$ at the nuclear precession frequencies.
Apart from the three spectral peaks, we find that $S(f)$ follows $f^{-1}$ dependence at $f<20\,\text{MHz}$. It suggests that the $f^{-1}$ noise background in the range of tens of MHz is the dominant limiting factor of the qubit control fidelity. The $f^{-\beta}$ dependence with $\beta=1$ differs from $\beta \sim 1.7$ for the nuclear spin diffusion noise[@Medford2012; @Malinowski2017a] at low frequencies (see Fig. \[fig:spectroscopy\]b) extracted from the data in Figs. \[fig:ramsey\] and \[fig:feedback\] (small-SEC regime). In addition, we confirmed that the amplitude of $S(f)$ is larger for a larger SEC (See Fig. S5 of Supplementary Information). We therefore conclude that the $f^{-1}$ spectrum arises from charge noise and SEC provided by the micromagnet stray field[@Yoneda2017]. Indeed, from a Ramsey measurement performed in the condition optimized for the large quality factor (large-SEC regime), we extract the low-frequency ($f<100\,\text{Hz}$) noise in line with the $f^{-1}$ dependence (Fig. \[fig:spectroscopy\]b). Similar $f^{-1}$ noise spectrum over seven decades of frequency has been observed in an isotopically purified $^{28}$Si device[@Yoneda2017]. Approximating $S(f)=A^{2}/f$, however, we find $A\sim 0.6\,\text{MHz}$ being two orders of magnitude larger than $A\sim 1.6\,\text{kHz}$ found in the $^{28}$Si device. It is also an order of magnitude larger than $A\sim 0.1\,\text{MHz}$ observed in a GaAs device without micromagnet[@Malinowski2017a]. The difference can be partly attributed to large SEC in the present device, as $S(f)$ is reduced by two orders of magnitude by decreasing SEC (see Fig. \[fig:spectroscopy\]b), at least at low frequencies. SEC also depends on the orbital energy splitting determined by confinement potential and effective mass. However, the influence of other factors on $A$, such as material properties and experimental setups, requires further investigations.
To summarize, we demonstrate $24$-fold enhancement of the dephasing time $T_2^*$ of a GaAs single electron spin qubit. The enhancement relies on suppressing the nuclear spin noise by feedback and is limited by classical control electronics that can be improved further. The feedback also boosts the qubit overall performance: We reach the quality factor of $Q=85\pm 8$ and the $\pi$-gate fidelity of $99.04\pm 0.23\,\%$. We find that, despite our device being a GaAs quantum dot, the ultimate fidelity with the feedback is not limited by nuclear-spin noise. The culprit is the $1/f$ charge noise leaking into the qubit through the micromagnet field gradient at megahertz frequencies.
Additional details of the feedback protocol
===========================================
Our feedback control protocol is implemented on a Xilinx ZedBoard device equipped with a coupled central processing unit and programmable logic. The device is interfaced with an AD-FMCOMMS2-EBZ peripheral board from Analog Devices which provides integrated RF demodulators, 12-bit digital-to-analog converters (DACs) with a sampling rate up to $122.88\,\text{M}$ samples per second, and local oscillators (LOs) operating at up to $6\,\text{GHz}$. The demodulators and DACs are used to digitize the RF charge-sensing signal for single-shot spin readouts, and one of the LOs generates the driving microwave at $f_\text{MW}$ for the EDSR. This system enables rapid switching of $f_\text{MW}$ conditioned on the spin measurement outcomes without overhead for the data transfer between different equipments. The off-resonant microwave burst is generated from a discrete signal generator and fed to the same signal path via a microwave switch. The whole setup is described in the Supplementary Information.
For the realtime feedback control, we first take $150$ single-shot data points of a Ramsey oscillation with varied intervals $t_\text{R}=2,4,\cdots 300\,\text{ns}$ in the “probe” step. Each measurement sequence shown in Fig. \[fig:ramsey\]b takes $T_\text{R}=31.71\,\mu\text{s}$, giving $T_\text{p}=150\,T_\text{R}\approx 4.8\,\text{ms}$ for one probe step. We update the value of ${f^\text{est}_\text{qubit}}$ based on $\delta f$ obtained from the Bayesian estimation[@Shulman2014; @Delbecq2015] and adjust the microwave frequency to $f_\text{MW}={f^\text{est}_\text{qubit}}+\Delta$ for the “target” experiment. We wait for $T_\text{w}$ before starting the target step, and we find that $T_\text{w}$ needs to be longer than $2\,\text{ms}$ to stabilize the microwave output. The time $T_\text{t}$ spent in the target step depends on the type of the experiment and it is $T_\text{t} = T_\text{p}$ in the case of the measurement shown in Fig. \[fig:feedback\]. Thus, we define the feedback latency between the probe and target steps as $\Delta t = T_\text{p}/2 + T_\text{w} + T_\text{t}/2$, which can be controlled by changing $T_\text{w}$.
We thank S. D. Bartlett for fruitful discussions. We thank RIKEN CEMS Emergent Matter Science Research Support Team and Microwave Research Group at Caltech for technical assistance. Part of this work was financially supported by CREST, JST (JPMJCR15N2, JPMJCR1675), the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan), JSPS KAKENHI Grants No. 26220710, No. 18H01819 and No. 16K05411, RIKEN Incentive Research Projects, The Murata Science Foundation Research Grant, and Q-LEAP project initiated by MEXT, Japan. T.O. acknowledges support from JSPS KAKENHI Grants No. 16H00817 and No. 17H05187, PRESTO (JPMJPR16N3), JST, Yazaki Memorial Foundation for Science and Technology Research Grant, Advanced Technology Institute Research Grant, Izumi Science and Technology Foundation Research Grant, TEPCO Memorial Foundation Research Grant, The Thermal & Electric Energy Technology Foundation Research Grant, The Telecommunications Advancement Foundation Research Grant, Futaba Electronics Memorial Foundation Research Grant, and MST Foundation Research Grant. A.D.W. and A.L. greatfully acknowledge support from Mercur Pr2013-0001, BMBF Q.Com-H 16KIS0109, TRR160, and DFH/UFA CDFA-05-06.
![Rabi noise spectroscopy. (a) Power spectral density $S(f)$ of the longitudinal noise in $f_\text{qubit}$ (red circles) extracted from the data in Fig. \[fig:control\]d. Vertical grey lines show Larmor precession frequencies for three nuclear species, $^{75}$As, $^{69}$Ga and $^{71}$Ga calculated with the micromagnet-induced field component of $B^\text{MM}_z=70\,\text{mT}$. Black and green lines are guides to the eye for $f^{-1}$ and $f^{2}$ dependence, respectively. The inset illustrates electron-nuclear spin coupling in an inhomogeneous magnetic field. Each nuclear spin is randomly oriented and precesses around a local magnetic field vector $\bm{B}(\bm{r}_k)$, leading to an oscillatory Overhauser field in the $z$ direction. (b) Comparison of $S(f)$ in (a) and those extracted from Ramsey measurements. The power spectral density of $\delta f$ in Ramsey measurements is calculated by the fast Fourier transform of $\delta f(t)$. The spectral density taken with small SEC and feedback off (blue) shows the $f^{-1.7}$ dependence (black dashed line) similar to those observed for nuclear spin diffusion noise[@Medford2012; @Malinowski2017a]. The spectral density is significantly suppressed with feedback on (orange) down to a level determined by the precision of the feedback control. The flat noise spectrum suggests that the residual low-frequency noise is uncorrelated. A peak at $2\,\text{Hz}$ is due to the vibration of the dilution refrigerator. The power spectral density increases as SEC is increased (red curve in the top left), and it follows the $f^{-1}$ dependence (black solid line) in line with $S(f)$ extracted from the Rabi spectroscopy in (a) (red curve in the bottom right). \[fig:spectroscopy\]](Fig4){width="90.00000%"}
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10.1103/PhysRevB.65.205309) [****, ()](https://doi.org/DOI
10.1103/PhysRevLett.88.186802) [****, ()](https://doi.org/10.1038/ncomms6156) [****, ()](https://doi.org/10.1103/PhysRevB.67.094510) [****, ()](https://doi.org/10.1103/PhysRevB.72.134519) [****, ()](https://doi.org/10.1103/PhysRevB.81.100511) [****, ()](https://doi.org/10.1103/PhysRevB.89.020503) [****, ()](https://doi.org/10.1103/PhysRevLett.108.086802) [****, ()](https://doi.org/10.1103/PhysRevLett.118.177702) [****, ()](https://doi.org/10.1073/pnas.1603251113) [****, ()](https://doi.org/10.1038/s41928-019-0234-1) [****, ()](https://doi.org/DOI 10.1038/nphys1053) [****, ()](https://doi.org/ARTN 047202) [****, ()](https://doi.org/10.1103/PhysRevLett.113.267601) [****, ()](https://doi.org/10.1063/1.4945592) [****, ()](https://doi.org/DOI 10.1103/PhysRevB.81.161308) [****, ()](https://doi.org/10.1103/PhysRevLett.101.236803) [****, ()](https://doi.org/10.1103/PhysRevA.74.032316) [****, ()](https://doi.org/10.1103/PhysRevB.73.205302) [****, ()](https://doi.org/10.1103/PhysRevLett.116.046802) [****, ()](https://doi.org/10.1038/nature25766) [****, ()](https://doi.org/10.1038/s41534-018-0105-z) [****, ()](https://doi.org/10.1017/CBO9781107415324.004) [****, ()](https://doi.org/10.1038/s41534-016-0003-1) [****, ()](https://doi.org/10.1088/0953-8984/27/15/154205) [****, ()](https://doi.org/10.1103/PhysRevLett.99.106803) [****, ()](https://doi.org/10.1103/PhysRevA.72.052113) [****, ()](https://doi.org/10.1038/ncomms3337) [****, ()](https://doi.org/10.1006/spmi.1996.0191) @noop [****, ()]{}
[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper focuses on interior penalty discontinuous Galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra which satisfies certain shape regularity conditions characterized in a recent paper by two of the authors in [@WangYe2012]. Such general meshes have important application in computational sciences. The usual $H^1$ conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation. However, the interior penalty discontinuous Galerkin method provides a simple and effective alternative approach which is efficient and robust. This article provides a mathematical foundation for the use of interior penalty discontinuous Galerkin methods in general meshes.'
author:
- 'Mu Lin[^1]'
- 'Junping Wang[^2]'
- 'Yanqiu Wang[^3]'
- 'Xiu Ye[^4]'
title: Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes
---
discontinuous Galerkin, finite element, interior penalty, second-order elliptic equations, hybrid mesh.
65N15, 65N30.
Introduction
============
Most finite element methods are constructed on triangular and quadrilateral meshes, or on tetrahedral, hexahedral, prismatic, and pyramidal meshes. To extend the idea of the finite element method into meshes employing general polygonal and polyhedral elements, one immediately faces the problem of choosing suitable discrete spaces on general polygons and polyhedrons. This issue has rarely been addressed in the past, partly because it can usually be circumvented by dividing the polygon or polyhedron into sub-elements using only one or two basic shapes. However, allowing the use of general polygonal and polyhedral elements does provide more flexibility, especially for complex geometries or problems with certain physical constraints. One of such example is the modeling of composite microstructures in material sciences. A well-known solution to this problem is the Voronoi cell finite element method [@Ghosh94; @Ghosh95; @Ghosh04; @Moorthy98], in which the mesh is composed of polygons or polyhedrons representing the grained microstructure of the given material. The main difficulty of constructing conforming finite element methods on Voronoi meshes is that, the finite element space has to be carefully chosen so that it is continuous along interfaces. Although the constructions on triangles, quadrilaterals, or three-dimensional simplexes are straight forward, it is not easy for general polygons and polyhedrons. Probably the only practically used solution is the rational polynomial interpolants proposed by Wachspress [@Wachspress75], in which rational basis functions are defined using distances from several “nodes”. An important constraint in the construction of the Wachspress basis is that, the rational basis functions need to be piecewise linear along the boundary of every element, in order to ensure $H^1$ conformity of the finite element space. This not only limits the approximation order of the entire Wachspress finite element space, but also complicates the construction. The Wachspress element has gained a renewed interest recently [@Dasgupta03; @Dasgupta03b; @Sukumar06]. However, as we have pointed out above, its construction is complicated and usually requires the aid of computational algebraic systems such as Maple.
Another practically important issue is to define finite element methods on hybrid meshes. Hybrid meshes are frequently used nowadays. It can handle complicated geometries, and can sometimes reduce the total number of unknowns. Another possible reason for using the hybrid mesh is that, some engineers argue that in three-dimensions, a hexahedral mesh yields more accurate solution than a tetrahedral mesh for the same geometry [@Yamakawa03; @Yamakawa09], as partly verified by numerical experiments. However, pure hexahedral meshes lack the ability of handling complicated geometries. Hence a hybrid mesh becomes a welcomed compromise between accuracy and flexibility. For conforming finite element methods based on hybrid meshes, continuity requirements on interfaces must be satisfied. Such a coupling is straight-forward for the $H^1$-conforming finite elements on a triangular-quadrilateral hybrid mesh. However, for three-dimensional meshes, high order finite elements, or other complicated finite element spaces, it usually requires special treatments.
An alternative solution, that can address both issues mentioned above, is to use the weak Galerkin method proposed in [@WangYe2012]. The weak Galerkin method uses discontinuous piecewise polynomials inside each element and on the interfaces to approximate the variational solution. In [@WangYe2012], the authors have proved optimal convergence of the weak Galerkin method for the mixed formulation of second order elliptic equations on very general polygonal and polyhedral meshes. Most of the existing error analysis of finite element methods assume triangular, quadrilateral, or some commonly-seen three-dimensional meshes. To our knowledge, it is the first time that optimal convergence for the finite element solutions has been rigorously proved in [@WangYe2012] for general meshes of arbitrary polygons and polyhedrons.
The discontinuous Galerkin method imposes the interface continuity weakly, and is known to be able to handle non-conformal, hybrid meshes as well as a variety of basis functions. There have been many research works in this direction, for example, nodal discontinuous Galerkin methods [@Bergot10; @Cohen00; @Hesthaven00] for hyperbolic conservation laws. However, we would like to point out that so far there has been no theoretical analysis on the convergence rate of discontinuous Galerkin method, on very general polygonal or polyhedral meshes yet. Motivated by the work in [@WangYe2012], here we would like to fill the gap. The objective of this paper is to establish the theoretical analysis of the interior penalty discontinuous Galerkin method [@Arnold02] for elliptic equations on very general meshes and discrete spaces.
The paper is organized as follows. In Section 2, we briefly describe the interior penalty discontinuous Galerkin method in an abstract setting. In Section 3, several assumptions on the discrete spaces are listed, which form a minimum requirement for the well-posedness and the approximation property of the discrete formulation. Abstract error estimations are given. In Section 4, we discuss choices of meshes and discrete spaces that satisfy the assumptions given in Section 3. Finally, numerical results are presented in Section 5.
The model problem and the interior penalty method
=================================================
Consider the model problem $$\label{eq:ellipticeq}
\begin{cases}
-\Delta u=f\qquad &\mbox{in }\Omega,\\
u=0 &\mbox{on }\partial\Omega,
\end{cases}$$ where $\Omega\in\mathbb{R}^d(d=2,3)$ is a closed domain with Lipschitz continuous boundary, and $f\in L^2(\Omega)$.
For any subdomain $K\subset \Omega$ with Lipschitz continuous boundary, we use the standard definition of Sobolev spaces $H^s(K)$ with $s\ge 0$ (e.g., see [@adams; @ciarlet] for details). The associated inner product, norm, and seminorms in $H^s(K)$ are denoted by $(\cdot,\cdot)_{s,K}$, $\|\cdot\|_{s,K}$, and $|\cdot|_{s,K}$, respectively. When $s=0$, $H^0(K)$ coincides with the space of square integrable functions $L^2(K)$. In this case, the subscript $s$ is suppressed from the notation of norm, semi-norm, and inner products. Furthermore, the subscript $K$ is also suppressed when $K=\Omega$. Finally, all above notations can easily be extended to any $e\subset \partial K$. For the $L^2$ inner product on $e$, we usually denote it as $\langle\cdot,\cdot\rangle_{e}$ in stead of $(\cdot,\cdot)_{e}$, as it can be replaced by the duality pair when needed.
For simplicity, we assume that $\Omega$ satisfy certain conditions such that Equation (\[eq:ellipticeq\]) has at least $H^{r}$ regularity with $r>3/2$, that is, the solution to Equation (\[eq:ellipticeq\]) satisfies $u\in H^{r}(\Omega)$ and $$\label{eq:regularity}
\|u\|_r \le C_R \|f\|.$$ This assumption is standard in the practice of interior penalty discontinuous Galerkin methods, as it ensures that the exact solution $u$ also satisfies the discontinuous Galerkin formulation, and thus the a priori error estimation can be easily derived in a Lax-Milgram framework. However, such a regularity assumption is not necessary in the practice of interior penalty methods. A well-known technique, which was first proposed by Gudi [@Gudi10], is to use a posteriori error estimation to derive an a priori error estimation for the interior penalty method, with only minimum regularity requirement $u\in H^1(\Omega)$. We believe that the same technique applies for the general polygonal and polyhedral meshes, as long as a working a posteriori error estimation is available. However, here we choose to completely skip this issue, as it is not the main purpose of this paper.
Assume that for all set $K$ discussed in this paper, including $\Omega$ itself, the unit outward normal vector $\bn$ is defined almost everywhere on $\partial K$. Note this is true for all polygonal and polyhedral elements with Lipschitz continuous boundaries. Since the exact solution $u\in H^{r}(\Omega)$ with $r>3/2$, it is clear that for any smooth function $v$ defined on $K$, $$(\nabla u, \, \nabla v)_K - \langle\nabla u\cdot\vn,\, v\rangle_{\partial K} = (f,\, v)_K,$$ where $(\cdot,\cdot)_K$ is the $L^2$-inner product in $L^2(K)$ and $\langle\cdot,\cdot\rangle_\pK$ is the $L^2$-inner product in $L^2(\pK)$
Let $\mathcal{T}_h$ be a partition of the domain $\Omega$ into non-overlapping subdomains/elements, each with Lipschitz continuous boundary. Here $h$ denotes the characteristic size of the partition, which will be defined in details later. The interior interfaces are denoted by $e = \bar{K_1}\cap\bar{K_2}$, where $K_1$, $K_2\in \mathcal{T}_h$. Boundary segments are similarly denoted by $e = \bar{K}\cap\partial\Omega$, where $K\in \mathcal{T}_h$. Denote by $\mathcal{E}_h$ the set of all interior interfaces and boundary segments in $\mathcal{T}_h$, and by $\mathcal{E}_h^0=\mathcal{E}_h\setminus\partial\Omega$ the set of all interior interfaces. For every $K\in\mathcal{T}_h$, let $|K|$ be the area/volume of $K$, and for every $e\in \mathcal{E}_h$, let $|e|$ be its length/area. Denote $h_e$ the diameter of $e\in \mathcal{E}_h$ and $h_K$ the diameter of $K\in \mathcal{T}_h$. Clearly, when $e\subset \partial K$, we have $h_e\le h_K$. Finally, define $h=\max_{K\in \mathcal{T}_h} h_K$ to be the characteristic mesh size.
Notice that $\mathcal{T}_h$ defined above is a very general mesh/partition on $\Omega$, as we do not specify the shape and conformal property of $K\in \mathcal{T}_h$. The interior penalty discontinuous Galerkin (IPDG) method can be extended to such a general mesh, without any modification of the formulation. However, to ensure its approximation rate, certain conditions must be imposed on $\mathcal{T}_h$ and the discrete function spaces. In this paper, we are interested in discussing the minimum requirements of such conditions. First, we shall give the formulation of the interior penalty discontinuous Galerkin method.
Let $V_K$ be a finite dimensional space of smooth functions defined on $K \in \mathcal{T}_h$. Define $$V_h=\{v\in L^2(\Omega):v|_K\in V_K,\textrm{ for all } K\in\mathcal{T}_h\},$$ and $$V(h)=V_h+\left( H_0^1(\Omega)\cap \prod_{K\in\mathcal{T}_h} H^{r}(K) \right),\qquad \textrm{where }r>\frac{3}{2}.$$ For any internal interface $e = \bar{K_1}\cap\bar{K_2} \in \mathcal{E}_h$, let $\bn_1$ and $\bn_2$ be the unit outward normal vectors on $e$, associated with ${K_1}$ and ${K_2}$, respectively. For $v\in V(h)$, define the average $\{\nabla v\}$ and jump $[v]$ on $e$ by $$\{\nabla v\} = \frac{1}{2}\left(\nabla v|_{K_1} + \nabla v|_{K_2} \right),\qquad
[v] = v|_{K_1} \bn_1 + v|_{K_2} \bn_2.$$ On any boundary segment $e= \bar{K}\cap\partial\Omega$, the above definitions of average and jump need to be modified: $$\{\nabla v\} = \nabla v|_{K},\qquad [v] = v|_{K} \bn_K,$$ where $\bn_K$ is the unit outward normal vector on $e$ with respect to $K$.
Define a bilinear form on $V(h)\times V(h)$ by $$\begin{aligned}
A(u,v) =& \sum_{K\in \mathcal{T}_h}(\nabla u,\nabla v)_K-\sum_{e\in\mathcal{E}_h}\langle\{\nabla u\},\, [v]\rangle_e \\
&\quad -\delta \sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [u]\rangle_e + \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \langle[u],\, [v]\rangle_e,
\end{aligned}$$ where $\delta = \pm 1,\, 0$ and $\alpha>0$. when $\delta=1$, the bilinear form $A(\cdot,\cdot)$ is symmetric. The constant $\alpha$ is usually required to be large enough, but still independent of the mesh size $h$, in order to guarantee the well-posedness of the discontinuous Galerkin formulation. Details will be given later.
It is clear that the exact solution $u$ to Equation (\[eq:ellipticeq\]) satisfies $$\label{eq:dg-exactsol}
A(u,v) = (f,v)\qquad\textrm{for all } v\in V_h,$$ as $[u]$ vanishes on all $e\in\mathcal{E}_h$. Hence the following interior penalty discontinuous Galerkin formulation is consistent with Equation (\[eq:ellipticeq\]): find $u_h \in V_h$ satisfying $$\label{eq:dg}
A(u_h,v) = (f,v)\qquad\textrm{for all } v\in V_h.$$
Finally, we would like to point out that the formulation (\[eq:dg\]) is computable, as long as each finite dimensional space $V_K$ has a clearly defined and computable basis.
Abstract theory
===============
Define a norm $\3bar\cdot\3bar$ on $V(h)$ as following: $$\begin{aligned}
\3bar v\3bar^2=\sum_{K\in\mathcal{T}_h}\|\nabla v\|_K^2+\sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2+\alpha\sum_{e\in\mathcal{E}_h}\frac{1}{h_e}\|[v]\|_e^2.\end{aligned}$$ By the Poincaré inequality, $\3bar\cdot\3bar$ is obviously a well-posed norm on $V(h)$.
Next, we give a set of assumptions, which form the minimum requirements guaranteeing the well-posedness and the approximation properties of the interior penalty discontinuous Galerkin method.
- (The trace inequality) There exists a positive constant $C_T$ such that for all $K\in\mathcal{T}_h$ and $\theta\in H^1(K)$, we have $$\label{eq:TraceIn}
\|\theta\|_{\partial K}^2\le C_{T}(h_K^{-1}\|\theta\|_K^2+h_K\|\nabla\theta\|_K^2).$$
- (The inverse inequality) There exists a positive constant $C_I$ such that for all $K\in\mathcal{T}_h$, $\phi\in V_K$ and $\phi\in \frac{\partial}{\partial x_i}V_K$ where $i=1,\ldots, d$, we have $$\label{eq:InverseIn}
\|\nabla\phi\|_K\le C_I\, h_K^{-1}\|\phi\|_K.$$
- (The approximability) There exist positive constants $s$ and $C_A$ such that for all $v\in H^{s+1}(\Omega)$, we have $$\label{eq:approximability}
\inf_{\chi_h\in V_h} \3bar v-\chi_h \3bar \le C_A \left(\sum_{K\in \mathcal{T}_h} h_K^{2s} \|v\|_{s+1,K}^2\right)^{1/2}.$$
The abstract theory of the interior penalty discontinuous Galerkin method can be entirely based on Assumptions [**[I1]{}**]{}-[**[I3]{}**]{}.
\[lem:wellposedness\] Assume [**[I1]{}**]{}-[**[I2]{}**]{} hold. The bilinear form $A(\cdot,\cdot)$ is bounded in $V(h)$, with respect to the norm $\3bar\cdot\3bar$. Indeed, $$A(u,v)\le \frac{1+\alpha}{\alpha} \3bar u\3bar\, \3bar v\3bar\qquad \textrm{for all } u,\, v\in V(h).$$ Furthermore, denote $C_1 = C_T(1+C_I)^2$. Then for any constant $0<C<1$ and $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, the bilinear form $A(\cdot,\cdot)$ is coercive on $V_h$. That is, $$A(v, v) \ge \frac{C}{1+C_1} \3bar v\3bar^2\qquad \textrm{for all } v\in V_h.$$
The boundedness of $A(\cdot,\cdot)$ follows immediately from the Schwarz inequality. Here we only prove the coercivity. First, notice that for all $v\in V_h$, by assumptions [**[I1]{}**]{}-[**[I2]{}**]{} and the fact that $h_e\le h_K$ for all $e\in \partial K\cap \mathcal{E}_h$, $$\begin{aligned}
\sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2 & \le \sum_{K\in\mathcal{T}_h} \left(\sum_{e\in \partial K\cap \mathcal{E}_h} h_e \|\nabla v\|_e^2\right) \\
&\le \sum_{K\in\mathcal{T}_h} h_K \|\nabla v\|_{\partial K}^2 \\
& \le \sum_{K\in\mathcal{T}_h} h_K \bigg( C_T(1+C_I^2) h_K^{-1} \|\nabla v\|_K^2 \bigg) \\
& = C_1 \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2.
\end{aligned}$$ Then, by the Schwarz inequality, the Young’s inequality and assumptions [**[I1]{}**]{}-[**[I2]{}**]{}, we have $$\begin{aligned}
\sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [v]\rangle_e &\le
\varepsilon \sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2 + \frac{1}{4\varepsilon} \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \\
&\le \varepsilon C_1 \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 + \frac{1}{4\varepsilon\alpha} \left(\alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \right),
\end{aligned}$$ where $\varepsilon$ is chosen to be $\frac{1-C}{(1+\delta)C_1}$ for any given constant $0<C<1$. Clearly, for such an $\varepsilon$, we have $1-(1+\delta)\varepsilon C_1 = C$ and $$1-\frac{1+\delta}{4\varepsilon\alpha} \ge C \quad \Longleftrightarrow \quad \alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}.$$ Combine the above and let $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, we have $$\begin{aligned}
A(v, v) &= \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 - (1+\delta) \sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [v]\rangle_e
+ \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \\
&\ge \bigg( 1-(1+\delta)\varepsilon C_1\bigg)\sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2
+ \bigg(1-\frac{1+\delta}{4\varepsilon\alpha}\bigg)\alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \\
&\ge C \left( \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 + \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \right)\\
&\ge \frac{C}{1+C_1} \3bar v\3bar^2.
\end{aligned}$$
Lemma \[lem:wellposedness\] guarantees the existence and uniqueness of the solution to Equation (\[eq:dg\]). In the rest of this paper, we shall always assume $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. By subtracting (\[eq:dg-exactsol\]) from (\[eq:ellipticeq\]), one gets the standard orthogonality property of the error, $$A(u-u_h,\, v_h) = 0\qquad\textrm{for all } v\in V_h.$$ Then clearly, for all $\chi_h\in V_h$, $$\begin{aligned}
\3bar \chi_h - u_h\3bar^2 &\le \frac{1+C_1}{C}A(\chi_h - u_h,\, \chi_h - u_h) \\
&= \frac{1+C_1}{C}A(\chi_h - u,\, \chi_h - u_h) \\
&\le \frac{(1+C_1)(1+\alpha)}{C\alpha} \3bar \chi_h - u\3bar \, \3bar \chi_h - u_h\3bar.
\end{aligned}$$ Then, using the triangle inequality, $$\begin{aligned}
\3bar u-u_h\3bar &\le \inf_{\chi_h\in V_h} \bigg( \3bar u-\chi_h\3bar + \3bar \chi_h-u_h\3bar \bigg) \\
&\le \bigg(1 + \frac{(1+C_1)(1+\alpha)}{C\alpha} \bigg) \inf_{\chi_h\in V_h}\3bar u-\chi_h\3bar.
\end{aligned}$$ Combine this with assumption [**[I3]{}**]{}, we get the following abstract error estimation:
Assume [**[I1]{}**]{}-[**[I3]{}**]{} hold, $C$ be a given constant in $(0,1)$ and $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. Then $$\3bar u-u_h\3bar \lesssim C_A\bigg(1 + \frac{(1+C_1)(1+\alpha)}{C\alpha} \bigg)
\left(\sum_{K\in \mathcal{T}_h} h_K^{2s} \|u\|_{s+1,K}^2\right)^{1/2}.$$
Finally, we derive the $L^2$ error estimation by using the standard duality argument. Let $\delta=1$, that is, the bilinear form $A(\cdot,\cdot)$ is symmetric. Consider the following problem $$\begin{cases}
-\Delta \phi=u-u_h\qquad &\mbox{in }\Omega,\\
\phi=0 &\mbox{on }\partial\Omega.
\end{cases}$$ Here again, we assume that the domain $\Omega$ satisfies certain condition such that $\phi$ has $H^r$ regularity, with $r>3/2$. Let $\phi_h\in V_h$ be an approximation to $\phi$ such that they satisfy Assumption [**[I3]{}**]{}. Clearly $$\begin{aligned}
\|u-u_h\|^2 &= (-\Delta \phi,\, u-u_h) = \sum_{K\in \mathcal{T}_h}(\nabla \phi,\nabla (u-u_h))_K-\sum_{e\in\mathcal{E}_h}\langle\{\nabla \phi\},\, [u-u_h]\rangle_e \\
&= A(\phi, \, u-u_h) = A(\phi-\phi_h,\, u-u_h)\\
&\le \frac{1+\alpha}{\alpha} \3bar \phi-\phi_h \3bar\, \3bar u-u_h\3bar \\
&\le \frac{1+\alpha}{\alpha}C_A \left(\sum_{K\in \mathcal{T}_h} h_K^{2\min\{r-1,s\}} \|\phi\|_{\min\{r,s+1\},K}^2\right)^{1/2} \3bar u-u_h\3bar.
\end{aligned}$$ This gives the following theorem
Assume [**[I1]{}**]{}-[**[I3]{}**]{} hold, $\delta=1$, $C$ be a given constant in $(0,1)$, $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, and the elliptic equation (\[eq:ellipticeq\]) has $H^r$ regularity with $r>3/2$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. Then $$\|u-u_h\| \le \frac{1+\alpha}{\alpha}C_AC_R h^{\min\{r-1,s\}} \3bar u-u_h\3bar.$$
Requirements on meshes and discrete spaces
==========================================
On triangular or quadrilateral meshes, the usual tool for proving assumptions [**[I1]{}**]{}-[**[I3]{}**]{} is to use a scaling argument built on affine transformations. However, on general polygons and polyhedrons, it is not clear how to define such affine transformations. The assumptions [**[I1]{}**]{}-[**[I3]{}**]{} were first validated in [@WangYe2012] for general polygonal and polyhedral meshes that satisfy a set of conditions introduced in [@WangYe2012]. Such conditions can be stated as follows.
All the elements of $\mathcal{T}_h$ are assumed to be closed and simply connected polygons or polyhedrons. We make the following shape regularity assumptions for the partition $\mathcal{T}_h$.
- Assume that there exist two positive constants $\rho_v$ and $\rho_e$ such that for every element $K\in\mathcal{T}_h$ and $e\in \mathcal{E}_h$, we have $$\begin{aligned}
\rho_vh_K^d\le |K|,\ \ \rho_eh_e^{d-1}\le |e|.\end{aligned}$$
- Assume that there exists a positive constant $\kappa$ such that for every element $K\in\mathcal{T}_h$ and $e\in \partial K\cap \mathcal{E}_h$, we have $$\begin{aligned}
\kappa h_K\le h_e.\end{aligned}$$
- Assume that for every $K\in\mathcal{T}_h,$ and $e\in\partial K \cap \mathcal{E}_h$, there exists a pyramid $P(e,K,A_e)$ contained in $K$ such that its base is identical with $e$, its apex is $A_e\in K$, and its height is proportional to $h_K$ with a proportionality constant $\sigma_e$ bounded away from a fixed positive number $\sigma^*$ from below. In other words, the height of the pyramid is given by $\sigma_eh_K$ such that $\sigma_e\ge \sigma^*>0.$ The pyramid is also assumed to stand up above the base $e$ in the sense that the angle between the vector ${\bf x}_e-A_e,$ for any ${\bf x}_e\in e$, and the outward normal direction of $e$ is strictly acute by falling into an interval $[0,\theta_0]$ with $\theta_0<\pi/2$.
- Assume that each $K\in\mathcal{T}_h$ has a circumscribed simplex $S(K)$ that is shape regular and has a diameter $h_{S(K)}$ proportional to the diameter of $K$; i.e., $h_{S(K)}\le\gamma_*h_K$ with a constant $\gamma_*$ independent of $K.$ Furthermore, assume that each circumscribed simplex $S(K)$ intersects with only a fixed and small number of such simplexes for all other elements $K\in\mathcal{T}_h.$
Under the above assumptions, the following results have been proved in [@WangYe2012]:
(The trace inequality). Assume [**[A1]{}**]{}-[**[A3]{}**]{} hold on a polygonal or polyhedral mesh. Then [**[I1]{}**]{} is true.
(The inverse Inequality). Assume [**[A1]{}**]{}-[**[A4]{}**]{} hold on a polygonal or polyhedral mesh and each $V_K$ is the space of polynomials with degree less than or equal to $n$. Then [**[I2]{}**]{} is true with $C_I$ depending on $n$, but not on $h_K$ or $|K|$.
\[lem:L2proj\] Assume [**[A1]{}**]{}-[**[A4]{}**]{} hold on a polygonal or polyhedral mesh and each $V_K$ is the space of polynomials with degree less than or equal to $n$. Let $Q_h$ be the $L^2$ projection onto $V_h$. Then for all $0\le s\le n$ and $v\in H^{s+1}(\Omega)$, $$\begin{aligned}
\sum_{K\in\mathcal{T}_h}\|v-Q_hv\|_K^2 &\le C_{Q0} h^{2(s+1)} \|v\|_{s+1}^2. \\
\sum_{K\in\mathcal{T}_h}\|\nabla (v-Q_hv)\|_K^2 &\le C_{Q1} h^{2s} \|v\|_{s+1}^2.
\end{aligned}$$
It is not hard to see that [**[I3]{}**]{} follows immediately from Lemma \[lem:L2proj\]. Indeed, notice that as long as [**[I1]{}**]{} and [**[I2]{}**]{} are true and $v\in H^r(\Omega)$ with $r>3/2$, we have $$\3bar v-Q_h v\3bar \le (1+C_1) \sum_{K\in\mathcal{T}_h}\|\nabla (v-Q_hv)\|_K^2 + \alpha\sum_{e\in\mathcal{E}_h}\frac{1}{h_e}\|[v-Q_h v]\|_e^2,$$ where $C_1=C_T(1+C_I)^2$. Next, notice that by [**[A2]{}**]{}, [**[I1]{}**]{} and Lemma \[lem:L2proj\], $$\begin{aligned}
\sum_{e\in\mathcal{E}_h}\frac{1}{h_e}\|[v-Q_h v]\|_e^2 &\le \sum_{K\in\mathcal{T}_h} \left(\sum_{e\in \partial K\cap \mathcal{E}_h} \frac{1}{h_e} \|(v-Q_h v)|_K\|_e^2\right) \\
& \le \sum_{K\in\mathcal{T}_h} \left(\sum_{e\in \partial K\cap \mathcal{E}_h} \frac{1}{\kappa h_K} \|(v-Q_h v)|_K\|_e^2\right) \\
& = \sum_{K\in\mathcal{T}_h} \frac{1}{\kappa h_K} \|(v-Q_h v)|_K\|_{\partial K}^2 \\
& \le \sum_{K\in\mathcal{T}_h} \frac{C_T}{\kappa h_K} \left(h_K^{-1} \|v-Q_h v\|_K^2 + h_K\|\nabla(v-Q_h v)\|_K^2\right) \\
&\le \frac{C_T}{\kappa} (C_{Q0} + C_{Q1}) h^{2s} \|v\|_{s+1}^2.
\end{aligned}$$ Combine the above, we have
(The aproximability) Assume [**[A1]{}**]{}-[**[A4]{}**]{} hold on a polygonal or polyhedral mesh and each $V_K$ is the space of polynomials with degree less than or equal to $n$. Then for all $\frac{1}{2}< s\le n$ and $v\in H^{s+1}(\Omega)$, there exists a constant $C_A$ independent of $h$ such that $$\inf_{\chi_h\in V_h} \3bar v-\chi_h \3bar \le C_A h^{s} \|v\|_{s+1}.$$ Here $s>\frac{1}{2}$ is added so that $\3bar v - \chi_h\3bar$ is well-defined.
Numerical Examples
==================
Finally, we present numerical results that support the theoretical analysis of this paper. We fix the coefficients $\delta = 1$ and $\alpha = 10$, since the purpose of the numerical experiments is to examine the accuracy of the interior penalty discontinuous Galerkin method on arbitrary polygonal meshes, not for different coefficients. Consider the Poisson’s equation on $\Omega = (0,1)\times(0,1)$ with the exact solution $u = sin(2\pi x)\cos(2\pi y)$. Clearly $u=0$ on $\partial\Omega$. For simplicity of the notation, we denote $$|u-u_h|_{1,h} = \left(\sum_{K\in\mathcal{T}_h} |\nabla(u-u_h)|_K^2\right)^{1/2}.$$
The first test is performed on a non-conformal triangular-quadrilateral hybrid mesh. The initial mesh and the mesh after one uniform refinement are given in Figure \[fig:mesh1\]. A sequence of uniform refinements are then applied to generate a set of nested meshes. Notice that the meshes are non-conformal and there are hanging nodes. However, the interior penalty discontinuous Galerkin method can deal with such meshes without special treatments. We solve the Poisson equation using the interior penalty discontinuous Galerkin formulation (\[eq:dg\]) on these meshes, where the local discrete spaces $V_K$ are taken to be $P_1$ polynomials on each $K\in \mathcal{T}_h$, no matter whether $K$ is a triangle or quadrilateral. The $H^1$ semi-norm and the $L^2$ norm of the errors are reported in Table \[tab:test1\] and Figure \[fig:test1\]. These errors are computed using a 5th order Gaussian quadrature on triangles. For quadrilateral elements, the errors can be conveniently computed by dividing the quadrilateral into two triangles and then applying the Gaussian quadrature. Our results show that the $H^1$ semi-norm has an approximate order of $O(h)$, while the $L^2$ norm has an approximate order of $O(h^2)$, as predicted by the theoretical analysis.
![Initial and refined mesh for test 1.[]{data-label="fig:mesh1"}](mesh1-1 "fig:"){width="6cm"}![Initial and refined mesh for test 1.[]{data-label="fig:mesh1"}](mesh1-2 "fig:"){width="6cm"}
------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ----------------
$h$ $\frac{1}{16}$ $\frac{1}{32}$ $\frac{1}{64}$ $\frac{1}{128}$ $\frac{1}{256}$ $O(h^r)$, $r=$
\[1mm\] $|u-u_h|_{1,h}$ 1.2006 0.5904 0.2917 0.1452 0.0725 1.0124
$\|u-u_h\|$ 0.0551 0.0159 0.0042 0.0011 0.0003 1.9270
------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ----------------
: Convergence rates for test 1.[]{data-label="tab:test1"}
![Convergence rates for test 1.[]{data-label="fig:test1"}](error1){width="8cm"}
In the second test, we consider a hybrid mesh containing mainly hexagons, but with a few quadrilaterals and pentagons. Indeed, it is derived by taking the dual mesh of a simple triangular mesh. In Figure \[fig:mesh2\], the initial triangular mesh and its dual mesh are shown. By refining the triangular mesh and computing its dual mesh, we get a sequence of hexagon hybrid meshes. Again, we solve the interior penalty discontinuous Galerkin formulation (\[eq:dg\]) on these hexagon hybrid meshes, with the local discrete spaces $V_K$ of $P_1$ polynomials. The $H^1$ semi-norm and the $L^2$ norm of the errors are reported in Table \[tab:test2\] and Figure \[fig:test2\]. Optimal convergence rates are achieved.
![The original triangular mesh and its dual mesh used in test 2.[]{data-label="fig:mesh2"}](mesh2-1 "fig:"){width="6cm"}![The original triangular mesh and its dual mesh used in test 2.[]{data-label="fig:mesh2"}](mesh2-2 "fig:"){width="6cm"}
------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ----------------
$h$ $\frac{1}{16}$ $\frac{1}{32}$ $\frac{1}{64}$ $\frac{1}{128}$ $\frac{1}{256}$ $O(h^r)$, $r=$
\[1mm\] $|u-u_h|_{1,h}$ 0.8139 0.3868 0.1894 0.0941 0.0470 1.0270
$\|u-u_h\|$ 0.0461 0.0129 0.0034 0.0009 0.0002 1.9393
------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ----------------
: Convergence rates for test 2.[]{data-label="tab:test2"}
![Convergence rates for test 2.[]{data-label="fig:test2"}](error2){width="8cm"}
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, [*Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite-element method*]{}, Comput. Methods Appl. Mech. Engrg., 121 (1995), pp. 373–409.
, [*Three dimensional Voronoi cell finite element model for microstructures with ellipsoidal heterogeneties*]{}, Computational Mechanics, 34 (2004), pp. 510–531.
, [*A new error analysis for discontinuous finite element methods for linear elliptic problems*]{}, Math. Comp., 79 (2010), pp. 2169–2189.
, [*Stable spectral methods on tetrahedral elements*]{}, SIAM J. Numer. Anal., 21 (2000), pp. 2352–2380.
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, [*A weak Galerkin mixed finite element method for second-order elliptic problems*]{}, [arXiv:1202.3655v1 \[math.NA\] 16 Feb 2012.]{}
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[^1]: Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204 (lxmu@ualr.edu).
[^2]: Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230 (jwang@nsf.gov). The research of Wang was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
[^3]: Department of Mathematics, Oklahoma State University, Stillwater, OK 74075 (yqwang@math.okstate.edu).
[^4]: Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204 (xxye@ualr.edu). This research was supported in part by National Science Foundation Grant DMS-1115097.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In the simplified setting of the Schwinger model we present a systematic study on the simulation of dynamical fermions by global accept/reject steps that take into account the fermion determinant. A family of exact algorithms is developed, which combine stochastic estimates of the determinant ratio with the exploitation of some exact extremal eigenvalues of the generalized problem defined by the ‘old’ and the ‘new’ Dirac operator. In this way an acceptable acceptance rate is achieved with large proposed steps and over a wide range of couplings and masses.'
author:
- |
Francesco Knechtli and Ulli Wolff[^1]\
Institut für Physik, Humboldt Universität\
Newtonstr. 15\
12489 Berlin, Germany
title: Dynamical fermions as a global correction
---
0.5 cm
HU-EP-03/12\
SFB/CCP-03-07
[^1]: e-mail: knechtli@physik.hu-berlin.de, uwolff@physik.hu-berlin.de
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Interface states in a 1-D photonic crystal heterostructure with multiple interfaces are examined. The heterostructure is a periodic network consisting of two different photonic crystals. In addition, the two crystals themselves are periodic, with one being made of alternating binary layers and the other being a quaternary crystal with a tunable layer. The second crystal can thus be smoothly transformed from one binary crystal to another. All individual photonic crystals in the superstructure have symmetric unit cells, as well as identical periods and optical path lengths. Therefore, as the tunable layer in the quaternary crystal expands, other layers will shrink. It is found that the behavior of the localized modes in the band gaps is dependent on whether there is an even or odd number of interfaces in the heterostructure. With certain sequences of all dielectric photonic crystals, topological states are shown to split in two, whereas for other heterostructures they are shown to vanish. Additional resonant modes appear depending on how many crystals are in the heterostructure. If the tunable layer is frequency dependent, the band gap can still support topological/resonant modes with some band gaps even supporting two separate groups.'
author:
- 'Nicholas J. Bianchi'
- 'Leonard M. Kahn'
title: 'Optical States in a 1-D Superlattice with Multiple Photonic Crystal Interfaces'
---
Introduction
============
A photonic crystal (PC) is a periodic array of dielectrics and/or conductors used to scatter light [@Yablonovitch; @John]. In a similar manner to how semiconductors control the passage of electrons, PCs possess passbands which allow photons in certain frequency ranges to propogate through the crystal and photonic band gaps (PBGs), which inhibit photon flow, producing regions of suppressed transmission. The existence of these pass and stop bands are governed by Bloch’s Thereom. Photonic heterostructure devices are comprised of multiple periodic components that can produce transmission properties and field localization not seen in isolated crystals [@Istrate1; @Istrate2]. Heterostructures with a single PC interface have been extensively studied. Examples of localized behavior are the surface or interface modes, also known as optical Tamm [@Tamm] states (OTSs). These modes can exist at a boundary only if their field amplitudes decay away as the distance from the boundary increases in either direction. This means the wavevectors must be imaginary. In the case of a PC, this occurs if the mode is trying to travel through a PBG. These modes have been found in a variety of photonic structures including 1-D [@Kavokin; @Vinogradov1; @Vinogradov2; @Gao] and 2-D [@Lin] PC interfaces, air-PC surfaces at oblique angles [@Feng], and PCs bordering media with a graded refractive index [@Zheng]. Tamm states have also been investigated in various systems containing a PC with a tunable cap layer adjacent to a uniform medium. Examples include PCs containing superconducting layers [@Abouti], systems containing metamaterials, both the PC layers [@Wang; @Barvestani] and the uniform medium [@Namdar], and systems with liquid crystal [@Hajian] and chiral [@Bashiri] cap layers. Note that in Ref. [@Feng], despite the PC being adjacent to a uniform medium with positive dielectric constant, surface modes can still form due to total internal reflection. The component of the wavevector parallel to the boundary, $k_\parallel$, is large enough to cause the normal component, $k_\bot$ to become imaginary. $$k_\bot = \sqrt{k^2 - k_\parallel^2} \label{wavevector}$$ A varient of OTSs are the Tamm plasmon-polaritons (TPP) formed at a boundary between a metal and a PC [@Kaliteevski; @Brand; @Zhou]. In order for a TPP to form, the condition, $$r_{\text{metal}}r_{\text{PC}}=1 \label{reflection}$$ must be satisfied. The reflection coefficent $r_{\text{metal}}$ describes the amplitude of the electric field, incident from the PC side of the interface, reflecting off the metallic surface. In the same manner, $r_{\text{PC}}$ describes the electric field amplitude from a wave incident from the metallic side reflecting off the PC surface of the interface. In the case described in Ref. [@Kaliteevski], the TPP is excited at a frequency below the plasma frequency of the metal, implying that $r_{\text{metal}}=-1$. Therefore, to ensure that Eq. \[reflection\] remains satisifed, $r_{\text{PC}}=-1$, implying that the higher index material in the PC should be adjacent to the metal. In Ref. [@Brand], the plasmon is produced above the plasma frequency. Since the permittivity of the metal is now positive, $r_{\text{metal}}$ flips sign. For the state to exist now, the sign of $r_{\text{PC}}$ must also flip, meaning that, in the PC, the low index material is adjacent to the material. Similar to Ref. [@Feng], the state is supported on the metallic side by total internal reflection.
If an interface is generated between two PCs with symmetric unit cells, localized states at the boundary can form that are governed by the bulk band structure of the two crystals. These states are referred to as topological interface states. Xiao *et.al.* [@Xiao] showed that their existence in a PBG can be predicted by ensuring that the imaginary parts of the surface impedances for the two crystals sum to zero in the selected gap. Their work established a relation between the sign of the impedance $Z$ for a PBG and sum of all Zak [@Zak] phases, $\theta^{\text{zak}}_m$, below the gap, where $m$ denote the (isolated) bands, $$\text{sign}(\text{Im}(Z^{(n)})) = (-1)^{n+l}\exp \left(i \sum_{m=1}^{n} \theta^{\text{zak}}_m \right) \label{Zak}$$ In Eq. , $n$ is the PBG where the impedance is calculated and $l$ denotes the number of points where two bands cross below band gap $n$. Due to the PC unit cells possessing inversion symmetry, all Zak phases can only take on the values of $\pi$ or $0$ [@Zak], and thus provide a useful measure for identifying topological states. Band gap $n$ contains a topological state if $Z_L+Z_R=0$, where the subscripts indicate the PCs to the left/right of the interface. Through control of $\theta^{\text{zak}}_m$, topological states have been demonstrated in both 1-D [@Choi; @Cai] and 2-D [@Yang] systems.
Heterostructures with multiple PC/PC or PC/metallic interfaces have more degrees of freedom due to the increased number of tunable parameters, as compared to a single interface system, leading to a much richer display of resonant states. Through the control of parameters within the heterostructure, several examples of coupling between resonant states have been demenstrated [@Zhou; @Fei; @Iorsh; @Durach; @Cox; @Hu]. As an extension to the work in Ref. [@Bianchi], the behavior of interface states is investigated in a heterostructure consisting of alternating binary and quaternary PCs. If the number of binary and quaternary crystals in the stucture are the same, then there is an odd number of interfaces. In this case if the first PC in the heterostructure is binary (quaternary), then, after the alternating pattern, the last will be quaternary (binary). The orginal topological state from the two crystal hetrostructure remains but is now accompanied by a sequence of resonant states on either side. The total number of states, including the original, is equal to the number of interfaces. For an even number of interfaces, there are two possible configurations. One possibility is to have the first and last PC of the structure be binary. In this case, it is found that the orginal topological state vanishes while the resonant states remain. The other case is to have the first and last crystals be quaternary. With only a single binary PC sandwiched between two quaternary PCs, the topological state splits. If more layers are added in this scenario, keeping the two ends quaternary, the split state is joined by resonant states.
Methods
=======
Our work was conducted using transfer matrix method (TMM) [@Yariv]. Keeping with Ref. [@Bianchi], all variables are made dimensionless for convenience. The lengths of the individual PC layers, $l_i$, are scaled to the unit cell period, $\Lambda$: $d_i=l_i/\Lambda$ and are such that $\Lambda$ and the optical path, $\Gamma$, for a unit cell are constant. In the heterostructure, shown as the middle image in Fig. \[PCM\], the periods for all the individual PCs are equal, as are the optical paths. The binary PCs are the gray regions and the quaternary PCs are the light blue regions. Since there is no fixed length scale, we set $\gamma=\Gamma/\Lambda$. For the quaternary PC, shown at the top of Fig. \[PCM\], the widths of layers $A$, in green, and $B$, in blue, can be expressed in terms of a free parameter, the width of the introduced layer, $d_C$, in orange, [@Bianchi],
$$d_A = \frac{\gamma - n_B - 2(n_C - n_B)d_C}{n_A - n_B} \label{eq:dA}$$
$$d_B = \frac{\gamma - n_A - 2(n_C - n_A)d_C}{n_B - n_A} \label{eq:dB}$$
Note that $d_C$ can only take on values in which both Eqs. and are non-negative. When $d_C$ reaches its maximum, the quaternary PC will become binary again, but with configuration, $...CBCBC...$, if $d_A$ tends to zero, or, $...ACACA...$, if $d_B$ tends to zero. For the special case, $\gamma=n_C$, both $d_A$ and $d_B$ will be zero when $d_C$ reaches its maximum; this will result in a uniform layer $C$. The lengths of the layers in the binary PC, displayed at the bottom of Fig. \[PCM\], are simply Eqs. and but with $d_C=0$ and thus do not change. The index of refraction of layer $j$ is $n^2_j = \epsilon_j \mu_j$, where $\epsilon_j$ and $\mu_j$ are the (relative) permitivites and permeabilites. In the binary and quaternary crystals, the $n_A$’s are the same and the $n_B$’s are the same, although $n_A \neq n_B$ [@Bianchi].
For the system described in Fig. \[PCM\], we only consider an electric field incident from the left, $E_{1+}$. The reflected field is $E_{1-}$ and the field that is transmitted through the entire structure is $E^{\prime}_{(N+1)+}$. To compute the transmission spectra for the system, first we must construct the transfer matrix, $M$, from the individual interface matrices, $I_j$, and propagation matrices, $P_j$, where the index, $j$, specifies the layer in question [@Orfanidis], $$I_j=
\frac{1}{\tau_j}
\begin{pmatrix}
1 & r_j \\[6pt]
r_j & 1 \\
\end{pmatrix}$$
$$P_j=
\begin{pmatrix}
e^{2 \pi i n_j d_j \xi} & 0 \\[6pt]
0 & e^{-2 \pi i n_j d_j \xi} \\
\end{pmatrix}$$
where $r_j$ and $\tau_j$ are the reflection and transmission coeffiecents, respectively, $$r_j = \frac{\mu_{j+1} n_{j} - \mu_{j} n_{j+1} }{\mu_{j+1} n_{j} + \mu_{j} n_{j+1} }$$ $$\tau_j = \frac{2 \mu_{j+1} n_{j} }{\mu_{j+1} n_{j} + \mu_{j} n_{j+1} }$$ In scaled variables, the phase argument, $i k_j l_j$ becomes $2 \pi i n_j d_j \xi$. The frequency, $f$, becomes $\xi=f\Lambda/c_0$, where $c_0$ is the speed of light in vacuum. The incident and scattered field are related by,
$$\begin{pmatrix}
E_{1+}\\[6pt]
E_{1-}\\
\end{pmatrix}
=
\begin{pmatrix}
M_{11} & M_{12} \\[6pt]
M_{21} & M_{22} \\
\end{pmatrix}
\begin{pmatrix}
E^{\prime}_{(N+1)+}\\[6pt]
0\\
\end{pmatrix}$$
where, $$M =
\begin{pmatrix}
M_{11} & M_{12} \\[6pt]
M_{21} & M_{22} \\
\end{pmatrix}
=
\prod_{j=1}^{N}I_j P_j I_{N+1}$$ The transmitted power is calculated via, $$T(\xi,d_C) = \abs{\frac{1}{M_{22}}}^2$$
Results
=======
In our first investigation, all layers of the heterostructures are assumed to be lossless dielectrics with no material dispersion. For both the binary and quaternary PCs, $\epsilon_A=6, \epsilon_B=\mu_A=\mu_B=1$. In the quaternary PC, $\epsilon_C=3$ and $\mu_C=1$. For simplicity, all PCs are given the same number of unit cells,$N_\Lambda$, periods and optical paths. In the following systems, $N_\Lambda=4$ and $\gamma=1.5$. With $d_C$ as a free parameter, $d_A$ and $d_B$ of the quaternary PC are described by Eqs. \[eq:dA\] & \[eq:dB\]. Since the physics of interface states is valid in any PBG, we will restrict ourselves to the $3^{\text{rd}}$ one since this is the lowest gap that produces such states with the described parameters. For convience, when describing individual PCs of the heterostructures, we will use $b$ for binary PC and $q$ for quaternary PC.
Transmission through the heterosturcture depends on the specific configuration of the binary and quaternary PCs. Fig. \[Trans\] displays nine different transmission examples. In Figs. -, the stucture is sandwiched between two binary PCs, while in Figs. -, the two endlayers are quaternary PCs. In both cases, the number of interfaces from left to right is 2, 4, and 6. Since there is an even number of interfaces in the heterostructure, a single topologial peak (Fig. ) is absent, even though Eq. \[Zak\] states that there is a change in the sign of surface impedance between the binary and quaternary components as $d_C$ increases from 0 to 0.341. For the transmission maps in the top row, the heterostructure has the form $bqb$, $bqbqb$ and $bqbqbqb$. It can be seen that the state in a single interface system splits into two sets of resonances, with one set below the original frequency and the other above. At $d_C=0$, all these states exist as pass band modes; however, as $d_C$ increases, they begin to wander into the band gap. As these resonant states appear at all values of $d_C$, they are not topological in nature, although after the impedance for the quaternary layers flips sign (see Eq. \[Zak\]), they appear to cluster together and move in a similar manner to the topological state in the single binary-quaternary interface heterostructure. The transmission for all these states remains at unity for all values of $d_C$.
When the heterostructure changes from $bqb$ to $bqbqb$, the two states themselves split into pairs such that these pairs (Fig. ) each have a higher and lower frequency state relative to their respective states in Fig. . This splitting is illustrated in Fig. \[T\_dC\]. A horizontal slice of Fig. at $d_C=0.25$ is considered, except now the states are plotted for varying thickness of the middle cystral. Each binary and quaternary represent 4 unit cells; $b/2$ represents 2 unit cells. In Fig. , we see two distant edge states (blue) in the absence of a middle $b$: $bqqb$. To see the two interface states, though, we must zoom into the cluttered middle region. These interface states are clearly seen in Fig. . When two binary unit cells are inserted in the center of the structure ($bq(b/2)qb$), there is now strong coupling between the two central states and the two edge states. The edge states rapidly move toward the central region. Inserting another two binary unit cells produces the familiar structure $bqbqb$ and the black transmission profile. Doubling the central region causes coupling of the states in each pair to weaken due to the increased distance bewteen the interface pairs $bqb$. This is seen in the magenta curve as the four peaks mostly merge into two, recovering Fig. at $d_C=0.25$. There is also a new pair of edge states in Fig. .
An important change occurs in the transmission behavior as $d_C$ increases if the sandwiching layers are quaternary. In Fig. , the heterostructure is $qbq$ and a topological state is observed in the upper half of the map; however, it splits into two seperate peaks since there are two interfaces. With additional layers, the structure becomes $qbqbq$ and $qbqbqbq$, shown in Fig. and Fig. respectively. Here the split topological state is strattled by resonant states that behave similarily to those described in Fig. \[T\_dC\]. To help understand why the topological state appears in the $qb...bq$ but not the $bq...qb$ configuration, it is helpful to examine the behavior of the band at small $d_C$. Recall that when Eq. \[Zak\] produces opposite signs for the isolated quaternary and binary PCs, the number of resonant states in the middle of the PBG must be equal to the number of interfaces in the composite heterostructure. For $bq...qb$, all the states remain separate (*i.e.* states do not merge). For example, let’s consider Fig. . Since there are 6 interfaces, the 6 states that enter the band gap are the 3 closest pass band states on either side of the gap. Compare this to Fig. , where only the two closest states on either side of the PBG wander into the gap when $d_C$ increases from 0. The third closest states to the PBG are seen merging and disappearing with other states in adjacent bands on the far left and far right of the map. As $d_C$ continues to increase, there are temporarily only 4 states. Therefore in order to have a total of 6 states, the split topological state must appear after the phase transition.
When the two endlayers are different, we get the sequence $bq...bq$. Figs. - are $bq$, $bqbq$, and $bqbqbq$ respectively. Since there are now an equal number of binary and quaternary crystals in the structure, there are an odd number of interfaces and reversing the order of the components ($bq\rightarrow qb$) will not change the transmission. Fig. is the familiar single topological state from heterostruture $bq$. For Fig. & , the addition of $bq$ layers produces resonant states that behave like those discussed previously.
\
\
In our second investigation, layer $C$ is given a permittivity with frequency dependence, in accordance with the Drude model of dispersion, $$\epsilon_C=1-\frac{\xi_p^2}{\xi^2+i g \xi} \label{Drude}$$ where $\xi_p$ and $g$ are the dimensionless plasma and collision frequencies. Eq. \[Drude\] is plotted in Fig. \[Epsilon\] with plasma frequency $\xi_p=2$ and negligible collision frequency $g=10^{-10}$. Therefore, layer $C$ acts as a metal. Layers $A$ and $B$ remain unchanged. Since the optical path in metal is not constant with frequency, the layer width defined in Eqs. \[eq:dA\] & \[eq:dB\] are given simplier forms, $$d_A = \frac{\gamma - n_B }{n_A - n_B}-d_C \label{eq:dA_2}$$
$$d_B = \frac{\gamma - n_A }{n_B - n_A}-d_C \label{eq:dB_2}$$
Now, $\gamma$ is only relevant when defining the layer widths before the metal is introduced. As $d_C$ increases, the band gap closing points are skewed towards higher frequencies due to the behavior of Eqs. \[eq:dA\_2\] & \[eq:dB\_2\]. As a concequence of this, toppological states in a single interface $bq$ system do not start and terminate at the closing points nor are they positioned near the center of the gap. An example of this behavior is shown in Fig. \[Trans\_Impedance\_bq\_M\]. In Fig. , the transmission map is plotted around $\xi=2$. The metallic layer, $d_C$ , follows the behavior in Fig. \[Epsilon\]. Note that we can have a case where one gap (top center) can support two states. The left state is much sharper than the right one. Also worth noting is that the two center states appear to cross the plasma frequency of the metallic layer without anything unusual happening. This is acceptable because the effective plasma frequency of the entire heterostructure is much lower than the plasma frequency of the metallic inclusion, so the effective permittivity of the heterosructure is positive in the region of these states [@Manzanares-Martinez]. This means that all visible gaps in Fig. are classifed as PBGs. There are also two distinct groups of Fabrey-Perot resonances. The brighter, more slanted triplets that largely encase the PBGs are caused by coupling among the 3 interfaces of the four unit cells in the quaternary PC. There is also a fainter vertical triplet of resonances between about $1.72<\xi<1.9$, that is caused by the three interfaces in the binary PC. As $d_C$ increases, the leftmost topological state eventually appears to turn into one of these resonances and the rightmost of these states breaks away to become the top-center topological state. The equation $\text{Im}(Z_b+Z_q)=0$ is plotted in Fig. , showing the exact location of those five topological states.
As in the all dielectric case, when there are multiple binary/quaternary interfaces, topological states can split; however, the split states are much closer together, meaning that they are more difficult to resolve. Transmission maps for the $qbq$ and $bqb$ configurations are displayed in Fig. \[Metal\_Split\_States\]. While they look very similar to each other and to the single interface system, some subtleties can be pointed out. Resonances in the $qbq$ system are much sharper compared to those in $bqb$. Also the splitting can be seen, although it is more pronounced in $qbq$. Cross sections of the lower center topological state for $d_C=0.1$ are shown in both structures in Fig. \[Cross\_Metal\] as the number of interface increases. In Fig. , the transmission is shown for a heterostructures sandwiched between two quaternary PCs. As the number of interfaces increases, each split state itself divides such that the total number equals the number of interfaces. Fig. zooms into the left cluster of states. If the heterostructure is bounded by binary PCs, shown in Fig. , the two central split states appear much closer together. As the number of interfaces increases these two eventually merge and the resultant peak decreases. In the plot, this occurs for six interfaces ($bqbqbqb$).This makes it appear that there is a missing state; however, similar to the all dielectric heterostructures $qb...bq$ (Figs. -), this merging occurs at lower values of $d_C$ as the number of interfaces increases. Therefore if the transmission cross section was taken for, say, $d_C=0.08$ rather than for $d_C=0.1$, then the central peak for structure $bqbqbqb$ would instead appear as a small doublet, bringing the total number of states to six.
To help understand what is happening within the heterostructure, it is benefical to compare the optical system to the more familar 1D coupled harmonic oscillator, shown in Fig. \[OM\]. The interfaces between the individual PCs act as identical masses and the PCs themselves can be thought of as the spring constants. Since there are two different PCs, two distinct spring constants are used. In this example, the constant $k$ corresponds to the binary PC while $\kappa$ corresponds to the quaternary PC or vice virsa. The topological state will split into a number of states corresponding to the number of interfaces. With an even number of interfaces, the central state vanishes and splits such that half are above the original frequency and half are below. Using this analogy with two interfaces, the lower of the two states is the symmetric state while the higher one is the antisymmetric state [@Thornton]. With an odd number of interfaces, the central state still splits as in the even case except now the original state remains.This splitting is shown in Fig. \[Split\]. Overall, the original and split frequencies can be related by an average,
$$\xi_0^{2}=\frac{1}{N}\sum_{i=1}^{N} \xi_i^{2} \label{xi_ave}$$
where $N$ is the number of PCs in the entire structure and the index, $i$, is summed through all frequencies after the splitting.
\
Conclusion
==========
We have described the evolution of resonant states in a photonic heterostructure composed of alternating binary and quaternary inversion symmetric photonic crystals as the quaternary crystal transforms from one binary to another. This was done by making the tunable layer in the quaternary crystal a free parameter. Two different heterostructures were given, one in which all components were dielectrics and one in which the free parameter was frequency dependent. For the all dielectric case, as shown in Fig. \[Trans\], the maximum number of resonant states in a PBG is equal to the number of PC interfaces for all configurations shown. All configurations (except Figs. and ) possess resonances that started as pass band states for $d_C=0$, but the central topological state can vanish, split in two, or remain intact, depending on if the heterostructure is $bq...qb$, $qb...bq$, or $bq...bq$ respectively. In Fig. \[Metal\_Split\_States\], it was shown that if the additional layer in the quaternary PC has frequency dependence, some of the split modes can become much more compressed and difficult to resolve, more so for $bqb$ than $qbq$. Without a constant optical path, the topological states in Figs. \[Trans\_Impedance\_bq\_M\] and \[Metal\_Split\_States\] do not start and end at PBG closing points, but rather on the edges. These results show that it is possible to generate sequences of resonant states in a binary-quaternary PC heterostructure solely through the manipulation of the geometry of a heterostructure. The fabrication of such a structure could be useful for filtering applications.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We demonstrate the absence of a DC Stark shift in an ytterbium optical lattice clock. Stray electric fields are suppressed through the introduction of an in-vacuum Faraday shield. Still, the effectiveness of the shielding must be experimentally assessed. Such diagnostics are accomplished by applying high voltage to six electrodes, which are grounded in normal operation to form part of the Faraday shield. Our measurements place a constraint on the DC Stark shift at the $10^{-20}$ level, in units of the clock frequency. Moreover, we discuss a potential source of error in strategies to precisely measure or cancel non-zero DC Stark shifts, attributed to field gradients coupled with the finite spatial extent of the lattice-trapped atoms. With this consideration, we find that Faraday shielding, complemented with experimental validation, provides both a practically appealing and effective solution to the problem of DC Stark shifts in optical lattice clocks.'
author:
- 'K. Beloy'
- 'X. Zhang'
- 'W. F. McGrew'
- 'N. Hinkley'
- 'T. H. Yoon'
- 'D. Nicolodi'
- 'R. J. Fasano'
- 'S. A. Schäffer'
- 'R. C. Brown'
- 'A. D. Ludlow'
title: 'Faraday-shielded, DC Stark-free optical lattice clock'
---
In the nearly seven decade-old quest to push the boundaries of atomic clock performance, and thus metrological capabilities in general, the elimination or precise evaluation of frequency shifts caused by external electromagnetic fields has been a persistent challenge [@Har52; @EssPar55]. In modern-day optical lattice clocks, AC Stark shifts due to lattice light and blackbody radiation are two prominent examples [@KatTakPal03; @LudBoyYe15]. DC Stark shifts, attributed to nearby electronics or patch charges on the clock apparatus, have been observed as large as $10^{-13}$ [@LodZawLor12] and pose a legitimate threat to state-of-the-art $10^{-18}$ clock performance (throughout, quoted shifts are understood to be in units of the clock frequency). Strategies to mitigate this threat include I) applying electric fields to measure and, if desired, cancel the stray-field shift [@LodZawLor12; @FalLemGre14; @BloNicWil14; @NicCamHut15; @NorCliMun17arXiv], or II) enclosing the atoms by equipotential conductive surfaces, furnishing them with a field-free environment [@BelHinPhi14; @UshTakDas14; @NemOhkTak16; @KolGroVog17]. DC Stark shifts have also been estimated from apparatus geometry and material properties [@PizThoRau17; @KimHeoLee17]. Recently, Rydberg atoms were demonstrated as an [*in situ*]{} probe of the stray field in an optical lattice clock [@BowHobHui17].
Here we identify a mechanism capable of compromising a Method I analysis, for which uncertainties at the $10^{-19}$ level have been reported. Using a simple model, we demonstrate how field gradients coupled with finite spatial extent of the lattice-trapped atoms can lead to appreciable clock error. Generally, the error scales with the measured stray-field shift. In principle, such error can be reduced by minimizing the stray field itself, which is precisely the objective of Method II. Unfortunately, practical constraints preclude surrounding the atoms with an ideal, continuous Faraday cage. Moreover, even conductive surfaces can acquire patch charges, a known concern for electrodes in ion clocks [@BerMilBer98]. Consequently, a residual shift may remain, and quantifying an upper bound may be challenging. Seemingly, an optimal solution combines the attributes of Methods I and II. We demonstrate this combined approach in an ytterbium optical lattice clock, with measurements confirming the absence of a stray-field shift at the $10^{-20}$ level.
Given a uniform static electric field $\mathbf{E}$, the clock acquires a frequency shift $\delta\nu=kE^2$, where $E=\left|\mathbf{E}\right|$ and $k$ is specific to the clock transition. Namely, $k\equiv-(\alpha_e-\alpha_g)/2h$, where $h$ is Planck’s constant and $\alpha_{g,e}$ are the static polarizabilities of the ground and excited clock states. To characterize blackbody radiation shifts, the coefficient $k$ has been accurately measured for both Yb and Sr clock transitions [@SheLemHin12; @MidFalLis12]. In practice, the lattice-trapped atoms have finite spatial extent, and the electric field may be nonuniform over this extent. Thus, a more complete representation of the clock shift is $\delta\nu=k\left\langle E^2\right\rangle$, where $\langle\cdots\rangle$ denotes an average over the atoms. Generally, $\mathbf{E}$ is composed of both stray and applied fields. Given some nonzero stray field, it is evident that a true null shift can only be achieved if the applied field identically cancels the stray field across the entire atomic extent.
To illustrate the role field gradients can play in Method I, we introduce a simple model that affords an analytical solution. The model is illustrated in Fig. \[Fig:modelquadcurve\](a) and amounts to a cylindrically symmetric boundary value problem for the fields. The vacuum apparatus is taken to be a hollow metallic cylinder sealed with glass windows. The cylinder is electrically grounded, while the windows carry uniformly distributed static charges $q_1$ and $q_2$ across their respective internal surfaces. The external surfaces are spanned by electrodes, to which opposite voltages $+V$ and $-V$ are applied. With the electrodes grounded ($V=0$), a stray field exists due to the charges. For $V\neq0$, the electrodes further introduce an applied field. A one-dimensional optical lattice aligned with the symmetry axis confines the atoms with negligible radial extent and Gaussian axial distribution $(2\pi s^2)^{-1/2}\exp\left(-z^2/2s^2\right)$, with $z$ being the distance from the center of the vacuum apparatus. The windows are separated by a distance $\ell$ and have diameter $d$, thickness $t$, and dielectric constant $\epsilon$. Expressions for the electric potential within the vacuum region can be found in the Supplemental Material (SM) [@SM].
![a) Section view of the clock model described in the text. b) Corresponding clock shift $\delta\nu(V)$, with the quantities $\delta\nu_0$, $\delta\nu^*$, and $\Delta\nu$ introduced in the text. For $k>0$ ($k<0$), the extremum is a minimum (maximum) and all three quantities are positive (negative). []{data-label="Fig:modelquadcurve"}](modelX.pdf){width="\linewidth"}
As demonstrated on a more general basis below, the clock shift has the functional form $$\delta\nu\left(V\right)=\delta\nu_0+aV+bV^2.
\label{Eq:dnuoneV}$$ The coefficients $a$ and $b$ are experimentally accessible parameters whose values may be determined by modulating $V$ and observing the clock response. Specifying the clock shift for any $V$ requires further knowledge of the stray-field shift $\delta\nu_0$. Towards this goal, we consider the extremum value of $\delta\nu\left(V\right)$, denoted $\delta\nu^*$. The stray-field shift $\delta\nu_0$, the extremum shift $\delta\nu^*$, and the difference between them $\Delta\nu\equiv \delta\nu_0-\delta\nu^*$ are depicted in Fig. \[Fig:modelquadcurve\](b). In contrast to $\delta\nu_0$ and $\delta\nu^*$, $\Delta\nu$ is accessible through modulation of $V$. Invoking elementary calculus with Eq. (\[Eq:dnuoneV\]), we find $\Delta\nu=a^2/4b$. Let us initially neglect the atomic extent, taking the limit $s\rightarrow0$. In this case, there exists a $V$ for which the applied field identically cancels the stray field at the atoms, resulting in a null clock shift. This necessarily coincides with the extremum of $\delta\nu\left(V\right)$, as any other $V$ yields a nonzero clock shift of definite sign (determined by $k$). This implies $\delta\nu^*=0$, and it follows that $\delta\nu_0$ may be inferred from $\Delta\nu$ according to $\delta\nu_0=\Delta\nu$.
The above reasoning breaks down for nonzero $s$, as we can no longer expect there to be a $V$ such that the applied field identically cancels the stray field over the entire atomic extent. Consequently, $\delta\nu^*$ plays the role of a frequency correction for the field gradients. We write $\delta\nu^*=\eta\Delta\nu$, motivated by the fact that $\delta\nu^*$ and $\Delta\nu$ scale similarly with the stray field. Namely, a uniform scaling of the stray charge leaves $\eta$ unchanged. The stray-field shift subsequently reads $\delta\nu_0=\left(1+\eta\right)\Delta\nu$. To leading order in $s$, we find $\eta=\zeta^2s^2/\mathcal{R}^2$, where $\mathcal{R}$ is an effective length whose expression is given in the SM [@SM] and $\zeta\equiv(q_1+q_2)/(q_1-q_2)$ quantifies the charge-symmetry between the windows. Choosing $d=150$ mm, $\ell=100$ mm, $t=10$ mm, and $\epsilon=3.8$, $\mathcal{R}$ evaluates to $\mathcal{R}=42$ mm. Further assuming $s=1$ mm and 25% more charge on one window than the other, we obtain $\eta\approx0.05$. The example above suggests that the frequency correction $\delta\nu^*$ may be non-negligible if $\delta\nu_0$ itself is appreciable. More specifically, for an optical lattice clock exhibiting a stray-field shift above $10^{-18}$, specification or cancellation of the shift at or below $10^{-18}$ may not be straightforward using Method I. There exists an additional burden in quantifying this correction or validating its neglect. Moreover, this effect could influence evaluations of other systematic effects. For example, lattice light shifts are typically characterized by varying lattice intensity, which may vary the atomic extent. Lastly, while our model suffices to demonstrate the potential importance of this effect, an actual system will inevitably be more complicated (lack symmetry in the apparatus, charge distribution, and atomic distribution; be an open-boundary system for the fields; contain dielectric surfaces in close proximity to the atoms; etc.). These complexities will presumably add to the difficulty of quantifying the correction due to field gradients.
From the preceding discussion, there is clear motivation for minimizing the stray-field shift. Recently, our group demonstrated an in-vacuum “shield” surrounding the lattice-trapped atoms in a Yb optical lattice clock [@BelHinPhi14]. An updated version is pictured in Fig. \[Fig:shield\]. The shield’s objective is twofold: provide a well-defined, near-ideal room-temperature blackbody radiation environment and suppress stray electric fields. The shield body is a single copper structure, internally coated with electrically-conductive carbon nanotubes. BK7 windows for optical access have an electrically-conductive, $\sim\!5$ nm-thick indium tin oxide (ITO) based coating. With the exception of two small apertures for atomic access, the copper body and ITO-coated windows collectively enclose the atoms. The design facilitates a Method I analysis. The windows are electrically isolated from the copper body using thin silicone spacers while being electrically connected to an external high-voltage source. We have independent control of the voltage on six windows, constituting three opposing pairs nominally aligned along mutually orthogonal axes. A seventh window is electrically connected to the copper body, which is permanently grounded.
![Faraday shield described in the text, with ITO coated windows. Voltages $-V_1$, $-V_2$, and $-V_3$ are assigned to the indicated windows, with voltages $+V_1$, $+V_2$, and $+V_3$ assigned to the respective opposing windows (unlabeled). The shield body is a single copper structure, internally coated with carbon nanotubes. PEEK plastic secures the windows to the body, suppresses radiative heat exchange with the environment, and hides functional components including electrical wires, resistance temperature detectors, and film heaters. For loading the optical lattice, a thermal beam of atoms enters the shield through an aperture (pictured bottom left), while a counterpropagating beam of slowing light enters through an opposing aperture. Bundled electrical wires (pictured bottom right) proceed to vacuum feedthroughs. For scale, the windows are 1 inch in diameter.[]{data-label="Fig:shield"}](shieldX.pdf){width="\linewidth"}
The model introduced above, Fig. \[Fig:modelquadcurve\](a), is quasi-one-dimensional (we refer to it as the 1D model below). That is, although the stray and applied fields are derived from a three-dimensional boundary value problem, only the symmetry axis is sampled by the atoms, with the symmetry ensuring that the fields align (or anti-align) along this axis. Practical clocks, such as ours, demand a more general theory. To this end, we allow multiple voltage variables $V_i$. We assign to each electrode in the system a linear combination of the $V_i$, defining the voltage applied to that electrode. The total electric field is $\mathbf{E}_0+\sum_i\mathbf{E}_i$, where $\mathbf{E}_0$ is the stray field (assumed independent of the $V_i$) and $\sum_i\mathbf{E}_i$ is the applied field with $\mathbf{E}_i\propto V_i$. The clock shift subsequently reads $$\delta\nu\left(V_1,V_2,\cdots\right)=\delta\nu_0+\sum_ia_iV_i+\sum_{ij}b_{ij}V_iV_j,
\label{Eq:dnumultiV}$$ where $\delta\nu_0=k\left\langle E_0^2\right\rangle$, $a_iV_i=2k\left\langle \mathbf{E}_0\cdot\mathbf{E}_i\right\rangle$, and $b_{ij}V_iV_j=k\left\langle \mathbf{E}_i\cdot\mathbf{E}_j\right\rangle$. Equation (\[Eq:dnumultiV\]) is a generalization of equation (\[Eq:dnuoneV\]) above. As before, we introduce the difference $\Delta\nu\equiv\delta\nu_0-\delta\nu^*$, where $\delta\nu^*$ is the extremum with respect to all $V_i$. In terms of the coefficients in Eq. (\[Eq:dnumultiV\]), $\Delta\nu$ reads [@myfootnote] $$\Delta\nu=\frac{1}{4}\sum_{ij}a_ia_j\left(b^{-1}\right)_{ij},
\label{Eq:DeltanumultiV}$$ where the $\left(b^{-1}\right)_{ij}$ are related to the $b_{ij}$ through matrix inversion (regarding the latter as elements of a matrix $b$ and the former as elements of its inverse $b^{-1}$). We partition the problem of specifying $\delta\nu_0$ into two parts according to the sum $\delta\nu_0=\Delta\nu+\delta\nu^*$. We initially focus on $\Delta\nu$, which represents the portion accessible through modulation of the $V_i$.
For our shield, we introduce a voltage variable for each opposing window pair, with voltages $-V_i$ and $+V_i$ assigned to the windows of each pair $i=1,2,3$ (refer to Fig. \[Fig:shield\]). Given these assignments, the corresponding $\mathbf{E}_i$ are nominally uniform and mutually orthogonal at the atoms. In the limit this is strictly true, the $b_{ij}$ with $i\neq j$ vanish and Eq. (\[Eq:DeltanumultiV\]) reduces to $$\Delta\nu=\sum_i\frac{a_i^2}{4b_{ii}},
\label{Eq:Deltanusimple}$$ indicating that contributions to $\Delta\nu$ can be evaluated independently for each “direction” and summed up. However, nonorthogonality or nonuniformity of the $\mathbf{E}_i$ must be considered. This could be assessed through independent means, such as geometrical considerations and field modeling. A more reliable estimate exploits the atoms themselves. Either way, once assessed, these effects can be treated perturbatively, as highlighted in the SM [@SM].
Alternatively, without claiming [*a priori*]{} knowledge about the $\mathbf{E}_i$, here we use the general Eq. (\[Eq:DeltanumultiV\]). We reserve Eq. (\[Eq:Deltanusimple\]) for future DC Stark shift assessments, where data from the present work can be leveraged to improve measurement efficiency and constrain deviations to this simple expression. We measure the induced shift $\delta\nu\left(V_1,V_2,V_3\right)-\delta\nu_0$ for various combinations of the arguments $V_i$. The $V_i$ define a “test” configuration, with the fully-grounded configuration serving as a common reference. Each measurement run involves interleaving interrogations for the two configurations (test and reference) and recording the frequency difference. Table \[Tab:results\] presents our data. We ascribe to each measurement a statistical uncertainty commensurate with the Allan deviation at the end of the run, $\sim\!1\times10^{-17}$. Voltage switching is enacted on the millisecond timescale, with spectroscopy initiated a few hundred milliseconds afterwards. Applied voltages are assessed with a voltage divider and found to be well-defined at the $2\times10^{-4}$ fractional level. This introduces negligible uncertainty, with the exception of one line in Table \[Tab:results\]. For this data, large shifts were induced, $-9\times10^{-14}$, with an uncertainty principally due to the applied voltage. All other induced shifts are at the $10^{-16}$ level. For each combination of $V_i$, measurements were performed under opposite polarity conditions.
[lcc]{} [(,,)]{} &\
\
[(,,)]{} & $-2.81(10)$ & $-2.78(10)$\
[(,,)]{} & $-2.76(10)$ & $-2.76(9)$\
[(,,)]{} & $-2.77(7)$ & $-2.67(7)$\
[(,,)]{} & $-915.2(4)$ & $-915.6(4)$\
[(,,)]{} & $-6.21(9)$ & $-6.09(9)$\
[(,,)]{} & $-5.15(9)$ & $-5.09(10)$\
[(,,)]{} & $-5.82(10)$ & $-5.92(11)$\
[(,,)]{} & $-5.60(10)$ & $-5.55(11)$\
[(,,)]{} & $-5.83(10)$ & $-5.94(10)$\
[(,,)]{} & $-5.49(10)$ & $-5.41(10)$
A cursory examination of Table \[Tab:results\] reveals no statistically-significant difference under any polarity reversal. This invariance immediately suggests $\Delta\nu\approx0$, though a more definitive analysis is clearly desired. Fitting Eq. (\[Eq:dnumultiV\]) to the data in Table \[Tab:results\] allows determination of the coefficients $a_i$ and $b_{ij}$, which can then be used to find $\Delta\nu$ via Eq. (\[Eq:DeltanumultiV\]). We implement a Monte Carlo protocol [@SM] to map probability distributions for the data (interpreted as uncorrelated Gaussian distributions) into a probability distribution for $\Delta\nu$, the result of which is presented in Fig. \[Fig:distrospectra\]. The distribution is clearly non-Gaussian and effectively constrains $\Delta\nu$ to negative values. The sign constraint is not surprising, considering all induced shifts are well-resolved negative (indicating $k<0$ with near certainty, in agreement with the known value [@SheLemHin12]). Based on this distribution, we assert a 68.3% confidence interval $-2.8\times10^{-20}<\Delta\nu<0$ and a 95.5% confidence interval $-6.7\times10^{-20}<\Delta\nu<0$.
![Main: Probability distributions for $\Delta\nu$ (yellow curve) and $\delta\nu_0$ (blue curve). The quantities are related by $\delta\nu_0=\left(1+\eta\right)\Delta\nu$; the $\delta\nu_0$ distribution is broadened relative to the $\Delta\nu$ distribution due to uncertainty from $\eta$ (see text). Visible “noise” stems from the Monte Carlo evaluation. Inset: Excitation fraction versus laser detuning for the clock transition, without ($V_3=0$) and with ($V_3=\pm2$ kV) an applied field. $V_1=V_2=0$ in each case. Detuning is referenced from the respective linecenter. A polarity-independent broadening accompanies the applied field.[]{data-label="Fig:distrospectra"}](distroX.pdf){width="\linewidth"}
In order to specify $\delta\nu_0$, we must further address $\delta\nu^*$. For any point near the shield’s center, an arbitrary applied field can, in principle, be constructed with an appropriate choice of the $V_i$. It is therefore possible to cancel an arbitrary stray field at that point. However, it is generally not possible to cancel the stray field over some extended volume. In complete analogy to the 1D model, $\delta\nu^*$ plays the role of a correction for field gradients.
To explore the role of gradients in our clock, we consider the atomic spectra in the Fig. \[Fig:distrospectra\] inset, obtained with one-second-long Rabi excitation. The blue trace shows the spectrum when all electrodes are grounded (i.e., normal operation), yielding a $\sim\!1$ Hz Fourier-limited linewidth. The red trace shows the spectrum with a large applied field ($V_1=V_2=0$, $V_3=+2$ kV). Because the lattice-trapped atoms are 1–2 mm from the symmetry plane between the charged windows, they experience a linear gradient from the applied field, resulting in the observed inhomogeneous broadening (with broadening attributed to noise in the applied voltage smaller by an order of magnitude). The magnitude of this gradient is corroborated by finite element analysis [@comsol]. Were a stray-field gradient also present, it could add to the applied-field gradient and the observed line broadening. Under opposite polarity conditions ($V_3=-2$ kV), the applied field and its gradient reverse. In this case, the stray-field gradient would subtract from the applied-field gradient, reducing the observed line broadening. The green trace shows the observed spectrum upon polarity reversal. Since this reversal does not change the observed spectral linewidth and amplitude, a bound can be placed on the stray-field gradient. This technique benefits from the large applied field at the atoms, which amplifies broadening from a stray-field gradient. For this axis of measurement, the constraint is $\left|\delta\nu^*\right|<2\times10^{-20}$.
While this technique could be repeated along the transverse lattice axes (where atomic extent is smaller and thus less sensitive to gradients) to constrain $\delta\nu^*$, here we exploit the fact that $\Delta\nu$ is essentially zero. As done previously for the 1D model, we write $\delta\nu^*=\eta\Delta\nu$, motivated by the fact that $\delta\nu^*$ and $\Delta\nu$ scale similarly with the stray field. While we lack a means to precisely evaluate $\eta$, the need is alleviated by our tight constraint on $\Delta\nu$. To investigate plausible values of $\eta$, we perform a finite element analysis of our shield plus atoms. Stray fields are introduced by applying patch voltages on internal shield surfaces (here $\eta$ is unaffected by a uniform scaling of the patch voltages). Within the physical constraint $\eta\geq0$, arbitrary values of $\eta$ can be manufactured. Larger values require increasingly fine-tuned conditions. To realize $\eta>1$, for instance, a high degree of symmetry is required between patch voltages on opposing sides of the shield; given a sufficiently symmetric arrangement, the atoms must then reside at a precise location. By examining various conditions, we take $\eta=1$ as a conservative upper limit for our clock. A smaller value could be argued, but there is little incentive to be more aggressive or meticulous in light of our tight constraint on $\Delta\nu$.
Finally, we assume complete ignorance of $\eta$ between zero and unity, assigning it a uniform probability distribution over this range. Combined with our results for $\Delta\nu$, we derive a probability distribution for $\delta\nu_0=\left(1+\eta\right)\Delta\nu$. Figure \[Fig:distrospectra\] presents the distribution, from which we assert a 68.3% confidence interval $-4.1\times10^{-20}<\delta\nu_0<0$ and a 95.5% confidence interval $-1.0\times10^{-19}<\delta\nu_0<0$. For comparison, the largest systematic uncertainties in our clock are presently at the $1\times10^{-18}$ level [@BelHinPhi14; @BroPhiBel17].
In conclusion, we have implemented Faraday shielding in an optical lattice clock and have constrained the stray-field DC Stark shift to below $10^{-19}$. In contrast to optical lattice clocks that lack Faraday shielding and exhibit nonzero stray-field shifts, our normal operation does not require regular spectroscopic monitoring of the shift, and there is no compromise to clock stability. Further measurements not part of this analysis, dispersed over multiple months and performed on independent Faraday-shielded clocks, have always yielded results consistent with zero stray-field shift. Here we have also identified a potential source of error in the measurement or cancellation of nonzero stray-field shifts attributed to field gradients. While exemplified for a Method I analysis, caution should generally be exercised. For example, the general approach put forth in Ref. [@BowHobHui17], employing Rydberg atoms, could also be susceptible to error from field gradients; this may especially be the case if the spatial sampling provided by the ballistic Rydberg atoms differs from that of the lattice-trapped clock atoms. By combining the distinct attributes of Method I (applied fields) and Method II (Faraday shielding), we have demonstrated an effective means for tackling the problem of DC Stark shifts in optical lattice clocks.
This work was supported by NIST, NASA Fundamental Physics, and DARPA QuASAR. The authors thank W. Zhang and D. R. Leibrandt for their careful reading of the manuscript. This work is a contribution of the National Institute of Standards and Technology/Physical Measurement Laboratory, an agency of the U.S. government, and is not subject to U.S. copyright.
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**Supplemental Material: Faraday-shielded, DC Stark-free optical lattice clock**
Analytical expressions for the model
====================================
Here we provide analytical expressions for the model described in the main text. For expressions here, we use $R=d/2$ and $L=\ell/2$ in favor of model parameters $d$ and $\ell$ of the main text. We also generalize the model somewhat by taking the charges $q_1$ and $q_2$ on the respective windows to be uniformly distributed over an area spanning a radial distance $p$ from the axis. Setting $p=R$ corresponds to charge spread out over the entire window surface, whereas $p\rightarrow0$ corresponds to a point charge on the axis. The former case is assumed in the main text. We also generalize the atomic distribution to $(2\pi)^{-3/2}s_\rho^{-2}s_z^{-1}\exp\left(-\rho^2/2s_\rho^2-z^2/2s_z^2\right)$, where $\rho$ and $z$ are the radial and axial coordinates from the center of the vacuum apparatus. The limit $s_\rho\rightarrow0$ is assumed in the main text, with the subscript on $s_z$ being omitted. The case $s_\rho=s_z$ corresponds to a spherically symmetric 3D Gaussian distribution. From the symmetry of the problem, we may write the total electric potential in the vacuum region as (Gaussian electromagnetic expressions) $$\Phi=\sum_nJ_0\left(x_{0n}\frac{\rho}{R}\right)
\left[
\alpha_n V\sinh\left(x_{0n}\frac{z}{R}\right)
+\beta_n \frac{\left(q_1-q_2\right)}{R}\sinh\left(x_{0n}\frac{z}{R}\right)
+\gamma_n\frac{\left(q_1+q_2\right)}{R}\cosh\left(x_{0n}\frac{z}{R}\right)
\right],$$ where $J_\nu(x)$ here and below are Bessel functions and $x_{\nu n}$ are the Bessel function zeros, $J_\nu\left(x_{\nu n}\right)=0$. The dimensionless coefficients $\alpha_n$, $\beta_n$, and $\gamma_n$ are given below in terms of the model parameters.
The clock shift is given in terms of the potential by $\delta\nu=(k/2)\left\langle\nabla^2\Phi^2\right\rangle$. Contributions to $\delta\nu$ are identified with $\delta\nu_0$, $aV$, or $bV^2$ according to their $V$-dependence. We expand $\nabla^2\Phi^2$ about the origin and use the fact that $\left\langle \rho^mz^n\right\rangle\propto s_\rho^ms_z^n$ if $m$ and $n$ are both even and $\left\langle \rho^mz^n\right\rangle=0$ otherwise. We subsequently form the combination $4(\delta\nu_0)(aV)^{-2}(bV^2)-1$, which is equivalent to $\eta$ from the main text. To leading order in $s_\rho$ and $s_z$, we find $$\eta=\frac{\zeta^2}{\mathcal{R}^2}\left(\frac{1}{4}s_\rho^2+s_z^2\right),$$ with $\zeta\equiv\left(q_1+q_2\right)/\left(q_1-q_2\right)$ and $$\mathcal{R}\equiv
R\frac{\sum_n\beta_nx_{0n}}{\sum_n\gamma_nx_{0n}^2}.$$ Omitted terms in the expression for $\eta$ are of order $s_\rho^ms_z^n$ with $m+n\geq4$. Whereas the coefficients $\beta_n$ and $\gamma_n$ appear at lowest order, the coefficient $\alpha_n$ does not.
Solving the boundary value problem for $\Phi$, the coefficients $\alpha_n$, $\beta_n$, and $\gamma_n$ are given by $$\begin{gathered}
\alpha_n=2\epsilon \frac{1}{x_{0n}}\frac{1}{J_1\left(x_{0n}\right)}
\left[
\frac{1}
{{\displaystyle\sinh\left(x_{0n}\frac{t}{R}\right)\cosh\left(x_{0n}\frac{L}{R}\right)
+\epsilon\,\sinh\left(x_{0n}\frac{L}{R}\right)\cosh\left(x_{0n}\frac{t}{R}\right)}}
\right],
\\
\beta_n=4\frac{1}{(p/R)}\frac{1}{x_{0n}^2}\frac{J_1{\displaystyle\left(x_{0n}\frac{p}{R}\right)}}{\left[J_1\left(x_{0n}\right)\right]^2}
\left[
\frac{{\displaystyle \sinh\left(x_{0n}\frac{t}{R}\right)}}
{{\displaystyle\sinh\left(x_{0n}\frac{t}{R}\right)\cosh\left(x_{0n}\frac{L}{R}\right)
+\epsilon\,\sinh\left(x_{0n}\frac{L}{R}\right)\cosh\left(x_{0n}\frac{t}{R}\right)}}
\right],
\\
\gamma_n=4\frac{1}{(p/R)}\frac{1}{x_{0n}^2}\frac{J_1{\displaystyle\left(x_{0n}\frac{p}{R}\right)}}{\left[J_1\left(x_{0n}\right)\right]^2}
\left[
\frac{{\displaystyle \sinh\left(x_{0n}\frac{t}{R}\right)}}
{{\displaystyle\sinh\left(x_{0n}\frac{t}{R}\right)\sinh\left(x_{0n}\frac{L}{R}\right)
+\epsilon\,\cosh\left(x_{0n}\frac{t}{R}\right)\cosh\left(x_{0n}\frac{L}{R}\right)}}
\right].\end{gathered}$$
Clock shift functional form {#Sec:validate}
===========================
For the 1D model, the simple functional form of the clock shift, Eq. ([1]{}), can be readily inferred from the superposition principle. One envisions the following two complementary problems: (i) inclusion of the stray charge, with zero electric potential on the boundary and (ii) omission of the stray charge, with non-zero electric potential on the boundary. The field in case (i) identifies with the stray field, while the field in case (ii) identifies with the applied field. The superposition of these fields gives the total field. Since in case (ii) there are no internal charges and $V$ amounts to a common scale factor for the electric potential on the boundary ($+V$ on the top electrode, $-V$ on the bottom electrode, zero on the side walls), the applied field is consequently proportional to $V$ at all points. The first and last terms in Eq. ([1]{}) are shifts due to the stray and applied fields, being quadratic in the respective field, while the middle term in Eq. ([1]{}) is a cross term that is linear in both.
While the functional form of Eq. ([1]{}) is evident for the 1D model, we likewise expect it to hold for our Faraday shield whenever $V$ represents a uniform scale factor for the electric potential on the boundary. To demonstrate this, we take $V$ to be the voltage on the top window, with all other windows grounded. We subsequently measure the induced frequency shift with values of $V$ ranging between $-2$ kV and $+2$ kV. All measurements are relative to the $V=0$ case. The resulting data is plotted in Fig. \[Fig:exptcurve\]. We perform a least squares fit of this data to the functional form of Eq. ([1]{}), $\delta\nu(V)-\delta\nu_0=aV+bV^2$, with $a$ and $b$ taken as free parameters. The resulting fit has a reduced-chi-squared $\chi^2_\mathrm{red}=1.3$, largely validating this functional form.
Equation ([2]{}) in the main text is a straightforward generalization of Eq. ([1]{}), exploiting the superposition principle to accommodate multiple voltage variables.
![Induced frequency shift measured relative to the grounded arrangement, $\delta\nu(V)-\delta\nu_0$. Here $V$ is the voltage applied to the top window, with the all other windows grounded. Blue circles: measured data. Red curve: fit of the data to $aV+bV^2$, with free parameters $a$ and $b$. The fit has a reduced-chi-squared $\chi^2_\mathrm{red}=1.3$.[]{data-label="Fig:exptcurve"}](curve.pdf){width="0.9\linewidth"}
Alternate window assignment
===========================
The shield windows are assigned voltages in terms of the variables $V_1$, $V_2$, and $V_3$ according to Fig. [2]{} of the main text, which is reproduced here as Fig. \[Fig:shield\](A). Here we introduce an alternate assignment, presented in Fig. \[Fig:shield\](B). Assignments A and B are related by a linear transformation of the variables, with the extremum $\delta\nu^*$ being invariant. It follows that $\Delta\nu$ is also invariant. For the purposes of the main text, the difference amounts to a “bookkeeping” choice for the data in Table [I]{}. The reason for introducing assignment B will be made clear in Section \[Sec:future\] below.
In the following section we describe our evaluation of $\Delta\nu$ from the data in Table [I]{} of the main text. The section can be read equally well from the perspective of assignment A or B. It should be understood that to apply the latter, the $V_i$ specifications given in Table [I]{} must be transformed accordingly.
![Assignments A and B specifying the window voltages in terms of variables $V_1$, $V_2$, and $V_3$. Applied voltages are explicitly given for three windows; opposing windows have opposite voltage. For simplicity, assignment A is used in the main text. Assignment B has attributes discussed in Section \[Sec:future\].[]{data-label="Fig:shield"}](shieldSM.pdf){width="0.7\linewidth"}
Monte Carlo evaluation of $\Delta\nu$
=====================================
To evaluate $\Delta\nu$ from the data in Table [I]{} of the main text, we first perform a least-squares fit of Eq. ([2]{}) to the full data set, treating the coefficients $a_i$ and $b_{ij}$ as fit parameters (conceptually, it helps to pull $\delta\nu_0$ to the left-hand side of the equation, being that it is data for $\delta\nu\left(V_1,V_2,V_3\right)-\delta\nu_0$ that is tabulated). The resulting fit has a reduced-chi-squared $\chi^2_\mathrm{red}=1.10$, with the fit coefficients and their uncertainties given in Table \[Tab:coeffs\]. Equation ([3]{}) could then be used to relate the coefficients to $\Delta\nu$. However, propagating uncertainty to $\Delta\nu$ is complicated by correlated and non-linear variations of the coefficients. To fully respect these intricacies, we use the following Monte Carlo procedure to map probability distributions for the original data into a probability distribution for $\Delta\nu$. 1) For each data point in Table I, a random value is pulled from the Gaussian distribution that it represents. An artificial data set is then generated by displacing the data points to these random values, while preserving error bars. 2) A least-squares fit of Eq. ([2]{}) to the artificial data set yields the coefficients $a_i$ and $b_{ij}$. Only the best-fit values are taken, with fit uncertainties being disregarded. 3) The coefficients are used in Eq. ([3]{}) to calculate $\Delta\nu$. Steps 1–3 are repeated, with the artificial data set being randomly generated each time. The observed scatter in $a_i$, $b_{ij}$, and $\Delta\nu$ are identified with probability distributions for the respective quantities. For each of the coefficients, the distribution is well-described by a Gaussian functional form with mean and standard deviation in excellent agreement with the respective entry in Table \[Tab:coeffs\]. The distribution for $\Delta\nu$ is presented in Fig. [3]{} of the main text.
[ccc]{} & A & B\
\
$a_1$ & $-0.011(16)$ & $-0.014(22)$\
$a_2$ & $-0.003(16)$ & $0.007(23)$\
$a_3$ & $0.10(13)$ & $0.10(13)$\
$b_{11}$ & $-0.715(9)$ & $-1.547(15)$\
$b_{22}$ & $-0.704(9)$ & $-1.291(16)$\
$b_{33}$ & $-228.85(7)$ & $-228.85(7)$\
$b_{12}$ & $-0.064(6)$ & $0.011(15)$\
$b_{13}$ & $-0.33(12)$ & $-0.82(17)$\
$b_{23}$ & $-0.49(12)$ & $-0.16(17)$
Future analyses {#Sec:future}
===============
We may desire to reassess the DC Stark shift in the future, either periodically or after some specific event (e.g., breaking vacuum). Acknowledging the symmetry $b_{ji}=b_{ij}$, there are nine coefficients between the $a_i$ and the $b_{ij}$. This implies a minimum of nine frequency measurements to evaluate $\Delta\nu$. Of the nine coefficients, however, only the $a_i$ depend on the stray field. So long as there has been no significant alteration to the geometry of the shield or distribution of lattice-trapped atoms within the shield, the $b_{ij}$ therefore remain fixed. As such, we can expedite future analyses by exploiting data from the present work to specify the $b_{ij}$. Three frequency measurements will be necessary to reassess the three $a_i$.
Briefly, we envision measurement of the $a_i$. For a given $i$, we set $V_{j\neq i}=0$ and use the result $$a_i=\frac{\left(\delta\nu_+-\delta\nu_-\right)}{2\mathcal{V}},$$ where $\delta\nu_\pm$ denotes the clock shift under the opposite polarity conditions $V_i=\pm\mathcal{V}$. Note that there is no need to implement the fully-grounded arrangement as a reference, as the quantity in parenthesis can be obtained from a single frequency difference measurement.
As a potential simplification for future analyses, we further consider applicability of Eq. ([4]{}) from the main text. Starting from the general Eq. ([3]{}), we expand to first order in the off-diagonal $b_{ij}$ to arrive at $$\Delta\nu\approx\sum_i\Delta\nu_i-\sum_{i\neq j}\mathrm{sign}(a_ia_j)\sqrt{\Delta\nu_i}\sqrt{\Delta\nu_j}\cos\theta_{ij},
\label{Eq:firstorder}$$ with $\Delta\nu_i\equiv a_i^2/4b_{ii}$ and $\cos\theta_{ij}\equiv b_{ij}/\left(\sqrt{b_{ii}}\sqrt{b_{jj}}\right)$. The $\mathrm{sign}$ function appearing in Eq. (\[Eq:firstorder\]) returns $\pm1$ according to the sign of the argument. We note the distinction $\sqrt{x}\sqrt{y}\neq\sqrt{xy}$; namely, for $x$ and $y$ both negative, $\sqrt{x}\sqrt{y}$ evaluates to $-\sqrt{xy}$. The $\Delta\nu_i$ and the $b_{ii}$ must all be of the same sign (the sign of $k$), which excludes imaginary results for the product $\sqrt{\Delta\nu_i}\sqrt{\Delta\nu_j}$ and for the $\cos\theta_{ij}$. The $\cos\theta_{ij}$ depend on the coefficients $b_{ij}$ and satisfy the physical relation $$\cos\theta_{ij}=\frac{\left\langle\mathbf{E}_i\cdot\mathbf{E}_j\right\rangle}
{\sqrt{\left\langle E_i^2\right\rangle\left\langle E_j^2\right\rangle}},$$ where we assume positive $V_i$ and $V_j$ for interpretation of the right-hand-side. For uniform fields, the parameter $\theta_{ij}$ is readily identified with the angle between $\mathbf{E}_i$ and $\mathbf{E}_j$. More generally, $\cos\theta_{ij}$ incorporates an average over the atoms. In the limit that the off-diagonal $\cos\theta_{ij}$ approach zero, $\Delta\nu$ is given exactly by $\Delta\nu=\sum_i\Delta\nu_i$, which is Eq. ([4]{}). Equation (\[Eq:firstorder\]) provides a means to assess validity of this simple relation on a case-to-case basis.
[ccc]{} & A & B\
\
$\cos\theta_{12}$ & $0.090(8)$ & $-0.008(10)$\
$\cos\theta_{13}$ & $0.026(9)$ & $0.044(9)$\
$\cos\theta_{23}$ & $0.039(9)$ & $0.010(10)$
We use the experimental data in Table I of the main text and the Monte Carlo technique discussed above to evaluate probability distributions for the off-diagonal $\cos\theta_{ij}$. In each case, the distribution is well-described by a Gaussian functional form. The results (mean and standard deviation) are presented in Table \[Tab:cos\].
Noting the technical limitation of $\pm2$ kV on any window, we find advantage in using assignment B for future analyses. For one, $\left|\mathbf{E}_1\right|$ and $\left|\mathbf{E}_2\right|$ can be made a factor of $\sim\!\sqrt{2}$ larger for B compared to A. Ultimately this implies a factor $\sim\!2$ tighter constraint on $\Delta\nu_1$ and $\Delta\nu_2$ for a given averaging time. In either case, $\left|\mathbf{E}_3\right|$ can be made much larger than $\left|\mathbf{E}_1\right|$ and $\left|\mathbf{E}_2\right|$, as the atoms are in closer proximity to the vertical windows than the horizontal windows. Consequently, $\Delta\nu_3$ can be constrained much tighter than $\Delta\nu_1$ and $\Delta\nu_2$ for a given averaging time. With $\Delta\nu_3$ tightly constrained (and presumably tightly constrained with respect to zero), the corrective terms of principal concern in Eq. (\[Eq:firstorder\]) scale as $\sqrt{\Delta\nu_1}\sqrt{\Delta\nu_2}\cos\theta_{12}$. Inspecting Table \[Tab:cos\], we see that $\cos\theta_{12}$ is suppressed by an order of magnitude for B compared to A. This is a result of the shield geometry. In particular, the internal copper surface is largely symmetric with respect to a vertical plane that passes through the center of the shield (where the atoms reside) and is normal to the axis defined by the two apertures. We conclude that assignment B is favorable for future analyses of the DC Stark shift, and we anticipate that the simple expression $\Delta\nu=\sum_i\Delta\nu_i$ will suffice for such analyses. By taking three frequency difference measurements $(\delta\nu_+-\delta\nu_-)$, averaged to a statistical uncertainty of $1\times10^{-17}$ for $i=1,2$ and $1\times10^{-16}$ for $i=3$, and applying the same conservative distribution for $\eta$ as in the main text, the stray-field shift can be constrained to within a few $10^{-19}$ at 95.5% confidence. With the stability afforded by our Yb clocks, this represents less than an hour of total measurement time.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The INTEGRAL Burst Alert System (IBAS) is the software for real time detection of Gamma Ray Bursts (GRBs) and the rapid distribution of their coordinates. IBAS has been running almost continuously at the INTEGRAL Science Data Center since the beginning of the INTEGRAL mission, yielding up to now accurate localizations for 12 GRBs detected in the IBIS field of view. IBAS is able to provide error regions with radii as small as 3 arcminutes (90% c.l.) within a few tens of seconds of the GRB start. We present the current status of IBAS, review the results obtained for the GRBs localized so far, and briefly discuss future prospects for using the IBAS real time information on other classes of variable sources.'
author:
- 'S. Mereghetti'
- 'D. Götz'
- 'J. Borkowski'
- 'M. Beck'
- 'A. von Kienlin'
- 'N. Lund'
title: 'The INTEGRAL Burst Alert System: Results and Future Perspectives'
---
\[2001/04/25 1.1 (PWD)\]
Introduction
============
A new era in the study of Gamma-ray Bursts (GRBs) started with the *BeppoSAX* observations leading to the discovery of their X–ray, optical and radio afterglows [@costa; @vanpa; @frail]. The great progress which occurred in the last few years in our understanding of GRBs has been possible thanks to extensive multi-wavelength observations of these unpredictable and rapidly fading events. In this respect, a quick derivation and distribution of accurate sky positions for GRBs is crucial. Here we review the contribution in this field obtained during the first 18 months of the INTEGRAL mission. We concentrate on the GRBs observed within the field of view of the IBIS instrument [@ibis]. Bursts observed with the SPI Anticoincidence Shield (ACS) are described elsewhere in these proceedings [@acs].
Thanks to its 72 hours orbit, the INTEGRAL satellite is in continuous contact with the ground stations during the observations. This has allowed us to implement a ground-based software, the INTEGRAL Burst Alert System (IBAS), for the search in near real time of GRBs [@ibas]. The IBAS software and its current performances are briefly described in Section 2. In Section 3 we summarize the main results on the twelve GRBs observed to date in the field of view of the INTEGRAL instruments. Finally, in Section 4 we describe the IBAS capability to provide real time information also on other classes of transient sources.
IBAS description and performances
=================================
A detailed description of IBAS is given in @ibas. Here we briefly remind the most salient features of the system.
As mentioned above, the search for GRBs is done on ground, at the INTEGRAL Science Data Centre [@isdc]. In fact, no on-board triggering system is present on INTEGRAL and the operating modes of the instruments do not change when a GRB occurs. Since, under nominal conditions, the telemetry data reach the ISDC without important delays, the IBAS programs can run in near real time. Such a ground based system offers some advantages with respect to systems operating on board satellites, e.g. a larger computing power and more flexibility for software and hardware upgrades. In fact, in the course of the first year after the launch of INTEGRAL several changes and additions have been done to the IBAS programs. The current configuration is based on two different methods to look for GRBs in the data from the IBIS lower energy detector ISGRI [@isgri].
In the first method the overall counting rate is monitored to look for significant excesses with respect to a running average of the background, in a way similar to traditional triggering algorithms used on-board previous satellites. Several different energy ranges and integration times (from 2 ms to 5.12 s) are sampled in parallel. A rapid imaging analysis is performed only when a significant counting rate excess is detected. Imaging allows to eliminate many false triggers caused, e.g. by instrumental effects or background variations that do not produce a point source in the reconstructed sky images. The second method is entirely based on imaging. Images of the sky are continuously produced (integration times of 10, 20, 40 and 100 s) and compared with the previous ones to search for new sources.
The GRB positions derived by IBAS are delivered via Internet to all the interested users. For the GRBs detected with high significance, this is done immediately by the software which sends *Alert Packets* using the UDP transport protocol. In case of events with lower statistical significance, the alerts are sent only to the members of the IBAS Localization Team, who perform further analysis and, if the GRB is confirmed, can distribute its position with an *Off-line Alert Packet*.
-------------- ------------- ----------------------- ------------------- ------------
GRB Approximate Delay$^{a}$ in External delivery References
duration position distribution of IBAS
\[s\] internal/public *Alert Packets*
021125 25 –$^{b}$ / 0.9 days OFF @021125D
021219 6 10 s / 5 hr OFF @021219D
030131 150 21 s / 2 hr ON @030131D
030227 20 35 s / 48 min OFF @030227D
030320 50 12 s / 6 hr ON @030320D
030501 40 30 s / 30 s ON @030501D
031203 30 18 s / 18 s ON @031203D
040106 60 12 s / 12 s ON @040106D
040223 250 210 s / 210 s ON @040223D
040323 20 30 s / 30 s ON @040323D
040403 35 21 s / 21 s ON @040403D
040422 8 17 s / 17 s ON @040422D
\[tab:spec\]
-------------- ------------- ----------------------- ------------------- ------------
$^{a}$ Computed from the GRB start time.
$^{b}$ The IBAS *Detector Programs* were in idle mode owing to the limited telemetry allocation for IBIS/ISGRI during this observation.
The first two months of operations after the INTEGRAL launch were devoted to the optimization of the IBAS parameters. Some changes in the algorithms were also required to adapt them to the in-flight data characteristics. Delivery of the *Alert Packets* to the external clients started on January 17, 2003. Since then it has always been enabled, except during the first calibration campaign on the Crab Nebula (12-28 February 2003), and a few very short interruption (few hours) due to maintenance reasons.
Up to now (April 2004), twelve GRBs have been discovered in the field of view of IBIS. Figure \[fov\] shows their positions in the fields of view of the INTEGRAL instruments. All of them were at off-axis angles too large to be seen with the OMC and JEM-X instruments.
The time and accuracy performances of the IBAS localizations for these bursts are summarized in Table 1 and illustrated in Figs. \[delays\] and \[erboxall\]. Note that at the beginning of the mission the in-flight instrument misalignment was not calibrated yet. Therefore, error radii as large as 20$'$ or 30$'$ were given. The systematic uncertainties could be reduced in the following months, leading to smaller error regions.
The time delay in the distribution of coordinates results from the sum of several factors. First of all there is a delay on board the satellite, which is variable and depends on the instrument. For IBIS/ISGRI data the average delay is about 5 s. Signal propagation to the ground station is negligible (maximum $\sim$0.6 s), but some time is required before the data are received at the ISDC. This is on average 3 s when the ESA ground station in Redu (Belgium) is used, or 6 s when the NASA Goldstone ground station is used. The time to detect the GRB depends on the algorithm which triggers. The delay between the trigger time and the GRB onset is of course dependent on the intensity and time profile of the event. The IBAS simultaneous sampling in different timescales should ensure a small delay in most cases, however in practice a minimum of $\sim$3 s is required to accumulate an image with enough statistics. Finally, the conversion to sky coordinates, comparison with list of known variable sources, *Alert Packet* construction and delivery require less than about 2 s. Of course, the above numbers assume nominal condition, i.e. no telemetry gaps, no saturation of the allocated telemetry, no missing auxiliary data files, etc...
As can be seen in Fig. \[delays\] all the burst detected by IBAS after April 2003 had very small error regions distributed within a few tens of seconds, often while the gamma-ray emission was still visible. Such a combination of high speed and small error region was never achieved before. Note that the 210 s delay in the localization of GRB 040223 was due do the particular light curve shape of this burst lasting about 4 minutes and with the brightest peak at the end.
-------------- -------------------------- --------------------------- --------------------- --------------------- ------------ -------------- ------------ -- --
GRB Peak Flux Peak Flux Fluence Power law Ref.$^{b}$ Afterglow Ref.$^{b}$
(20-200 keV) (20-200 keV) (20-200 keV) photon index$^{a}$
\[ph cm$^{-2}$s$^{-1}$\] \[erg cm$^{-2}$s$^{-1}$\] \[erg cm$^{-2}$\]
021125 22 2 $\times10^{-6}$ 7.4$\times 10^{-6}$ 2.2/3.7 (1) –
021219 3.7 3.5$\times10^{-7}$ 9$\times 10^{-7}$ 1.3$\rightarrow$2.5 (2) –
030131 1.9 1.7$\times10^{-7}$ 7$\times 10^{-6}$ $\sim2$ (3) opt. (3,4)
030227 1.1 1.6$\times10^{-7}$ 7.5$\times 10^{-7}$ 1.9 (5) opt./X (6,5)
030320 5.7 5.4$\times10^{-7}$ 1.1$\times 10^{-5}$ 1.3$\rightarrow$1.9 (7) –
030501 2.7 3$\times10^{-7}$ 3$\times 10^{-6}$ 1.75 (8) –
031203 1.2 1.3$\times10^{-7}$ $^{c}$ $^{c}$ (9) radio/opt./X (10,11,12)
040106 0.6 6.5$\times10^{-8}$ $^{c}$ $^{c}$ (13) opt.?/X (14,15)
040223 0.4 3$\times10^{-8}$ $^{c}$ $^{c}$ (16) X (17)
040323 1.7 2.2$\times10^{-7}$ $^{c}$ $^{c}$ (18) opt.? (19)
040403 0.4 3$\times10^{-8}$ $^{c}$ $^{c}$ (20)
040422 2.7 2.5$\times10^{-7}$ $^{c}$ $^{c}$ (21)
\[tab:spec\]
-------------- -------------------------- --------------------------- --------------------- --------------------- ------------ -------------- ------------ -- --
$^{a}$ The two values for GRB 021125 are for the ranges 20-200 keV (ISGRI) and 170-500 keV (PICsIt). The arrow indicates time evolution.
$^{b}$ References: (1) [@021125P]; (2) @021219P; (3) @030131P; (4) @030131O; (5) @030227P; (6) @030227O; (7) @030320P; (8) @030501P; (9) @031203D; (10) @031203R; (11) @031203O; (12) @031203X; (13) @040106D; (14) @040106O; (15) @040106X; (16) @040223D; (17) @040223X; (18) @040323D; (19) @040323O; (20) @040403D; (21) @040422D;
$^{c}$ Results not yet published
The accuracy of the localizations derived by the IBAS on-line programs (and distributed in the automatic *Alert Packets*) can be estimated based on the IBAS triggers caused by known sources. Figure \[acc\] shows the distribution of the differences between the true and derived coordinates based on $\sim$24,000 triggers from Sco X-1, Cyg X-1, Vela X-1 and other sources with well known positions. From the curves shown in Fig. \[acc\] we can estimate the location accuracy as a function of the source signal to noise ratio (S/N). This is reported in Fig. \[acc90\], where it can be seen that the 90% c.l. error radius is smaller than 2.5$'$ for S/N$>$10 (the current threshold for the automatic delivery of *Alert Packets* containing the GRB coordinates is S/N=8). Note that the above discussion refers to the on-line imaging analysis which is based on simplified algorithms. In general, the GRB error regions can be further reduced by the more sophisticated interactive analysis performed off-line.
Results on Gamma-ray bursts
===========================
The main properties of the twelve INTEGRAL GRBs are summarized in Table 2. In Fig. \[bigplot\] we show the light curves of the first six bursts, as measured with the IBIS/ISGRI instrument. The results for these six bursts, including their spectral analysis, have been published in the references given in Table 2. Most of the spectral information for these bursts is based on IBIS/ISGRI data. Although the nominal range of the instrument is 15 keV - 1 MeV, the coverage above $\sim$200-300 keV is actually limited by the small statistics. Thus most of the spectra are well described by a power-law over the $\sim$15-200 keV range (see photon index values in Table 2). Only for GRB 030131 a curved spectrum gives a better fit, with parameters of the Band model $\alpha$=1.4, $\beta$=3 and E$_o$=70 keV [@030131P]. We remark that the spectral response of IBIS in the partially coded field of view is not completely calibrated yet. Therefore, the published GRB spectra should be considered as preliminary. No evidence for spectral lines has been seen so far in the SPI spectra, which are generally in good agreement with the IBIS/ISGRI ones.
Thanks to the good sensitivity of IBIS/ISGRI, it has been possible to study the time evolution of the spectra even for relatively faint bursts. A typical hard to soft variation has been clearly seen in GRB 021219 and GRB 030320 [@021219P; @030320P], and with lower significance in GRB 030227 and GRB 030131 [@030227P; @030131P].
The distribution of peak fluxes for the INTEGRAL bursts is shown in Fig. \[lognlogp\], but of course the small number of events does not allow for the moment a meaningful interpretation of such a LogN-LogP function.
As can be seen in Fig. \[t90\] all the bursts detected so far belong to the long duration class ($>$2 s). The optimization of the IBAS programs and parameters for short bursts required some time, leading, in the first months after the launch, to a bias in favor of long GRBs. However, since the Summer of 2003 IBAS has also a good sensitivity also for short events, as it has been demonstrated by the real time detection of many weak bursts lasting $\sim$0.1-0.2 s from SGR 1806-20 [@sgr1806]. Therefore, we expect to obtain a rapid localization also for a short burst in the coming months.
INTEGRAL spends most of the time pointing at Galactic targets. As a consequence the majority of the detected bursts are at low Galactic latitude (see Fig. \[pos\]). This is not an ideal situation for what concerns their follow-up observations at other wavelengths. Nevertheless, thanks to the rapid dissemination of their coordinates, X-ray and/or optical/IR afterglows have been reported for six INTEGRAL GRBs.
IBAS and other variable sources
===============================
The IBAS programs are sensitive to several kinds of variable/transient events, in addition to GRBs. Most of the triggers caused by bright and highly variable sources, like e.g. Sco X-1 or Cyg X-1, represent just a disturbance to the main IBAS task. However, some triggers are due to potentially interesting phenomena. These include type I and type II X-ray bursts from Low Mass X-ray Binaries (see Fig. \[burst\]), bursts from Soft Gamma-ray Repeaters, outbursts from known and unknown transients. As for GRBs, the information on the occurrence of these events is derived in real time (i.e. within few tens of seconds). To avoid alerting the community of users mainly interested in GRBs and to comply with the INTEGRAL data rights rules, all the IBAS *Alert Packets* with derived coordinates consistent with the positions in a list of known sources are not distributed (except for Soft Gamma-ray Repeaters, whose alerts are being distributed since January 2004).
We give here just two examples of interesting cases. One is the new transient hard X-ray source IGR J17544–2619, discovered by INTEGRAL in the Galactic bulge region. The second example is the very high state reached by the high mass X-ray binary pulsar Vela X-1 on November 28, 2003 [@velax1]. In both cases the IBAS programs triggered in real time, while the public announcements were given respectively 14 hours and 4 days later [@igr; @velax1D]. Rapid follow-up at other wavelengths would have provided very useful information: the first outburst from IGR J17544–2619 (the one during which it was discovered) lasted only two hours, while the 15-40 keV flux of Vela X-1 increased from 0.5 to 7 Crabs during the first 5400 s before decreasing in the following hours.
In general, these “secondary” IBAS results are used to complement the quick-look analysis performed at the ISDC. However, the real time information which is potentially available through IBAS is not fully exploited yet. We plan to implement these capabilities in the future, possibly also including the use of JEM-X data. Robotic optical/IR telescopes operating on ground, as well as satellites with a rapid reaction time, such as Swift [@swift], could benefit from the IBAS alert messages to perform follow-up observations of X-ray transients with an unprecedented rapidity.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been partly funded by ASI. JB was supported by grant 2P03C00619p02 from KBN.
Bazzano, A. & Paizis A. 2002, GCN Circ. n. 1706 Beckmann V., Borkowski J., Courvoisier T.J.-L., et al. 2003, A&A 411, L327 Borkowski J., Götz D. & Mereghetti S. 2003, GCN Circ. n. 1836 Castro-Tirado A.J., Gorosabel J, Guziy S., et al. 2003, A&A 411, L315 Costa E., Frontera F., Heise J., et al. 1997, Nature 387, 783 Courvoisier T.J.-L., Walter R., Beckmann V., et al. 2003, A&A 411, L53 Ehle M. et al. 2004, GCN Circ. n. 2508 Fox D.W., Price P.A., Heter T., et al. 2003, GCN Circ. n. 1857 Frail D.A., Kulkarni S.L., Nicastro S.R., et al. 1997, Nature 389, 261 Frail D. et al. 2004, GCN Circ. n. 2473 Gal-Yam A. et al. 2004, GCN Circ. n. 2555 Gehrels N. 2001, in [*Gamma-Ray Bursts in the Afterglow Era*]{}, eds. E. Costa, F. Frontera & J. Hjorth, Berlin, Heidelberg, Springer, p. 357 Götz D. & Mereghetti S. 2004a, GCN Circ. n. 2506 Götz D. & Mereghetti S. 2004b, GCN Circ. n. 2526 Götz D., Mereghetti S., Hurley K., et al. 2003a, A&A 409, 831 Götz D., Borkowski J. & Mereghetti S. 2003b, GCN Circ. n. 1895 Götz D., Mereghetti S., Beck M., et al. 2003c, GCN Circ. n. 2459 Götz D., Mereghetti S., Beck M., et al. 2004a, GCN Circ. n. 2525 Götz D., Mereghetti S., Mirabel F. & Hurley K. 2004b, A&A 417, L45 Götz D., Mereghetti S., Borkowski J., et al. 2004c, GCN Circ. n. 2560 Krivonos R. et al. 2003, ATEL n. 211 Lebrun F., Leray J.P., Lavocat P., et al. 2003, A&A 411, L141 Malaguti G., Bazzano A., Beckmann V., et al. 2003, A&A 411, L307 Masetti N. et al. 2004, GCN Circ. n. 2515 Mereghetti S. & Götz D., GCN Circ. n. 2460 Mereghetti S., Götz D., Borkowski J., et al. 2003a, A&A 411, L291 Mereghetti S., Götz D. & Borkowski J. 2002b, GCN Circ. n. 1731 Mereghetti S., Götz D., Beckmann V., et al. 2003c, A&A 411, L311 Mereghetti S., Götz D., Tiengo A., et al. 2003d, ApJ 590, L73 Mereghetti S., Götz D. & Borkowski J., et al. 2003e, GCN Circ. n. 1941 Mereghetti S., Götz D., Borkowski J., Shaw S. & Courvoisier T. 2003f, GCN Circ. n. 2183 Mereghetti S., Götz D., Beck M. & Borkowski J. 2004a, GCN Circ. n. 2505 Mereghetti S., Götz D., Beck M. et al. 2004b, GCN Circ. n. 2551 Mereghetti S., Götz D., Borkowski J. et al. 2004c, GCN Circ. n. 2572 Prochaska J.X. et al. 2004, astro-ph/0402085 Rau A., et al. 2004, these proceedings Staubert R. et al. 2004, these proceedings Sunyaev R. et al. 2003, ATEL n.190 Tiengo A., Mereghetti S. & De Luca A. 2004, GCN Circ. n. 2548 Ubertini P., Lebrun F., Di Cocco G., et al. 2003, A&A 411, L131 van Paradijs J., Groot P.J., Galama T., et al. 1997, Nature 386, 686 von Kienlin A., Beckmann V., Covino S., et al. 2003b, A&A 411, L321 Watson D. et al. 2004, ApJ 605, L101
| {
"pile_set_name": "ArXiv"
} |
---
bibliography:
- 'cite.bib'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Consider a discrete-time linear time-invariant descriptor system $Ex(k+1)=Ax(k)$ for $k \in \mathbb Z_{+}$. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system $\hat E x(k+1)= \hat A x(k)$ with $\text{rank}(\hat E)=r$. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.'
author:
- 'Nicolas Gillis[^1] Michael Karow[^2] Punit Sharma[^3]'
bibliography:
- 'GilKS18b.bib'
title: 'A note on approximating the nearest stable discrete-time descriptor system with fixed rank'
---
**Keywords.** stability radius, linear discrete-time descriptor system, stability
Introduction
============
In [@OrbNV13; @GilKS18a], authors have tackled the problem of computing the nearest stable matrix in the discrete case, that is, given an unstable matrix $A$, find the smallest perturbation $\Delta_A$ with respect to Frobenius norm such that $\hat A = A+\Delta_A$ has all its eigenvalues inside the unit ball centred at the origin. In this paper, we aim to generalize the results in [@GilKS18a] for matrix pairs $(E,A)$, where $E,A\in \R^{n,n}$. The matrix pair $(E,A)$ is called *regular* if $\operatorname{det}(\lambda E-A)\neq 0$ for some $\lambda \in \mathbb C$, which we denote $\operatorname{det}(\lambda E-A) \not\equiv 0$, otherwise it is called *singular*. For a regular matrix pair $(E,A)$, the roots of the polynomial $\operatorname{det}(z E-A)$ are called *finite eigenvalues* of the pencil $zE-A$ or of the pair $(E,A)$. A regular pair $(E,A)$ has *$\infty$ as an eigenvalue* if $E$ is singular. A regular real matrix pair $(E,A)$ can be transformed to *Weierstraß canonical form* [@Gan59a], that is, there exist nonsingular matrices $W, T \in \C^{n,n}$ such that $$E=W{\left[\begin{array}{cc}}I_q& 0\\0&N{\end{array}\right]}T \quad \text{and}\quad A=W {\left[\begin{array}{cc}}J &0\\0&I_{n-q}{\end{array}\right]}T,$$ where $J \in \C^{q,q}$ is a matrix in *Jordan canonical form* associated with the $q$ finite eigenvalues of the pencil $z E-A$ and $N \in \C^{n-q,n-q}$ is a nilpotent matrix in Jordan canonical form corresponding to $n-q$ times the eigenvalue $\infty$. If $q < n$ and $N$ has degree of nilpotency $\nu \in \{1,2,\ldots\}$, that is, $N^{\nu}=0$ and $N^i \neq 0$ for $i=1,\ldots,\nu-1$, then $\nu$ is called the *index of the pair* $(E,A)$. If $E$ is nonsingular, then by convention the index is $\nu=0$; see for example [@Meh91; @Var95]. The matrix pair $(E,A) \in (\R^{n,n})^2$ is said to be *stable* (resp. *asymptotically stable*) if all the finite eigenvalues of $zE-A$ are in the closed (resp. open) unit ball and those on the unit circle are semisimple. The matrix pair $(E,A)$ is said to be *admissible* if it is regular, of index at most one, and stable.
The various distance problems for linear control systems is an important research topic in the numerical linear algebra community; for example, the distance to bounded realness [@AlaBKMM11], the robust stability problem [@Zho11], the stability radius problem for standard systems [@Bye88; @HinP86] and for descriptor systems [@ByeN93; @DuLM13], the nearest stable matrix problem for continuous-time systems [@OrbNV13; @GilS17; @MehMS17; @GugL17] and for discrete-time systems [@OrbNV13; @NesP17; @GP2018; @GilKS18a], the nearest continuous-time admissible descriptor system problem [@GilMS17], and the nearest positive real system problem [@GilS17b]. For a given unstable matrix pair $(E,A)$, the discrete-time nearest stable matrix pair problem is to solve the following optimization problem $$\label{mainprob}
\inf_{(\hat E,\hat A)\in\mathcal S^{n,n}} {\|E-\hat E\|}_F^2+{\|A-\hat A\|}_F^2,\tag{$\mathcal{P}$}$$ where $\mathcal S^{n,n}$ is the set of admissible pairs of size $n \times n$. This problem is the converse of stability radius problem for descriptor systems [@ByeN93; @DuLM13] and the discrete-time counter part of continuous-time nearest stable matrix pair problem [@GilMS17]. Such problems arise in systems identification where one needs to identify a stable matrix pair depending on observations [@OrbNV13; @GilS17]. This is a highly nonconvex optimization problem because the set $\mathcal S^{n,n}$ is unbounded, nonconvex and neither open nor closed. In fact, consider the matrix pair $$\label{eq:ex1}
(E,A)=\Bigg(
{\left[\begin{array}{ccc}}1&0&0\\0&0&0\\0&0&0 {\end{array}\right]},~{\left[\begin{array}{ccc}}1/2&0&2\\0&1&0\\0&0&1 {\end{array}\right]}\Bigg).$$ The pair $(E,A)$ is regular since $\text{det}(\lambda E-A)=\text{det}(\lambda -1/2)\not\equiv 0$, of index one, and stable with the only finite eigenvalue $\lambda_1=1/2$. Thus $(E,A) \in \mathcal S^{3,3}$. Let $$\label{eq:ex1perturb}
(\Delta_E,\Delta_A)=\Bigg(
{\left[\begin{array}{ccc}}0&0&0\\0&\epsilon_1&\epsilon_2\\0&0&0 {\end{array}\right]},{\left[\begin{array}{ccc}}0&0&0\\0&0&0\\0&0&-\delta {\end{array}\right]}\Bigg),$$ and consider the perturbed pair $(E+\Delta_E,A+\Delta_A)$. If we let $\delta=\epsilon_1=0$ and $\epsilon_2>0$, then the perturbed pair is still regular and stable as the only finite eigenvalue $\lambda_1=1/2$ belongs to the unit ball, but it is of index two. For $\epsilon_2=\delta=0$ and $0<\epsilon_1<1$, the perturbed pair is regular, of index one but has two finite eigenvalues $\lambda_1=1/2$ and $\lambda_2=1/\epsilon_1 >1$. This implies that the perturbed pair is unstable. This shows that $\mathcal S^{3,3}$ is not open. Similarly, if we let $\epsilon_1=\epsilon_2=0$ and $\delta >0$, then as $\delta \rightarrow 1$ the perturbed pair becomes non-regular. This shows that $\mathcal S^{3,3}$ is not closed. The nonconvexity of $\mathcal S^{n,n}$ follows by considering for example $$\label{eq:nonconvex}
\Sigma_1=\Big(I_2,\underbrace{{\left[\begin{array}{cc}}0.5 & 2\\ 0& 1 {\end{array}\right]}}_{A}\Big), \quad \Sigma_2=\Big(I_2,\underbrace{{\left[\begin{array}{cc}}0.5 & 0\\ -2& 1{\end{array}\right]}}_{B}\Big),$$ where $\Sigma_1,\Sigma_2 \in \mathcal S^{2,2}$, while $\gamma \Sigma_1 + (1-\gamma)\Sigma_2 \notin \mathcal S^{2,2}$ for $\gamma=\frac{1}{2}$, since $\frac{1}{2} \Sigma_1+\frac{1}{2} \Sigma_2$ has two eigenvalues 0.75$\pm$0.96$i$ outside the unit ball. Therefore it is in general difficult to work directly with the set $\mathcal S^{n,n}$. We explain in Section \[reform\] the difficulty in generalizing the results in [@GilKS18a] for problem .
In this paper, we consider instead a *rank-constrained nearest stable matrix pair problem*. For this, let $r (<n) \in \mathbb Z_{+}$ and let us define a subset $\mathcal S_r^{n,n}$ of $\mathcal S^{n,n}$ by $$\mathcal S_r^{n,n} :=\left\{(\hat E,\hat X)\in \mathcal S^{n,n}:~\text{rank}(\hat E)=r\right\}.$$ For a given unstable matrix pair $(E,A)$, the rank-constrained nearest stable matrix pair problem requires to compute the smallest perturbation $(\Delta_E,\Delta_A)$ with respect to Frobenius norm such that $(E+\Delta_E,A+\Delta_A)$ is admissible with $\text{rank}(E+\Delta_E)=r$, or equivalently, we aim to solve the following optimization problem $$\label{restprob}
\inf_{(\hat E, \hat A)\in\mathcal S_r^{n,n}} {\|E-\hat E\|}_F^2+{\|A-\hat A\|}_F^2 \tag{$\mathcal{P}_r$}.$$ The problem is also nonconvex as the set $\mathcal S_r^{n,n}$ is nonconvex. To solve , we provide a simple parametrization of $\mathcal S_r^{n,n}$ in terms of a matrix quadruple $(T,W,U,B)$, where $T,W\in \R^{n,n}$ are invertible, $U\in \R^{r,r}$ is orthogonal, and $B\in \R^{r,r}$ is a positive semidefinite contraction, see Section \[reform\]. This parametrization results in an equivalent optimization problem with a feasible set onto which it is easy to project, and we derive a block coordinate descent method to tackle it; see Section \[sec:algo\]. We illustrate the effectiveness of our algorithm over several numerical examples in Section \[sec:numexp\].
#### Notation
Throughout the paper, $X^T$ and $\|X\|$ stand for the transpose and the spectral norm of a real square matrix $X$, respectively. We write $X\succ 0$ and $X\succeq 0$ $(X \preceq 0)$ if $X$ is symmetric and positive definite or positive semidefinite (symmetric negative semidefinite), respectively. By $I_m$ we denote the identity matrix of size $m \times m$.
Reformulation of problem {#reform}
=========================
As mentioned earlier the set $\mathcal S_r^{n,n}$ is nonconvex. It is also an unbounded set which is neither open nor closed. Consider $$\tilde\Sigma_1=\Big({\left[\begin{array}{cc}}I_r & 0\\ 0& 0 {\end{array}\right]},{\left[\begin{array}{cc}}A_1 & 0\\ 0& I_{n-r} {\end{array}\right]}\Big), \quad
\tilde\Sigma_2=\Big({\left[\begin{array}{cc}}I_r & 0\\ 0& 0 {\end{array}\right]},{\left[\begin{array}{cc}}B_1 & 0\\ 0& I_{n-r} {\end{array}\right]}\Big),$$ where $A_1={\left[\begin{array}{cc}}A &0\\0& I_{r-2}{\end{array}\right]}$, $B_1={\left[\begin{array}{cc}}B &0\\0& I_{r-2}{\end{array}\right]}$, and $A$ and $B$ are defined as in . We have that $\tilde\Sigma_1,\tilde\Sigma_2 \in \mathcal S_r^{n,n}$ because $(I_r,A_1)$ and $(I_r,B_1)$ are stable. Moreover $\frac{1}{2} \tilde\Sigma_1+\frac{1}{2} \tilde\Sigma_2 \notin \mathcal S_r^{n,n}$ as it has two eigenvalues 0.75$\pm$0.96$i$ outside the unit ball hence $\mathcal S_r^{n,n}$ is non-convex. To show that $\mathcal S_r^{n,n}$ is neither open nor closed, let $(E,A)$ and $(\Delta_E,\Delta_A)$ be as defined in and , and consider $$(\tilde E,\tilde A)=\Big({\left[\begin{array}{cc}}I_{r-1} & 0\\ 0& E {\end{array}\right]},{\left[\begin{array}{cc}}I_{r-1} & 0\\ 0& A {\end{array}\right]}\Big)$$ and the perturbation $$(\Delta_{\tilde E},\Delta_{\tilde A})=\Big({\left[\begin{array}{cc}}I_{r-1} & 0\\ 0& \Delta_E {\end{array}\right]},{\left[\begin{array}{cc}}I_{r-1} & 0\\ 0& \Delta_A {\end{array}\right]}\Big).$$ By using similar arguments as in the case of $\mathcal S^{n,n}$ one can show that $\mathcal S_r^{n,n}$ is neither open nor closed. Therefore it is difficult to compute a global solution to problem and to work directly with the set $\mathcal S_r^{n,n}$. For this reason, we reformulate the rank-constrained nearest stable matrix pair problem into an equivalent problem with a relatively simple feasible set. In order to do this , we derive a parametrization of admissible pairs into invertible, symmetric and orthogonal matrices. We first recall a result from [@GilKS18a] that gives a characterization for stable matrices.
[[@GilKS18a Theorem 1]]{}\[thm:stabmatchar\] Let $A\in \R^{n,n}$. Then $A$ is stable if and only if $A=S^{-1}UBS$ for some $S,U,B \in \R^{n,n}$ such that $S\succ 0$, $U^TU=I_n$, $B\succeq 0$, and $\|B\|\leq 1$.
We note that, in the proof of Theorem \[thm:stabmatchar\], only the invertibility of matrix $S$ is needed and the condition of symmetry on $S$ can be relaxed. We found that this relaxation on matrix $S$ does not make any difference on the numerical results in [@GilKS18a]. The only gain is that the projection of $S$ on the set of positive definite matrices takes some time and that can be avoided. Therefore, we rephrase the definition of a SUB matrix in [@GilKS18a] and the corresponding characterization of stable matrices as follows.
\[thm:newstabmatchar\] Let $A\in \R^{n,n}$. Then $A$ is stable if and only if $A$ admits a SUB form, that is, $A=S^{-1}UBS$ for some $S,U,B \in \R^{n,n}$ such that $S$ is invertible, $U^TU=I_n$, $B\succeq 0$, and $\|B\|\leq 1$.
\[thm:reform1\] Let $E,A\in \R^{n,n}$ be such that $\text{rank}(E)=r$. Then $(E,A)$ is admissible if and only if there exist matrices $T,W \in \R^{n,n}$, $S,U,B\in \R^{r,r}$ such that the matrices $T,W,S$ are invertible, $U^TU=I_r$, $B\succeq 0$, $\|B\|\leq 1$ such that $$\label{eq:firstreform}
E=W \begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}T, \quad
\text{and}
\quad
A=W\begin{bmatrix} S^{-1}UBS & 0 \\ 0 & I_{n-r}\end{bmatrix}T.$$
For a regular index one pair $(E,A)$, there exist invertible matrices $W,T\in \R^{n,n}$ such that $$E=W \begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}T \qquad \text{ and } \qquad
A=W\begin{bmatrix} \tilde A & 0 \\ 0 & I_{n-r}\end{bmatrix}T,$$ see [@Dai89]. Further, the finite eigenvalues of $(E,A)$ and $\tilde A$ are same because $\text{det}(\lambda E-A)=0$ if and only if $\text{det}(\lambda I_r-\tilde A)=0$. Thus by stability of $(E,A)$ and Theorem \[thm:newstabmatchar\], it follows that $\tilde A$ admits a SUB form, that is, there exist $S,U,B \in \R^{r,r}$ such that $S$ is invertible, $U^TU=I_r$, $B\succeq 0$, $\|B\|\leq 1$, and $\tilde A =S^{-1}UBS$.\
Conversely, it is easy to see that any matrix pair $(E,A)$ in the form is regular and of index one. The stability of $(E,A)$ follows from Theorem \[thm:newstabmatchar\] as the matrix $S^{-1}UBS$ is stable.
If the matrix $E$ is nonsingular, then Theorem \[thm:reform1\] can be further simplified as follows.
\[eq:nonsingcase\] Let $E,A \in \R^{n,n}$, and let $E$ be nonsingular. Then $(E,A)$ is admissible if and only if there exist matrices $S,U,B\in \R^{n,n}$ such that $A=S^{-1}UBSE$, where $S$ is invertible, $U^TU=I_n$, $B\succeq 0$, and $\|B\|\leq 1$.
Since $E$ is nonsingular, the matrix pair $(E,A)$ can be equivalently written as a standard pair $(I_n,AE^{-1})$, and then stability of $(E,A)$ can be determined by the eigenvalues of $AE^{-1}$. That means, $(E,A)$ is stable if and only if $AE^{-1}$ is stable. Thus from Theorem \[thm:newstabmatchar\], $AE^{-1}$ is stable if and only if $AE^{-1}$ admits a SUB form, that is, $AE^{-1}=S^{-1}UBS$ for some $S,U,B \in \R^{n,n}$ such that $S$ is invertible, $U^TU=I_n$, $B\succeq 0$ and $\|B\|\leq 1$.
We note that, for a standard pair $(I_n,A)$ (with $E=I_n$), Theorem \[thm:reform1\] coincides with Theorem \[thm:newstabmatchar\] as in this case $W$ and $T$ can be chosen to be the identity matrix which yields $A=S^{-1}UBS$. A similar result also holds for asymptotically stable matrix pairs which can be seen as a generalization of [@GilKS18a Theorem 2].
Let $E,A\in \R^{n,n}$ be such that $\text{rank}(E)=r$. Then $(E,A)$ is regular, of index one and asymptotically stable if and only if there exist matrices $T,W \in \R^{n,n}$, $S,U,B\in \R^{r,r}$ such that the matrices $T,W,S$ are invertible, $U^TU=I_r$, $B\succeq 0$, $\|B\|< 1$ such that $$\label{eq:secondreform}
E=W \begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}T, \quad
\text{and}
\quad
A=W\begin{bmatrix} S^{-1}UBS & 0 \\ 0 & I_{n-r}\end{bmatrix}T.$$
The proof follows is similar to that of Theorem \[thm:reform1\] by using [@GilKS18a Theorem 2] instead of Theorem \[thm:newstabmatchar\].
Note that the matrix $S$ is invertible in Theorem \[thm:reform1\] and therefore it can be absorbed in $W$ and $T$. The advantage is that this reduces the number of variables in the corresponding optimization problem.
\[thm:reform2\] Let $E,A\in \R^{n,n}$ be such that $\text{rank}(E)=r$. Then $(E,A)$ is admissible if and only if there exist invertible matrices $T,W \in \R^{n,n}$, and $U,B\in \R^{r,r}$ with $U^TU=I_r$, $B\succeq 0$ and $\|B\|\leq 1$ such that $$\label{eq:secreform}
E=W \begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}T, \quad
\text{and}
\quad
A=W\begin{bmatrix} UB & 0 \\ 0 & I_{n-r}\end{bmatrix}T.$$
In view of Corollary \[thm:reform2\], the set $\mathcal S_r^{n,n}$ of restricted rank admissible pairs can be characterized in terms of matrix pairs , that is, $$\begin{aligned}
&\mathcal S_r^{n,n} = \Bigg \{\left(W {\left[\begin{array}{cc}} I_r & 0 \\ 0 & 0 {\end{array}\right]}T,
W{\left[\begin{array}{cc}} UB & 0 \\ 0 & I_{n-r}{\end{array}\right]}T\right):~\text{invertible}~ T,W \in \R^{n,n},\\
& \hspace{6cm} U,B\in\R^{r,r},U^TU=I_r,B\succeq 0,\|B\|\leq 1 \Bigg \}.\end{aligned}$$ This parametrization changes the feasible set and the objective function in problem as $$\label{eq:reform_prob}
(\mathcal{P}_r) \quad = \quad \inf_{W,T \in \R^{n,n},\,UB\in \R^{r,r},\,U^TU=I_r,\, \|B\|\leq 1} \; f(W,T,U,B),$$ where $$f(W,T,U,B)=
{\left\|E-W {\left[\begin{array}{cc}} I_r & 0 \\ 0 & 0 {\end{array}\right]}T\right\|}_F^2+
{\left\|A-W{\left[\begin{array}{cc}} UB & 0 \\ 0 & I_{n-r}{\end{array}\right]}T\right\|}_F^2 .$$ An advantage of this reformulation over is that it is relatively easy to project onto the feasible set of . This enables us to use standard optimization schemes to solve it, see Section \[sec:algo\].
As mentioned in [@GilMS17], for the standard pair $(I_n,A)$ making $A$ stable without perturbing the identity matrix gives an upper bound to the solution of $({\mathcal P}_n)$, because $$\label{eq:numupbound}
\inf_{(M,X)\in \mathcal S_n^{n,n}} {\|I_n-M\|}_F^2+{\|A-X\|}_F^2\leq \inf_{(I_n,X)\in \mathcal S_n^{n,n}} {\|A-X\|}_F^2
=\inf_{(I_n,S^{-1}UBS)\in \mathcal S_n^{n,n}} {\|A-S^{-1}UBS\|}_F^2.$$ Note that the right hand side infimum in is the distance of $A$ from the set of stable matrices [@GilKS18a]. It is demonstrated in our numerical experiments that (as expected) the inequality in is strict. We also note that similar arguments do not extend to the solution of problem , when $r < n$. In this case, the distance of $A$ from the set of stable matrices is not an upper bound for the solution of , see Section \[sec:numexp\]. We close the section with a remark that emphasizes the difficulty in solving over .
[In view of Corollary \[thm:reform2\], the set $\mathcal S^{n,n}$ of admissible pairs can be written as $$\begin{aligned}
\mathcal S^{n,n}
&=&\bigcup_{r=1}^n \mathcal S_r^{n,n}.\end{aligned}$$ Hence we have that $$\eqref{mainprob} = \min_{r=1,2,\ldots,n}\eqref{restprob}.$$ To compute a solution of , a possible way is therefore to solve $n$ rank-constrained problems . For $n$ large, this would be rather costly as it makes the corresponding algorithm for $n$ times more expensive than for . However, in practice, the rank $r$ has to be chosen close to the (numerical) rank of $E$ so that it can be estimated from the input data. Also, as we will see in Section \[sec:numexp\], the error tends to change monotonically with $r$ (first it decreases as $r$ increases –unless $r=1$ is the best value– and then increases after having achieved the best value for $r$) which could also be used to avoid computing the solutions for all $r$. ]{}
Algorithmic solution for {#sec:algo}
=========================
To solve , we use a block coordinate descent method and optimize alternatively over $W$, $T$ and $(U,B)$. For $T$, $U$ and $B$ fixed, the optimal $W$ can be computed using least squares, and similarly for the optimal $T$. Note that the least squares problem in $W$ (resp. $T$) can be solved independently for each row (resp. each column) To update $(U,B)$ for $W$ and $T$ fixed, we use the fast gradient method from [@GilKS18a] (it can be easily adapted by fixing $S$ to the identity and modifying the gradients).
An initialization $W \in \mathbb{R}^{n \times n}, T \in \mathbb{R}^{n \times n}, U \in \mathbb{R}^{r \times r}, B \in \mathbb{R}^{r \times r}$.
An approximate solution $(W,T,U,B)$ to .
$W \leftarrow \operatorname{argmin}_{Y} f(Y,T,U,B)$; *% Least squares problem*
$T \leftarrow \operatorname{argmin}_{X} f(W,X,U,B)$; *% Least squares problem*
Apply a few steps of the fast gradient method from [@GilKS18a] on $$\min_{(U,B) \text{ s.t. } U^TU=I_r, \|B\| \leq 1} f(W,T,U,B)$$ to update $(U,B)$.
Initialization
--------------
For simplicity, we only consider one initialization scheme in this paper which is similar to the one that performed best in [@GilKS18a]. However, it is important to keep in mind that Algorithm \[bcd\] is sensitive to initialization and that coming up with good initialization schemes is a topic of further research.
We take $W=T=I_n$ and $(U,B)$ as the optimal solution of $$\min_{(U,B) \text{ s.t. } U^TU=I_r, \|B\| \leq 1} {\|A_{1:r,1:r}-UB\|}_F^2.$$ In this particular case, it can be computed explicitly using the polar decomposition of $A_{1:r,1:r}$ [@GilKS18a].
Numerical experiments {#sec:numexp}
=====================
In this section, we apply Algorithm \[bcd\] on several examples. As far as we know, there does not exist any other algorithm to stabilize matrix pairs (in the discrete case) hence we cannot compare it to another technique. However, when $E=I_n$, we will compare to the fast gradient method of [@GilKS18a] which provides a nearby stable matrix (but does not allow to modify $E$). Our code is available from <https://sites.google.com/site/nicolasgillis/> and the numerical examples presented below can be directly run from this online code. All tests are preformed using Matlab R2015a on a laptop Intel CORE i7-7500U CPU @2.7GHz 24Go RAM. Algorithm \[bcd\] runs in $O(n^3)$ operations per iteration, including projections onto the set of orthogonal matrices, the resolution of the least squares problem and all necessary matrix-matrix products. Hence Algorithm \[bcd\] can be applied on a standard laptop with $n$ up to a thousand (each iteration on the specified laptop takes about 10 seconds for $r=n$).
Grcar matrix
------------
Let us first consider the pair $(I_n,A)$ where $A$ is the Grcar matrix of dimension $n$ and order $k$ [@GilS17]. For $n = 10$ and $k=3$, the nearest stable matrix found in [@GilKS18a] has relative error ${\|A-\hat A\|}_F^2 = 3.88$. Applying Algorithm \[bcd\] with $r=n$, we obtain a matrix pair $(\hat E, \hat A)$ such that ${\|A-\hat A\|}_F^2 + {\|E-\hat E\|}_F^2 = 1.88$. Figure \[fig:grcar\] displays the evolution of the error (left) and the eigenvalues of the solutions (right).
![ (Left) Evolution of the error $\|E-\hat E\|_F^2 + \|A-\hat A\|_F^2$ for the Grcar matrix of dimension 10 and order 3 in the matrix and matrix pair cases (in the matrix case, $\hat E = I_n$). (Right) Location of the eigenvalues of $A$, and of the solutions in the matrix case and in the matrix pair case. \[fig:grcar\]](figure/Grcar-n10k3.png){width="\textwidth"}
We observe that allowing $\hat E$ to be different than the identify matrix allows the matrix pair $(\hat E, \hat A)$ to be much closer to $(I_n,A)$ and have rather different eigenvalues.
#### Effect of the dimension $n$
Let us perform the same experiment as above except that we increase the value of $n$. Table \[tab:grcarn\] compares the error of the nearest stable matrix and of the nearest stable matrix pair. As $n$ increases, the nearest stable matrix pair allows to decrease the error of approximation.
--------------- --------- --------- ---------- ---------- ----------
$n= 5$ $n= 10$ $n= 20$ $n= 50$ $n= 100$
(30 s.) (60 s.) (120 s.) (300 s.) (600 s.)
Stable matrix 1.76 3.88 15.89 68.18 160.00
Stable pair 1.16 1.88 3.02 8.69 20.41
--------------- --------- --------- ---------- ---------- ----------
: Comparison of the error for Grcar matrices $A$ of order $k=3$ and $E = I_n$ for different values of $n$.[]{data-label="tab:grcarn"}
#### Effect of $r=\operatorname{rank}(\hat E)$ and $\operatorname{rank}(E)$
Let us now perform more extensive numerical experiments on the Grcar matrix of dimension $n=10$ of order $3$. Let us fix $0 \leq p \leq n-1$ and define $E(i,i) = 1$ for $i > p$ otherwise $E(i,j) = 0$ (that is, $E$ is the identity matrix where $p$ diagonal entries have been set to zero) with $\operatorname{rank}(E) = n-p$. Table \[tab:grcarerr\] gives the error of the solution obtained by Algorithm \[bcd\] for $r=1,2,\dots,n$.
$\operatorname{rank}(E)$ $r= 1$ $r= 2$ $r= 3$ $r= 4$ $r= 5$ $r= 6$ $r= 7$ $r= 8$ $r= 9$ $r=10$
-------------------------- ---------- ---------- ---------- -------- ---------- ---------- -------- -------- ---------- ----------
10 9.02 8.05 7.09 6.20 5.44 4.63 3.94 3.16 2.16 **1.88**
9 8.04 7.08 6.13 5.33 4.61 3.83 3.16 2.16 1.57 **1.36**
8 7.05 6.10 5.17 5.16 4.16 3.16 2.16 1.46 **1.37** 1.42
7 6.05 5.13 4.22 4.16 2.83 2.17 1.57 1.52 **1.44** 1.44
6 5.07 4.17 3.27 3.16 2.04 **1.32** 1.34 1.69 1.69 1.91
5 4.09 3.24 2.34 1.76 **1.20** 1.52 1.30 1.56 1.74 3.13
4 3.12 2.26 1.49 1.25 **1.19** 1.24 1.30 1.68 2.96 2.95
3 2.13 1.31 **0.69** 1.19 1.23 1.29 1.69 2.83 2.83 2.83
2 1.15 **0.41** 1.06 1.22 1.27 1.27 1.91 2.79 2.79 2.79
1 **0.17** 0.81 1.21 1.21 1.36 1.22 2.71 2.72 2.72 4.33
: Comparison of the error for Grcar matrices $A$ with $k=3$, and $E(i,i) = 1$ for $i > p$ otherwise $E(i,j) = 0$.[]{data-label="tab:grcarerr"}
We observe that
- In 6 out of the 10 cases, using $r = \operatorname{rank}(E)$ provides the best solution. In 3 out of the 10 cases, using $r = \operatorname{rank}(E)+1$ provides the best solution, and in one case $r = \operatorname{rank}(E)+2$ provides the best solution. This illsutartes the fact that the best value for $r$ should be close to the (numerical) rank of $E$. (Of course, since we use a single initialization, there is no guarantee that the error in Table \[tab:grcarerr\] is the smallest possible.)
- In all cases, the error behaves monotonically, that is, it increases as the value of $r$ goes away from the best value.
The two observations above could be used in practice to tune effectively the value of $r$: start from a value close to the numerical rank of $E$, then try nearby values until the error increases.
Table \[tab:grcartim\] gives the computational time for the different cases. We use the following stopping criterion: $$e(i) - e(i+1) < 10^{-8} e(i),$$ where $e(i)$ is the error obtained at the $i$th iteration, and a time limit of 60 seconds.
$\operatorname{rank}(E)$ $r= 1$ $r= 2$ $r= 3$ $r= 4$ $r= 5$ $r= 6$ $r= 7$ $r= 8$ $r= 9$ $r=10$
-------------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- --------
10 0.59 1.00 0.72 0.56 2.44 2.20 2.08 43.92 13.41 49.16
9 0.23 0.58 0.44 2.34 2.53 17.89 3.25 24.19 43.34 47.72
8 0.22 0.89 0.42 0.36 13.16 1.16 16.34 3.67 8.09 11.02
7 0.16 0.81 0.77 0.61 13.63 10.20 60 60 60 60
6 0.19 0.67 0.73 1.19 27.42 60 60 60 60 60
5 0.17 0.83 0.59 4.22 60 60 39.17 60 60 60
4 0.25 1.20 0.39 29.11 60 21.80 60 60 40.81 60
3 0.14 0.28 0.13 31.08 60 60 60 55.08 60 60
2 0.06 0.17 17.50 60 60 60 60 60 60 60
1 0.02 23.56 26.61 48.11 60 60 37.95 60 60 60
: Time in seconds to compute the solution obtained in Table \[tab:grcarerr\]. The time limit is 60 seconds. []{data-label="tab:grcartim"}
We observe that the algorithm converges much faster when $r$ is small. This can be partly explained by the smaller number of variables, being $2n^2 + 2r^2$.
Scaled all-one matrix
---------------------
In this section, we perform a similar experiment than in the previous section with $A = \alpha ee^T$ where $e$ is the vector of all ones, which is an example from [@GP2018]. For $\alpha > 1/n$, the matrix is unstable. For $1/n \leq \alpha \leq 2/n$, the nearest stable matrix is $ee^T/n$.
Let us take $n=10$ and $\alpha = 2/n = 0.2$ for which the nearest stable matrix is $A = 0.1 ee^T$ with error 1. The nearest stable matrix pair computed by Algorithm \[bcd\] is given by $A = 0.15 ee^T$ and $E = I_n + 0.05 ee^T$ with error $\frac{1}{2}$. As for the Grcar matrix, allowing $E$ to be different from the identity matrix allows to reduce the error in approximating $(I_n,A)$ significantly (by a factor of two).
[^1]: Department of Mathematics and Operational Research, Faculté Polytechnique, Université de Mons, Rue de Houdain 9, 7000 Mons, Belgium; `nicolas.gillis@umons.ac.be`. N. Gillis acknowledges the support of the ERC (starting grant n$^\text{o}$ 679515) and F.R.S.-FNRS (incentive grant for scientific research n$^\text{o}$ F.4501.16).
[^2]: TU Berlin, Institut f${\rm \ddot{u}}$r Mathematik, Stra[ß]{}e des 17. Juni 136, 10623 Berlin, Germany; `karow@math.tu-berlin.de. `
[^3]: Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India; `punit.sharma@maths.iitd.ac.in`.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The role played by a kinetic barrier originated by out-of-plane step edge diffusion, introduced in \[Leal *et al.*, [J. Phys. Condens. Matter **23**, 292201 (2011)](https://doi.org/10.1088/0953-8984/23/29/292201)\], is investigated in the Wolf-Villain and Das Sarma-Tamborenea models with short range diffusion. Using large-scale simulations, we observe that this barrier is sufficient to produce growth instability, forming quasiregular mounds in one and two dimensions. The characteristic surface length saturates quickly indicating a uncorrelated growth of the three-dimensional structures, which is also confirmed by a growth exponent $\beta=1/2$. The out-of-plane particle current shows a large reduction of the downward flux in the presence of the kinetic barrier enhancing, consequently, the net upward diffusion and the formation of three-dimensional self-assembled structures.'
address: 'Departamento de Estatística, Física e Matemática, Campus Alto Paraopeba, Universidade Federal de São João Del-Rei, 36420-000, Ouro Branco, MG, Brazil'
author:
- 'Anderson J. Pereira'
- 'Sidiney G. Alves'
- 'Silvio C. Ferreira'
title: 'Effects of a kinetic barrier on limited-mobility interface growth models'
---
=1
Introduction
============
A rich variety of morphologies can be observed during far-from-equilibrium growth processes and many of them with potential for technological applications [@michely2004islands; @Evans2006; @barabasi; @meakin]. Growth instability can induce three-dimensional mound-like patterns in different types of films such as metals [@Jorritsma; @Caspersen; @Han], inorganic [@Johnson; @Tadayyon] and organic [@Zorba; @Hlawacek] semiconductors materials to cite only a few examples. Such a growth instability has been mainly attributed to the presence of Ehrlich-Schwoebel (ES) step barriers [@Ehrlich; @Schwoebel] that reduce the rate with which atoms move downwardly on the edges of terraces leading to net uphill flows. Growth instabilities can also emerge from topologically induced uphill currents which depend on the crystalline structure [@Kanjanaput2010] or from fast diffusion on terrace edges [@Murty2003; @Pierre-Louis1999] among other mechanisms [@Evans2006; @michely2004islands]. The existence of ES barriers is supported by molecular dynamic simulations [@Yang].
{width="0.35\linewidth"} {width="0.35\linewidth"}
Discrete solid-on-solid (SOS) growth models constitute an important approach to investigate the dynamic of kinetic roughening and morphological properties of interfaces. The rules are easily implemented in a discrete space (lattices) rid of overhangs and bulk voids. The role played by ES barriers has been investigated in models with thermally activated diffusion [@Evans2006; @michely2004islands] being the Clark-Vvedenski (CV) model [@CV_PRL; @Clarke1988] one of the simplest examples, in which any surface adatom can move according to an Arrhenius diffusion coefficient $D\sim
\exp(-E/k_B T)$ [@barabasi] where $E$ is an energy activation barrier to be overcome in a diffusion hopping. An ES barrier can be included as an additional activation energy for diffusion at the edges of terraces [@Evans2006]. The effects of a step barrier of purely kinetic origin, namely simple diffusion, were investigated in an epitaxial growth model with thermally activated diffusion [@Leal_JPCM]. In this model, a particle performing an interlayer movement through steps with more than one monolayer has to diffuse along the columns, perpendicularly to the substrate, instead of attaching directly at the bottom or top of a terrace. This kinetic barrier reduces downhill currents and three-dimensional structures in the form of mounds are obtained at short-time scales even in the case of weak ES barriers where the conventional rule would not lead to mound formation.
Simple models with limited mobility can be used to investigate kinetic roughening [@barabasi; @meakin]. Wolf-Villain (WV) [@WV] and Das Sarma-Tamborenea (DT) [@DT] models, introduced to investigate molecular-beam-epitaxy (MBE) growth, are benchmarks of this class and have been intensively investigated [@Smilauer; @Milan; @Huang; @HaselwandterPRL; @HaselwandterPRE; @Sarma; @Punyindu; @wvbogo; @Xun; @Luis2019]. A variation of the CV model with limited mobility has been considered [@Aarao2010; @Aarao2013] and many features of the original model have been reproduced with this simplified version [@To2018]. Effects of a step barrier were investigated in both WV [@Rangdee] and DT [@DasSarma_SC] models introducing two additional probabilities for downward and upward interlayer diffusion with the former larger than the latter, and mound formation was observed in both models. WV and DT models without step barrier were investigated in several lattices [@Chatraphorn2001; @Kanjanaput2010] and it was found that the WV model can present topologically induced mound morphologies on some lattices but not in others while no clear evidence for three-dimensional structures was observed for DT. In one-dimension, it is widely accepted that both DT and WV models asymptotically produce self-affine surfaces belonging to nonlinear MBE [@Luis2019] and Edwards-Wilkinson [@Vvedensky] universality classes, respectively.
It was reported that a kinetic barrier alone does not induce mound morphologies in thermally activated CV-like models [@Leal_JPCM] but, instead, they exhibit kinetic roughening with exponents consistent with the nonlinear MBE universality class [@DT; @Villain; @LSarma]. Therefore, given the simplicity of limited-mobility growth models and the non-trivial effects of topologically induced uphill currents in DT and WV models, one would wonder how they respond to a barrier of purely kinetic origin. In order to fill this gap, we investigate WV and DT models with the introduction of the kinetic barrier proposed in Ref. [@Leal_JPCM]. We observed mounds in both models in 1+1 and 2+1 dimensions, being much more evident for WV model. The surface coarsening ceases quickly with the saturation of the characteristic surface length and regimes of uncorrelated mound growth are asymptotically observed. Analysis of the out-of-plane currents shows a large reduction of the downhill flux of particles, enhancing surface instabilities and mound formation.
The remaining of the paper is organized as follows. The model implementation details are presented in section \[sec:model\]. In section \[results\], we discuss the results obtained in the simulations. Our conclusions and some perspectives are drawn in the section \[conclusion\].
Models {#sec:model}
======
In all investigated models, the particles are randomly deposited on a $d$-dimensional lattice of linear size $L$ with periodic boundary conditions under the SOS condition. Results presented in this work correspond to regular chains in $d=1$ and square lattices in $d=2$. Other lattices were tested and the central conclusions remain unaltered. The height of the interface at site $i$ and time $t$ is represented by $h_i(t)$ and the initial condition is given by $h_i(0)=0$ such that the initial interface is flat.
In the WV model with a kinetic barrier [investigated in the present work]{}, the growth rule is implemented as follows. At each time step, a position $i$ is randomly chosen. A location $i'$ with the largest number of bonds that a new deposited adatom would have is determined within a set containing $i$ and its nearest-neighbors. If the initial position corresponds to the largest number of bonds ($i'\equiv i$), it is chosen as the deposition place and the simulation runs to the next step. In case of multiple options, one is chosen at random. Otherwise, the particle tries to diffuse to the neighbor $i'$ with a probability given by [@Leal_JPCM] $$\label{prob}
P_{\delta h}(i,i')=
\left\{
\begin{array}{cl}
1, & \textrm{if } ~ |\delta h|<2\\
\frac {1}{|\delta h|}, & \textrm{if } ~ |\delta h|\geq 2
\end{array}
\right.$$ where $\delta h = h_i-h_{i'}$. With probability $1 - P_{\delta h}(i,i')$ the particle remains at the site $i$. It is important to mention that Eq. is obtained [ assuming that the adatom first moves to top kink of the terrace and then start a unbiased one-dimensional random-walk normally to the initial substrate, stopping the movement if it either arrives at the bottom or return to top of the terrace. The result is the]{} solution of a non-directed one-dimensional random walk with absorbing boundaries separated by a distance $|\delta h|$ [@Shehawey]; see Fig. 1 of Ref. [@Leal_JPCM] for further details of this diffusion rule. This diffusion attempt is successively applied $N_s$ times (representing a $N_s$ diffusive steps) departing from the last position of the adatom. A unit time is defined as the deposition of $L^d$ particles.
The implementation of the DT model with kinetic barrier is similar. The difference is that diffusion to the nearest-neighbors are performed only if the adatom does not have lateral bounds and any neighbor with a number of bonds higher than 1 can be chosen with equal chance as the target site.
{width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"}\
{width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"}
![\[hh\_corr\_time\]Main panels: Height-height correlation function for the WV model at distinct times indicated in the legends for (a) one- and (b) two-dimensional substrates. The number of steps is $N_s=1$. The averages were computed over 100 independent runs. Insets: Correlation functions averaged over 1 and 10 samples for the original WV model at time $t=10^5$ showing that the oscillations observed in single samples are not due to regular structures.](gamma_wv1d.pdf "fig:"){width="0.8\linewidth"}\
![\[hh\_corr\_time\]Main panels: Height-height correlation function for the WV model at distinct times indicated in the legends for (a) one- and (b) two-dimensional substrates. The number of steps is $N_s=1$. The averages were computed over 100 independent runs. Insets: Correlation functions averaged over 1 and 10 samples for the original WV model at time $t=10^5$ showing that the oscillations observed in single samples are not due to regular structures.](gamma_wv2dNs1.pdf "fig:"){width="0.8\linewidth"}
Results
=======
The one-dimensional simulations were carried out on chains with up to $L=2^{14}$ sites and evolution times of up to $t = 10^7$. In the two-dimensional case, the simulations were done in systems of size up to $L=2^{10}$ and time up to $t = 10^6$. The averages were performed over $100$ independent runs.
Figures \[snapshot\_1d\] and \[snapshot\_2d\] show interfaces obtained in simulations in one- and two-dimensional substrates, respectively. Surfaces for the original WV and DT models without and with ($N_s=1$ or $N_s=10$) kinetic barriers are compared. In both dimensions, the irregular morphologies without a characteristic length observed in the original versions change to structures separated by valleys that present a well-defined characteristic length. We also observe that an increase in the value of $N_s$ reduces valley deepness and increases the characteristic width of the mounds. The effects of the kinetic barrier seem to be stronger in two- than one-dimension. A remarkable change in the profiles happens when just one hop to nearest-neighbors is allowed in the DT model with kinetic barrier, as can been seen in Fig. \[snapshot\_1d\](e). Surfaces become columnar with a high aspect ratio (height/width). Such a behavior is reminiscent of the very strict rule for diffusion in DT when a single lateral bound is enough to irreversibly stick the adatom on a site. In the WV case, where diffusion happens more readily, mound morphologies with quasiregular structures emerge more clearly.
![\[hh\_corr\_Ns\] Main plot: Height-height correlation function dependence with the parameter $N_s$ (indicated in the legends) for the WV model in two-dimensional substrates at a time $t=10^5$. Inset: Same as the main plot for one dimension at a time $t=10^7$. Curves correspond to averages over 100 independent samples.](gamma_wv2d1d.pdf){width="0.8\linewidth"}
A standard tool to characterize the morphology of interfaces in growth process is the height-height correlation function defined as [@Evans2006; @Murty2003; @Chatraphorn2001] $$\Gamma({\mathbf}r) = \left\langle \tilde h({\mathbf}x) \tilde h({\mathbf}x + {\mathbf}r) \right\rangle_x,$$ here $\tilde h({\mathbf}x)$ is the height interface at position ${\mathbf}x$ relative to the mean height and $\left\langle \ldots \right\rangle_x$ denotes an average over the surface. The height-height correlation for ${\mathbf}r = 0$ is related to the interface width by $$\sqrt{\left\langle\Gamma(0)\right\rangle} = w$$ here $\left\langle\ldots\right\rangle$ denotes an average over independent runs. A self-affine interface is characterized by a height-height correlation function that goes monotonically to zero while those characterized by mounds exhibit oscillatory behavior around $0$. In the latter case, the first zero of $\Gamma(r)$, denoted by $\xi$, is a characteristic lateral length of the mounds in the surface.
Figure \[hh\_corr\_time\] shows the height-height correlation function for the WV model with kinetic barrier in one- and two-dimensional substrates. The curves clearly exhibit oscillatory behavior even for averages over 100 independent samples. Conversely, the irregular oscillatory behavior observed for the original WV model shown in insets of Fig. \[hh\_corr\_time\] is lumped after averaging. Therefore, interfaces obtained with kinetic barrier are characterized by the formation of quasiregular mound structures differently from those obtained using the original model that exhibits irregular structures within the intervals of size and time we investigated. These plots also show a coarsening of the mounds represented by the first minimum displacement at the early growth times.
![Main plot: Correlation function for the DT model in two-dimensional substrates for distinct times shown in the legends and fixed $N_s=10$. Inset: Correlation function for DT model in two dimensions at a fixed time $t=10^5$ and different values of $N_s$ shown in legends. Curves correspond to averages over 100 independent samples.[]{data-label="h_corr_DT"}](gamma_dt2d_ts_Ns.pdf){width="0.8\linewidth"}
The effect of the parameter $N_s$ in WV model is shown in Fig. \[hh\_corr\_Ns\]. As indicated by the interface profiles shown in Figs. \[snapshot\_1d\] and \[snapshot\_2d\], the characteristic lateral length increases with $N_s$ in both dimensions. The correlation function for DT model follows a qualitative similar dependence with $N_s$, as can be seen in Fig. \[h\_corr\_DT\] where the effects of time and number of diffusion steps in the correlation function of the DT model are shown. However, the mounds are much less evident than those obtained in the WV model. However, the correlation functions still present the typical oscillatory behavior of mounded structures that is preserved after the averaging over 100 independent samples. Besides, the typical width of the mounds in the DT model are much smaller than those of WV. It is important to note that the correlation function of the original DT model also presents an irregular behavior as does the WV model.
![\[width\]Time evolution of the interface width $w$ for WV (main panels) and DT (insets) models grown on (a) one- and (b) two-dimensional substrates. Both simulations with the kinetic barrier (using $N_s$ values indicated in the legend) and the original version are shown. In (a), dashed and solid lines are power-laws with exponents $3/8$ and $1/2$, respectively, in both main panels and insets. In (b), the slopes of the dashed and solid lines are $1/4$ and $1/2$, respectively.](w_wvdt_1d.pdf "fig:"){width="0.8\linewidth"}\
![\[width\]Time evolution of the interface width $w$ for WV (main panels) and DT (insets) models grown on (a) one- and (b) two-dimensional substrates. Both simulations with the kinetic barrier (using $N_s$ values indicated in the legend) and the original version are shown. In (a), dashed and solid lines are power-laws with exponents $3/8$ and $1/2$, respectively, in both main panels and insets. In (b), the slopes of the dashed and solid lines are $1/4$ and $1/2$, respectively.](w_wvdt_2d.pdf "fig:"){width="0.8\linewidth"}
Figure \[width\] shows the time evolution of the interface width for both models in one and two dimensions. The main panels and insets present the results for the WV and DT models, respectively, including or not the kinetic barrier. The interface width is expected to scale as $w\sim t^\beta$ where $\beta$ is the growth exponent [@barabasi]. The short time dynamics of both WV and DT models is well described by the linear version of the MBE equation [@Villain; @LSarma] $$\frac{\partial h}{\partial t} = -\nu \nabla^{4}h + \lambda \nabla^{2} (\nabla h)^{2} + \eta,
\label{Eq1}$$ with $\lambda=0$ where $\eta$ is a non-conservative Gaussian noise [@Villain; @LSarma; @DasSarma1992]. This result is confirmed in Fig. \[width\] where the short time behavior is consistent with the growth exponents $\beta=3/8$ in $d=1$ and $\beta=1/4$ in $d=2$ expected for the linear MBE universality class [@barabasi]. It is worth to mention that these models may undergo crossovers to different universality classes in the asymptotic, depending on the dimension and model [@wvbogo; @Vvedensky; @Xun; @Chen2017; @AaraoReis2004b; @Punyindu]. The curves in Fig. \[width\] are consistent with crossovers to different universality classes at long times. One expects that DT is asymptotically consistent with the non-linear MBE equation with $\lambda>0$ [@Xun; @Luis2019; @Luis2017], for which $\beta\approx 1/3$ and $1/5$ in $d=1$ and $d=2$, respectively[^1], while crossovers to the Edwards-Wilkinson universality class with $\beta=1/4$ in $d=1$ and $\beta=0$ (logarithmic growth) in $d=2$ are expected for the WV model [@wvbogo; @Vvedensky]. The simulations with the kinetic barrier, however, departs from the original dynamics after a transient which increases with the diffusion of particles. For long times, an evolution consistent with an uncorrelated growth [described by $\frac{\partial h}{\partial t}=\eta$, characterized by a growth exponent $\beta=1/2$ [@barabasi], is observed.]{} [This observation can be rationalized as follows. At long times, mounds interact weakly since the kinetic barrier reduces drastically inter-mound diffusion. Consider the idealized case of plateaus of size $L_0$ with an infinity barrier at their edges. A particle initially adsorbed on the top of a plateau will never slide down to its bottom. So, the probability that this plateau receives $R$ particles after one unity of time (deposition of $L$ particles) is a binomial distribution $$P(R)=\binom{L}{R} p^R (1-p)^{L-R}\simeq
\frac{1}{\sqrt{2\pi L_0}} e^{-\frac{(R-L_0)^2}{2L_0}},
\label{eq:PR}$$ where $p=L_0/L$ is the probability that a particle is deposited on this terrace and $1\ll L_0\ll L$ is assumed in the Gaussian limit in right-hand side of Eq. . We argue that this situation is similar to the weakly interacting mound observed in our simulations.]{}
![Characteristic length of mounds $\xi$ for WV (main plots) and DT (insets) models with and without the kinetic barrier in (a) one- and (b) two-dimensional substrates for different values of the parameter $N_s$ indicated in the legend.\[xi\]](xi_wvdt_1d.pdf "fig:"){width="0.8\linewidth"}\
![Characteristic length of mounds $\xi$ for WV (main plots) and DT (insets) models with and without the kinetic barrier in (a) one- and (b) two-dimensional substrates for different values of the parameter $N_s$ indicated in the legend.\[xi\]](xi_wvdt_2d.pdf "fig:"){width="0.8\linewidth"}
In addition, as can be seen in Fig. \[xi\], the characteristic lateral lengths of simulations with kinetic barrier saturate after an initial transient in values that increase with the parameter $N_s$ while the models without barrier present coarsening with $\xi\sim t^{1/z}$ [@barabasi]. The saturation implies that the aspect ratio (height/width) of the mounds remains increasing with time and the surface does not present slope selection forming columnar growth. This property is also reflected in the asymptotic interface width scaling as $w\sim t^{1/2}$. As explained previously, it can be interpreted as an uncorrelated evolution of the columns, in which the 1/2 exponent comes out. The results shown in the insets of Figs. \[width\] and \[xi\] corroborate that the DT model presents the same behavior of the WV model despite of the mounds are less evident in the former.
![\[current\_1d\] Evolution of the out-of-plane current for (a) WV and (b) DT models grown in one-dimensional substrates. Models with the kinetic barrier using $N_s =1$, $2$ and $10$ steps (indicated in the legend) and the original version are shown. ](jz_wv1d.pdf "fig:"){width="0.8\linewidth"}\
![\[current\_1d\] Evolution of the out-of-plane current for (a) WV and (b) DT models grown in one-dimensional substrates. Models with the kinetic barrier using $N_s =1$, $2$ and $10$ steps (indicated in the legend) and the original version are shown. ](jz_dt1d.pdf "fig:"){width="0.8\linewidth"}
Instability and mound formation can be investigated considering the surface currents [@Siegert1994; @Krug1993]; see [@Krug1997] for details. In this work, we investigated the out-of-plane component of the current defined as [@Leal_Jstat] $$J_z = \frac{1}{N}\sum_{(i,j)} \mathrm{sgn}(\delta h) D(i,j) P_{\delta h}(i,j)$$ where $\mathrm{sgn}(x)=1$ for $x>0$, $\mathrm{sgn}(x)=-1$ for $x<0$, and $\mathrm{sgn}(0)=0$ is the definition of sign function, $P_{\delta h}(i,j)$ is given by Eq. , and $D(i,j)$ is the rate of hopping attempts from site $i$ to $j$ and depend on the investigated model. The sum runs over all $N$ pairs of nearest-neighbors of the lattice. Let $n_i$ be the number of lateral bonds of site $i$ and $n^\text{max}_i$ the largest number of bonds among the nearest-neighbors of $i$. For the WV model, $D(i,j)$ is given by $$D(i,j)=
\left\{
\begin{array}{cl}
1/q_i^\text{WV}, & \textrm{if } ~ n_j = n^\text{max}_i \textrm{ and } n_i < n^\text{max}_i\\
0, & \textrm{otherwise.}
\end{array}
\right.,$$ where $q_i^\text{WV}$ is the number of nearest-neighbors with $n^\text{max}_i$ lateral bonds. We can express $D(i,j)$ for the DT as $$D(i,j)=
\left\{
\begin{array}{cl}
1/q_i^\text{DT}, & \textrm{if } ~ n_j > 0 \textrm{ and } n_i = 0\\
0, & \textrm{otherwise.}
\end{array}
\right.,$$ where $q_i^{DT}$ is number of nearest-neighbors with at least one lateral bond. The quantity $J_z$ is the average interlayer diffusion rate per site.
=\_20pt\^3pt
-- --------- --------- -------- --------------------
WV DT WV DT
-0.0034 -0.0042 -0.015 -5$\times 10^{-5}$
-0.0011 -0.0052 -0.019 -3$\times 10^{-4}$
-0.014 -0.0053 -0.020 -5$\times 10^{-4}$
-0.090 -0.050 -0.047 -0.030
-- --------- --------- -------- --------------------
: \[fit\_parameters\] Parameters $J_\infty$ obtained in the regression using the Eq. (\[current\_fit\]) in the in the last decade of data of the out-plane current curves ($t>10^6$ for $d=1$ and $t>10^5$ for $d=2$).
The currents for simulations in $d=1$ are presented in Fig. \[current\_1d\]. All versions in both $1+1$ and $2+1$ dimensions are characterized by a current with a downward (negative) flux with the intensity decreasing monotonically. Considering the last decade of time, we estimated the current $J_\infty$ for $t\rightarrow\infty$ using a regression with a simple allometric function in the form. $$J_z = J_\infty + a t^{-\gamma}\label{current_fit},$$ where $a$ and $\gamma$ are parameters. In all cases with step barrier, we obtained asymptotic small negative currents with a non-universal value of $\gamma$. The results can be seen in table \[fit\_parameters\]. The currents for the standard models are considerably larger than in the cases with barrier. The values for the DT model with barrier are very small indicating that this current could be actually null in the asymptotic limit as observed in thermally activated diffusion models with ES step barriers [@Leal_Jstat]. In the case of the WV model, the current values may indicate the same asymptotic behavior, but our present accuracy does not allow a conclusion on this issue.
Conclusions {#conclusion}
===========
In this work, we investigate the effects of a purely kinetic barrier caused by the out-of-plane step edge diffusion [@Leal_JPCM] on limited-mobility growth models. The cases of studies were the benchmark models of Wolf-Villain [@WV] and Das Sarma-Tamborenea [@DT]. Large-scale simulations were performed considering one- and two-dimensional substrates. It was observed that the introduction of the kinetic barrier induces the formation of quasiregular mound structures differently from those obtained with the original models that forms irregular (self-affine) structures in the interface. The kinetic barrier stabilizes the mound width, leading to the formation of quasiregular structures. The interface width in models with kinetic barriers has an initial regime similar to the original models. However, a growth exponent very close to $\beta = 1/2$ is observed for asymptotically long times. Also, the characteristic lateral length saturates after a transient that depends on the number of steps that an adatom can perform before irreversibly stick in a position. These results are consistent with mounds evolving independently. The dynamics in both one- and two-dimensional substrates are characterized by a strong reduction of downward current with respect to the original models. The downward flux have an intensity decreasing monotonically to a asymptotic value that seems to be null for DT model and small for WV, being the latter possibly still subject to strong crossover effects in the present analysis.
A central contribution of this work is to show that a very simple mechanism neglected in previous analysis, in which particles also diffuse in the direction perpendicular to the substrate, is able to change markedly the surface morphology of basic growth models with limited mobility. Our results are qualitatively very similar to those obtained when an explicit step barrier, with a smaller probability to move downward, is considered [@Rangdee]. Particularly, asymptotic mound morphology has been reported for limited mobility models in $d=2$ without barriers with the application of the noise reduction method [@Chatraphorn2001]. Our results corroborate this scenario since a small perturbation induces mound instability in this kind of processes while it alone does not produce mounds in models with thermally activated diffusion [@Leal_Jstat].
We expect that the concepts investigated in this work will be applied to more sophisticated models and aid the understanding of pattern formation in film growth and the production of self-assembled structures for technological applications.
SGA and SCF thank the financial support of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Pesquisa do Estado de Minas Gerais (FAPEMIG). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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10.1051/jp1:1991114) [****, ()](http://link.aps.org/doi/10.1103/PhysRevLett.66.2348) [****, ()](https://iopscience.iop.org/article/10.1088/0305-4470/33/49/301) [****, ()](\doibase 10.1103/PhysRevLett.69.3762) [****, ()](http://dx.doi.org/10.1016/j.physa.2016.08.057
http://linkinghub.elsevier.com/retrieve/pii/S037843711630574X) [****, ()](https://link.aps.org/doi/10.1103/PhysRevE.70.031607) [****, ()](http://link.aps.org/doi/10.1103/PhysRevE.95.042801) [****, ()](http://link.aps.org/doi/10.1103/PhysRevLett.78.1082) [****, ()](http://link.aps.org/doi/10.1103/PhysRevLett.73.1517) [****, ()](https://link.aps.org/doi/10.1103/PhysRevLett.70.3271) [****, ()](http://www.tandfonline.com/doi/abs/10.1080/00018739700101498) [****, ()](http://stacks.iop.org/1742-5468/2011/i=09/a=P09018?key=crossref.72e1a82e5e15653a11bd7c44b2902efa)
[^1]: The exponents $\beta=1/3$ and $1/5$ are predictions of the one-loop renormalization group [@Villain; @LSarma]. Two-loop calculations [@Janssen], however, predict corrections where the growth exponents are slightly smaller than these values.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'One of the most important properties influencing the chemical behavior of an element is the energy released with the addition of an extra electron to the neutral atom, referred to as the electron affinity (EA). Among the remaining elements with unknown EA is astatine, the purely radioactive element 85. Astatine is the heaviest naturally occurring halogen and its isotope $^{211}$At is remarkably well suited for targeted radionuclide therapy of cancer. With the At$^-$ anion being involved in many aspects of current astatine labelling protocols, the knowledge of the electron affinity of this element is of prime importance. In addition, the EA can be used to deduce other concepts such as the electronegativity, thereby further improving the understanding of astatine’s chemistry. Here, we report the first measurement of the EA for astatine to be **[2.41578(7)]{}** eV. This result is compared to state-of-the-art relativistic quantum mechanical calculations, which require incorporation of the electron-electron correlation effects on the highest possible level. The developed technique of laser-photodetachment spectroscopy of radioisotopes opens the path for future EA measurements of other radioelements such as polonium, and eventually super-heavy elements, which are produced at a one-atom-at-a-time rate.'
author:
- 'David Leimbach$^{1,2,3*}$, Julia Sundberg$^2$, Yangyang Guo$^4$, Rizwan Ahmed$^{5}$, Jochen Ballof$^{1,6}$, Lars Bengtsson$^2$, Ferran Boix Pamies$^1$, Anastasia Borschevsky$^4$, Katerina Chrysalidis$^{1,3}$, Ephraim Eliav$^{11}$, Dmitry Fedorov$^{7}$, Valentin Fedosseev$^1$, Oliver Forstner$^{8,9}$, Nicolas Galland$^{10}$, Ronald Fernando Garcia Ruiz$^1$, Camilo Granados$^1$, Reinhard Heinke$^3$, Karl Johnston$^1$, Agota Koszorus$^1$, Ulli Köster$^{13}$, Moa K. Kristiansson$^{14}$, Yuan Liu$^{15}$, Bruce Marsh$^1$, Pavel Molkanov$^{7}$, Lukáš F. Pašteka$^{12}$, Joao Pedro Ramos$^1$, Eric Renault$^{10}$, Mikael Reponen$^{16}$, Annie Ringvall-Moberg$^{1,2}$, Ralf Erik Rossel$^1$, Dominik Studer$^3$, Adam Vernon$^{17}$, Jessica Warbinek$^{2,3}$, Jakob Welander$^2$, Klaus Wendt$^3$, Shane Wilkins$^1$, Dag Hanstorp$^2$ and Sebastian Rothe$^1$'
bibliography:
- 'bib.bib'
title: The electron affinity of astatine
---
CERN, Geneva, Switzerland
Department of Physics, University of Gothenburg, Gothenburg, Sweden
Institut für Physik, Johannes Gutenberg-Universität, Mainz, Germany
Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Groningen, The Netherlands
National Centre for Physics (NCP), Islamabad, Pakistan
Institut für Kernchemie, Johannes Gutenberg-Universität, Mainz, Germany
Petersburg Nuclear Physics Institute - NRC KI, Gatchina, Russia
Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität Jena, Germany
Helmholtz-Institut Jena, Jena, Germany
CEISAM, Université de Nantes, CNRS, Nantes, France
School of Chemistry, Tel Aviv University, Tel Aviv, Israel
Department of Physical and Theoretical Chemistry & Laboratory for Advanced Materials, Faculty of Natural Sciences, Comenius University, Bratislava, Slovakia
Institut Laue-Langevin, Grenoble, France
Department of Physics, Stockholm University, Stockholm, Sweden
Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
Department of Physics, University of Jyväskylä, Jyväskylä, Finland
School of Physics and Astronomy, The University of Manchester, Manchester, UK
Introduction {#introduction .unnumbered}
============
Chemistry is all about molecule formation through the creation or destruction of chemical bonds between atoms and relies on an in-depth understanding of the stability and properties of these molecules. Most of these properties can be traced back to the molecule’s constituents, the atoms. Thus, the intrinsic characteristics of chemical elements are of crucial importance in the formation of chemical bonds. The electron affinity (EA), one of the most fundamental atomic properties, is defined as the amount of energy released when an electron is added to a neutral atom in the gas phase. Large EA values characterize electronegative atoms, i.e. atoms that tend to attract shared electrons in chemical bonds. Hence, the EA informs about the subtle mechanisms in bond making between atoms, and it also reveals information about molecular properties such as the dipole moment or the molecular stability. Since the attraction from the nucleus is efficiently screened by the core electrons, the value of the EA is mainly determined by electron-electron correlation. Hence, negative ions are excellent systems to benchmark theoretical predictions that go beyond the independent particle model.\
The EA also enters into the definition of several concepts, notably the chemical potential within the purview of conceptual density functional theory (DFT), promoted by Robert G. Parr[@Parr1978], and the chemical hardness which is the core of the hard and soft acids and bases (HSAB) theory, introduced by Ralph G. Pearson in the early 1960s[@Pearson1963]. Robert S. Mulliken used the EA in combination with the ionization energy (IE), the minimum amount of energy required to remove an electron from an isolated neutral gaseous atom, to develop a scale for quantifying the electronegativity of the elements[@Mulliken1934]. The usefulness of these concepts for chemists, especially in the field of reactivity, has been amply demonstrated in recent decades[@Geerlings2003; @Chattaraj2006].\
The atomic IEs, which essentially are determined by the Coulomb attraction between the electrons and the nucleus, show a specific and well understood variation along the periodic table of elements. Starting from lowest values in the lower left corner at the heaviest alkalines, a mostly steady trend towards higher values is observed both towards ligther elements with similar chemical behaviour in one column and along rows to the right side of the chart with halogenes and noble gases, with only few exceptions. Conversely, the EAs display comparably strong irregularities and variations across the periodic table, as shown in Fig. \[fig: PT\].
A number of elements such as all the noble gases do not form stable negative ions at all, and thus have negative EAs. The group of elements with the largest EAs are the halogens. As in most other groups of elements, no monotonic trend is observed here when progressing along the rows of the periodic table, with chlorine exhibiting the largest EA () of all elements[@AndHauHot99; @Thorium].\
The EA of the heaviest naturally occurring elements in the halogen group, astatine, has not been measured to date. Indeed, little is known of the chemistry of this rare element: it is not only one of the rarest of all naturally occurring elements[@asimov], but the minute amounts that can be produced artificially prevent the use of conventional spectroscopic tools. For instance, while astatine was discovered in the 1940s[@Corson1940; @Thornton2019], it is only recently that the IE of astatine was measured through a sophisticated on-line laser-ionization spectroscopy experiment at CERN-ISOLDE[@Rothe_2013].\
However, the EA(At) has been predicted with various quantum mechanical methods[@SiFis18; @FinPet19; @Mitin2006Two-componentMethods; @LiZhaAnd12; @Borschevsky2015IonizationAt; @Sergentu2016; @ChaLiDon10]. Hence, an experimental determination of EA(At) is of fundamental interest, both to test sophisticated atomic theories and to gain precise knowledge about the chemical properties of this element. The measurement of the EA(At) is also of practical interest regarding the envisaged medical applications of astatine, since its chemical compounds are currently studied for use in cancer treatment: $^{211}$At, available in nanogram quantities only through synthetic production methods, is a most promising candidate for radiopharmaceutical applications via targeted alpha therapy (TAT)[@Zalutsky2011; @MulfordTAT; @teze:in2p3-01529705], due to its favorable half-life of about and its cumulative $\alpha$-particle emission yield of . However, in order to successfully develop efficient radiopharmaceuticals, a better understanding of astatine’s basic chemical properties is required[@Wilbur2013].\
The interest in the experimental determination of the EA notably lies in current labelling protocols that aim at binding astatine to tumor-targeting biomolecules: in many cases, the chemical reactions involve an aqueous astatine solution in which the astatide anion (At$^-$) readily forms. In addition, a current problem for the investigated $^{211}$At-radiopharmaceuticals is the significant *in vivo* de-labelling, releasing At$^-$ that could damage healthy tissues and organs of the patient[@teze:in2p3-01529705; @Vaidyanathan2008; @Wilbur2008]. In order to describe these reaction kinetics as well as the stability of astatine compounds, knowledge of the electron binding energy of the atomic anion, i.e. the EA, is required.\
In this paper, we present the first experimental determination of the electron affinity of astatine. The measured value is then compared to independent results from state-of-the-art relativistic quantum mechanical calculations carried out alongside the measurement.
Results {#results .unnumbered}
=======
Laser photodetachment of astatine.
----------------------------------
Due to its scarcity and short half-life, artificial production of astatine is required to perform any experiment on this element. Thus, a laser photodetachment threshold spectrometer was coupled to an on-line isotope separator at the CERN-ISOLDE radioactive ion beam facility[@Catherall_2017]. Here, At$^-$ ions were produced through nuclear spallation reactions of thorium nuclei, induced by a bombardement of highly energetic proton projectiles and subsequently ionized in a negative surface ion source coupled to a mass separator (see Fig. \[fig:RIB\] in the Methods section). A negative ion beam of $^{211}$At was extracted and superimposed with a laser beam in the GANDALPH spectrometer (Fig. \[fig:GANDALPH\]). The yield of neutral atoms produced in the photodetachment process, ${\mathrm{At}^- + \mathrm{h}\nu} \rightarrow \mathrm{At} + \mathrm{e}^-$, was recorded as a function of the photon energy $\mathrm{h}\nu$, where $\nu$ is the laser frequency and $\mathrm{h}$ is Planck’s constant.
\[h\]
The general behavior of the photodetachment cross section $\sigma$ just above the threshold is described by Wigner’s law[@Wigner1948OnThresholds]: $\sigma= a+b\cdot E^{l+1/2}$, where $a$ is the background level, $b$ the strength of the photodetachment process, $l$ the orbital angular momentum quantum number of the outgoing electron, $E=E_\mathrm{photon}-\mathrm{EA}$ is the energy of the ejected electron and $E_\mathrm{photon}=\mathrm{h}\nu$ the photon energy.\
In At$^-$, the electron is detached from a $p$-state. Close to the threshold, the angular momentum of the outgoing electron will then be $l=0$ due to the selection rules ($\Delta l= \pm 1$) and the centrifugal barrier preventing the emission of a $d$-wave electron ($l=2$)[@Pegg_2004]. The ion is a closed shell system with no internal structure. The ground state $6p^{5}~^{2}P_{3/2}$ of the $^{211}$At atom, on the other hand, with a total angular momentum of $J={3/2}$ and nuclear spin $I=9/2$, is split into four hyperfine levels. This splitting was recently measured with high precision by Cubiss *et al.*[@Cubiss_2018]. The relative strengths of these four photodetachment channels are given by the multiplicity of the final hyperfine structure levels, i.e. $2F+1$, where $F=I+J$ is the total angular momentum of the atom, spanning from $|I - J|$ to $|I + J|$, i.e. 3,4,5,6[@HanGus92]. The energy dependence of the cross section for photodetachment of astatine near the threshold can be described by the function
$$\sigma(E_\mathrm{photon}) = a + b \sum_{F = 3}^6 (2F+1) \sqrt{E_\mathrm{photon} - (\mathrm{EA} + \mathrm{E_{hfs,F}})}~\Theta\mathopen{}\left(E_\mathrm{photon}- (\mathrm{EA} + \mathrm{E_{hfs,F}})\right)
\label{eq:cross_section2}$$
where $\Theta\mathopen{}\left(E- (\mathrm{EA} + \mathrm{E_{hfs,F}})\right)$ is a heaviside function and $\mathrm{E_{hfs,F}}$ is the energy of the hyperfine levels of the $^{211}$At atomic ground state, differing by less than between the contributing levels.\
\[h\]
The photon energy (i.e. laser frequency) was scanned from below the threshold to well above all four hyperfine levels in the ground state of $^{211}$At. In total, six threshold scans were performed with laser and ion beam co- and counter-propagating, respectively. Fig. \[fig:Unshifted\_EA\] shows the measured neutralization cross section $\sigma(E_\mathrm{photon})$ as a function of the photon energy, corrected for the Doppler shift, for the sum of all threshold scans with co-propagating ion and laser beams.\
The statistical error of the measurement is dominated by the laser bandwidth of , corresponding to . The contribution to the statistical uncertainty from all other effects are smaller than , as discussed further in the Methods section, and can hence be neglected. Systematic errors arise due to instabilities of the ion beam energy and the determination of the photon energy. The combined systematic error of photon energy and beam energy is estimated to be smaller than by comparing two reference measurements of stable $^{127}$I which were performed before and after the experiment on astatine, under the same experimental conditions.\
Including both systematic and statistical errors, the resulting value of EA(At), calculated by the geometric mean of the photodetachment thresholds measured in the co- and counter-propagating geometries, was determined to be .
Theoretical Calculation.
------------------------
Alongside the measurements, state-of-the-art calculations of the electron affinities of astatine and of its lighter homologue, iodine ($^{127}$I) were carried out. The results for EA(I) served to assess the performance and the expected accuracy of the computational method. The calculations were carried out with the DIRAC15 program package[@DIRAC15] using the single reference coupled cluster approach in the framework of the Dirac-Coulomb Hamiltonian (DC-CCSD(T)), which is considered to be extremely powerful for treatment of heavy many-electron systems. Large, saturated basis sets[@Dya06] were used in these calculations, and extrapolation to the complete basis set limit was performed. The correction from perturbative to the full triple excitations, +$\Delta$T, and the contribution of the perturbative quadruple excitations, +(Q), were evaluated[@PasEliBor17]. To further improve the precision we have also accounted for the Breit interaction and the quantum electrodynamics (QED) contributions; the latter were calculated using the model Lamb shift operator (MLSO) of Shabaev *et al.*[@ShaTupYer15]. Further computational details can be found in the Methods section. The contributions of higher order excitations and Breit and QED corrections are added to the DC-CCSD(T) EAs to obtain the final values.
\[h\]
Method EA(I)/eV EA(At)/eV
------------------ ------------------------------------------------ -----------------------------
DC-CCSD(T) 3.040 2.401
+$\Delta$T(Q) 0.008 0.007
+Breit 0.003 0.003
+QED 0.003 0.003
**Final theor.** **3.055(16)** **2.414(16)**
Exp. 3.059 0463(38)[@Pel_ez_2009] **[2.41578(7)]{}**
The computational scheme outlined above was previously applied to the determination of the IE and the EA of gold, yielding accuracy[@PasEliBor17]. Using our knowledge of the magnitude of the various effects, we are able to set a conservative uncertainty of on the computed values (see Methods section for further details). Hence, the expected value of the EA(At) from the theoretical calculations is . The results for iodine and astatine, including the break-down of the various higher order contributions are presented in Tab. \[table1\] and compared to the experimental value. The final result of the electron affinity calculation for iodine lies within of the measured value of [@Pel_ez_2009].
Discussion {#discussion .unnumbered}
==========
Over the years, many attempts were made to calculate the EA of astatine. However, the high atomic number and thus the need of refined treatments of relativity as well as the dominance of the electron correlation effects made this a challenging task. With the given uncertainties, our computed value is in excellent agreement with the experiment. This clearly demonstrates that careful, systematic, and as complete as possible inclusion of higher-order correlation and relativistic contributions is necessary for achieving benchmark accuracy. Hence, our measured EA(At) represents a sharp test for assessing theoretical methods used to study the chemistry of heavy and super-heavy elements. For a more detailed comparison of our computed results with previous theoretical investigations we refer the reader to the Methods section.\
Our result of the EA of astatine, , indicates that among the halogen elements, astatine has the lowest EA. On the other hand, its EA remains larger than the measured values of all elements in other groups of the periodic table. Therefore, this value is consistent with the propensity of halogens to complete their valence shell on gaining one extra electron. For the halogen elements, the significance of large EAs is the strong tendency to form anions in aqueous solution. In fact, the redox potential associated with the formation of At$^-$ is primarily determined by its EA, and to a lesser extent by the difference of Gibbs’ free energy of solvation between the anion and the neutral atom. In addition to the EA, the IE contributes also to the determination of the nature of elemental forms of astatine in aqueous solutions: the Pourbaix (potential/pH) diagram of astatine shows coexistence of the and ions, whose dominance domains are governed by the redox potential E$^\circ$(/), which directly depends on the EA(At) and IE(At)[@Champion2010; @Champion2011].\
The usefulness of the EA for a better understanding of astatine’s chemistry is also shown through the deduction of the electronegativity, softness, hardness, and the electrophilicity index, which are shown in Tab. \[tab:theory\_comp\], together with the respective definitions. The electronegativity of astatine is determined to be $\chi _M= $ according to the Mulliken scale, which is significantly lower than that of hydrogen ($\chi _M=$ ), supporting the calculated bond polarization towards the hydrogen atom in the molecule[@Pilm2014; @SAUE1996]. Hence, it must be named hydride instead of hydrogen halide as opposed to all other halogen-hydrogen molecules, where the halogen is usually the negatively charged atom.
Property Definition Value
------------------- ------------------------------------ ---------------
Electron affinity $EA$
Ionization energy $IE$ [@Rothe_2013]
Electronegativity $\chi_M =\frac{IE+EA}{2}$
Hardness $\eta =\frac{IE-EA}{2}$
Softness $S=\frac{1}{2\eta}$
Electrophilicity $\omega = \frac{\chi^2_M}{2 \eta}$
Additionally, the intermediate value of $\chi _M$(At) between the electronegativities reported for boron () and carbon () atoms, allows us to anticipate different polarizations for At-B and At-C bonds. This simple analysis is of high relevance for the use of astatine in nuclear medicine. The applications in targeted alpha therapy are currently hindered by the rapid de-astatination of carrier-targeting agents that occurs *in vivo*. In radiosynthetic protocols[@Vaidyanathan2008; @Wilbur2008], most reported biomolecules of interest have been labeled with $^{211}$At by formation of At-C or At-B bonds. The greater stability observed *in vivo* for the At-B bonds could be related to the polarization of those bonds towards the astatine atom[@AYED2016156]. The electrophilicity index is particularly relevant in view of the currently prevalent approach for the $^{211}$At-radiolabelling, which is to bind astatine to carrier molecules through an electrophilic substitution[@Vaidyanathan2008; @Wilbur2008]. In addition, recent studies have illustrated how the electrophilicity of the astatine atom modulates the ability of astatinated compounds to form stabilizing molecular interactions known as halogen bonds[@Graton2018; @GaMontavon2018]. The moderate value of hardness, $\eta$(At) = , is consistent with the observed high affinity of astatine in direct attachment experiments with proteins bearing soft sulfur donor groups[@VISSER1981905], according to the hard and soft (Lewis) acids and bases (HSAB) theory ($\eta$(S) = for the S atom[@Pearson1988]).\
The list of chemical descriptors presented in Tab. \[tab:theory\_comp\] represents a significant advance over the computed data reported by Paul Geerlings and co-workers[@Giju2005], and may be regarded as basic properties which will serve as the foundation for the design and the assessment of innovative astatine radiopharmaceuticals by theoretical and experimental chemists.
Conclusion {#conclusion .unnumbered}
==========
We have carried out the first measurement of the electron affinity of astatine and determined it to be EA(At)$=$. In addition, relativistic calculations carried out alongside the experiment are in excellent agreement with the experimental results, supporting the reliability and accuracy of both the experimental technique and the theoretical description. The EA of astatine is thus an excellent case for benchmarking theoretical models in atomic physics since it requires a full relativistic many-body treatment that also includes Breit and QED effects. These theoretical models can then be applied to the chemistry of elements heavier than astatine.\
By combining the present result with the recent measurement of the ionization energy of astatine[@Rothe_2013], we were able to determine several fundamental chemical properties of this element: namely the electronegativity, softness, hardness and electrophilicity. For instance, it can be concluded from our results, that in the astatine-hydrogen molecule, in contrary to other hydrogen halides, the hydrogen atom is more electro-negative than the halide. Hence, according to chemical nomenclature this molecule should be called astatine hydride rather than hydrogen astatide.\
As $^{211}$At is a promising candidate for targeted alpha therapy, these properties have direct implications for its use in cancer treatments. Most of $^{211}$At-radiopharmaceuticals suffer from *in vivo* release of astatide (At$^{-}$) and the development of radiosynthetic procedures so far is severely hampered by the limited knowledge of the chemical properties of this element. Hence, the new information about astatine’s chemical properties presented here will be of great importance in the development of innovative radio-labelling protocols.\
Finally, the on-line technique presented in this work enables further EA measurements of artificially produced, short-lived radioactive elements with high precision. Furthermore, our theoretical methods were demonstrated to be capable of accurately treating heavy elements with a high number of electrons, paving the way for both experimental and theoretical studies of superheavy elements.
Negative astatine ions.
-----------------------
Astatine isotopes were produced at the CERN-ISOLDE radioactive ion beam facility[@Catherall_2017]. A proton beam with an energy of provided by the CERN accelerator complex impinged onto a thick Th/Ta mixed foil target, which was resistively heated to . A schematic view of this process is given in Fig. \[fig:RIB\]. The reaction products diffused from the target matrix and effused into an ISOLDE-MK4 negative surface ion source[@VOSICKI1981307], comprised of a hot tantalum transfer tube and a surface ionizer pellet heated to .
\[h\]
Thermionic electrons emitted from the hot surface were deflected with a permanent magnetic field and absorbed in a dedicated electron collector. Negative ions produced on the hot surface were accelerated across a extraction potential and thereafter directed through the ISOLDE general purpose mass separator magnet (GPS). The resolution of the mass separator was sufficient to select a single isobar, which in our case was $^{211}$At.\
In order to ensure stable astatine beam intensity throughout the experiments, the pulsed proton impact on the target was distributed equidistant in time with an average current of about . An average ion current of about () of $^{211}$At$^-$ was measured using a Faraday cup (FC) inserted in the beam path just before the experimental chamber.
Laser setup.
------------
The phototodetachment experiment was performed using a part of the ISOLDE RILIS (Resonance Ionization Laser Ion Source) laser system which normally serves for production of positively charged ion beams[@Fedosseev_2017]. In particular, laser radiation tunable in the range of to ( to ) was generated by a commercial dye laser (*Sirah Laser- und Plasmatechnik GmbH Credo Dye*) operated with an ethanol solution of Coumarin 503 dye. This laser was pumped by the third harmonic output () of a pulsed Nd:YAG INNOSLAB laser (*EdgeWave GmbH, model CX16III-OE*) with a pulse repetition rate. Beam delivering optics comprising a set of lenses and mirrors were installed to transport the dye laser beam from the RILIS laboratory to the GANDALPH photodetachment apparatus over a distance of about . In the laser-ion beam interaction region, the laser power was in the range of 20-. Typical values of the spectral bandwidth and pulse duration emitted by the dye laser were and , respectively. The laser radiation frequency was scanned in the range of - (-), determined according to earlier theoretical predictions of the EA(At)[@Borschevsky2015IonizationAt]. The photon energy of the laser radiation was measured continuously using a wavelength meter (WS7 model from HighFinesse/Ångstrom).
Collinear laser photodetachment threshold spectroscopy with GANDALPH.
---------------------------------------------------------------------
The Gothenburg ANion Detector for Affinity measurements by Laser PHotodetachment (GANDALPH), illustrated in Fig. \[fig:GANDALPH\], is a detector designed for measurements of the EA of radioactive elements by collinear laser photodetachment[@iodine128; @Leimbach_EMIS]. Electrostatic beam steering and ion optical elements are used to superimpose a continuous negative ion beam with a pulsed laser beam within the interaction region of the GANDALPH spectrometer, which is defined by two apertures of diameter placed apart. The experimental layout allows both co- and counter-propagating geometries for laser and ion beams.\
When a negative ion absorbs a photon of sufficient energy, its extra electron can be detached, creating a fast moving neutral atom. The Doppler shift resulting from the velocity of the ion beam in reference to the detector and laser rest frame, can be eliminated to all orders by taking the geometrical mean of the measurements which are recorded in co- and counter-propagating geometry of the laser and the ion beam, respectively.\
Subsequent to the interaction region, all charged particles are deflected into either a FC or a channel electron multiplier (CEM), allowing for continuous monitoring of the ion beam intensity. Neutral atoms proceed forward and impinge on a target made of a graphene coated quartz plate[@Warbinek; @iodine128; @Hanstorp_1992].\
Secondary electrons created by the impact of the neutral atoms on the target are extracted and deflected into a second CEM (*DeTech Channeltron XP-2334*), placed off-axis and biased with a potential of . The signal originating from the CEM is amplified with a fast pulse amplifier (*FAST TA2000B-2*) by a factor of 40 and fed into a gated photon counter (*Stanford Research Systems SR400*) connected to a computer. A data acquisition cycle is triggered by the signal of the photoelectrons resulting from the laser pulse impinging on the glass plate target. Due to the time of flight from the interaction region to the glass plate, the neutral atoms created in the photodetachment proccess arrive in the time window - after the photon impact. Hence, the data acquisition is set to record the signal within this time window after the trigger. Background measurements are performed simultaneously by setting a second measurement gate of the same width but delayed by microseconds after the laser pulse.\
We estimate the transmission from the FC positioned in the chamber in front of GANDALPH to the detectors placed after the interaction region to be $\approx$1%, calculated from the initial intensity of before the setup and the ion velocity (), derived from $E_{kin}= \frac{1}{2}mv^2$. This means that there were only 0.1 ions on average in the interaction region. Nevertheless, we observed a photodetachment signal as high as of neutralized $^{211}$At in the GANDALPH beam-line when the photon energy was tuned well above the photodetachment threshold. Under these conditions, the combined neutralization and detection efficiency for an ion in the interaction region, which was illuminated by the repitition rate pulsed laser light, was .
Accuracy of EA measurements.
----------------------------
The uncertainty in our experiment is dominated by the laser bandwidth of , corresponding to [@Fedosseev_2017]. In addition, there are several minor effects contributing to the uncertainty: for an surface ionizer, as used in this experiment, the energy spread has been determined to be of the order of [@Kashihira1977SourceIons]. This implies a velocity spread of the ions which is compressed due to the acceleration over a high potential in the subsequent ion beam extraction process[@Kaufman1976High-resolutionBeams]. The compressed velocity spread of the ions is given by the expression $\Delta v={\Delta W} / {\sqrt{2mW}}$, where $m$ is the ion mass, $\Delta W$ the energy spread of the ions and $W$ the kinetic energy of the ion beam[@HANSTORP1995165]. The velocity spread of the ion beam can be converted to a spread of the frequency of the laser light of $\Delta \nu = {\Delta v} /{\lambda}$ seen by the ions. This results in a frequency Doppler broadening of only a few MHz in the fast ion beam. The divergence of the ion and laser beams and the interaction time will also contribute to the broadening. However, this accumulates to uncertainties of less than . Consequently, the uncertainties arising from these minor effects could be ignored and only the laser bandwidth of needs to be considered.\
In addition to these statistical errors, some systematic uncertainties arise: the Doppler shift due to the velocity difference of ions and photons is very large but it can, as described above, be eliminated to all orders by performing the experiment with both co- and counter-propagating laser and ion beams and calculating the geometrical mean to determine the Doppler-free threshold. Hence, the Doppler shift does not contribute to the uncertainty of the result, barring slight potential angle misalignment of maximum as defined by the apertures. However, uncertainties of the ion beam energy and the wavelength calibration could potentially affect the results. Such drifts were estimated to be smaller than by comparing two reference scans on stable $^{127}$I which were performed with the same setup before and after the measurements on astatine.
Computational details.
----------------------
To achieve an optimal accuracy in the DC-CCSD(T) calculations, all electrons of iodine and astatine were correlated, and all virtual orbitals with energies below were included in the virtual space. Fully uncontracted correlation-consistent all-electron relativistic basis sets of Dyall were used[@Dya06]. In order to obtain accurate results for the EA, high quality description of the region removed from the nucleus (that will contain the added electron) is important. We have thus augmented the basis sets with two diffuse functions for each symmetry block. Finally, we performed an extrapolation to the complete basis set (CBS) limit, using the scheme of Halkier *et al.*[@HalHelJor99] for the DHF values and the CBS(34)[@HelKloKoc97] scheme for the correlation contribution. In the DC-CCSD(T) calculations, the finite size of the nucleus was taken into account and modelled by a Gaussian charge distribution within the DIRAC15 program package[@VisDya97].\
Full triple and perturbative quadruple (Q) contributions were calculated in a limited correlation space with the valence $6s$ and $6p$ electrons and a virtual orbital energy cutoff of . It has been previously demonstrated that higher-order correlation is dominated by the valence contributions[@PasEliBor17], and thus this correlation space is deemed sufficient. The valence vXz basis sets of Dyall[@Dya06] were used, and extrapolated to the CBS limit as above. These calculations were performed using the program package MRCC[@MRCC; @KalSur01; @BomStaKal05; @KalGau05; @KalGau08] linked to DIRAC15. Full Q contributions evaluated at the v2z level were below for both systems and were thus omitted.\
Due to the non-instantaneous interaction between particles being limited by the speed of light in the relativistic framework, a correction to the two-electron part of $H_\text{DC}$ is added, in the form of the zero-frequency Breit interaction calculated within the Fock-space coupled cluster approach (DCB-FSCC), using the Tel Aviv atomic computational package [@TRAFS-3C]. To account for the QED corrections, we applied the model Lamb shift operator (MLSO) of Shabaev and co-workers[@ShaTupYer15] to the atomic no-virtual-pair many-body DCB Hamiltonian. This model Hamiltonian uses the Uehling potential and an approximate Wichmann–Kroll term for the vacuum polarization (VP) potential[@Blo72] as well as local and non-local operators for the self-energy (SE), the cross terms (SEVP) and the higher-order QED terms[@ShaTupYer13]. The implementation of the MLSO formalism in the Tel Aviv atomic computational package allows us to obtain the VP and SE contributions beyond the usual mean-field level, namely at the DCB-FSCC level.\
The three remaining sources of error in these calculations are the basis set incompleteness, the neglect of even higher excitations beyond (Q), and the higher-order QED contributions. The first of these is the largest. We have extrapolated our results to the complete basis set limit, and as the associated error, we take half the difference between the CBS result and the doubly augmented ae4z (d-aug-ae4z) basis set value which is . We assume that the effect of the higher excitations should not exceed the (Q) contribution of , and that the error due to the incomplete treatment of the QED effects is not larger than the vacuum polarization and the self energy contributions of . Combining the above sources of error and assuming them to be independent, the total conservative uncertainty estimate on the calculated EA of At is , dominated by the basis set effects.
Comparison to previous theoretical results.
-------------------------------------------
Some recent calculations, including our final theoretical value of the EA of At (labelled CBS - DC - CCSDT(Q) + Breit + QED) are compared to the experimental value in Tab. \[tableII\]. Of particular interest is the recent multi-configurational Dirac-Hartree-Fock (MCDHF) study of Si and Fischer[@SiFis18]. Including the Breit and the QED corrections and extrapolating systematically in terms of included configurations, they obtained an EA for iodine () in excellent agreement with experiment. However, the analogous result for At () lies outside the uncertainty of our experiment.
[ l S l ]{} Method & EA(At)/eV& Ref.\
CBS-DC-CCSDT(Q)+Breit+QED & 2.414(16) & this work\
MCDHF+SE corr.[^1] & 2.38(2) & [@ChaLiDon10]\
MCDHF & 2.416 & [@LiZhaAnd12]\
DC-CCSD(T)+Breit+QED & 2.412 & [@Borschevsky2015IonizationAt]\
MCDHF+Extrap.+Breit+QED [^2] & 2.3729(46) & [@SiFis18]\
CBS-DC-CCSD(T)+Gaunt+QECBS-DC-CCSD(T)+Gaunt+QEDD& 2.423(13) & [@FinPet19]\
Experiment & 2.41578(5) & this work\
More recently, another very accurate calculation of the EA of At (and other heavy *p*-block elements) was carried out by Finney and Peterson[@FinPet19], using an approach similar to that employed in this work. They obtained an EA of , which is in very good agreement with both the measurement and the prediction of this work. The difference between the two theoretical results is mainly due to the number of correlated electrons (all 85 in the present calculation vs. 25 in Ref.[@FinPet19]), the use of the Gaunt correction (instead of Breit) in Ref.[@FinPet19] and the lack of the higher excitations in earlier work.
Data availability
-----------------
The data-sets generated and/or analyzed during the current study are available from the corresponding authors on reasonable request.
References
----------
We thank the ISOLDE technical team and the operators for their work converting ISOLDE to a negative ion machine. The Swedish Research Council is acknowledged for financial support. We would also like to thank the Center for Information Technology of the University of Groningen for their support and for providing access to the Peregrine high performance computing cluster. N.G and E.R. acknowledge the French National Agency for Research for grants called Programme d$'$Investissements d$'$Avenir (ANR-11-EQPX-0004, ANR-11-LABX-0018). Y.L. acknowledges support from the Office of Nuclear Physics, U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 654002 and by the innovative training network fellowship under grant No 642889. L.F.P is grateful for the support from the Slovak Research and Development Agency (APVV-15-0105) and the Scientific Grant Agency of the Slovak Republic (1/0777/19). R.H. acknowledges support by the Bundesministerium für Bildung und Forschung (BMBF, Germany) under the consecutive projects 05P12UMCIA, 05P15UMCIA and 05P18UMCIA. This work was also supported by the FNPMLS ERC Consolidator Grant no. 64838 and the FWO-Vlaanderen (Belgium) and the GOA 15/010 grant from KU Leuven. We would like to acknowledge Kevin Patrice Moles for his help with the design of Fig. \[fig:GANDALPH\] and \[fig:RIB\].
V.F., N.G., D.H., E.R., S.R., J.S. conceived the experiment and wrote the proposal; L.B., D.H, D.L., A.R.M., S.R., J.S., J.Wa., J.We. designed and constructed GANDALPH; L.B., D.H., R.H., M.K.K., D.L., A.R.M, S.R., J.S., J.We. setup and operated GANDALPH; R.A., K.C., D.F., R.G.-R., C.G., R.H., A.K., B.A.M., P.M., M.R., S.R., D.S., M.R.,A.V., S.W. setup and operated the laser system; J.B., F.B.P., D.L., J.P.R., S.R. operated the target and ion source; K.C.,O.F., R.G.-R., D.H., R.H., D.L., Y.L., A.R.M., M.R., R.R., S.R., D.S., J.S., J.Wa., K.W. participated in data taking; D.H., D.L., S.R., J.S., J.We. analyzed the data; A.B., N.G., Y.G., E.E., L.F.P., E.R. performed calculations; N.G., D.H., D.L., B.A.M., U.K., S.R., J.S. wrote the manuscript draft; A.B., V.F., N.G., D.H., S.R., K.W. coordinated the project and/or supervised the participants; all authors contributed to the discussion of the manuscript.
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to D. Leimbach (email: davidleimbach@posteo.de).
[^1]: Multiconfigurational Dirac-Fock (MCDF) results corrected using experimental data.
[^2]: MCDF results extrapolated to complete active space limit
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The size estimates approach for Electrical Impedance Tomography (EIT) allows for estimating the size (area or volume) of an unknown inclusion in an electrical conductor by means of one pair of boundary measurements of voltage and current. In this paper we show by numerical simulations how to obtain such bounds for practical application of the method. The computations are carried out both in a 2–D and a 3–D setting.'
address:
- '$^\vartriangle$ Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, Trieste, Italy.'
- '$^\circ$ Dipartimento di Strutture, Università della Calabria, Rende (CS), Italy.'
- '$^\triangledown$ Dipartimento di Georisorse e Territorio, Università degli Studi di Udine, Udine, Italy.'
- '$^\star$ Dipartimento di Architettura e Pianificazione, Università degli Studi di Sassari, Alghero, Italy.'
date: 'December 22, 2006'
title: |
Computing volume bounds of inclusions\
by EIT measurements$^\ast$
---
Introduction {#sec:introduction}
============
EIT is aimed at imaging the internal conductivity of a body from current and voltage measurements taken at the boundary. It is well known, [@l:a88], [@l:m01], that, even in the ideal situation in which all possible boundary measurements are available, the correspondence *boundary data* $\rightarrow$ *conductivity* is highly (exponentially) unstable. As a consequence it is evident that, in practice, it is impossible to distinguish high resolution features of the interior from limited and noisy boundary data, [@l:av].
Motivated by applications, a line of investigation pursued by many authors, [@l:fr], [@l:frg], [@l:fri], [@l:ai], [@l:fks], [@l:aip], [@l:isak], [@l:isak-libro], has been the one of limiting the analysis to cases in which one seeks an unknown interior inclusion embedded in an otherwise known (may be even homogeneous) conductor, and whose conductivity is assumed to differ from the background.
Even in this restricted case, and even when full boundary data are available, the instability remains of exponential type [@l:dcr].
It is therefore reasonable to further restrict the goal and attempt to evaluate some parameters expressing the size (area, volume) of the inclusion, disregarding its precise location and shape, having at our disposal one pair of boundary measurements of voltage and current. This approach, which can be traced back to [@l:fr], has been well developed theoretically, [@l:ar98], [@l:kss], [@l:ars], [@l:amr03], see also [@l:ike] and [@l:amr04] for the analogous treatment in the linear elasticity framework. In order to describe such type of results we need first to introduce some notation.
We denote by $\Omega$ a bounded domain in ${\mathbb{R}}^n$, $n=2,3$, representing an electrical conductor. The boundary $\partial
\Omega$ of $\Omega$ is assumed of Lipschitz class, with constants $r_0$, $M_0$, that is the boundary can be locally represented as a graph of a Lipschitz continuous function with Lipschitz constant $M_0$ in some ball of radius $r_0$. When no inclusion is present in the conductor we assume that it is homogeneous and we pose its conductivity $\sigma(x)\equiv 1$. When the conductor contains an unknown inclusion $D$ of different conductivity, say $k>0$, $k
\neq 1$ the overall conductivity in the conductor will be given by $\sigma(x)=1+(k-1)\chi_D(x)$. Here and in what follows it is assumed that $D$ is strictly contained in $\Omega$. More precisely, for a given $d_0> 0$, $$\label{eq:2.condition_d0}
\textrm{dist}(D, \partial \Omega) \geq d_0.$$ Let $\varphi \in H^{- \frac{1}{2}}(\partial \Omega)$, $\int_{\partial \Omega} \varphi =0$, be an applied current density on $\partial \Omega$. The induced electrostatic potential $u \in
H^1(\Omega)$ is the solution of the Neumann problem $$\label{eq:2.Neumann_pbm_with_incl}
\left\{ \begin{array}{ll}
{\textrm{div}\,}((1+(k-1) \chi_D) \nabla u)=0, &
\mathrm{in}\ \Omega ,\\
& \\
\nabla u \cdot \nu= \varphi, &
\mathrm{on}\ \partial \Omega,
\end{array}\right.$$ where $\nu$ denotes the outer unit normal to $\partial \Omega$.
When $D$ is the empty set, that is when the inclusion is absent, the reference electrostatic potential $u_0 \in H^1(\Omega)$ satisfies the Neumann problem $$\label{eq:2.Neumann_pbm_without_incl}
\left\{ \begin{array}{ll}
\Delta u_0=0, &
\mathrm{in}\ \Omega ,\\
& \\
\nabla u_0 \cdot \nu= \varphi, &
\mathrm{on}\ \partial \Omega.
\end{array}\right.$$
In both cases and , the solutions $u$ and $u_0$ are determined up to an additive constant.
Let us denote by $W$, $W_0$ the powers required to maintain the current density $\varphi$ on $\partial \Omega$ when $D$ is present or it is absent, respectively. Namely $$\label{eq:2.def_W}
W=\int_{\partial \Omega} u \varphi = \int_{\Omega}(1+(k-1)\chi_D)|
\nabla u|^2,$$ $$\label{eq:2.def_W0}
W_0=\int_{\partial \Omega} u_0 \varphi = \int_{\Omega}|\nabla u_0|^2.$$ The size estimate approach developed in [@l:ar98], [@l:kss], [@l:ars], [@l:amr03], tells us that the measure $|D|$ of $D$ can be bounded from above and below in terms of the quantity $\left|\frac{W_0-W}{W_0}\right|$ which we call the normalized power gap. More precisely the following bounds hold, see [@l:amr03 Theorem 2.3].
\[theo:size-estim-EIT-general\] Let $D$ be any measurable subset of $\Omega$ satisfying . Under the above assumptions, if $k > 1$ we have $$\label{eq:2.size-estim-EIT-more-conduct}
\frac {1} {k-1} C^{+}_{1}
\frac{W_0-W}{W_0}
\leq
|D|
\leq
\left (
\frac{k}{k-1}
\right )^{ \frac{1}{p} }
C^{+}_{2}
\left (
\frac{W_0-W}{W_0}
\right )^{ \frac{1}{p} }.$$ If, conversely, $k < 1$, then we have $$\label{eq:2.size-estim-EIT-less-conduct}
\frac {k} {1-k} C^{-}_{1}
\frac{W-W_0}{W_0}
\leq
|D|
\leq
\left (
\frac{1}{1-k}
\right )^{ \frac{1}{p} }
C^{-}_{2}
\left (
\frac{W-W_0}{W_0}
\right )^{ \frac{1}{p} },$$ where $C^{+}_{1}$, $C^{-}_{1}$ only depend on $d_0$, $|\Omega|$, $r_0$, $M_0$, whereas $p>1$, $C^{+}_{2}$, $C^{-}_{2}$ only depend on the same quantities and, in addition, on the *frequency of $\varphi$* $$\label{eq:2.frequency}
F[\varphi] = \frac{\|\varphi \|_{H^{ -\frac{1}{2} }(\partial
\Omega)}}{\|\varphi \|_{H^{-1}(\partial \Omega)}}.$$
When it is a priori known that the inclusion $D$ is not too small (if it is at all present), a situation which often occurs in practical applications, stronger bounds apply.
\[theo:size-estim-EIT-fat-incl\] Under the above hypotheses, let us assume, in addition, that $$\label{eq:2.fat-inclusion}
|D| \geq m_0,$$ for a given positive constant $m_0$. If $k > 1$ we have $$\label{eq:2.size-estim-EIT-more-conduct-fat-incl}
\frac {1} {k-1} C^{+}_{1}
\frac{W_0-W}{W_0}
\leq
|D|
\leq
\frac{k}{k-1}
C^{+}_{2}
\frac{W_0-W}{W_0}
.$$ If, conversely, $k < 1$, then we have $$\label{eq:2.size-estim-EIT-less-conduct-fat-incl}
\frac {k} {1-k} C^{-}_{1}
\frac{W-W_0}{W_0}
\leq
|D|
\leq
\frac{1}{1-k}
C^{-}_{2}
\frac{W-W_0}{W_0},$$ where $C^{+}_{1}$, $C^{-}_{1}$ only depend on $d_0$, $|\Omega|$, $r_0$, $M_0$, whereas $C^{+}_{2}$, $C^{-}_{2}$ only depend on the same quantities and, in addition, on $m_0$ and $F[\varphi]$.
Theorem \[theo:size-estim-EIT-fat-incl\] can be easily deduced from Theorem \[theo:size-estim-EIT-general\] by the arguments sketched in [@l:abfmrt04 Appendix].
One of the goals of the present paper is to test the applicability of such bounds by numerical simulations with the following purposes:
*i) provide practical evaluations of the constants $C_1^{\pm}$, $C_2^{\pm}$ appearing in the above inequalities , , , ;*
*ii) when, due to special geometric configurations, it is possible to compute theoretically such constants, compare such theoretical values with those obtained by simulations;*
*iii) show that such upper and lower bounds deteriorate as the frequency $F[\varphi]$ increases.*
The other goal of this paper is to perform similar kinds of numerical simulations when the so-called *complete model* of EIT is adopted. We recall that this model is aimed at an accurate description of the boundary measurements suitable for medical applications, and was introduced in [@l:cing] and subsequently developed in [@l:pbp] and [@l:sci]. In this model, the metal electrodes behave as perfect conductors and provide a low-resistance path for current. An electrochemical effect at the contact between the electrodes and the body results in a thin, highly resistive, layer. The impedance of this layer is characterized by a positive quantity $z_l$ on each electrode $e_l$, $l=1,...,L$, which is called *surface impedance*.
Denoting by $I_l$ the current applied to each $e_l$, the resulting boundary condition on each electrode $e_l$ becomes $$\label{eq:2.bound-cond}
u+z_l \nabla u \cdot \nu = U^l, \quad \quad \hbox{on } e_l,$$ where the unknown constant $U^l$ is the voltage which can be measured at the electrode $e_l$.
We assume, as before, that the reference conductor has conductivity $\sigma \equiv 1$ and that an unknown inclusion $D$ of conductivity $\sigma \equiv k$, with $k>0$ and $k \neq 1$, is strictly contained in $\Omega$. Therefore, the electrostatic potential $u$ inside the conductor is determined, up to an additive constant, as the solution to the following problem $$\label{eq:2.Phys-Neumann_pbm_with_incl}
\left\{ \begin{array}{ll}
{\textrm{div}\,}((1+(k-1) \chi_D) \nabla u)=0, &
\mathrm{in}\ \Omega ,\\
u+z_l \nabla u \cdot \nu = U^l, &
\mathrm{on}\ e_l, \ 1 \leq l \leq L, \\
\nabla u \cdot \nu= 0, &
\mathrm{on}\ \partial \Omega \setminus \cup_{l=1}^L e_l, \\
\int_{e_l} \nabla u \cdot \nu = I_l, &
\ 1 \leq l \leq L,
\end{array}\right.$$ where the so-called current pattern $I=(I_1, ..., I_L)$ is subject to the conservation of charge condition $\sum_{l=1}^L I_l=0$, and the unknown constants $U^l$ are the components of the so-called voltage pattern $U=(U^1, ..., U^L)$.
When the inclusion is absent, the electrostatic potential $u_0$ induced by the same current pattern $I$ is determined, up to an additive constant, as the solution of the following problem $$\label{eq:2.Phys-Neumann_pbm_without_incl}
\left\{ \begin{array}{ll}
\Delta u_0=0, &
\mathrm{in}\ \Omega ,\\
u_0+z_l \nabla u_0 \cdot \nu = U_0^l, &
\mathrm{on}\ e_l, \ 1 \leq l \leq L, \\
\nabla u_0 \cdot \nu= 0, &
\mathrm{on}\ \partial \Omega \setminus \cup_{l=1}^L e_l, \\
\int_{e_l} \nabla u_0 \cdot \nu = I_l, &
\ 1 \leq l \leq L,
\end{array}\right.$$ where, as before, the $U_0^l$ are unknown constants in the direct problem .
We shall assume that the sets $e_1,...,e_L$, representing the electrodes, are open, pairwise disjoint, connected subsets of $\partial \Omega$ and, in addition, $$\label{eq:2.cond-electrodes}
\textrm{dist}(e_l,e_k) \geq \delta_1 > 0 \quad \hbox{for every }
l,k, \ l \neq k.$$
The surface impedance $z_l$ on $e_l$, $l=1,...,L$, is assumed to be real valued and to satisfy the following bounds $$\label{eq:2.cond-impedance}
0<m\leq z_l\leq M, \quad \hbox{for every } l=1,...,L.$$
In this formulation, the powers $W$ and $W_0$ become $$\label{eq:2.def_W-phys-model}
W=\sum_{i=1}^L I_i U^i,$$ $$\label{eq:2.def_W0-phys-model}
W_0=\sum_{i=1}^L I_i U_0^i.$$ Size estimates like those of Theorems \[theo:size-estim-EIT-general\], \[theo:size-estim-EIT-fat-incl\] were obtained for the complete model in [@l:ar04]. In particular we have
\[theo:size-estim-Phys-EIT-general\] Let $D$ be any measurable subset of $\Omega$ satisfying and let $W$, $W_0$ be given by , . Then, inequalities , hold for $k > 1$ and $k < 1$, respectively, where the constants $C^{+}_{1} $, $C^{-}_{1}$ only depend on $d_0$, $|\Omega|$, $r_0$, $M_0$, and $C^{+}_{2}$, $C^{-}_{2}$ and $p>1$ only depend on the same quantities and, in addition, on $\delta_1$, $M$ and $m$.
Also in this case, the size estimates of $|D|$ can be improved to the form , when condition is satisfied.
In Section \[sec:num-EIT\] we consider the standard EIT setting. We start by describing the finite element setup used in our numerical simulations in Section \[subsec:nummodel\]. Next (as a warmup) we consider a two-dimensional model in Section \[subsec:2D\].
In Section \[subsec:3D\] we consider the three-dimensional case and we discuss all items i), ii), iii) introduced above. In particular we observe that, comparing the results as the frequency $F[\varphi]$ increases, we have quite rapidly a serious deterioration of the bounds. This poses a severe warning on the limitations that have to be taken into account in the choice of the boundary current profile $\varphi$.
Section \[sec: num-phys-EIT\] is devoted to simulations with the complete EIT model. In this case it is reasonable analyze the case when only two electrodes, one positive and one negative, are attached to the surface of the conductor. In this case, the frequency function is not available from the data since we are not prescribing the boundary current $\nabla
u\cdot\nu_{|\partial\Omega}$ but only the current pattern, which is a 2-electrode configuration, is just the pair $(1,-1)$. In place of the frequency function, the parameters that may influence the constants in the volume bounds are: the width of the electrodes and the distance between them. We perform various experiments to test such variability.
Numerical simulations for the EIT model {#sec:num-EIT}
=======================================
Numerical model {#subsec:nummodel}
---------------
The numerical model is based on the discretization of the energy functional $J: H^1(\Omega, {\mathbb{R}}^n) \rightarrow {\mathbb{R}}$ $$\label{eq:energy_cont}
J(u) = \frac{1}{2} \int_\Omega (1 + (k-1) \chi_D ) \nabla u
\cdot \nabla u - \int_{\partial \Omega} \varphi u ,$$ associated to the variational formulation of problem (\[eq:2.Neumann\_pbm\_with\_incl\]). The energy functional (\[eq:energy\_cont\]) has been discretized by using the High Continuity (HC) technique already presented in [@l:ari85] and [@l:bft04] in the context of linear elasticity. Accordingly, for 2–D problems the electric potential on the $e$–th finite element can be represented as $$\label{eq:HCdispl}
u_e = \sum_{i,j=1}^{3} \phi_i(\xi_1) \phi_j(\xi_2) u_{ij},$$ whereas for the 3–D case it assumes the form $$\label{eq:HCdispl3D}
u_e = \sum_{i,j,l=1}^{3} \phi_i(\xi_1) \phi_j(\xi_2) \phi_l
(\xi_3) u_{ijl},$$ where the coordinates $\xi_r$, $r=1,...,n$, span the unitary element domain $[-\frac{1}{2},\frac{1}{2}]^n$, $n=2,3$, and $u_{ij}$, $u_{ijl}$ are the HC parameters involved in the field interpolation on the generic element. The shape functions $\phi_i(\xi_r)$ are defined as $$\label{eq:HCfun}
\left\{
\begin{split}
\phi_1(\xi_r)=& \frac{1}{8} - {\frac{1}{2}}\xi_r + {\frac{1}{2}}\xi_r^2, \\
\phi_2(\xi_r)=& \frac{3}{4} - \xi_r^2, \\
\phi_3(\xi_r)=& \frac{1}{8} + {\frac{1}{2}}\xi_r + {\frac{1}{2}}\xi_r^2.
\end{split}
\right.$$
![HC interpolation in the 1–D case: nodes, parameters and shape functions.[]{data-label="fig:HCmono"}](HCmono.eps){width="10cm"}
![HC mesh in the 2–D case: nodes for boundary and inner elements.[]{data-label="fig:HCbi"}](HCbi.eps){width="12cm"}
The 1–D case illustrated in Figure \[fig:HCmono\] shows the meaning of the HC parameters. They allow to define the slopes of the interpolated function at the end points of the element. On the same figure one can see also the positions of the HC nodes and the shape functions (\[eq:HCfun\]).
Figure \[fig:HCbi\] shows a typical structured mesh on a rectangular domain and the nodes used for the approximation of the potential field in the 2–D case. For elements with a side lying on the boundary, in order to easily impose the Neumann boundary conditions, special shape functions are used. In practice, the external HC nodes are translated onto the boundary $\partial
\Omega$ and the related HC parameters have the meaning of function values (see again Figure \[fig:HCbi\]). In this case the shape functions relative to a [*left*]{} boundary ($\xi_r=-\frac{1}{2}$) and a [*right*]{} boundary ($\xi_r=\frac{1}{2}$) of the finite element are $$\label{eqn:HCfunLR}
{\rm left:} \left\{
\begin{split}
\phi_1(\xi_r)\;\; =& \frac{1}{4} - \xi_r + \xi_r^2, \\
\phi_2(\xi_r)\;\; =& \frac{5}{8} + {\frac{1}{2}}\xi_r - \frac{3}{2}
\xi_r^2, \\
\phi_3(\xi_r)\;\; =& \frac{1}{8} + {\frac{1}{2}}\xi_r + {\frac{1}{2}}\xi_r^2;
\end{split}
\right. \qquad\; {\rm right:} \left\{
\begin{split}
\phi_1(\xi_r)\;\; =& \frac{1}{8} - {\frac{1}{2}}\xi_r + {\frac{1}{2}}\xi_r^2, \\
\phi_2(\xi_r)\;\; =& \frac{5}{8} - {\frac{1}{2}}\xi_r - \frac{3}{2}
\xi_r^2, \\
\phi_3(\xi_r)\;\; =& \frac{1}{4} + \xi_r + \xi_r^2.
\end{split}
\right.$$
Further details about the HC interpolation can be found in [@l:ari85] and [@l:bft04]. This interpolation technique, which can be considered as a particular case of the Bézier interpolation, has the main advantage of reproducing potential fields of $C^1$ smoothness with a computational cost equivalent to a $C^0$ interpolation.
By or , the potential field $u$ on each element $e$ takes the compact form $$\label{eq:HCinterp_mat}
u_e = \mathbf{N}_e \mathbf{w}_e.$$ The one–row matrix $\mathbf{N}_e$ collects the shape functions of the HC interpolation, whereas the components of the vector $\mathbf{w}_e$ are the nodal parameters of the underlying element. With this notation, the gradient of the potential field is given by $$\label{eq:HCinterp_grad}
\nabla u_e = \nabla \mathbf{N}_e \mathbf{w}_e.$$
We remark that the dimensions of the matrices $\mathbf{N}_e$, $\nabla \mathbf{N}_e$ and vector $\mathbf{w}_e$ are $1 \times 9$, $2 \times 9$ and $9 \times 1$ for the 2–D case and $1 \times 27$, $3\times 27$ and $27\times1$ for the 3–D case.
By and , the discrete form of becomes $$\label{eq:EIT_var_form_discr}
J( \mathbf{w}_e ) = \sum_e \left( \frac{1}{2} \int_{{\Omega}_e} (1
+ (k-1) \chi_D) (\nabla \mathbf{N}_e \mathbf{w}_e)
\cdot (\nabla \mathbf{N}_e \mathbf{w}_e) - \int_{{\partial {\Omega}}_e}
\varphi \mathbf{N}_e \mathbf{w}_e \right) ,$$ or, in a compact form, $$\label{eq:EIT_min_w}
J(\mathbf{w}_e) = \sum_e \left( \mathbf{w}_e^T \mathbf{K}_e
\mathbf{w}_e - \mathbf{w}_e^T \mathbf{p}_e \right).$$ The latter equation provides the definition of the matrix and vector associated to $e$–th element $$\label{eq:EIT_mat}
\left\{
\begin{split}
\mathbf{K}_e = & \int_{{\Omega}_e} (1 + (k-1) \chi_D) (\nabla
\mathbf{N}_e)^T \nabla \mathbf{N}_e, \\
\mathbf{p}_e = & \int_{{\partial {\Omega}}_e} \varphi \mathbf{N}_e,
\end{split}
\right.$$ which can be used to assemble, by using standard techniques, the system of equations to solve.
Two–dimensional case {#subsec:2D}
--------------------
Numerical analysis has been performed on a square conductor $\Omega$ of side $l$ under the two current density fields $\varphi$ illustrated in Figure \[fig:Test numerici\_2D\]. The domain $\Omega$ has been discretized with a mesh of $21 \times 21$ HC finite elements and for both Test $T_1$ and Test $T_2$ of Figure \[fig:Test numerici\_2D\] we have considered an inclusion $D$ with conductivity $k=0.1$ or $k=10$.
![Square conductor considered in 2–D numerical simulations for the EIT model and applied current density fields: Test $T_1$ (a), Test $T_2$ (b).[]{data-label="fig:Test numerici_2D"}](T1_2D.eps "fig:"){width="5.5cm"}\
![Square conductor considered in 2–D numerical simulations for the EIT model and applied current density fields: Test $T_1$ (a), Test $T_2$ (b).[]{data-label="fig:Test numerici_2D"}](T2_2D.eps "fig:"){width="5.5cm"}\
A first series of experiments has been carried out by considering all possible square inclusions with side ranging from $1$ to $5$ elements, that is the size of inclusion has been kept lower than $6\%$ of the total size of the conductor. The results are collected in Figures \[fig:T1\_2D\_pos\] and \[fig:T2\_2D\_pos\] for different values of the minimum distance $d_0$ between the inclusion $D$ and the boundary of $\Omega$.
 ($21 \times 21$ FE mesh): $k=0.1
$ (a), $k=10$ (b).[]{data-label="fig:T1_2D_pos"}](T1_2D_pos_f01_piu_teoriche.eps "fig:"){width="6cm"}\
 ($21 \times 21$ FE mesh): $k=0.1
$ (a), $k=10$ (b).[]{data-label="fig:T1_2D_pos"}](T1_2D_pos_f10_piu_teoriche.eps "fig:"){width="6cm"}\
 ($21 \times 21$ FE mesh): $k=0.1
$ (a), $k=10$ (b).[]{data-label="fig:T2_2D_pos"}](T2_2D_pos_f01.eps "fig:"){width="6cm"}\
 ($21 \times 21$ FE mesh): $k=0.1
$ (a), $k=10$ (b).[]{data-label="fig:T2_2D_pos"}](T2_2D_pos_f10.eps "fig:"){width="6cm"}\
From Figures \[fig:T1\_2D\_pos\](a) and \[fig:T2\_2D\_pos\](a), which refer to the case $k=0.1$, one can note that the upper bound of $|D|$ is rather insensitive to the choice of $d_0$, whereas the lower bound in improves as $d_0$ increases. The converse situation occurs when the inclusion is made by material of higher conductivity, see Figures \[fig:T1\_2D\_pos\](b) and \[fig:T2\_2D\_pos\](b).
As a second class of experiments, we have considered inclusions of general shape on a FE mesh of $15 \times 15$ HC elements. More precisely, each inclusion is the union of elements having at least a common side and being at least $d_0=2$ elements far from the boundary $\partial \Omega$. Results are collected in Figures \[fig:T1\_2D\_shp\] and \[fig:T2\_2D\_shp\].
The straight lines drawn in Figures \[fig:T1\_2D\_pos\] and \[fig:T1\_2D\_shp\] correspond to the theoretical size estimates for test $T_1$ of Figure \[fig:Test numerici\_2D\](a). For both cases $k=0.1$ and $k=10$ we have $$\label{eq:3.theor-size-T1}
\frac{1}{9} \frac{|W-W_0|}{W_0} \leq \frac{|D|}{|\Omega|}\leq \frac
{10}{9} \frac{|W-W_0|}{W_0}.$$ The comparison with the region of the plane $\left(\frac{|D|}{|\Omega|}, \frac{|W-W_0|}{W_0} \right)$ covered by the corresponding numerical experiments confirms, as already remarked in [@l:abfmrt04] in the context of linear elasticity, that practical applications of the size estimates approach lead to less pessimistic results with respect to those obtained via the theoretical analysis.
 ($21 \times 21$ FE mesh, $d_0=2
$): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_2D_shp"}](T1_2D_shp_f01_piu_teoriche.eps "fig:"){width="6cm"}\
 ($21 \times 21$ FE mesh, $d_0=2
$): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_2D_shp"}](T1_2D_shp_f10_piu_teoriche.eps "fig:"){width="6cm"}\
 ($21 \times 21$ FE mesh, $d_0=2
$): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_2D_shp"}](T2_2D_shp_f01.eps "fig:"){width="6cm"}\
 ($21 \times 21$ FE mesh, $d_0=2
$): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_2D_shp"}](T2_2D_shp_f10.eps "fig:"){width="6cm"}\
Three–dimensional case {#subsec:3D}
----------------------
The first part of this subsection is devoted to the extension to the 3–D case of the numerical simulations given in \[subsec:2D\]. In the second part, we shall investigate on the effect of the oscillation character of the Neumann data on the upper bound of size inclusion.
Similarly to the 2–D case, a first series of numerical simulations has been performed on an electrical conductor of cubic shape, of side $l$, with the two current density fields illustrated in Figure \[fig:Test numerici\_3D\]. In both cases, a mesh of $20 \times 20 \times 20$ finite elements has been considered when performing simulations in presence of cubic inclusions. The results are illustrated in Figures \[fig:T1\_3D\_pos\] and \[fig:T2\_3D\_pos\]. Figure \[fig:T1\_3D\_pos\] contains also the straight lines corresponding to the theoretical size estimates for test $T_1$ of Figure \[fig:Test numerici\_3D\], that is $$\label{eq:3.theor-size-T1-3D}
\frac{1}{9} \frac{|W-W_0|}{W_0} \leq \frac{|D|}{|\Omega|}\leq \frac
{10}{9} \frac{|W-W_0|}{W_0}.$$
![Cubic conductor considered in 3–D numerical simulations for the EIT model and applied current density fields: Test $T_1$ (a) and Test $T_2$ (b).[]{data-label="fig:Test numerici_3D"}](T1_3D.eps "fig:"){width="5cm"}\
![Cubic conductor considered in 3–D numerical simulations for the EIT model and applied current density fields: Test $T_1$ (a) and Test $T_2$ (b).[]{data-label="fig:Test numerici_3D"}](T2_3D.eps "fig:"){width="5cm"}\
![Influence of $d_0$ for cubic inclusions in test $T_1$ of Figure \[fig:Test numerici\_3D\] ($20 \times 20 \times 20$ FE mesh): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_3D_pos"}](T1_3D_pos_f01_piu_teoriche.eps "fig:"){width="6cm"}\
![Influence of $d_0$ for cubic inclusions in test $T_1$ of Figure \[fig:Test numerici\_3D\] ($20 \times 20 \times 20$ FE mesh): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_3D_pos"}](T1_3D_pos_f10_piu_teoriche.eps "fig:"){width="6cm"}\
![Influence of $d_0$ for cubic inclusions in test $T_2$ of Figure \[fig:Test numerici\_3D\] ($20 \times 20 \times 20$ FE mesh): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_3D_pos"}](T2_3D_pos_f01.eps "fig:"){width="6cm"}\
![Influence of $d_0$ for cubic inclusions in test $T_2$ of Figure \[fig:Test numerici\_3D\] ($20 \times 20 \times 20$ FE mesh): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_3D_pos"}](T2_3D_pos_f10.eps "fig:"){width="6cm"}\
In order to deal with inclusions of general shape, however, the numerical experiments require some restrictions to reduce the computer time. A rough estimate of the computational cost can be obtained noting that the numerical effort is essentially due to the decomposition of the matrix associated to the linear system and to the computation of its solution. Denoting by $m$ the number of the equations and by $b$ the half bandwidth of the matrix, the decomposition requires $m(b-1)$ multiplications and $m b (b-1)$ additions, whereas the computation of the solution involves $m b$ multiplications.
Therefore, for each given inclusion in a $20\times20\times20$ FE mesh, a linear system of $10648$ ($b=1015$) equations has to be solved, requiring a computer time of approximately $86$ s working on an Opteron $2.4$ GHz computer. Since the number of all possible inclusions formed by $n_i$ elements on a mesh of $n_e \times n_e
\times n_e$ is $\frac{n_e^3 !}{n_i ! (n_e^3-n_i)!}$, the way to calculate all the possible case is practically impossible. Indeed by considering that the $20\times20\times20$ is formed by $8000$ elements and that, if the ratio $|D|/|{\Omega}|$ is less than $6\%$ that is $480$ elements, the number of cases to analyze is $69.1183\times 10^{785}$.
In order to reduce the computer time significantly we have considered a $7\times 7 \times 7$ mesh generating a system of $729$ equations. Despite of this, the number of possible cases to consider still remains very high; for instance, for inclusions formed by $5$ elements, one should solve about $3.8\times 10^{10}$ linear systems. Therefore, we decided to restrict our analysis to inclusions satisfying the following additional hypotheses:
i) the inclusion is the union of elements having at least one common face and it is formed by starting from a generic element inside an octant of the cube (this last assumption is not really restrictive due to the symmetries of the problem);
ii) $d_0=1$.
For inclusions formed by $1,...,7$ elements, we have considered all possible inclusions satisfying the limitations $i)$ and $ii)$, whereas for inclusions formed by $8,...,17$ elements we have considered a random sample because of the high computational cost. For these cases, the ratio between the sample dimension and that of all the data approximately spans between $20\%$ for inclusions formed by $8$ elements and $0.01\%$ for inclusions formed by $17$ elements. The results are presented in Figures \[fig:T1\_3D\_shp\] and \[fig:T2\_3D\_shp\] for Test $T_1$ and Test $T_2$, respectively. In Figure \[fig:T1\_3D\_shp\], the straight lines corresponding to the theoretical bounds for Test $T_1$ are also drawn. As already remarked in the treatment of the 2–D case, the theoretical analysis leads to rather pessimistic results with respect to those obtained by the numerical simulations, especially when the inclusion is softer than the surrounding material.
 ($7 \times 7
\times 7$ FE mesh, $d_0=1 $): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_3D_shp"}](T1_3D_shp_f01_piu_teoriche.eps "fig:"){width="6cm"}\
 ($7 \times 7
\times 7$ FE mesh, $d_0=1 $): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_3D_shp"}](T1_3D_shp_f10_piu_teoriche.eps "fig:"){width="6cm"}\
 ($7 \times 7
\times 7$ FE mesh, $d_0=1 $): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_3D_shp"}](T2_3D_shp_f01.eps "fig:"){width="6cm"}\
 ($7 \times 7
\times 7$ FE mesh, $d_0=1 $): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_3D_shp"}](T2_3D_shp_f10.eps "fig:"){width="6cm"}\
The Neumann data considered in the above experiments give raise to potential fields inside the conductor with nonvanishing gradient. In the general case, when the gradient of the solution may vanish, we expect, accordingly to Theorems \[theo:size-estim-EIT-general\], \[theo:size-estim-EIT-fat-incl\], that the upper bounds deteriorate as the frequency $F[\varphi]$ given by increases. Since $F[\varphi]$ is a ratio which measures the frequency of oscillation of $\varphi$, we are interested to investigate on the effectiveness of size estimates approach for oscillating Neumann data.
In particular, the numerical simulations have been carried out for the cubic electrical conductor considered before and choosing the following Neumann data: $$\label{eq:3.oscill-Neu-data}
\left.
\begin{array}{crl}
\varphi = & - \cos \frac{n \pi x}{l} & \qquad \hbox
{on } z=0 , \\
\varphi = & \cos \frac{n \pi x}{l} & \qquad \hbox
{on } z=l , \\
\varphi = & 0 & \qquad
\hbox{elsewhere on } \partial \Omega,
\end{array}
\right\}
\quad \hbox{for } n=0, 1, 2 .$$
Case $n=0$ has been already discussed at the beginning of this paragraph and corresponds to the simple case in which the gradient of the unperturbed solution $u_0$ does not vanish in $\Omega$.
The two other cases are examples of Neumann data with higher frequency $F[\varphi]$. More precisely, the corresponding solutions $u_0$ have critical lines of equation $$\left\{ x= \frac{l}{n} \left( \frac{1}{2}+i \right), z= \frac{l}{n} \left( \frac{1}{2}+j \right) \right\}, \quad i,j = 0, ..., n-1.$$
The mesh employed is made by $20 \times 20 \times 20$ HC finite elements. The analysis has been focussed on cubic inclusions having volume up to $6 \%$ of the total volume of the specimen and conductivity $k=0.1$ and $k=10$. The numerical results in case $n=1$ and $n=2$ are presented in Figures \[fig:cos\_1\] and \[fig:cos\_2\], respectively. The numerical results show that the lower bound in size estimates , improves as $d_0$ increases, whereas the upper bound of $|D|$ is rather insensitive to the choice of $d_0$.
Theoretical estimates for cases $n=1$ and $n=2$ of are given by $$\label{eq:cos_est}
\begin{split}
\hbox{for } & k>1: \\
& \frac{\tanh \frac{n\pi}{2}}{ n\pi(k-1) } \frac{W_0-W}{W_0}
\leq
\frac{|D|}{|\Omega|}
\leq
\frac{1}{C_n} \frac{k}{k-1} \frac{\tanh \frac{n\pi}{2}}{n\pi}
\frac{W_0-W}{W_0}; \\
\hbox{for } & k<1: \\
&\frac{k}{n\pi(1-k)} \tanh \frac{n\pi}{2} \frac{W-W_0}{W_0}
\leq
\frac{|D|}{|\Omega|}
\leq \frac{1}{C_n} \frac{1}{1-k} \frac{\tanh \frac{n\pi}{2}}{n
\pi}
\frac{W-W_0}{W_0},
\end{split}$$ where $$C_n= \frac{10}{n\pi \cosh^2 \frac{n\pi}{2} } \left ( \sinh
\frac{n\pi}{20} - \sin \frac{n\pi}{20} \right ), \quad \quad
n=1,2.$$ The theoretical estimates are indicated in Figures \[fig:cos\_1\] and \[fig:cos\_2\]. The slope of the straight line corresponding to the upper bound is so high that it practically coincides with the vertical axis, at least for the portion of graph near the origin considered in this study. The theoretical lower bound gives, for a fixed power gap, values significantly less than those obtained in the numerical experiments.
![Cubic electrical conductor with Neumann data as in case $n=1$ of : lower and upper bound of the power gap for different values of $d_0$ ($k=0.1$ (a) and $k=10$ (b)) on a $20 \times 20 \times 20$ mesh.[]{data-label="fig:cos_1"}](Tk0k1_20x20x20_f=01_piu_teoriche.eps "fig:"){height="6cm"}\
![Cubic electrical conductor with Neumann data as in case $n=1$ of : lower and upper bound of the power gap for different values of $d_0$ ($k=0.1$ (a) and $k=10$ (b)) on a $20 \times 20 \times 20$ mesh.[]{data-label="fig:cos_1"}](Tk0k1_20x20x20_f=10_piu_teoriche.eps "fig:"){height="6cm"}\
![Cubic electrical conductor with Neumann data as in case $n=2$ of : lower and upper bound of the power gap for different values of $d_0$ ($k=0.1$ (a) and $k=10$ (b)) on a $20 \times 20 \times 20$ mesh.[]{data-label="fig:cos_2"}](Tk0k2_20x20x20_f=01_piu_teoriche.eps "fig:"){height="6cm"}\
![Cubic electrical conductor with Neumann data as in case $n=2$ of : lower and upper bound of the power gap for different values of $d_0$ ($k=0.1$ (a) and $k=10$ (b)) on a $20 \times 20 \times 20$ mesh.[]{data-label="fig:cos_2"}](Tk0k2_20x20x20_f=10_piu_teoriche.eps "fig:"){height="6cm"}\
Numerical simulations for the complete EIT model {#sec: num-phys-EIT}
================================================
Numerical model {#subsec:nummodel-phys}
---------------
In this case, by using the same notation introduced in Section \[sec:num-EIT\], the energy functional $J:H^1({\Omega})\times\mathbb{R}^L \rightarrow \mathbb{R}$ related to the variational formulation of problem (\[eq:2.Phys-Neumann\_pbm\_with\_incl\]) is given by $$\label{eq:Phis_EIT_var_form}
J(u, U^l) = \frac{1}{2} \int_{\Omega}(1 + (k-1) \chi_D) \nabla u
\cdot \nabla u +
\frac{1}{2} \sum_{l=1}^{L} \frac{1}{z_l} \int_{{\partial {\Omega}}_l} (u-
U^l)^2 - \sum_{l=1}^{L} I_l U^l .$$
Using HC interpolation for the potential field $u$ and with the notation introduced in Section \[sec:num-EIT\], the discrete energy functional becomes $$\label{eq:Phis_EIT_var_form_discr}
\begin{split}
J(\mathbf{w}_e, U^l) & = \frac{1}{2} \sum_e \int_{{\Omega}_e} (1 +
(k-1) \chi_D) (\nabla \mathbf{N}_e \mathbf{w}_e) \cdot (\nabla
\mathbf{N}_e \mathbf{w}_e) + \\
& + \frac{1}{2} \sum_{l=1}^{L} \frac{1}
{z_l} \sum_{\hat{e}} \int_{({\partial {\Omega}}_l)_e} (\mathbf{N}_{e} \mathbf{w}
_{e} - U^l)^2 - \sum_{l=1}^{L} I_l U^l ,
\end{split}$$ or $$\label{eq:Phis_EIT_compact}
\begin{split}
J(\mathbf{w}_e, U^l) & = \frac{1}{2} \sum_e \mathbf{w}_e^T \mathbf
{K}_e \mathbf{w}_e + \\
& + \frac{1}{2} \sum_{l=1}^
{L} \frac{1}{z_l} \sum_{\hat{e}} ( \mathbf{w}_e^T \mathbf{K}_{ll}
\mathbf{w}_e + (U^l)^2 - 2 \mathbf{w}_e^T \mathbf{K}_{el} U^l) -
\sum_{l=1}^{L} I_l
U^l,
\end{split}$$ having used the compact notation $$\label{eq:Phis_EIT_mat}
\begin{split}
\mathbf{K}_e = & \int_{{\Omega}_e} (1 + (k-1) \chi_D) (\nabla
\mathbf{N}_e)^T \nabla \mathbf{N}_e, \\
\mathbf{K}_{ll} = & \int_{({\partial {\Omega}}_l)_e} \mathbf{N}_e^T
\mathbf{N}_e, \\
\mathbf{K}_{el} = & \int_{({\partial {\Omega}}_l)_e} \mathbf{N}_e^T.
\end{split}$$ We remark that the second sum in the right hand side of and , that on $\hat{e}$, is extended only to the elements under the electrodes.
Collecting the unknown parameters representing the potential field in $\mathbf{w}$, those of the electrodes in $\mathbf{U}$ and the current pattern in $\mathbf{I}$, by a standard method of assembling we obtain the following linear system $$\label{eq:Phis_EIT_system_mat_form}
\begin{bmatrix}
\mathbf{K}_{ww} & -\mathbf{K}_{wU} \\
-\mathbf{K}_{wU}^T & \mathbf{K}_{UU} \\
\end{bmatrix} \begin{bmatrix}
\mathbf{w} \\
\mathbf{U} \\
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} \\
\mathbf{I} \\
\end{bmatrix} ,$$ which can be efficiently solved taking advantage of the particular structure of coefficient matrix.
Results for 3–D cases {#subsec:phys-3D}
---------------------
The analysis has been restricted to the case of two electrodes located on the boundary of a cubic electrical conductor of side $l$, see Figure \[fig:EIT\_prototype\]. The specimen has been discretized by a mesh of $17 \times 17 \times 17$ cubic HC finite elements and the numerical experiments have been carried out on cubic inclusions only, with volume up to $6\%$ of the total volume and conductivity value $k=0.1$ or $k=10$. The surface impedance takes a constant value such that $\zeta=\frac{z \sigma}{l}=0.2$ on both electrodes, according to properties of human skin reported in literature, see, for instance, [@l:ssbs].
In test $T_1$ of Figure \[fig:EIT\_prototype\], the electrodes cover completely two opposite faces of the specimen, whereas in Test $T_2$ one electrode coincides with a face of $\partial \Omega$ and the other is a square, formed by one or nine surface finite elements, and it is located in central position of the opposite face. Finally, in Test $T_3$, two electrodes are placed on the same face of the conductor $\Omega$ in a symmetric way respect to middle lines of the face. The electrodes are separated by three finite elements and their dimensions are equal to the element size.
![Cubic conductor considered in 3–D numerical simulations for the physical EIT model and location of the electrodes: test $T_1$ (a), test $T_2$ (b) and test $T_3$ (c).[]{data-label="fig:EIT_prototype"}](T1_EIT.eps "fig:"){width="4cm"}\
![Cubic conductor considered in 3–D numerical simulations for the physical EIT model and location of the electrodes: test $T_1$ (a), test $T_2$ (b) and test $T_3$ (c).[]{data-label="fig:EIT_prototype"}](T2_EIT.eps "fig:"){width="4cm"}\
![Cubic conductor considered in 3–D numerical simulations for the physical EIT model and location of the electrodes: test $T_1$ (a), test $T_2$ (b) and test $T_3$ (c).[]{data-label="fig:EIT_prototype"}](T3_EIT.eps "fig:"){width="4cm"}\
The numerical results for Test $T_1$ are presented in Figure \[fig:T1\_reg\] for $k=0.1$ and $k=10$, respectively, and for varying values of $d_0$. For both cases $k=0.1$ and $k=10$, the theoretical size estimates are given by $$\frac{1}{9} \left ( \frac{l+2z}{l} \right )
\frac{|W-W_0|}{W_0} \leq \frac{|D|}{|\Omega|}\leq \frac{10}{9}
\left ( \frac{l+2z}{l} \right )
\frac{|W-W_0|}{W_0}$$ and, again, they lead to a rather pessimistic evaluation of the upper and lower bounds.
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$ ): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_reg"}](T1_17x17x17_f=01_piu_teoriche.eps){width="6cm"}
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$ ): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T1_reg"}](T1_17x17x17_f=10_piu_teoriche.eps){width="6cm"}
Concerning Test $T_2$, Figure \[fig:T2\_reg\_1\] shows the results when the small electrode coincides with one surface finite element, whereas Figure \[fig:T22\_reg\_3\] refers to the case of a $3 \times 3$ finite elements electrode. One can notice that in all the four cases, the upper bound is not really influenced by the value of $d_0$. Moreover, the inaccuracy in determining the lower bound of the angular sector, is probably due to the fact that the present analysis is restricted to the special class of cubic inclusions.
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $1 \times 1$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_reg_1"}](T2_17x17x17_f=01.eps){width="6cm"}
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $1 \times 1$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_reg_1"}](T2_17x17x17_f=10.eps){width="6cm"}
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $3 \times 3$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T22_reg_3"}](T22_17x17x17_f=01.eps){width="6cm"}
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $3 \times 3$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T22_reg_3"}](T22_17x17x17_f=10.eps){width="6cm"}
A comparison between Figure \[fig:T2\_reg\_1\] and Figure \[fig:T22\_reg\_3\] suggests that better upper bounds can be obtained by enlarging the size of the small electrode. Moreover, from Figures \[fig:T2\_reg\_1\] and \[fig:T22\_reg\_3\] it appears clearly that the lower bound significantly improves as the distance $d_0$ between the inclusion $D$ and the boundary of $\Omega$ increases. This property has been further investigated by increasing only the distance $d_{03}$ of the inclusion $D$ from the face of the conductor containing the small electrode. Figure \[fig:T2\_rvar\] shows the results of simulations in the case of a single finite element electrode and a comparison with Figure \[fig:T2\_reg\_1\] suggests that the improvement of the lower bound is mainly due to the greater distance from the electrode.
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $1 \times 1$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_rvar"}](T2_17x17x17_f=01_r3.eps){width="6cm"}
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $1 \times 1$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T2_rvar"}](T2_17x17x17_f=10_r3.eps){width="6cm"}
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $1 \times 1$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T3_reg"}](T3_17x17x17_f=01.eps){width="6cm"}
 ($17 \times 17 \times 17$ FE mesh, $\zeta=0.2$, $1 \times 1$ FE electrode): $k=0.1$ (a), $k=10$ (b).[]{data-label="fig:T3_reg"}](T3_17x17x17_f=10.eps){width="6cm"}
Finally, the results of the numerical simulations for Test $T_3$ are presented in Figure \[fig:T3\_reg\]. In this case, the lower bound improves as the distance $d_0$ between the inclusion $D$ and the boundary of $\Omega$ increases, whereas the upper bound is indistinguishable from the vertical axis.
Conclusions {#sec:conclusions}
===========
We have tested by numerical simulations the approach of *size estimates* for EIT. We could perform experiments in the 2–D setting with a large varieties of shapes of inclusions and we found quite satisfactory bounds, which in some cases are markedly better than those derived theoretically.
In the 3–D case, we had to limit the variety of shapes of the test inclusions since the growth of their degree of freedom conflicts with the limitations on computer time. We showed that good volume bounds hold when the boundary data $\varphi$ is *well-behaved* in terms of its frequency, whereas they rapidly deteriorate as the frequency increases.
For the complete EIT model we have also made tests in a 3–D setting and compared the bounds in terms of the size of the electrodes, their relative distance and their a-priori assumed distance from the inclusion $D$. We have shown that we obtain good bounds when the electrodes are not too small and when $D$ is sufficiently away from them.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Consider a continuous word embedding model. Usually, the cosines between word vectors are used as a measure of similarity of words. These cosines do not change under orthogonal transformations of the embedding space. We demonstrate that, using some canonical orthogonal transformations from SVD, it is possible both to increase the meaning of some components and to make the components more stable under re-learning. We study the interpretability of components for publicly available models for the Russian language (RusVectōrēs, fastText, RDT).'
author:
- |
Alexey Zobnin\
National Research University Higher School of Economics,\
Faculty of Computer Science,\
azobnin@hse.ru
bibliography:
- 'paper.bib'
title: 'Rotations and Interpretability of Word Embeddings: the Case of the Russian Language'
---
Introduction
============
Word embeddings are frequently used in NLP tasks. In vector space models every word from the source corpus is represented by a dense vector in $\mathbb{R}^d$, where the typical dimension $d$ varies from tens to hundreds. Such embedding maps similar (in some sense) words to close vectors. These models are based on the so called distributional hypothesis: similar words tend to occur in similar contexts [@harris1954distributional]. Some models also use letter trigrams or additional word properties such as morphological tags.
There are two basic approaches to the construction of word embeddings. The first is count-based, or explicit [@levy2014linguistic; @dhillon2015eigenwords]. For every word-context pair some measure of their proximity (such as frequency or PMI) is calculated. Thus, every word obtains a sparse vector of high dimension. Further, the dimension is reduced using singular value decomposition (SVD) or non-negative sparse embedding (NNSE). It was shown that truncated SVD or NNSE captures latent meaning in such models [@landauer1997solution; @murphy2012learning]. That is why the components of embeddings in such models are already in some sense canonical. The second approach is predict-based, or implicit. Here the embeddings are constructed by a neural network. Popular models of this kind include word2vec [@mikolov2013efficient; @mikolov2013distributed] and fastText [@bojanowski2016enriching].
Consider a predict-based word embedding model. Usually in such models two kinds of vectors, both for words and contexts, are constructed. Let $N$ be the vocabulary size and $d$ be the dimension of embeddings. Let $W$ and $C$ be $N \times d$-matrices whose rows are word and context vectors. As a rule, the objectives of such models depend on the dot products of word and context vectors, i. e., on the elements of $WC^T$. In some models the optimization can be directly rewritten as a matrix factorization problem [@levy2014neural; @cotterell2017explaining]. This matrix remains unchanged under substitutions $W \mapsto W S, \quad C \mapsto C {S^{-1}}^T$ for any invertible $S$. Thus, when no other constraints are specified, there are infinitely many equivalent solutions [@fonarev2017riemannian].
Choosing a good, not necessarily orthogonal, post-processing transformation $S$ that improves quality in applied problems is itself interesting enough [@mu2017all]. However, only word vectors are typically used in practice, and context vectors are ignored. The cosine distance between word vectors is used as a similarity measure between words. These cosines will not change if and only if the transformation $S$ is orthogonal. Such transformations do not affect the quality of the model, but may elucidate the meaning of vectors’ components. Thus, the following problem arises: *what orthogonal transformation is the best one for describing the meaning of some (or all) components?*
It is believed that the meaning of the components of word vectors is hidden [@gladkova2016intrinsic]. But even if we determine the “meaning” of some component, we may loose it after re-training because of random initialization, thread synchronization issues, etc. Many researchers [@luo2015online; @ruseti2016using; @andrews2016compressing; @jang2017elucidating] ignore this fact and, say, work with vector components directly, and only some of them take basis rotations into account [@tsvetkov2016correlation]. We show that, generally, re-trained model differ from the source model by almost orthogonal transformation. This leads us to the following problem: *how one can choose the canonical coordinates for embeddings that are (almost) invariant with respect to re-training?*
We suggest using well-known plain old technique, namely, the singular value decomposition of the word matrix $W$. We study the principal components of different models for Russian language (RusVectōrēs, RDT, fastText, etc.), although the results are applicable for any language as well.
Related Work
============
Interpretability of the components have been extensively studied for topic models. In [@chang2009reading; @lau2014machine] two methods for estimating the coherence of topic models with manual tagging have been proposed: namely, word intrusion and topic intrusion. Automatic measures of coherence based on different similarities of words were proposed in [@aletras2013evaluating; @nikolenko2016topic]. But unlike topic models, these methods cannot be applied directly to word vectors. There are lots of new models where interpretability is either taken into account by design [@luo2015online] (modified skip-gram that produces non-negative entries), or is obtained automagically [@andrews2016compressing] (sparse autoencoding).
Lots of authors try to extract some predefined significant properties from vectors: [@jang2017elucidating] (for non-negative sparse embeddings), [@tsvetkov2016correlation] (using a CCA-based alignment between word vectors and manually-annotated linguistic resource), [@rothe2016word] (ultradense projections).
Singular vector decomposition is the core of count-based models. To our knowledge, the only paper where SVD was applied to predict-based word embedding matrices is [@mu2017all]. In [@arora2017simple] the first principal component is constructed for sentence embedding matrix (this component is excluded as the common one).
Word embeddings for Russian language were studied in [@kutuzov2015texts; @Kutuzov2015; @panchenko2015russe; @arefyev2015evaluating].
Theoretical Considerations
==========================
Singular value decomposition
----------------------------
Let $m \ge n$. Recall [@jolliffe2002principal] that a singular value decomposition (SVD) of an $m\times n$-matrix $M$ is a decomposition $M = U \Sigma V^T$, where $U$ is an an $m \times n$ matrix, $U^T U = I_{n}$, $\Sigma$ is a diagonal $n \times n$-matrix, and $V$ is an $n \times n$ orthogonal matrix. Diagonal elements of $\Sigma$ are non-negative and are called singular values. Columns of $U$ are eigenvectors of $M M^T$, and columns of $V$ are eigenvectors of $M^T M$. Squares of singular values are eigenvalues of these matrices. If all singular values are different and positive, then SVD is unique up to permutation of singular values and choosing the direction of singular vectors. Buf if some singular values coincide or equal zero, new degrees of freedom arise.
Invariance under re-training
----------------------------
Learning methods are usually not deterministic. The model re-trained with similar hyperparameters may have completely different components. Let ${M_1}$ and ${M_2}$ be the word matrices obtained after two separate trainings of the model. Let these embeddings be similar in the sense that cosine distances between words are almost the same, i. e., ${M_1}{M_1}^T \approx {M_2}{M_2}^T$. Suppose also that singular values of each ${M_i}$ are different and non-zero. Then one can show that ${M_1}$ and ${M_2}$ differ only by the (almost) orthogonal factor. Indeed, left singular vectors in SVD of ${M_i}$ are eigenvectors of ${M_i}{M_i}^T$. Hence, matrices $U$ and $\Sigma$ in SVD of ${M_1}$ and ${M_2}$ can be chosen the same. Thus, ${M_2}\approx {M_1}Q$, where $Q Q^T = I_d$. Here $Q$ can be chosen as $V_1 V_2^T$ where $V_i$ are matrices of right singular vectors in SVD of ${M_i}$.
Interpretability measures
-------------------------
One of traditional measures of interpretability in topic modeling looks as follows [@newman2010automatic; @lau2014machine]. For each component, $n$ most probable words are selected. Then for each pair of selected words some co-occurrence measure such as PMI is calculated. These values are averaged over all pairs of selected words and all components. The other approaches use human markup. Such measures need additional data, and it is difficult to study them algebraically. Also, unlike topic modeling, word embeddings are not probabilistic: both positive and negative values of coordinates should be considered.
Let all word vectors be normalized and $W$ be the word matrix. Inspired by [@nikolenko2016topic], where vector space models are used for evaluating topic coherence, we suggest to estimate the interpretability of $k$th component as $$ \operatorname{interp}_k W = \sum_{i,j=1}^N W_{i,k} W_{j,k} \left(W_i \cdot W_j \right).$$ The factors $W_{i,k}$ and $W_{j,k}$ are the values of $k$th components of $i$th and $j$th words. The dot product $\left(W_i \cdot W_j\right)$ reflects the similarity of words. Thus, this measure will be high if similar words have similar values of $k$th coordinates.
What orthogonal transformation $Q$ maximizes this interpretability (for some, or all components) of $WQ$? In matrix terms, $$\operatorname{interp}_k W =(W^T W W^T W)_{k, k},$$ and $$\operatorname{interp}_k WQ = \left(Q^T W^T W W^T W Q\right)_{k,k}$$ because $Q$ is orthogonal. The total interpretability over all components is $$\begin{gathered}
\sum_{k=1}^d \operatorname{interp}_k WQ = \sum_{k=1}^d \left(Q^T W^T W W^T W Q \right)_{k,k} = \\
= \operatorname{tr}Q^T W^T W W^T W Q = \operatorname{tr}\left(W^T W W^T W\right) = \sum_{k=1}^d \operatorname{interp}_k W,\end{gathered}$$ because $\operatorname{tr}Q^T X Q = \operatorname{tr}Q^{-1} X Q = \operatorname{tr}X$. It turns out that *in average* the interpretability is constant under any orthogonal transformation. But it is possible to make the first components more interpretable due to the other components. For example, $$(Q^T W^T W W^T W Q)_{1, 1} = \left(q^T W^T W q\right)^2$$ is maximized when $q$ is the eigenvector of $W^T W$ with the largest singular value, i. e., the first right singular vector of $W$ [@jolliffe2002principal]. Let’s fix this vector and choose other vectors to be orthogonal to the selected ones and to maximize the interpretability. We arrive at $Q = V$, where $V$ is the right orthogonal factor in SVD $W = U \Sigma V^T$.
Experiments
===========
\[singular\_values\]
{width="12cm"}
\[overlapping\]
{width="12cm"}
\[alignment\_shifts\]
{width="12cm"}
{width="12cm"}
Canonical basis for embeddings
------------------------------
We train two fastText skipgram models on the Russian Wikipedia with default parameters. First, we normalize all word vectors. Then we build SVD decompositions[^1] of obtained word matrices and use $V$ as an orthogonal transformation. Thus, new “rotated” word vectors are described by the matrix $WV = U \Sigma$. The corresponding singular values are shown in Figure 1, they almost coincide for both models (and thus are shown only for the one model). For each component both in the source and the rotated models we take top 50 words with maximal (positive) and bottom 50 words with minimal (negative) values of the component. Taking into account that principal components are determined up to the direction, we join these positive and negative sets together for each component.
We measure the overlapping of these sets of words. Additionally, we use the following alignment of components: first, we look for the free indices $i$ and $j$ such that $i$th set of words from the first model and $j$th set of words from the second model have the maximal intersection, and so on. We call the difference $i - j$ the alignment shift for the $i$th component. Results are presented in Figures 2 and 3. We see that at least for the first part of principal components (in the rotated models) the overlapping is big enough and is much larger that that for the source models. Moreover, these first components have almost zero alignment shifts. Other principal components have very similar singular values, and thus they cannot be determined uniquely with high confidence.
Normalized interpretibility measures for different components (calculated for 50 top/bottom words) for the source and the rotated models are shown in Fig. 4.
Principal components of different models
----------------------------------------
We took the following already published models:
- RusVectōrēs[^2] lemmatized models (actually, word2vec) trained on different Russian corpora [@KutuzovKuzmenko2017];
- Russian Distributional Thesaurus[^3] (actually, word2vec skipgram) models trained on Russian books corpus [@Panchenko:17:RDT];
- fastText[^4] model trained on Russian Wikipedia [@bojanowski2016enriching].
For each model we took $n = 10000$ or $n = 100000$ most frequent words. Each word vector was normalized in order to replace cosines with dot products. Then we perform SVD $W = U \Sigma V^T$ and take the matrix $W V = U \Sigma$. For each of $d$ components we sort the words by its value and choose top $t$ “positive” and bottom $t$ “negative” words ($t=15$ or 30). For clarity, every selection was clustered into buckets with the simplest greedy algorithm: list the selected words in decreasing order of frequency and either add the current word to some cluster if it is close enough to the word (say, the cosine is greater than $0.6$), or make a new cluster. The cluster’s vector is the average vector of its words. Intuitively, the smaller the number of clusters, the more interpretable the component is. Similar approach was used in [@ramrakhiyani2017measuring].
Tables in the Appendix show the top “negative” and “positive” words of the first principal components for different models. We underline that principal components are determined up to the direction, and thus the separation into “negative” and “positive” parts is random. The full results are available at <https://alzobnin.github.io/>. We cluster these words as described above; different clusters are separated by semicolons. We see the following interesting features in the components:
- stop words: prepositions, conjunctions, etc. (RDT 1, fastText 1; in RusVectōrēs models they are absent just because they were filtered out before training);
- foreign words with separation into languages (fastText 2, web 2), words with special orthography or tokens in broken encoding (not presented here);
- names and surnames (RDT 8, fastText 3, web 3), including foreing names (fastText 9, web 6);
- toponyms (not presented here) and toponym descriptors (web 7);
- fairy tale characters (fastText 6);
- parts of speech and morphological forms (cases and numbers of nouns and adjectives, tenses of verbs);
- capitalization (in fact, first positions in the sentences) and punctuation issues (e. g., non-breaking spaces);
- Wikipedia authors and words from Wikipedia discussion pages (fastText 5);
- other different semantic categories.
We also made an attempt to describe obtained components automatically in terms of common contexts of common morphological and semantic tags using MyStem tagger and semantic markup from Russian National Corpus. Unfortunately, these descriptions are not as good as desired and thus they are not presented here.
Conclusion
==========
We study principal components of publicly available word embedding models for the Russian language. We see that the first principal components indeed are good interpretable. Also, we show that these components are almost invariant under re-learning. It will be interesting to explore the regularities in canonical components between different models (such as CBOW versus Skip-Gram, different train corpora and different languages [@smith2017offline]. It is also worth to compare our intrinsic interpretability measure with human judgements.
Acknowledgements
================
The author is grateful to Mikhail Dektyarev, Mikhail Nokel, Anna Potapenko and Daniil Tararukhin for valuable and fruitful discussions.
Appendix {#appendix .unnumbered}
========
Top/bottom words for the first few principal components for different Russian models {#topbottom-words-for-the-first-few-principal-components-for-different-russian-models .unnumbered}
------------------------------------------------------------------------------------
Note misspellings in 4a.
[^1]: With numpy.linalg.svd it took up to several minutes for 100K vocabulary.
[^2]: <http://rusvectores.org/ru/models/>
[^3]: <https://nlpub.ru/Russian_Distributional_Thesaurus>
[^4]: <https://github.com/facebookresearch/fastText/blob/master/pretrained-vectors.md>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras to model the exponential is a model of differential linear logic. Thus, in this semantics, proof denotations have natural derivatives. We give several examples of these derivatives.'
author:
- 'James Clift, Daniel Murfet'
title: Cofree coalgebras and differential linear logic
---
ł[|]{}
Introduction
============
The idea of taking derivatives of programs is an old one [@paige §2] with many manifestations, including automatic differentiation of algorithms computing real-valued functions [@autodiff] and incremental computation [@incdiff]. However, these approaches are limited to restricted classes of computations, and it is only recently with the development of the differential $\lambda$-calculus by Ehrhard-Regnier [@difflambda] and its refinement differential linear logic [@blutecs; @ehrhard-survey] that derivatives have been defined for general higher-order programs. As with ordinary calculus, the aim of these theories is to assign to a program $P$ another program $\partial P$ (the derivative) which computes the change in the output of $P$ resulting from an infinitesimal change to its input. Here we give ourselves arbitrary $\mathbb{C}$-linear combinations of programs (meaning $\lambda$-terms or proofs in linear logic) as a starting point so that “small” changes to the input make sense.
This paper is about the semantics of differential linear logic, following [@blutecs]. The aim is to explain how the natural semantics of intuitionistic linear logic in vector spaces [@hyland; @murfet_ll] is already a model of differential linear logic. The key point is that tangent vectors and derivatives appear as soon as we introduce cofree coalgebras to model the exponential, which shows that the differential structure is intrinsic to the algebra of linear logic.
To see this, let ${\llbracket - \rrbracket}$ denote semantics in vector spaces and suppose we are given a proof $\pi$ in linear logic computing a function from inputs of type $A$ to outputs of type $B$: $$\varwidth{.9\textwidth}\centering\leavevmode
\AxiomC{$\pi$}
\noLine\UnaryInfC{$\vdots$}
\def\extraVskip{5pt}
\noLine\UnaryInfC{${!} A \vdash B$\,.}
\DisplayProof\endvarwidth$$ The space of inputs to ${\llbracket \pi \rrbracket}$ is ${\llbracket A \rrbracket}$, and a small change in the input starting from $P \in {\llbracket A \rrbracket}$ is a tangent vector $\nu$ at $P$, viewing ${\llbracket A \rrbracket}$ as a smooth manifold or a scheme. This is equivalent to the data of a linear map @C+2pc[ (\[\]/\^2)\^\* & [A ]{} ]{} where $\mathbb{C}[\varepsilon]/\varepsilon^2$ is the ring of dual numbers (this bijection is reviewed in Appendix \[section:tangent\_vectors\]). If ${\llbracket {!} A \rrbracket}$ is the universal cocommutative counital coalgebra mapping to ${\llbracket A \rrbracket}$ then there is a unique lifting of this linear map to a morphism of coalgebras \[eq:toucan\] @C+2pc[ (\[\]/\^2)\^\* & [ A ]{}. ]{} Similarly the linear map ${\llbracket \pi \rrbracket}: {\llbracket {!} A \rrbracket} {\longrightarrow}{\llbracket B \rrbracket}$ lifts to a morphism of coalgebras ${\llbracket {!} A \rrbracket} {\longrightarrow}{\llbracket {!} B \rrbracket}$ which may be composed with to give a morphism of coalgebras \[eq:toucan2\] @C+2pc[ (\[\]/\^2)\^\* & [ A ]{} & [ B ]{} ]{} which, in turn, defines a tangent vector at the point ${\llbracket \pi \rrbracket}\ket{\emptyset}_P \in {\llbracket B \rrbracket}$, where $\ket{\emptyset}_P$ is the point of ${\llbracket {!} A \rrbracket}$ corresponding to $P$. The tangent vector gives the infinitesimal variation of the output of $\pi$ on the input $P$, when the input is varied in the direction of $\nu$.
The formal statement is that for any algebraically closed field $k$ of characteristic zero the semantics of intuitionistic linear logic in $k$-vector spaces defined using cofree coalgebras is model of differential linear logic (Theorem \[main\_theorem\]). We refer to this as the *Sweedler semantics*, since the explicit description of this universal coalgebra is due to him [@sweedler; @murfet_ll]. The proof is elementary and we make no claim here to technical novelty; the link between the symmetric coalgebra and differential calculus is well-known. Perhaps our main contribution is to give several detailed examples showing how to compute these derivatives. We do this with the aim of reinforcing the fact that differentiating programs, even higher-order ones, is a natural thing to do.\
We conclude this introduction with a sketch of one such example and a comparison of our work to other semantics of differential linear logic. To elaborate a little more on the notation: for any type $A$ of linear logic (which for us has only connectives $\otimes, \multimap, !$) there is a vector space ${\llbracket A \rrbracket}$, and for any proof $\pi$ of $A \vdash B$ there is a linear map ${\llbracket \pi \rrbracket}: {\llbracket A \rrbracket} {\longrightarrow}{\llbracket B \rrbracket}$. In particular every proof $\xi$ of type $A$ has a denotation ${\llbracket \xi \rrbracket} \in {\llbracket A \rrbracket}$, and the promotion of $\xi$ has for its denotation a vector $\ket{\emptyset}_{{\llbracket \xi \rrbracket}} \in {\llbracket !A \rrbracket}$, see [@murfet_ll §5.3].
For any binary sequence $S \in \{0,1\}^*$ there is an encoding of $S$ as a proof $\underline{S}$ of type $$\textbf{bint}_A = {!}(A \multimap A) \multimap \big({!}(A \multimap A) \multimap (A \multimap A)\big)\,.$$ Repetition of sequences can be encoded as a proof $$\varwidth{.9\textwidth}\centering\leavevmode
\AxiomC{${\underline{\mathrm{repeat}}}$}
\noLine\UnaryInfC{$\vdots$}
\def\extraVskip{5pt}
\noLine\UnaryInfC{${!} \textbf{bint}_A \vdash \textbf{bint}_A$\,.}
\RightLabel{\scriptsize $\multimap R$}
\DisplayProof\endvarwidth$$ The denotation is a linear map ${\llbracket {!}\textbf{bint}_A \rrbracket} {\longrightarrow}{\llbracket \textbf{bint}_A \rrbracket}$ sending $\ket{\emptyset}_{{\llbracket \underline{S} \rrbracket}}$ to ${\llbracket \underline{SS} \rrbracket}$. The derivative of ${\underline{\mathrm{repeat}}}$ according to the theory of differential linear logic is another a proof $$\varwidth{.9\textwidth}\centering\leavevmode
\AxiomC{$\partial\, {\underline{\mathrm{repeat}}}$}
\noLine\UnaryInfC{$\vdots$}
\def\extraVskip{5pt}
\noLine\UnaryInfC{${!} \textbf{bint}_A, \textbf{bint}_A \vdash \textbf{bint}_A$\,}
\RightLabel{\scriptsize $\multimap R$}
\DisplayProof\endvarwidth$$ which can be derived from ${\underline{\mathrm{repeat}}}$ by new deduction rules called codereliction, cocontraction and coweakening (see Section \[section:coder\]). We prove in Section \[section:bint\] that the denotation of this derivative in the Sweedler semantics is the linear map $$\begin{gathered}
{\llbracket \partial\, {\underline{\mathrm{repeat}}} \rrbracket}: {\llbracket {!}\textbf{bint}_A \rrbracket} \otimes {\llbracket \textbf{bint}_A \rrbracket} {\longrightarrow}{\llbracket \textbf{bint}_A \rrbracket}\,,\\
\ket{\emptyset}_{{\llbracket \underline{S} \rrbracket}} \otimes {\llbracket \underline{T} \rrbracket} \longmapsto {\llbracket \underline{ST} \rrbracket} + {\llbracket \underline{TS} \rrbracket}\end{gathered}$$ whose value on the tensor $\ket{\emptyset}_{{\llbracket \underline{S} \rrbracket}} \otimes {\llbracket \underline{T} \rrbracket}$ we interpret as the derivative of the repeat program at the sequence $S$ in the direction of the sequence $T$. This can be justified informally by the following calculation using an infinitesimal $\varepsilon$ $$\begin{aligned}
(S + \varepsilon T)( S + \varepsilon T) = SS + \varepsilon( ST + TS ) + \varepsilon^2 TT,\end{aligned}$$ which says that varying the sequence infinitesimally from $S$ in the direction of $T$ causes a variation of the repetition in the direction of $ST + TS$.\
The Sweedler semantics is far from the first semantics of differential linear logic: basic examples include the categories of sets and relations [@blutecs §2.5.1] and suplattices [@blutecs §2.5.2]. The motivating examples using topological spaces and differentiable functions are the Köthe and finiteness space semantics of Ehrhard [@ehrhard-kothe; @ehrhard-finiteness] and the semantics of Blute-Ehrhard-Tasson [@blutecon] based on the theory of convenient vector spaces [@frolicher]. These papers explain that the geometric “avatar” of the exponential connective of linear logic is the functor sending a space $X$ to the space of distributions on $X$ (for a precise statement, see Remark \[remark:distr\]). This remarkable analogy between logic and geometry deserves further study. One obstacle is that it seems difficult to compute examples of denotations and their derivatives in the convenient vector space setting of [@blutecon]. For example the coproduct [@blutecon p.12] is defined by extension to a Mackey closure, and is rather implicit.
Conceptually the Sweedler semantics is similar to these examples in that the exponential is modelled by a space of distributions (with finite support) but it is purely algebraic and there are simple explicit formulas for all the structure maps. Moreover in the algebraic approach the differential structure emerges naturally from the exponential structure, rather than being “baked in”. The downside is that the smoothness of proof denotations in our semantics is obscured; in particular, in the case $k = \mathbb{C}$ some extra work is required to see the relation between our differential structure and the derivatives in the usual sense.\
For background material on linear logic and its semantics see [@girard_llogic; @girard_prooftypes; @mellies]. The formal theory of coalgebras is simpler over algebraically closed fields, which explains why we use $k = \mathbb{C}$ in our examples, but this is not really important: one could work over $k = \mathbb{R}$ by taking $\mathbb{C}$-points into account in the explicit description of the cofree coalgebra.\
*Acknowledgements.* Thanks to Kazushige Terui, who stimulated this project by asking if the cofree coalgebra gave a model of differential linear logic.
Main Theorem
============
Let $k$ be an algebraically closed field of characteristic zero and ${\mathcal{V}}$ the category of $k$-vector spaces. This is a model of intuitionistic linear logic, as observed in [@hyland p.5] and developed in detail in [@murfet_ll]. Our notation is as in [@murfet_coalg] and [@murfet_ll §5], see Appendix \[section:background\] for a review.
For a vector space $V$ we denote by ${!} V$ the universal cocommutative counital coalgebra mapping to $V$, with $d: {!} V {\longrightarrow}V$ the universal map. The map $V \mapsto {!} V$ extends to a comonad on ${\mathcal{V}}$ with natural transformation $\delta: {!} {\longrightarrow}{!} {!}$. The coproduct and counit are respectively denoted $\Delta: {!} V {\longrightarrow}{!} V \otimes {!} V$ and $w: {!} V {\longrightarrow}k$. Given vector spaces $V,W$ we write $\sigma_{V,W}: V \otimes W {\longrightarrow}W \otimes V$ for the swap map $\sigma_{V,W}(x \otimes y) = y \otimes x$.
By [@blutecs Proposition 2.6] to equip ${\mathcal{V}}$ with the coalgebra modality $({!}, \delta, d, \Delta, w)$ as a differential category, we need to define a deriving transformation [@blutecs Definition 2.5].
A *deriving transformation* for $({\mathcal{V}}, {!}, \delta, d, \Delta, w)$ is a family of morphisms $$D_V: {!} V \otimes V {\longrightarrow}{!} V$$ natural in $V$, satisfying the following properties for all $V$:
- $w \circ D = 0$, that is, @C+2pc[ V V \^-[D]{} & [!]{} V \^-[w]{} & k ]{} = 0.
- $\Delta \circ D = (1 \otimes D) \circ (\Delta \otimes 1) + (D \otimes 1) \circ (1 \otimes \sigma) \circ (\Delta \otimes 1)$, that is, @C+2pc[ [!]{} V V \^-[D]{} & [!]{} V \^- & [!]{} V V ]{} is equal to the sum $$\begin{gathered}
\xymatrix@C+2pc{{!} V \otimes V \ar[r]^-{\Delta \otimes 1} & {!} V \otimes {!} V \otimes V \ar[r]^-{1 \otimes D} & {!} V \otimes {!} V} \quad +\\
\xymatrix@C+2pc{{!} V \otimes V \ar[r]^-{\Delta \otimes 1} & {!} V \otimes {!} V \otimes V \ar[r]^-{1 \otimes \sigma_{{!}V, V}}_{\cong} & {!} V \otimes V \otimes {!} V \ar[r]^-{D \otimes 1} & {!} V \otimes {!} V}\end{gathered}$$
- $d \circ D = a \circ (w \otimes 1)$, that is, @C+2pc[ [!]{} V V \^-[D]{} & [!]{} V \^-[d]{} & V ]{} = @C+2pc[ [!]{} V V \^-[w 1]{} & k V \^-[a]{}\_- & V ]{} where $a(\lambda \otimes x) = \lambda x$.
- $\delta \circ D = D_{{!} V} \circ (\delta \otimes D) \circ (\Delta \otimes 1)$, that is, @C+2pc[ [!]{} V V \^-[D]{} & [!]{} V \^- & [!]{}[!]{} V ]{} is equal to \[eq:D4final\] @C+2pc[ [!]{} V V \^-[1]{} & [!]{} V V V \^-[D]{} & [!]{}[!]{} V V \^-[D\_[[!]{}V]{}]{} & [!]{}[!]{} V. ]{}
We refer to [@blutecs §2.2] for an explanation of these axioms. Briefly, (D.1) says the derivative of constant maps is zero, (D.2) is the product rule, (D.3) says the derivative of a linear map is constant, and (D.4) is the chain rule. Clearly the rules specify how to commute $D$ past the structural maps $\delta, d, \Delta, w$. Here $d$ stands for the dereliction rule in linear logic, $\Delta$ for contraction and $w$ for weakening. The map $\delta$ stands for promotion, since for a linear map $\phi: {!} V {\longrightarrow}W$ the unique lifting to a morphism of coalgebras $\Phi: {!} V {\longrightarrow}{!} W$ can be obtained as the composite @C+2pc[ [!]{} V \^- & [!!]{} V \^-[[!]{} ]{} & [!]{} W. ]{}
\[defn:D\] We define the $k$-linear map $D_V: {!} V \otimes V {\longrightarrow}V$ by \[defn:DV\] D\_V( \_P ) = \_P.
See Remark \[remark:justify\] for a justification of this definition from the point of view of [@murfet_coalg].
\[main\_theorem\] $D_V$ is a deriving transformation for any vector space $V$.
We split the proof into a series of lemmas. We prefer to give the proofs without first choosing a basis of $V$, but if one is willing to do so, then the connection between these identities and the usual rules of calculus follows from writing the formula for the coproduct $\Delta$ as a kind of Taylor expansion; see for example [@seiler (B.65)].
(D.1) holds for $V$.
This is clear, since the counit $w: {!} V {\longrightarrow}V$ vanishes on $\ket{\omega_1,\ldots,\omega_t}_P$ if $t > 0$.
(D.2) holds for $V$.
Setting $\nu_0 = \nu$ we have $$\begin{aligned}
\Delta D \big( \ket{\nu_1,\ldots,\nu_s}_P \otimes \nu \big) &= \Delta \ket{\nu, \nu_1, \ldots, \nu_s }_P\\
&= \sum_{I \subseteq \{0,1,\ldots,s\}} \ket{\nu_I}_P \otimes \ket{\nu_{I^c}}_P\\
&= \sum_{0 \in I} \ket{\nu_I}_P \otimes \ket{\nu_{I^c}}_P + \sum_{0 \notin I} \ket{\nu_I}_P \otimes \ket{\nu_{I^c}}_P \\
&= \sum_{J \subseteq \{1,\ldots,s\}}\ket{\nu, \nu_J}_P \otimes \ket{\nu_{J^c}}_P + \sum_{J \subseteq \{1,\ldots,s\}} \ket{\nu_J}_P \otimes \ket{\nu, \nu_{J^c}}_P \\
&= \sum_{J \subseteq \{1,\ldots,s\}} \Big\{ D\big( \ket{\nu_J}_P \otimes \nu \big) \otimes \ket{\nu_{J^c}}_P + \ket{\nu_J}_P \otimes D\big( \ket{\nu_{J^c}}_P \otimes \nu \big) \Big\}\end{aligned}$$ as claimed, where for $I \subseteq \{0,1,\ldots,s\}$ we write $I^c$ for $\{0,\ldots,s\} \setminus I$ and for $J \subseteq \{1,\ldots,s\}$, we write $J^c$ for $\{1,\ldots,s\} \setminus J$.
(D.3) holds for $V$.
We have $$d D\big( \ket{\nu_1,\ldots,\nu_s}_P \otimes \nu \big) = d\ket{\nu,\nu_1,\ldots,\nu_s}_P = \delta_{s=0} \nu$$ while $$a( w \otimes 1 )\big( \ket{\nu_1,\ldots,\nu_s}_P \otimes \nu \big) = w\ket{\nu_1,\ldots,\nu_s}_P \cdot \nu = \delta_{s=0} \nu\,.$$
(D.4) holds for $V$.
The trivial case is, with $Q = \ket{\emptyset}_P$, $$\delta D( \ket{\emptyset}_P \otimes \nu ) = \delta\ket{\nu}_P = \big|\, \ket{\nu}_P \big\rangle_Q$$ and on the other side $$\begin{aligned}
D_{{!} V}( \delta \otimes D )( \Delta \otimes 1)( \ket{\emptyset}_P \otimes \nu ) &= D_{{!} V}( \delta \otimes D)( \ket{\emptyset}_P \otimes \ket{\emptyset}_P \otimes \nu )\\
&= D_{{!} V}( \ket{\emptyset}_Q \otimes \ket{\nu}_P )\\
&= \big|\, \ket{\nu}_P \big\rangle_Q\,.\end{aligned}$$ Now we consider the case $s > 0$. Putting $\nu_0 = \nu$ and writing ${\mathcal{P}}_T$ for the set of partitions of $T$ we have $$\delta D\big( \ket{\nu_1,\ldots,\nu_s}_P \otimes \nu \big) = \sum_{X \in {\mathcal{P}}_{\{0,1,\ldots,s\}}} \Big| \otimes_{x \in X} \ket{\nu_{x}}_P \Big\rangle_Q$$ where for a partition $X = \{ x_1,\ldots,x_t \}$ the notation means $$\Big| \otimes_{x \in X} \ket{\nu_{x}}_P \Big\rangle_Q = \Big|\, \ket{\nu_{x_1}}_P, \ldots, \ket{\nu_{x_t}}_P \Big\rangle_Q\,.$$ See Appendix \[section:background\] for the definition of $\ket{\nu_x}$ when $x$ is a set. There is a surjective function $$\begin{gathered}
\theta: {\mathcal{P}}_{\{0,1,\ldots,s\}} {\longrightarrow}{\mathcal{P}}_{\{1,\ldots,s\}}\\
\theta( X ) = \big\{ x \setminus \{0\} \l x \in X \text{ and } x \neq \{0\} \big\}\end{gathered}$$ and given a partition $X = \{ x_1,\ldots,x_t \}$ of $\{1,\ldots,s\}$, $$\begin{aligned}
\theta^{-1}(X) &= \Big\{ \{ x_1 \cup \{0\}, x_2, \ldots, x_t \},\\
&\quad\{ x_1, x_2 \cup \{0\}, \ldots, x_t \},\\
&\quad\ldots,\\
&\quad\{ x_1, x_2, \ldots, x_{t-1}, x_t \cup \{0\} \}\\
&\quad\{ x_1, x_2, \ldots, x_t, \{0\} \} \Big\}\,.\end{aligned}$$ With this in mind we have, writing $\otimes_{x' \neq x} \ket{\nu_{x'}}_P$ for the list of $\ket{\nu_{x'}}_P$ as $x'$ ranges over elements of $X \setminus \{x\}$, that $\delta D\big( \ket{\nu_1,\ldots,\nu_s}_P \otimes \nu \big)$ is equal to \[eq\_written72\] \_[X \_[{1,…,s}]{}]{} { \_[x X]{} | \_P, \_[x’ x]{} \_P\_Q + | \_P, \_[x X]{} \_P \_Q }. Note that when $X = \big\{ \{ 1,\ldots,s \} \big\}$ the summand is $$\Big|\, \ket{\nu,\nu_1,\ldots,\nu_s}_P \Big\rangle_Q + \Big|\, \ket{\nu}_P, \ket{\nu_1,\ldots,\nu_s}_P \Big\rangle_Q\,.$$ On the other hand, the right hand side of the (D.4) identity is $$\begin{aligned}
&D_{{!} V}(\delta \otimes D)(\Delta \otimes 1)\big( \ket{\nu_1,\ldots,\nu_s}_P \otimes \nu \big)\\
&= \sum_{I \subseteq \{1,\ldots,s\}} D_{{!}V}(\delta \otimes D)\Big( \ket{\nu_I}_P \otimes \ket{\nu_{I^c}}_P \otimes \nu \Big)\\
&= \sum_{I \subseteq \{1,\ldots,s\}} D_{{!} V}\Big( \delta \ket{\nu_I}_P \otimes \ket{\nu, \nu_{I^c}}_P \Big)\\
&= D_{{!} V}\Big( \delta\ket{\emptyset}_P \otimes \ket{\nu, \nu_1, \ldots, \nu_s }_P \Big)\\
&\qquad + \sum_{\emptyset \subset I} \sum_{Y \in {\mathcal{P}}_I} D_{{!}V}\Big( \big| \otimes_{y \in Y} \ket{\nu_y}_P \big\rangle_Q \otimes \ket{\nu, \nu_{I^c}}_P \Big)\\
&= \Big|\, \ket{ \nu, \nu_1,\ldots,\nu_s }_P \Big\rangle_Q + \sum_{\emptyset \subset I}\sum_{Y \in {\mathcal{P}}_I} \Big|\, \ket{\nu, \nu_{I^c}}_P\,, \otimes_{y \in Y} \ket{\nu_y}_P \Big\rangle_Q\\
&= \Big|\, \ket{ \nu, \nu_1,\ldots,\nu_s }_P \Big\rangle_Q + \sum_{Y \in {\mathcal{P}}_{\{1,\ldots,s\}}} \Big|\, \ket{\nu}_P\,, \otimes_{y \in Y} \ket{\nu_y}_P \Big\rangle_Q\\
&\qquad+ \sum_{\emptyset \subset I \subset \{1,\ldots,s\}} \sum_{Y \in {\mathcal{P}}_I} \Big|\, \ket{\nu, \nu_{I^c}}_P\,, \otimes_{y \in Y} \ket{\nu_y}_P \Big\rangle_Q\end{aligned}$$ which matches since the last sum can be rewritten as $$\sum_{\substack{X \in {\mathcal{P}}_{\{1,\ldots,s\}}\\ \text{ with } |X| > 1}} \sum_{x \in X} \Big|\, \ket{\nu, \nu_x}_P\,, \otimes_{x' \neq x} \ket{\nu_{x'}}_P \Big\rangle_Q\,.$$
${\mathcal{V}}$ is a differential category.
This follows from [@blutecs Proposition 2.6].
Codereliction, cocontraction, coweakening {#section:coder}
-----------------------------------------
An alternative formulation of the differential structure in differential linear logic is in terms of *codereliction, cocontraction* and *coweakening* maps; see [@fiore] and [@blutecon §5.1]. This has the advantage of providing an appealing symmetry to the formulation of the syntax. In this section we briefly sketch the definition of these maps in the Sweedler semantics. Throughout linear logic means intuitionistic linear logic with the connectives ${!}, \otimes, \multimap$.
First we recall the canonical commutative Hopf structure on ${!} V$ of [@sweedler §6.4]. Given vector spaces $V_1,V_2$ then (see [@sweedler Remark 2.19]) there is an isomorphism of coalgebras $$\begin{gathered}
\Psi: {!} V_1 \otimes {!} V_2 {\longrightarrow}{!} (V_1 \oplus V_2)\,,\\
\ket{\nu_1,\ldots,\nu_s}_P \otimes \ket{\omega_1,\ldots,\omega_t}_Q \longmapsto \ket{\nu_1,\ldots,\nu_s,\omega_1,\ldots,\omega_t}_{(P,Q)}\,.\end{gathered}$$ Using this and the definitions in [@sweedler], it is easy to check that the product $\nabla$ is $$\begin{gathered}
\nabla: {!} V \otimes {!} V {\longrightarrow}{!} V\,,\\
\ket{\nu_1,\ldots,\nu_s}_P \otimes \ket{\omega_1,\ldots,\omega_t}_Q \longmapsto \ket{\nu_1,\ldots,\nu_s,\omega_1,\ldots,\omega_t}_{P+Q}\,,\end{gathered}$$ while the antipode $S$ is $$\begin{gathered}
S: {!} V {\longrightarrow}{!} V\,,\\
\ket{\nu_1,\ldots,\nu_s}_P \longmapsto \ket{-\nu_1,\ldots,-\nu_s}_{-P}\end{gathered}$$ and the unit $u: k {\longrightarrow}{!} V$ is $u(1) = \ket{\emptyset}_0$. By [@sweedler Theorem 6.4.8] these maps make ${!} V$ into a commutative (and cocommutative) Hopf algebra. In the terminology of [@ehrhard-survey] the map $\nabla$ is the *cocontraction* map and $u$ is the *coweakening* map (the antipode seems not to have a formal role in differential linear logic). Finally,
The *codereliction* $\bar{d}$ is the composite $$\xymatrix@C+2pc{
V \cong V \otimes k \ar[r]^-{1 \otimes u} & V \otimes {!} V \ar[r]^-{D} & {!} V
}$$ which is given by $\nu \mapsto \ket{\nu}_0$.
Note that we can recover $D$ as $$\begin{gathered}
\xymatrix@C+2pc{
{!} V \otimes V \ar[r]^-{1 \otimes \bar{d}} & {!} V \otimes {!} V \ar[r]^-{\nabla} & {!} V
}\\
\ket{\nu_1,\ldots,\nu_s}_P \otimes \nu \mapsto \ket{\nu_1,\ldots,\nu_s}_P \otimes \ket{\nu}_0 \mapsto \ket{\nu,\nu_1,\ldots,\nu_s}_P\,.\end{gathered}$$ It seems more convenient to model differentiation syntactically using the codereliction, cocontraction and coweakening maps, rather than the deriving transformation $D$ itself. We briefly sketch how this works, following [@ehrhard-survey]. In the sequent calculus for linear logic one introduces three new deduction rules “dual” to dereliction, contraction and weakening: $$\AxiomC{$\Gamma, {!}A, \Delta \vdash B$}
\LeftLabel{(Codereliction): }
\RightLabel{\scriptsize coder}
\UnaryInfC{$\Gamma, A, \Delta \vdash B$}
\DisplayProof$$ $$\AxiomC{$\Gamma, !A, \Delta \vdash B$}
\LeftLabel{(Cocontraction): }
\RightLabel{\scriptsize coctr}
\UnaryInfC{$\Gamma, !A, !A, \Delta \vdash B$}
\DisplayProof$$ $$\AxiomC{$\Gamma, !A, \Delta \vdash B$}
\LeftLabel{(Coweakening): }
\RightLabel{\scriptsize coweak}
\UnaryInfC{$\Gamma, \Delta \vdash B$}
\DisplayProof$$ together with new cut-elimination rules [@ehrhard-survey §1.4.3].
\[defn:derivative\_proof\] Given a proof $\pi$ of ${!} A \vdash B$ in linear logic, the *derivative* $\partial \pi$ is the proof
whose denotation is, by our earlier remark, the composite @C+2pc[ [!]{} [A ]{} [A ]{} \^-[D]{} & [!]{} [A ]{} \^- & [B ]{}. ]{}
\[remark:totem\] Given $\pi$ as above we have the function [@murfet_ll Definition 5.10] \_[nl]{}: [A ]{} [B ]{}, P \_P, and for $P, \nu \in {\llbracket A \rrbracket}$ we interpret the vector D( \_P ) = \_P [B ]{} as the derivative of ${\llbracket \pi \rrbracket}_{nl}$ at the point $P$ in the direction of $\nu$. Here we implicitly identify ${\llbracket A \rrbracket}$ with the tangent space $T_P{\llbracket A \rrbracket}$ and ${\llbracket B \rrbracket}$ with the tangent space $T_{{\llbracket \pi \rrbracket}_{nl}(P)} {\llbracket B \rrbracket}$. This interpretation is justified by the following elaboration of the remarks in the Introduction.
Let $\operatorname{prom}(\pi)$ denote the proof which is the promotion of $\pi$, which has for its denotation the unique morphism of coalgebras ${\llbracket \operatorname{prom}(\pi) \rrbracket}: {!} {\llbracket A \rrbracket} {\longrightarrow}{!} {\llbracket B \rrbracket}$ with $d \circ {\llbracket \operatorname{prom}(\pi) \rrbracket} = {\llbracket \pi \rrbracket}$. Let $\Psi: (k[\varepsilon]/\varepsilon^2)^* {\longrightarrow}{!} {\llbracket A \rrbracket}$ be the morphism of coalgebras as in corresponding to the tangent vector $\nu$ at a point $P \in {\llbracket A \rrbracket}$. Then the morphism of coalgebras \[eq:prompiafterpsi\] [() ]{} : (k\[\]/\^2)\^\* [!]{} [B ]{} has the following values, writing $Q = {\llbracket \pi \rrbracket}_{nl}(P)$, we have by [@murfet_coalg Theorem 2.22] $$\begin{aligned}
{\llbracket \operatorname{prom}(\pi) \rrbracket}\Psi(1) &= {\llbracket \operatorname{prom}(\pi) \rrbracket} \ket{\emptyset}_P = \ket{\emptyset}_Q\,,\\
{\llbracket \operatorname{prom}(\pi) \rrbracket}\Psi(\varepsilon^*) &= {\llbracket \operatorname{prom}(\pi) \rrbracket} \ket{\nu}_P = \Big|\, {\llbracket \pi \rrbracket}\ket{\nu}_P \Big\rangle_Q\,.\end{aligned}$$ Under the bijection of Section \[section:tangent\_vectors\] the morphism of coalgebras therefore corresponds to the tangent vector ${\llbracket \pi \rrbracket}\ket{\nu}_P \in {\llbracket B \rrbracket}$ at $Q$.
It is easy using the formulas for $\nabla, D$ to check that the $\nabla$-rule of [@blutecs §4.3] is satisfied:
The diagram @C+2pc[ V V V \_-[D 1]{} \^-[1 ]{} & V V \^-[D]{}\
[!]{} V V \_- & [!]{} V ]{} commutes.
This, together with [@blutecs Theorem 4.12], shows that ${\mathcal{V}}$ with the comonad ${!}$ and deriving transformation $D$ is a model of the differential calculus in the sense of [@blutecs Definition 4.11].
${\mathcal{V}}$ is a categorical model of the differential calculus.
Examples {#section:examples}
========
In this section we give various examples of proofs $\pi$ and the derivatives ${\llbracket \pi \rrbracket} \circ D$ of their denotations, according to Definition \[defn:derivative\_proof\]. We recall briefly the definition of the semantics ${\llbracket - \rrbracket}$ in the category ${\mathcal{V}}$ of $k$-vector spaces from [@hyland] and [@murfet_ll §5.1, §5.3]. For a propositional variable $x$ the denotation ${\llbracket x \rrbracket}$ is any finite-dimensional vector space, and $$\begin{aligned}
{\llbracket A \multimap B \rrbracket} &= \Hom_k({\llbracket A \rrbracket},{\llbracket B \rrbracket})\,,\\
{\llbracket A \otimes B \rrbracket} &= {\llbracket A \rrbracket} \otimes {\llbracket B \rrbracket}\,,\\
{\llbracket {!}A \rrbracket} &= {!} {\llbracket A \rrbracket}\,,\end{aligned}$$ where ${!} V$ denotes the universal cocommutative counital coalgebra mapping to $V$.
The encoding of integers and binary sequences in linear logic is based on the following encoding of the composition rule.
For any formula $A$ let $C^1_A$ denote the proof
We define recursively for $n > 1$ a proof $C^n_A$ of $A, (A \multimap A)^n \vdash A$, where $(A \multimap A)^n$ denotes a sequence of $n$ copies of $A \multimap A$, to be
For $n \ge 1$ let $\comp^n_A$ denote the proof
We define $\comp^0_A$ to be the proof
If $V = {\llbracket A \rrbracket}$ and $\alpha_i \in {\llbracket A \multimap A \rrbracket} = \operatorname{End}_k(V)$ for $1 \le i \le n$ then \[order\_comp\] [\^n\_A ]{}( \_1 \_n ) = \_n \_1, while ${\llbracket \comp^0_A \rrbracket} = 1_V$.
Church numerals {#section:church}
---------------
The type of *integers on $A$* [@girard_llogic §5.3.2] is: $$\inta_A = {!}( A \multimap A ) \multimap (A \multimap A)\,.$$ For $n \ge 0$ we define the Church numeral $\underline{n}_A$ to be the proof
Generally we omit the final step, since it is irrelevant semantically. In the case $n = 0$ the ${!}(A \multimap A)$ is introduced on the left by a weakening rule.
The proof $\underline{2}_A$ (see e.g. [@murfet_ll Example 5.9]) is
From now on $A$ is fixed and we write $\underline{n}$ for $\underline{n}_A$. Let $V = {\llbracket A \rrbracket}$ so ${\llbracket A \multimap A \rrbracket} = \operatorname{End}_k(V)$. In the notation of Remark \[remark:totem\], there is a function [ ]{}\_[nl]{}: \_k(V) \_k(V).
For $n \ge 0$ and $\alpha \in \operatorname{End}_k(V)$, we have ${\llbracket \underline{n} \rrbracket}\ket{\emptyset}_\alpha = \alpha^n$ so ${\llbracket \underline{n} \rrbracket}_{nl}(\alpha) = \alpha^n$.
This is an easy exercise, see [@murfet_ll] for the case $n = 2$.
The derivative $\partial\, \underline{n}$ of Definition \[defn:derivative\_proof\] is a proof of ${!}(A \multimap A), A \multimap A \vdash A \multimap A$ and for $\alpha, \nu \in \operatorname{End}_k(V)$ the value of its denotation ${\llbracket \partial\, \underline{n} \rrbracket} = {\llbracket \underline{n} \rrbracket} \circ D$ on $\ket{\emptyset}_\alpha \otimes \nu$, that is, the derivative of $\underline{n}$ at $\alpha$ in the direction of $\nu$, is ${\llbracket \underline{n} \rrbracket} \ket{\nu}_\alpha$.
\[lemma:nderiv\] ${\llbracket \underline{n} \rrbracket}\ket{\nu}_\alpha = \sum_{i = 1}^{n} \alpha^{i-1} \nu \alpha^{n-i}$.
This may be computed using the formulas of [@murfet_ll p.19]. For example, in the case $n = 2$ the image of $\ket{\nu}_\alpha$ under ${\llbracket \underline{n} \rrbracket}$ is given by $$\begin{aligned}
\ket{\nu}_\alpha
&\xmapsto{\makebox[1cm]{\scriptsize ctr}} \ket{\nu}_\alpha \otimes \ket{\emptyset}_\alpha + \ket{\emptyset}_\alpha \otimes \ket{\nu}_\alpha
\\&\xmapsto{\makebox[1cm]{\scriptsize $2 \times$ \text{der}} } \nu \otimes \alpha + \alpha \otimes \nu
\\&\xmapsto{\makebox[1cm]{\scriptsize $ - \circ -$} } \alpha \circ \nu + \nu \circ \alpha\,,\end{aligned}$$ as claimed.
When $k = \mathbb{C}$, $V$ is $r$-dimensional and $\varphi = {\llbracket \underline{n} \rrbracket}_{nl}$, the vector ${\llbracket \underline{n} \rrbracket}\ket{\nu}_\alpha$ agrees with the image of $\nu$ under the usual tangent map of the smooth map $\varphi$ $$\xymatrix@C+2pc{
M_r(\mathbb{C}) \cong T_\alpha \operatorname{End}_k(V) \ar[r]^-{T_\alpha \varphi} & T_{\alpha^n}\operatorname{End}_k(V) \cong M_r(\mathbb{C})\,.
}$$ This justifies in this case the interpretation of ${\llbracket \underline{n} \rrbracket}\ket{\nu}_\alpha$ as the derivative.
Binary integers {#section:bint}
---------------
The type of *binary integers on $A$* [@girard_complexity §2.5.3] is: $$\binta_A = {!}( A \multimap A) \multimap ({!}( A \multimap A) \multimap ( A \multimap A)).$$ Given a sequence $S \in \{0,1\}^*$ we define a proof $\underline{S}_A$ of $\binta_A$ as follows. Let $l \ge 0$ be the length of $S$. The proof tree for $\underline{S}_A$ matches that of the Church numeral $\underline{l}$ up to the step where we perform contractions, that is, $$\label{bint_Upto}
\varwidth{.9\textwidth}\centering\leavevmode
\AxiomC{$\comp^l_A$}
\noLine\UnaryInfC{$\vdots$}
\noLine\UnaryInfC{$(A \multimap A)^{l} \vdash A \multimap A$}
\doubleLine\RightLabel{\scriptsize $n \times$ der}
\UnaryInfC{${!}(A \multimap A)^l \vdash A \multimap A$}
\DisplayProof\endvarwidth$$ We match each copy of ${!}(A \multimap A)$ on the left with the corresponding position in $S$, and using a series of contractions we identify all copies corresponding to a position in which $0$ appears in $S$, and likewise all copies corresponding to positions with a $1$. After these contractions, there will be two copies of ${!}(A \multimap A)$ on the left (the first being by convention the remnant of all the $0$-associated copies) unless $S$ contains only $0$’s or only $1$’s. In this case we use further a weakening rule to introduce the “missing” ${!}(A \multimap A)$, giving finally the desired proof $\underline{S}_A$:
In the final right $\multimap R$ introduction rules, the second copy of ${!}(A \multimap A)$ (associated with the $1$’s in $S$) is moved across the turnstile first. If $S$ is the empty sequence, then $l = 0$ and the proof is a pair of weakenings on the left followed by the $\multimap R$ introduction rules.
For the rest of this section $A$ is fixed and we write $\underline{S}$ for $\underline{S}_A$.
The proof $\underline{001}$ is
where the colouring indicates which copies of ${!}(A \multimap A)$ are contracted. Using , [ ]{}(\_\_) = [\^3\_A ]{}(\_\_\_) = .
Generalising the calculation of Section \[section:church\] we now describe the derivatives of binary integers. The general formula computes, for $S \in \{0,1\}^*$, the linear operator $${\llbracket \underline{S} \rrbracket}\big( \ket{\alpha_1,\ldots,\alpha_r}_{\gamma} \otimes \ket{\beta_1,\ldots,\beta_s}_{\delta} \big) \in \operatorname{End}_k(V)\,.$$ Informally, this operator is described by inserting $\gamma$ for $0$ and $\delta$ for $1$ in (the reversal of) $S$, and then summing over all ways of replacing $r$ of the $\gamma$’s in this composite with $\alpha_i$’s, and $t$ of the $\delta$’s with $\beta_j$’s. Let $\operatorname{Inj}(P,Q)$ denote the set of injective functions $P {\longrightarrow}Q$, and write $[s] = \{1,\ldots,s\}$.
\[lemma:derivative\_bint\] Let $S = a_l a_{l-1} \cdots a_1$ with $a_i \in \{0,1\}$ be a binary sequence, and set $$N_0 = \{ j \l a_j = 0 \}\,, \qquad N_1 = \{ j \l a_j = 1 \}\,.$$ Then we have \[eq:formula\_derivative\_bint\] [ ]{}( \_ \_ ) = \_[f (\[s\],N\_0)]{} \_[g (\[r\],N\_1)]{} \^[f,g]{}\_1 \^[f,g]{}\_l, where $$\Gamma^{f,g}_i = \begin{cases}
\gamma & i \in N_0 \setminus \operatorname{Im}(f)\,,\\
\delta & i \in N_1 \setminus \operatorname{Im}(g)\,,\\
\alpha_j & \text{if } i \in \operatorname{Im}(f) \text{ and } f(j) = i\,,\\
\beta_j & \text{if } i \in \operatorname{Im}(g) \text{ and } g(j) = i\,.
\end{cases}$$ In particular this vanishes if $s > |N_0|$ or $r > |N_1|$.
This is clear, since ${\llbracket \underline{S} \rrbracket}$ applies $n = |N_0|$ coproducts to $\ket{\alpha_1,\ldots,\alpha_s}_{\gamma}$ yielding $$\sum_{\substack{J_1,\ldots,J_n\\ \text{pairwise disjoint, s.t.} \\ J_1 \cup \cdots \cup J_n = \{1,\ldots,s\}}} \ket{\alpha_{J_1}}_\gamma \otimes \cdots \otimes \ket{\alpha_{J_n}}_\gamma\,,$$ to which the dereliction operator $d^{\otimes n}$ is applied, which annihilates those tuples $(J_1,\ldots,J_n)$ where any $J_i$ contains more than one element. The resulting sum is over $f \in \operatorname{Int}([s],N_0)$ and each summand is $\gamma \otimes \cdots \otimes \alpha_{\sigma(1)} \otimes \cdots \otimes \gamma \otimes \cdots \alpha_{\sigma(s)} \otimes \cdots \otimes \gamma$ for a permutation $\sigma$. The same is true of $\ket{\beta_1,\ldots,\beta_r}_{\delta}$, and after the two resulting tensors are intertwined the final step is compose all the operators, yielding .
For $S = 001$ we have $$\begin{aligned}
{\llbracket \underline{001} \rrbracket}\big( \ket{\alpha}_\gamma \otimes \ket{\emptyset}_\delta \big) &= \delta \circ \alpha \circ \gamma + \delta \circ \gamma \circ \alpha\,,\\
{\llbracket \underline{001} \rrbracket}\big( \ket{\alpha_1,\alpha_2}_\gamma \otimes \ket{\emptyset}_\delta \big) &= \delta \circ \alpha_1 \circ \alpha_2 + \delta \circ \alpha_2 \circ \alpha_1\,,\\
{\llbracket \underline{001} \rrbracket}\big( \ket{\emptyset}_\gamma \otimes \ket{\beta}_\delta \big) &= \beta \circ \gamma \circ \gamma\,,\\
{\llbracket \underline{001} \rrbracket}\big( \ket{\alpha}_\gamma \otimes \ket{\beta}_\delta \big) &= \beta \circ \alpha \circ \gamma + \beta \circ \gamma \circ \alpha\,,\\
{\llbracket \underline{001} \rrbracket}\big( \ket{\alpha_1,\alpha_2}_\gamma \otimes \ket{\beta}_\delta \big) &= \beta \circ \alpha_1 \circ \alpha_2 + \beta \circ \alpha_2 \circ \alpha_1\,.\end{aligned}$$ and zero for all other inputs.
More interestingly we can also compute the derivatives of proofs of ${!}\binta_A \vdash \binta_A$. In what follows $A$ is fixed and $E = A \multimap A$.
The proof $\repeat$ is
which repeats a binary sequence in the sense that the cutting it against the promotion of $\underline{S}$ is equivalent under cut-elimination to $\underline{SS}$. In particular, ${\llbracket \repeat \rrbracket}\ket{\emptyset}_{{\llbracket \underline{S} \rrbracket}} = {\llbracket \underline{SS} \rrbracket}$.
Given $S, T \in \{0,1\}^*$ the derivative of $\repeat$ at $S$ in the direction of $T$ is \_[[ ]{}]{} [\_A ]{} = \_k( [!]{} \_k(V) \_k(V), \_k(V)), and as promised in the Introduction:
${\llbracket \repeat \rrbracket}\ket{{\llbracket \underline{T} \rrbracket}}_{{\llbracket \underline{S} \rrbracket}} = {\llbracket \underline{ST} \rrbracket} + {\llbracket \underline{TS} \rrbracket}$.
The value of the left-hand side on a tensor $\ket{\alpha_1,\ldots,\alpha_s}_{\gamma} \otimes \ket{\beta_1,\ldots,\beta_r}_{\delta}$ is computed by reading the proof-tree for $\repeat$ from bottom to top: $$\begin{aligned}
\ket{{\llbracket \underline{T} \rrbracket}}_{{\llbracket \underline{S} \rrbracket}}
&\xmapsto{\makebox[1cm]{\scriptsize\text{ctr}}} \ket{{\llbracket \underline{T} \rrbracket}}_{{\llbracket \underline{S} \rrbracket}} \otimes \ket{\emptyset}_{{\llbracket \underline{S} \rrbracket}} + \ket{\emptyset}_{{\llbracket \underline{S} \rrbracket}} \otimes \ket{{\llbracket \underline{T} \rrbracket}}_{{\llbracket \underline{S} \rrbracket}}
\\&\xmapsto{\makebox[1cm]{\scriptsize$2\times$ der}} {\llbracket \underline{T} \rrbracket} \otimes {\llbracket \underline{S} \rrbracket} + {\llbracket \underline{S} \rrbracket} \otimes {\llbracket \underline{T} \rrbracket}
\\&\xmapsto{\makebox[1cm]{\scriptsize${2\times} {R \multimap}$}} \ket{\alpha_1,\ldots,\alpha_s}_{\gamma} \otimes \ket{\beta_1,\ldots,\beta_r}_{\delta} \otimes \big({\llbracket \underline{T} \rrbracket} \otimes {\llbracket \underline{S} \rrbracket} + {\llbracket \underline{S} \rrbracket} \otimes {\llbracket \underline{T} \rrbracket}\big)
\\&\xmapsto{\makebox[1cm]{\scriptsize$2\times$ ctr}} \sum_{I,J} \ket{\alpha_I}_{\gamma} \otimes \ket{\beta_J}_{\delta} \otimes \ket{\alpha_{I^c}}_{\gamma} \otimes \ket{\beta_{J^c}}_{\delta} \otimes \big( {\llbracket \underline{T} \rrbracket} \otimes {\llbracket \underline{S} \rrbracket} + {\llbracket \underline{S} \rrbracket} \otimes {\llbracket \underline{T} \rrbracket} \big)
\\&\xmapsto{\makebox[1cm]{}} \sum_{I,J} {\llbracket \underline{S} \rrbracket}\big( \ket{\alpha_I}_{\gamma} \otimes \ket{\beta_J}_{\delta} \big) \circ {\llbracket \underline{T} \rrbracket}\big( \ket{\alpha_{I^c}}_{\gamma} \otimes \ket{\beta_{J^c}}_{\delta} \big)
\\& \qquad\qquad + \sum_{I,J} {\llbracket \underline{T} \rrbracket}\big( \ket{\alpha_I}_{\gamma} \otimes \ket{\beta_J}_{\delta} \big) \circ {\llbracket \underline{S} \rrbracket}\big( \ket{\alpha_{I^c}}_{\gamma} \otimes \ket{\beta_{J^c}}_{\delta} \big)\end{aligned}$$ which agrees with ${\llbracket \underline{ST} \rrbracket} + {\llbracket \underline{TS} \rrbracket}$ on $\ket{\alpha_1,\ldots,\alpha_s}_{\gamma} \otimes \ket{\beta_1,\ldots,\beta_r}_{\delta}$ by Lemma \[lemma:derivative\_bint\].
Multiplication {#section:mult}
--------------
The multiplication of Church numerals is encoded by a proof $\mult_A$ of ${!} \inta_A, \inta_A \vdash \inta_A$, see for example [@girard_complexity §2.5.2]. To construct the proof tree it will be convenient to introduce the following intermediate proof $\gamma$, writing $E = A \multimap A$ as above:
Then ${\llbracket \gamma \rrbracket}: {!}\operatorname{End}_k(V) \otimes {!}{\llbracket \inta_A \rrbracket} \to {!}\operatorname{End}_k(V)$ is a morphism of coalgebras such that $${\llbracket \gamma \rrbracket}(t \otimes \vacu_\alpha) = \vacu_{\alpha(t)} \qquad \text{and} \qquad {\llbracket \gamma \rrbracket}(t \otimes \ket{\nu}_\alpha) = \ket{\nu(t)}_{\alpha(t)},$$ for $\alpha, \nu \in {\llbracket \inta_A \rrbracket} = \Hom_k({!}\operatorname{End}_k(V), \operatorname{End}_k(V))$ and $t \in {!} \operatorname{End}_k(V)$. The proof $\mult_A$ is
Let $l,m,n \ge 0$ be integers. We write $\mult_A(-,n)$ for the proof of ${!} \inta_A \vdash \inta_A$ obtained from the above by cutting against the proof $\underline{n}$ of $\vdash \inta_A$. The derivative of this proof at $\alpha = {\llbracket \underline{l} \rrbracket}$ in the direction of $\nu = {\llbracket \underline{m} \rrbracket}$ is the element of ${\llbracket \inta_A \rrbracket}$ given on $t \in {!} \operatorname{End}_k(V)$ by $$\begin{aligned}
{\llbracket \mult_A \rrbracket}\big(\ket{\nu}_\alpha \otimes {\llbracket \underline{n} \rrbracket}\big)(t)
= {\llbracket \underline{n} \rrbracket}\big(\ket{\nu(t)}_{\alpha(t)}\big)
= \sum_{i = 1}^{n} \alpha(t)^{i-1} \nu(t) \alpha(t)^{n-i}\end{aligned}$$ using Lemma \[lemma:nderiv\] in the last step. When $t = \vacu_x$ for $x \in \operatorname{End}_k(V)$, this evaluates to $$\sum_{i = 1}^{n} \alpha(t)^{i-1} \nu(t) \alpha(t)^{n-i}
= \sum_{i = 1}^{n} x^{l(i-1)} x^m x^{l(n-i)}
= n x^{l(n-1)+m}.$$ This result agrees with a more traditional calculus approach using limits: $$\begin{aligned}
&\lim_{h \to 0} \frac{{\llbracket \mult_A \rrbracket}(\vacu_{{\llbracket \underline{l} \rrbracket} + h {\llbracket \underline{m} \rrbracket}} \otimes {\llbracket n \rrbracket})\vacu_x - {\llbracket \mult_A \rrbracket}(\vacu_{{\llbracket \underline{l} \rrbracket}} \otimes {\llbracket \underline{n} \rrbracket})\vacu_x}{h}\\
&= \lim_{h \to 0} \frac{{\llbracket \underline{n} \rrbracket}\vacu_{x^l + h x^m} - {\llbracket \underline{n} \rrbracket}\vacu_{x^l}}{h} \\
\\&= \lim_{h \to 0} \frac{(x^l + h x^m)^n - x^{ln}}{h}
\\&= nx^{l(n-1)+m}.\end{aligned}$$ This agreement between limits and the derivatives of Definition \[defn:derivative\_proof\] holds more generally, but will be discussed elsewhere.
The cofree coalgebra {#section:background}
====================
Let $k$ be an algebraically closed field of characteristic zero and ${\mathcal{V}}$ the category of $k$-vector spaces (possibly infinite dimensional). It is straightforward to check that the universal counital cocommutative coalgebra mapping to a finite-dimensional vector space $V$ (i.e. the cofree coalgebra) is the space $\Hom_k^{\cont}(\operatorname{Sym}(V^*),k)$ of linear functionals on the symmetric algebra $\operatorname{Sym}(V^*)$ which vanish on an ideal of finite codimension [@hyland]. Using the Chinese remainder theorem one sees further that this coalgebra breaks into a sum of pieces each isomorphism to a symmetric algebra, that is, that there is an isomorphism of coalgebras \[eq:isocontnu\] \_k\^((V\^\*),k) \_[P V]{} \_P(V) where $\operatorname{Sym}_P(V)$ is the symmetric algebra with its usual coproduct structure. Following the tradition of linear logic, we denote this universal coalgebra ${!} V$. The explicit formulas for the coproduct $\Delta_V$, counit $w_V$ and universal morphism $d_V: {!} V {\longrightarrow}V$ are given in [@murfet_coalg] and [@murfet_ll §5.2]. The universality of $d_V$ means more precisely that for any cocommutative coalgebra $C$ there is an isomorphism $$\begin{gathered}
\Hom_{\operatorname{Coalg}(k)}(C, {!} V) \cong \Hom_{k}(C, V)\,,\\
\Phi \longmapsto d_V \circ \Phi\end{gathered}$$ where the left hand side denotes morphisms of $k$-coalgebras.
We use the following notation from [@murfet_coalg; @murfet_ll]: the image in the summand $\operatorname{Sym}_P(V)$ of ${!} V$ of a tensor $\nu_1 \otimes \cdots \otimes \nu_s \in V^{\otimes s}$ is denoted $$\ket{\nu_1,\ldots,\nu_s}_P = \overline{\nu_1 \otimes \cdots \otimes \nu_s} \in \operatorname{Sym}_P(V) \subseteq {!} V\,.$$ By definition $\ket{\nu_{\tau(1)}, \ldots, \nu_{\tau(s)}}_P$ is the same element of ${!} V$ for any permutation $\tau \in \mathfrak{S}_s$. The identity $1 \in \operatorname{Sym}_P(V)$ is denoted $\ket{\emptyset}_P$. Given a subset $C = \{i_1,\ldots,i_t\} \subseteq \{1,\ldots,s\}$ we write $\ket{\nu_C}_P$ for $\ket{\nu_{i_1},\ldots,\nu_{i_t}}_P$. By [@murfet_coalg Theorem 2.22] the unique morphism of coalgebras $\delta: {!} V {\longrightarrow}{!} {!} V$ lifting the identity on ${!} V$, that is, satisfying $d_{{!} V} \circ \delta = 1_{{!} V}$, is given for $P, \nu_1,\ldots,\nu_s \in V$ by the formula \_P = \_ | \_P, …, \_P \_Q where $Q = \ket{\emptyset}_P$. As a special case $\delta\ket{\emptyset}_P = \ket{\emptyset}_Q$. By construction the tuple $({!}, \delta, d, \Delta, w)$ is a coalgebra modality on ${\mathcal{V}}$ in the sense of [@blutecs Definition 2.1].
Tangent vectors {#section:tangent_vectors}
---------------
There is a well-known connection between the cofree coalgebra and tangent vectors, arising from the identification of the former with a coalgebra of distributions on $V$. We review this point of view at the end of Section \[section:residues\]. But first we recall the approach to this connection of [@murfet_ll Appendix B] which emphasises the ring of dual numbers $k[\varepsilon]/(\varepsilon^2)$. In algebraic geometry this ring represents tangent vectors, in the sense that there is a bijection between morphisms of $k$-schemes $\operatorname{Spec}(k[\varepsilon]/\varepsilon^2) {\longrightarrow}X$ and pairs $(P, \nu)$ consisting of a closed point $P \in X$ and a tangent vector $\nu \in T_P X = ({\mathfrak{m}}/{\mathfrak{m}}^2)^*$ where ${\mathfrak{m}} \subseteq {\mathcal{O}}_{X,P}$ is the maximal ideal.
When $X = \operatorname{Spec}(\operatorname{Sym}(V^*))$ for a finite-dimensional vector space $V$, the closed points of $X$ are in bijection with vectors in $V$, and the tangent space $T_P X$ is canonically isomorphic to $V$. Hence a morphism $\Phi: \operatorname{Spec}(k[\varepsilon]/(\varepsilon^2)) {\longrightarrow}X$ is determined by two vectors $P, \nu \in V$. This data is naturally associated with the element $$\ket{\nu}_P \in {!} V$$ as follows. Let ${\mathcal{T}} = (k[\varepsilon]/(\varepsilon^2))^*$ with its natural coalgebra structure. Then $$\begin{aligned}
\Hom_{\operatorname{Sch}/k}\!\big(\operatorname{Spec}(k[\varepsilon]/(\varepsilon^2)), X\big) &\cong \Hom_{\operatorname{Alg}(k)}\!\big(\operatorname{Sym}(V^*), k[\varepsilon]/(\varepsilon^2)\big)\\
&\cong \Hom_{k}(V^*, k[\varepsilon]/(\varepsilon^2))\\
&\cong \Hom_{k}({\mathcal{T}}, V)\\
&\cong \Hom_{\operatorname{Coalg}(k)}({\mathcal{T}}, {!} V)\,.\end{aligned}$$ Under these bijections the scheme morphism $\Phi$ corresponds to the linear map : V\^\* k\[\]/(\^2), (f) = f(P) 1 + \_( f )|\_[x=P]{} and to the morphism of coalgebras \[eq:app\_psiman\] : [!]{} V, ( 1 ) = \_P, ( \^\* ) = \_P. This justifies the identification of $\ket{\nu}_P \in {!} V$ with the tangent vector $\nu$ at $P \in X$.
Under the isomorphism the element $\ket{\nu}_P$ corresponds to the functional taking the derivative of a function $f \in \operatorname{Sym}(V^*)$ in the direction $\nu$ at $P$. One way to talk about such distributions for arbitrary fields is the framework of local cohomology and residues, which we now recall from [@murfet_coalg].
Local cohomology and residues {#section:residues}
-----------------------------
For a finite-dimensional vector space $V$ of dimension $n$ with $R = \operatorname{Sym}(V^*)$, one proves using local duality [@murfet_coalg Theorem 2.6] that there is an isomorphism \[eq:otherbangV\] \_[P V]{} H\^[n]{}\_[P]{}(R, \^n\_[R/k]{}) \_k\^((V\^\*),k) , where $H^n_P$ denotes local cohomology at $P$ [@residuesduality]. This isomorphism is defined by sending a class $\tau$ in the local cohomology at $P$ to the functional $f \mapsto \Res_P(f \tau)$ where $\Res_P$ denotes a generalised residue and $f \tau$ the action by $R$ on local cohomology. The connection between and is given by an isomorphism $H^{n}_{P}(R, \Omega^n_{R/k}) \cong \operatorname{Sym}_P(V)$ identifying the identity $\ket{\emptyset}_P$ in $\operatorname{Sym}_P(V)$ with the class of the meromorphic differential form [@murfet_coalg Definition 2.9] H\^[n]{}\_[P]{}(R, \^n\_[R/k]{}). It is easy to see that $$\label{eq:residue_differentiates}
\Res_P\Big( f \ket{\nu}_P \Big) = \partial_{\nu}( f )|_{x = P}\,,$$ and more generally that [@murfet_coalg Lemma 2.13] $$\label{eq:residue_differentiates2}
\Res_P\Big( f \ket{\nu_1,\ldots,\nu_s}_P \Big) = \partial_{\nu_1} \cdots \partial_{\nu_s}( f )|_{x = P}\,.$$ Thus we may identify elements of ${!} V$ with functionals on the space of polynomial functions, given by taking derivatives at points of $V$.
When $k = \mathbb{C}$ with $V = \mathbb{C} \nu$ and $z = \nu^*$ the generator of $R = \mathbb{C}[z]$, this is nothing but the Cauchy integral formula since we have \_P = , \_P = and the Cauchy formula says $$f'(P) = \frac{1}{2 \pi i} \oint_\gamma \frac{f(z)}{(z-P)^2} {\operatorname{d}\!{z}}\,.$$
\[remark:distr\] When $k = \mathbb{R}$ this agrees with the analytic theory of distributions, since by [@friedlander Theorem 3.2.1] the $\mathbb{C}$-vector space of distributions on the real manifold $V$ supported at a point $P$ is spanned by the functions $$f \longmapsto \partial_{\nu_1} \cdots \partial_{\nu_s}(f)|_{x=P}$$ as $s \ge 0$ and $\nu_1,\ldots,\nu_s$ varies over all sequences in $V$. So in this case we can identify the coalgebra ${!} V \otimes_{\mathbb{R}} \mathbb{C}$ with the space of distributions on $V$ with finite support.
In the semantics of differential linear logic defined using finiteness spaces [@ehrhard-finiteness] and convenient vector spaces [@blutecon] the space ${!} V$ is a closure of the linear span of Dirac distributions (in our notation, $\ket{\emptyset}_P$) on $V$. More precisely, if $V$ is a finite-dimensional convenient vector space then ${!} V$ consists of distributions of compact support, which are obtained as limits of Dirac distributions. For example, see [@blutecs Theorem 5.7] for the limit defining the distribution $\ket{v}_0$ in our notation. There is a very similar role for Dirac distributions in the Coherent Banach space semantics of linear logic in [@girard_banach §3.2].
It is interesting to note that functional programs extended with Dirac distributions have already been considered in the literature on automatic differentiation; see [@nilsson]. For an abstract categorical theory of distributions via monads, see [@kock].
Any cocommutative coalgebra is the direct limit of finite-dimensional coalgebras, and the category of finite-dimensional cocommutative coalgebras is isomorphic to the category of zero-dimensional schemes over $k$. This is taken as the starting point of one approach to noncommutative geometry which has been influential in the study of $A_\infty$-algebras, where one posits that an arbitrary coalgebra is the coalgebra of distributions on a “noncommutative space” [@kontnc p.15], [@kontnc2; @lebruyn].
\[remark:justify\] Definition \[defn:D\] is justified as follows: adding $\nu$ to a ket, under the residue, contributes a partial derivative in the direction of $\nu$ by . Stating this in a different way, observe that there is a canonical map $\iota: V {\longrightarrow}{\mathcal{D}}(R)$ sending $\nu \in V$ to the differential operator $\partial_\nu$ and we have a $k$-linear map @C+2pc[ H\^n\_P(R, \^n\_[R/k]{}) V \^-[1 ]{} & H\^n\_P(R, \^n\_[R/k]{}) (R) \^-[a]{} & H\^n\_P(R, \^n\_[R/k]{}) ]{} where $a$ denotes the action of the ring ${\mathcal{D}}(R)$ on local cohomology [@murfet_coalg Lemma 2.7]. These maps assemble in the colimit to give .
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a model for the origin of the extended law of star formation in which the surface density of star formation ($\Sigma_{\rm SFR}$) depends not only on the local surface density of the gas ($\Sigma_{g}$), but also on the stellar surface density ($\Sigma_{*}$), the velocity dispersion of the stars, and on the scaling laws of turbulence in the gas. We compare our model with the spiral, face-on galaxy NGC 628 and show that the dependence of the star formation rate on the entire set of physical quantities for both gas and stars can help explain both the observed general trends in the $\Sigma_{g}-\Sigma_{\rm SFR}$ and $\Sigma_{*}-\Sigma_{\rm SFR}$ relations, but also, and equally important, the scatter in these relations at any value of $\Sigma_{g}$ and $\Sigma_{*}$. Our results point out to the crucial role played by existing stars along with the gaseous component in setting the conditions for large scale gravitational instabilities and star formation in galactic disks.'
author:
- |
Sami Dib$^{1}$[^1], Sacha Hony$^{2}$ Guillermo Blanc$^{3,4,5}$\
$^{1}$Universidad de Atacama, Copayapu 485, Copiapó, Chile\
$^{2}$Institut für Theoretische Astrophysik, Zentrum für Astronomie der Universität Heidelberg, Albert-Überle-Stra[ß]{}e 2, 69120 Heidelberg, Germany.\
$^{3}$Observatories of the Carnegie Institution for Science, 813 Santa Barbara St, Pasadena, CA, 91101, USA\
$^{4}$Departamento de Astronomía, Universidad de Chile, Camino del Observatorio 1515, Las Condes, Santiago, Chile\
$^{5}$Centro de Astrofísica y Tecnologías Afines (CATA), Camino del Observatorio 1515, Las Condes, Santiago, Chile\
date: 'Accepted XXX. Received XXX'
title: 'The extended law of star formation: the combined role of gas and stars'
---
\[firstpage\]
galaxies: star formation - galaxies: kinematics and dynamics - galaxies: stellar content - ISM: structure - galaxies: ISM - galaxies: evolution
INTRODUCTION {#motiv}
============
The star formation rate (SFR) is the quantity that describes how galaxies convert their gas reservoirs into stars per unit time. Quantifying the dependence of the SFR on the global properties of galaxies as well as on the local conditions within galaxies is essential towards understanding their observed properties and their dynamical and chemical evolution across cosmic time. Traditionally, observational studies have sought the correlation between the surface density of star formation ($\Sigma_{\rm SFR}$) and the surface density of the gas $\Sigma_{g}=\Sigma_{\ion{H}{i}}+\Sigma_{{\rm H_{2}}}$, where $\Sigma_{\ion{H}{i}}$ and $\Sigma_{{\rm H_{2}}}$ are the surface densities of the neutral and molecular hydrogen, respectively. The emerging picture from all of these works is that $\Sigma_{\rm SFR} \propto \Sigma_{g}^{n}$ with $n \approx 1.4$ (e.g., Schmidt 1959; Kennicutt 1998; Bigiel et al. 2008; Blanc et al. 2009). Other studies found that the surface density of star formation scales linearly or sub-linearly ($n \lesssim 1$) with the surface density of molecular hydrogen traced by CO lines or with the surface density of molecules that trace higher density gas such as HCN (e.g., Gao & Solomon 2004; Shetty et al. 2013; Liu et al. 2016). Several ideas have been proposed in order to explain the origin of the star formation scaling relations. The earliest scenarios proposed that stars form as a result of gravitational instabilities (GI) in the gaseous component of galactic disks over a timescale which is the local free-fall time of the gas and which is given by $t_{ff,g} \propto \rho_{g}^{-0.5}$, where $\rho_{g}$ is the local gas volume density. For a constant scale height of the disk, $\rho_{g} \propto \Sigma_{g}$ and thus $\Sigma_{\rm SFR} \propto \Sigma_{g}/t_{ff,g} \propto \Sigma_{g}^{1.5}$ (e.g., Madore 1977). Wong & Blitz (2002) argued that the value of the star formation law slope is related to the value of the molecular fraction $f_{{\rm H_{2}}}=\Sigma_{{\rm H_{2}}}/\Sigma_{g}$ and Blitz & Rosolowsky (2006) showed that $f_{{\rm H_{2}}}$ can be related to the pressure of the interstellar medium. It was also suggested that the value of $n$ is related to the width of the density probability distribution function of the interstellar gas and to the threshold density that is associated with the gas tracer (Tassis 2007; Wada & Norman 2007). Escala (2011) argued that a correlation exists between the largest mass-scale for structures not stabilised by rotation and the SFR. Other groups (e.g., Krumholz & McKee 2005; Padoan & Nordlund 2011; Hennebelle & Chabrier 2011; Federrath 2013; Kraljic et al. 2014) explored ideas based on the role of turbulent fragmentation in GMCs and in which the SFR is a function of the dynamical properties of the clouds. Meidt et al. (2013) argued that the star formation rate in molecular clouds in M51 may correlate with the intensity of the dynamical pressure the clouds are subjected to. The role of feedback coupled to turbulent fragmentation and its effects on the regulation of the SFR on galactic scales have been included in a number of models (e.g., Dopita 1985; Dopita & Ryder 1994; Dib et al. 2011a,b; Dib 2011a,b; Renaud et al. 2012; Dib et al. 2013; Orr et al. 2017).
It is however necessary to include stars in the treatment of GI on large scales in galactic disks, since in most disk galaxies, the stellar surface density is observed to be a factor $\approx 10-100$ larger than the gas surface density (e.g., Leroy et al. 2008). The role of existing stars in determining the development of gravitational instabilities has been investigated in a limited number of studies. Jog & Solomon (1984a,b) explored the characteristics of the gravitational instability in a two fluid medium (gas and stars) in which both components interact gravitationally with each other and are treated each as an isothermal gas with specific velocity dispersions. One of their main conclusions is that even when each fluid component is gravitationally stable, the joint fluid system may be gravitationally unstable. Rafikov (2001) expanded the study of Jog & Solomon to the case where the stars are treated as a collisionless component. Setting stars aside, Romeo et al. (2010) investigated the role of turbulent motions on the stability of galactic disks. They described interstellar turbulence using scaling laws that relate the size of a region to the gas surface density ($\Sigma_{g}$) and gas velocity dispersion ($\sigma_{g}$). Romeo & Agertz (2014) investigated the development of GI for various regimes of turbulence (i.e., different dependence of $\Sigma_{g}$ and $\sigma_{g}$ on the physical scale). In parallel, Romeo & Wiegert (2011) and Romeo & Falstad (2013) proposed a derivation of the effective Toomre $Q$ parameter (Toomre 1964) for multicomponent disks of stars and gas and taking into account the effects of disk thickness. Shadmehri & Khajenabi (2012) and Hoffman & Romeo (2012) coupled aspects of the analysis of Jog & Solomon (1984a) to that of Romeo et al. (2010) and investigated the linear growth rate of the GI in a gas+star galactic disk while at the same time accounting for the turbulent nature of the gas. On the observational side, Shi et al. (2011) showed that the scatter in the $\Sigma_{g}-\Sigma_{\rm SFR}$ relation may be reduced if $\Sigma_{\rm SFR}$ is a function that depends on both $\Sigma_{g}$ and $\Sigma_{*}$. When describing $\Sigma_{\rm SFR}$ as the product of two power law functions of the gas and stellar surface densities ($\Sigma_{\rm SFR} \propto \Sigma_{g}^{\alpha}~\Sigma_{*}^{\beta}$). They obtained $\alpha=0.8\pm0.01$ and $\beta=0.63\pm0.01$ from the combined measurements on sub-galactic scales (scales of $\approx 750$ pc) of 12 nearby galaxies, with a non-negligible galaxy-to-galaxy scatter when the data of each galaxy is fitted individually (see aslo Westfall et al. 2014). Rahmani et al. (2016) performed a similar study for the Andromeda galaxy, and showed that these exponents may well depend on the distance from the centre of the galaxy. It is important to mention that the description of the extended law of star formation as being the product of two power-laws (for gas and stars) is an empirical one, and possibly is an over-simplification of the physical processes that may be connecting the gas and stellar properties to the star formation rate.
However, in all of these above mentioned works, the origin of the dependence of the surface density of star formation on the local properties of the gas and stars has not been explicitly quantified. In this work, we examine the role of GI in a two fluid medium (gas and stars) and investigate the quantitative relationship between the surface density of the star formation rate and the surface densities and velocity dispersions of the stellar and turbulent gaseous components[^2]. The basis of our model is that the fastest growing mode of the GI is the one that is directly connected to the star formation rate. In § \[framework\] we recall the basic equations that lead to the derivation of the wavelength of the fastest growing mode in a stellar+turbulent gas disk ($\lambda_{\rm SF}$), and to the quantitative dependence of $\Sigma_{\rm SFR}$ on $\lambda_{\rm SF}$ and other gas and stellar structural and dynamical properties. In § \[applicationngc628\] we make a detailed comparison between the predictions of the $\Sigma_{\rm SFR}$ from our model and the observed values for the face-own, spiral galaxy NGC 628. We also discuss how including the effects of stellar feedback can affect, and in fact improve, the matching between the models and the observations. In § \[conclusions\] we conclude.
Theoretical framework {#framework}
=====================
Derivation of the most unstable mode {#mostunstable}
------------------------------------
The initial analytical formalism follows that of Jog & Solomon (1984a) for the two fluid approach, Romeo et al. (2010) concerning the inclusion of the turbulent motions of the gas, and Shadmehri & Khajenabi (2012) who combined both aspects. We recall here some of the basic assumptions. Both gas and stars in the disk are treated as isothermal fluids with velocity dispersions $\sigma_{g}$ and $\sigma_{*}$ and their unperturbed surface densities are given by $\Sigma_{g}$ and $\Sigma_{*}$, respectively. The scale-heights of the gaseous and stellar components are given by $h_{g}$ and $h_{*}$, respectively. Starting from the perturbed and coupled hydrodynamical gas-stars equations, and a solution for the perturbed quantities that has the functional form $\rm {exp}\left[i \left(k r+\omega t \right)\right]$, Jog & Solomon (1984a) derived the dispersion relation that describes the growth rate of the instability in the linear regime, $\omega$. This is given by the following biquadratic equation:
$${\omega}^{4}-{\omega}^{2}\left(\alpha_{*}+\alpha_{g} \right)+ \left(\alpha_{*}\alpha_{g}-\beta_{*}\beta_{g} \right)=0,
\label{eq1}$$
where
$$\alpha_{*} = \kappa^{2}+k^{2} \sigma_{*}^{2} - 2\pi G k \Sigma_{*} \frac{1}{1+k h_{*}},
\label{eq2}$$
$$\alpha_{g} = \kappa^{2}+k^{2} \sigma_{g}^{2} - 2\pi G k \Sigma_{g} \frac{1}{1+k h_{g}},
\label{eq3}$$
$$\beta_{*} = 2\pi G k \Sigma_{*} \frac{1}{1+k h_{*}}
\label{eq4}$$
$$\beta_{g} = 2\pi G k \Sigma_{g} \frac{1}{1+ k h_{g}},
\label{eq5}$$
where $\kappa$ is the epicyclic frequency, and $1/(1+k h_{g})$ and $1 / (1+ k k_{*})$ are the reduction factors due to the gas and stellar disks thickness, respectively (Vandevoort 1970; Romeo 1992). The solutions to Eq. \[eq1\] are given by
$$\omega^{2}\left(k\right)=\frac{1}{2}\left[\left(\alpha_{*}+\alpha_{g}\right) \pm \sqrt{\left(\alpha_{*}+\alpha_{g}\right)^{2}-4\left(\alpha_{*}\alpha_{g}-\beta_{*}\beta_{g}\right)} \right].
\label{eq6}$$
Only one of these roots allows for unstable modes to grow. This is given by
$$\omega_{-}^{2}\left(k\right)=\frac{1}{2}\left[\left(\alpha_{*}+\alpha_{g}\right) - \sqrt{\left(\alpha_{*}+\alpha_{g}\right)^{2}-4\left(\alpha_{*}\alpha_{g}-\beta_{*}\beta_{g}\right)} \right].
\label{eq7}$$
inserting back the expressions of $\alpha_{*}$, $\alpha_{g}$, $\beta_{*}$, $\beta_{g}$ from Eqs. \[eq2\]-\[eq5\] into Eq. \[eq7\] and working in the limit where $h_{*} k \lesssim 1$ and $h_{g} k \lesssim 1$ i.e., in the limit of the thin disk approximation in which case the perturbations have a length-scale that is of the order, or larger, than the gaseous and stellar scales heights, then Eq. \[eq7\] becomes:
$$\begin{aligned}
\omega_{-}^{2}\left(k\right) &= \kappa^{2} + \frac{\left(\sigma_{*}^{2}+\sigma_{g}^{2}\right)}{2} k^{2} - \pi G \left(\Sigma_{*}+\Sigma_{g}\right) k \nonumber \\
& - \frac{k}{2} {\biggr[} \left(\sigma_{*}^{2}-\sigma_{g}^{2}\right)^{2} k^{2}+ 4 \pi G \left(\sigma_{*}^{2}-\sigma_{g}^{2}\right)\left(\Sigma_{g}-\Sigma_{*}\right) k+ 4 \pi^{2} G^{2} \left(\Sigma_{*}+\Sigma_{g}\right)^{2} {\biggr]^{1/2}},
\label{eq8}\end{aligned}$$
which is independent of both $h_{*}$ and $h_{g}$. While this assumption is not explicitly necessary if $h_{g}$ and $h_{*}$ are known, the advantage of applying the thin disk approximation is that these two scales heights are generally poorly constrained for face-on disk galaxies. Under the plausible assumption that the gaseous component is turbulent, the surface density and velocity dispersion of the gas are scale dependent and are assumed to follow Larson type scaling relations (Larson 1981), and are given by:
$$\Sigma_{g} = \Sigma_{g0} \left(\frac{k}{k_{0}}\right)^{-a},
\label{eq9}$$
and
$$\sigma_{g} = \sigma_{g0}\left(\frac{k}{k_{0}}\right)^{-b},
\label{eq10}$$
where $a$ and $b$ are descriptive of the nature of turbulent motions, and $\Sigma_{g0}$ and $v_{g0}$ are the surface density and velocity dispersion on the scale of the spatial resolution of the observations (i.e., $2\pi/k_{0}$), respectively. Replacing Eqs. \[eq9\]-\[eq10\] in Eq. \[eq8\] yields
$$\begin{aligned}
\omega_{-}^{2}\left(k\right) &= \kappa^{2} + \frac{\sigma_{*}^{2}} {2} k^{2}+ \frac {\sigma_{g0}^{2}} {2} \left(\frac{k}{k_{0}}\right)^{-2b} k^{2}
- \pi G \left(\Sigma_{*}+\Sigma_{g0}\left(\frac{k}{k_{0}}\right)^{-a}\right) k \nonumber \\
& - \frac{k}{2} {\biggr[} \left(\sigma_{*}^{2}-\sigma_{g0}^{2} \left(\frac{k}{k_{0}}\right)^{-2 b}\right)^{2} k^{2}+ 4 \pi G \left(\sigma_{*}^{2}-\sigma_{g0}^{2} \left(\frac{k}{k_{0}}\right)^{-2b}\right)\left(\Sigma_{g0} \left(\frac{k}{k_{0}}\right)^{-a} - \Sigma_{*}\right) k+ \nonumber \\
& 4 \pi^{2} G^{2} \left(\Sigma_{*}+\Sigma_{g0} \left(\frac{k}{k_{0}}\right)^{-a}\right)^{2} {\biggr]^{1/2}}.
\label{eq11}\end{aligned}$$
We posit that the fastest growing mode is directly linked to the star formation rate. The fastest growing mode, $k_{\rm SF}$, can be obtained by requiring that
$$\frac{d \omega_{-}^{2} \left(k_{\rm SF}\right)} {d k}=0,
\label{eq12}$$
which is an equation that can be solved numerically. It is interesting to note that Eq. \[eq12\] possesses always a positive, non-zero root, for any positive values of the exponents $a$ and $b$ when $ a < 1/2$ and $b < 1/2$. These values are the typical upper limits measured for $a$ and $b$ in all phases of the interstellar gas. The full analytical expression of Eq. \[eq12\] is of little direct interest here and is given in Appendix \[appa\]. It also implies that the SFR is independent of the galactic rotation (i.e., no dependence of $\kappa$). This is consistent with the findings of Dib et al. (2012) who found no correlation between the star formation levels in Galactic molecular clouds and the degree of shear the clouds are subjected to. Following the method of Dib et al. (2012), Thilliez et al. (2014) reached a similar conclusion for molecular clouds in the Large Magellanic Cloud. It is useful to point out that our definition of the characteristic length scale of the most unstable mode ($\lambda_{\rm SF}=2\pi/k_{\rm SF}$), which we associate with star formation, is different from the one used by Romeo & Falstad (2013) (see also Fathi al. 2015). The latter authors define the characteristic length scale as being the scale at which the effective Toomre parameter drops below unity.
Connection to the SFR {#connectionsfr}
---------------------
The SFR can be directly related to the length scale of the most unstable mode $\lambda_{\rm SF}$. The mass of the gas that is associated with $\lambda_{\rm SF}$ is given by:
$$M_{\rm SF} = \bar{\rho} V_{\rm SF},
\label{eq13}$$
where $\bar{\rho}$ is the average density within the mass $M_{\rm SF}$, and $V_{\rm SF}$ is the volume of the gravitationally unstable gas. In the limit of $k_{\rm SF} h_{g} \lesssim 1$ as adopted above, $V_{\rm SF}$ is given by $V_{\rm SF}=\pi \lambda_{\rm SF}^{2} 2h_{g}$ and the volume density of the gas can be replaced by the gas surface density. Thus, Eq. \[eq13\] becomes:
$$M_{\rm SF} = \frac{\Sigma_{g}} {2 h_{g}} \pi \lambda_{\rm SF}^{2} 2h_{g} = \Sigma_{g} \pi \lambda_{\rm SF}^{2}.
\label{eq14}$$
The theoretical star formation rate is then given by:
$${\rm SFR}_{th} \approx \epsilon_{ff} \frac{M_{\rm SF}} {t_{ff}},
\label{eq15}$$
where $t_{ff}$ is the free fall time of the unstable mass reservoir, and $\epsilon_{ff}$ is the efficiency of the star formation process per unit free fall time. We approximate $t_{ff}$ with $1/\sqrt{G \rho_{mp0}}$, where $\rho_{mp0}$ is the gas volume density at the mid-plane. The mid-plane volume density can be written as (e.g., Krumholz & McKee 2005):
$$\rho_{mp0} \approx \frac{\pi G \phi_{P} \Sigma_{g0}^{2}}{2 \sigma_{g0}^{2}},
\label{eq16}$$
where $\Sigma_{g0}$ and $\sigma_{g0}$ carry the same meaning as in §. \[mostunstable\] and with $\phi_{P}$ being a term of order unity that describes the contribution of stars to the mid plane pressure. An approximation of $\phi_{P}$ is given by (e.g., Elmegreen 1989):
$$\phi_{P} \approx 1+\frac{\Sigma_{*}}{\Sigma_{g0}} \frac{\sigma_{g0}}{\sigma_{*}}.
\label{eq17}$$
With these approximations $t_{ff}$ can be written as:
$$t_{ff} = \sqrt{\frac{2}{\pi}} \frac{1}{G} \frac{\sigma_{g0}}{\Sigma_{g0}} \left(1+\frac{\Sigma_{*}}{\Sigma_{g0}}\frac{\sigma_{g0}}{\sigma_{*}}\right)^{-1/2}.
\label{eq18}$$
Combining Eq. \[eq14\] and Eq. \[eq18\] yields the expression for the ${\rm SFR}_{th}$:
$${\rm SFR}_{th} = \epsilon_{ff} \frac{\pi^{3/2}} {2^{1/2}} G \lambda_{\rm SF}^{2} \frac{\Sigma_{g0}^{2}}{\sigma_{g0}} \left(1+\frac{\Sigma_{*}}{\Sigma_{g0}}\frac{\sigma_{g0}}{\sigma_{*}}\right)^{1/2}.
\label{eq19}$$
The theoretical estimate of the surface density of the star formation rate is then simply given by:
$$\Sigma_{{\rm SFR},th}=\frac{{\rm SFR}_{th}} {S},
\label{eq20}$$
where $S$ is the surface area covered by the beam size in the observations.
Application to NGC 628 {#applicationngc628}
======================
We test our model by comparing its predictions to the face-on, spiral galaxy NGC 628. The values of $\Sigma_{\ion{H}{i}}$ and $\sigma_{\ion{H}{i}}$ for NGC 628 are derived from the moment 0 and moment 2 maps of the Nearby Galaxy Survey (THINGS; Walter et al. 2008). The spatial resolution (i.e., beam size) for these observations at the distance of NGC 628 are 750 pc, thus, the surface area of the resolution element in NGC 628 used in this work is $S=750\times750$ pc$^{2}$. As in Shi et al. (2011), the values of $\Sigma_{{\rm H_{2}}}$ are derived from the moment 0 CO $J=1-0$ BIMA SONG survey (Helfer et al. 2003), and the stellar surface density is taken from the SIRTF Nearby Galaxies Survey (SINGS; Kennicutt et al. 2003). Since the gas is ubiquitously present in the galaxy, we approximate the velocity dispersion of the gas as being the velocity dispersion of the gas, $\sigma_{g} \approx \sigma_{\ion{H}{i}}$. Measurements of the stellar velocity dispersions in nearby galaxies are scarce. Yet, the VENGA survey has made such measurements, with selected mosaics, for a sample of nearby galaxies, including NGC 628 (Blanc et al. 2013). We use the same local observational estimates of $\Sigma_{\rm SFR}$ for NGC 628 (hereafter $\Sigma_{{\rm SFR},obs}$) as in Shi et al. (2011) which are based on a combination of [*GALEX*]{} far-UV measurements (Gil de Paz et al. 2007) and [*Spitzer*]{} 24 $\micron$ (Kennicutt et al. 2003) and which have a $3-\sigma$ lower limit of $10^{-4}$ M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$.
For each resolution element in NGC 628 with measurements in the VENGA survey, we estimate the value of $\lambda_{\rm SF}$ by solving Eq. \[eq12\]. We then use Eq. \[eq19\] and Eq. \[eq20\] to evaluate the theoretical values of the SFR (${\rm SFR}_{th}$) and the surface density of the star formation rate, $\Sigma_{{\rm SFR},th}$. The number of resolution elements in NGC 628 that simultaneously have $\sigma_{*}$ measurements in VENGA as well as measured values of $\Sigma_{{\rm SFR},obs}$ is 91. Fig. \[fig1\] displays the distribution function of $\lambda_{\rm SF}$ for these pointings, obtained for $a=b=1/3$. The values of $a=1/3$ and $b=1/3$ are consistent with average values of these quantities derived using cold intensity fluctuations (Lazarian & Pogosyan 2000; Elmegreen et al. 2001; Begum et al. 2006; Dutta et al. 2009). The distribution in Fig. \[fig1\] peaks at $\approx 850$ pc and is positively skewed towards larger values, and we argue in App. \[appb\] that this result is not dependent on the spatial resolution of the observations. While there are no accurate estimates of the vertical scales heights of gas and stars in NGC 628[^3], the values of $\lambda_{\rm SF}$ are large enough such that the condition $\lambda_{\rm SF} \gtrsim 2 \pi h_{g}$ and $\lambda_{\rm SF} \gtrsim 2 \pi h_{*}$ seems to be reasonably fulfilled for almost all resolution elements. The values of ${\rm SFR}_{th}$ and $\Sigma_{{\rm SFR},th}$ are then derived following the formalism given in §. \[connectionsfr\] with an assigned value of $\epsilon_{ff}=0.008$. A value of $0.008$ for $\epsilon_{ff}$ is consistent with the Galaxy-wide average value of $\approx 0.006$ (McKee & Tan 2007; Murray 2011), and with the average value of $\epsilon_{ff} \approx 0.01$ found in numerical simulations (e.g., Semenov et al. 2016, see Fig. \[fig2\] in their paper). This value is a factor $\approx 10$ smaller than the average value measured on the scale of giant molecular clouds (GMCs) in the Galaxy (Murray 2011). This is expected since the gas is denser and more gravitationally bound in GMCs than the spatially averaged gas densities on scales of 750 pc (as are the observations of NGC 628) or on entire galactic scales.
Fig. \[fig2\] displays the distribution function of the ratio of the theoretical to observational star formation rates $ \left({\rm SFR}_{th}/{\rm SFR}_{obs}\right)$. The dispersion in this distribution is $\approx 0.3$ dex. Fig. \[fig3\] displays the scatter plots in the $\Sigma_{g}-\Sigma_{\rm SFR}$ space (left column) and in the $\Sigma_{*}-\Sigma_{\rm SFR}$ space (right column). The observations are shown with the red open triangles, and the theoretical estimates are shown with the black open diamonds (top) and as a closed contours containing $68\%$ of the theoretical points (bottom). A noticeable aspect of Fig. \[fig3\] is that in the low surface density regime ($\Sigma_{g} \lesssim 10-15$ M$_{\odot}$ pc$^{-2}$), the model matches perfectly the data, both in terms of the dependence of $\Sigma_{\rm SFR}$ on $\Sigma_{g}$ and $\Sigma_{*}$ and in terms of the level of dispersion at any given value of $\Sigma_{g}$ and $\Sigma_{*}$. At higher surface densities ($\Sigma_{g} \gtrsim 15$ M$_{\odot}$ pc$^{-2}$), the theoretical estimates of $\Sigma_{\rm SFR}$ are larger than the observed ones by factors of $\approx 2-5$. It is important to note that our formalism does not account explicitly for the effects of feedback from massive stars, which are more important at higher surface densities where more massive clusters can form. The increased effect of feedback at high surface densities leads to a more rapid expulsion of the gas from the clusters and to a reduction of the star formation efficiency per unit time (Dib 2011a). Dib (2011a), showed that in the surface density regime relevant for this work ($1$ M$_{\odot}$ pc$^{-2} \lesssim \Sigma_{g} \lesssim 50$ M$_{\odot}$ pc$^{-2}$), the value of the star formation efficiency per free-fall time ($\epsilon_{ff}$) decreases by a factor of $\approx 4$ going from low to higher gas surface densities, and this is valid for any given value of the gas phase metallicity. Using a scaling of $\epsilon_{ff}$ as a function of $\Sigma_{g}$ ($\epsilon_{ff} \propto \Sigma_{g}^{-0.34}$) (Dib 2011a), and fixing the value of $\epsilon_{ff}=0.008$ at $\Sigma_{g}=1$ M$_{\odot}$ yr$^{-1}$, we make a new estimate of $\Sigma_{{\rm SFR},th}$. The distribution function of the ratio ${\rm SFR}_{th}/{\rm SFR}_{obs}$ in the presence of the effects of feedback is displayed in Fig. \[fig4\]. While the distribution in Fig. \[fig4\] does not peak at unity (because of the arbitrary choice of fixing $\epsilon_{ff}=0.008$ at $\Sigma_{g}=1$ M$_{\odot}$ yr$^{-1}$), the inclusion of a correction due to feedback removes the positive skewness of the distribution (i.e., at high surface densities) and leads to a quasi symmetric dispersion around each side of the observations. Fig. \[fig5\] displays the corresponding scatter plots for $\Sigma_{\rm SFR}$ versus $\Sigma_{g}$ and $\Sigma_{*}$ (left and right panels, respectively). The figure shows that the inclusion of feedback in the treatment of GI in a star+gas galactic disk is necessary in order to better match the observed dependence of $\Sigma_{\rm SFR}$ on both $\Sigma_{g}$ and $\Sigma_{*}$.
CONCLUSIONS
===========
In this work, we explore the dependence of the surface density of star formation in galactic disks on the gas and stellar surface densities and velocity dispersions. We treat both gas and stars as an isothermal fluid and use the linear stability analysis of the gravitationally coupled hydrodynamical equations in order to derive the wavelength of the most unstable mode of the gravitational instability (GI) ($\lambda_{\rm SF}$). We find that the latter quantity is a function of the stellar surface density, the gas surface density, the velocity dispersion of stars, and the scaling laws of turbulence in the gas phase. When applying our model to the face-on, spiral galaxy NGC 628, for which all the required observational data are available, we find that the distribution of $\lambda_{\rm SF}$ for the ensemble of resolution elements for which the required stellar+gas data is available peaks at $\approx 850$ pc and is skewed towards higher values (with a tail of the distribution up to $\approx 2.5$ kpc; see Fig. \[fig1\]). Gravitational instabilities on such large scales are likely to determine the rate of giant molecular cloud (GMC) formation. In turn, stars form in GMCs with a distribution of the star formation efficiencies that depend on the distribution of GMC masses, and on the distributions of their internal physical and dynamical properties coupled to a regulation provided by stellar feedback (e.g., Padoan & Nordlund 2011; Dib et al. 2013). It is therefore reasonable to assume that reservoirs of gas that become gravitationally unstable on large scales are correlated with the star formation rate (SFR) on these scales. For a given set of physical conditions in each resolution elements of NGC 628, we derive the theoretical value of the SFR under the assumption that the fastest growing mode of the gas+star GI is directly linked to the SFR. The theoretical surface density of the star formation rate ($\Sigma_{\rm SFR,th}$) is obtained by dividing the SFR by the physical surface area of the surface element in the observations. The only free parameters of the models are the exponents of the turbulence scaling laws of the gas (i.e., $a$, and $b$ which are the exponents of the gas surface density- and velocity dispersion size relations, see Eq. \[eq10\] and Eq. \[eq11\]), and the star formation efficiency per unit free-fall time, $\epsilon_{ff}$. The values of $a$ and $b$ and $\epsilon_{ff}$ are fixed at $a=b=1/3$ and $\epsilon_{ff}=0.8\%$, respectively. These values of $a$ and $b$ are appropriate for the description of the structure and velocity dispersion of the cold neutral hydrogen in the disk galaxies. A fixed value of $\epsilon_{ff}$ serves only as a normalisation, and does not affect neither the shapes of the $\Sigma_{g}-\Sigma_{\rm SFR}$ and $\Sigma_{*}-\Sigma_{\rm SFR}$ relations, nor the amount of scatter at any fixed value of $\Sigma_{g}$ or $\Sigma_{*}$.
We find an encouraging match between the theoretical estimates of the surface density of star formation $\Sigma_{{\rm SFR},th}$ from our model and the observational values for NGC 628 ($\Sigma_{{\rm SFR},obs}$), both in terms of the shapes of the $\Sigma_{g}-\Sigma_{\rm SFR}$ and $\Sigma_{*}-\Sigma_{\rm SFR}$ scatter relations and in terms of the dispersion of the data points at fixed values of $\Sigma_{g}$ or $\Sigma_{*}$. The model-observations matching is further improved if the value of $\epsilon_{ff}$ is taken to decrease with increasing gas surface density as earlier suggested by Dib (2011a,b). The origin of the dependence of $\epsilon_{ff}$ on $\Sigma_{g}$ is attributed to the effects of feedback in the pre-supernova phase in stellar clusters. More massive clusters are more likely to form at higher surface densities. Gas expulsion from more massive clusters occurs on shorter timescales than in lower mass clusters (Dib et al. 2013), and the rapid expulsion of gas results in a faster quenching of star formation and to a reduction of the star formation efficiency per unit time. Our model opens a new path towards a better understanding of the dependence of the star formation rate in galaxies on the local stellar and gas properties. Higher spatial and spectral resolution observations will allow us to further constrain the model and will also help reduce the number of free parameters by directly measuring the scaling laws of turbulence.
Acknowledgments
===============
We thank the referee Alessandro Romeo for a careful reading of the paper which helped clarify some aspects of the text, Sophia Lianou and Matthew Orr for useful comments on a draft version of this paper, and Shi Yong for sharing some of the observational data. S. D. acknowledges the support provided by a Marie Curie Intra European Fellowship under the European Community’s Seventh Framework Program FP7/2007-2013 grant agreement no 627008, during the early phase of this work. S. H. acknowledges financial support from DFG program HO 5475/2-1. G. B. is supported by CONICYT/FONDECYT programa de iniciación Folio 11150220. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
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GOVERNING EQUATION FOR $k_{\rm SF}$ {#appa}
====================================
The derivation of the wavenumber of the fastest growing mode of the instability, $k_{\rm SF}$, is achieved using Eq. \[eq12\], where $\omega_{-}^{2}$ is given by Eq. \[eq11\]. There exist an analytical expression for the general equation of $k_{\rm SF}$ which is given by:
$$\begin{aligned}
& \sigma_{*}^{2} k_{\rm SF}+\sigma_{g0}^{2} (1-b) \left(\frac{1}{k_{0}} \right)^{-2b} k_{\rm SF}^{1-2b}- \pi G \Sigma_{*} - \pi G \Sigma_{g0} \left( \frac{k_{\rm SF}}{k_{0}}\right)^{-a}\nonumber\\
& - {\biggr[} \left(\sigma_{*}^{2} -\sigma_{g0}^{2} \left(\frac {k_{\rm SF}}{k_{0}}\right)^{-2b} \right)^{2} k_{\rm SF}^{2} + 4 \pi G \left(\Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a}-\Sigma_{*} \right) \left(\sigma_{*}^{2}-\sigma_{g0}^{2}\left(\frac{k_{\rm SF}}{k_{0}} \right)^{-2b}\right) k_{\rm SF}\nonumber\\
& + 4\pi^{2} G^{2} \left(\Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a} +\Sigma_{*}\right)^{2}{\biggr]^{1/2}} \nonumber\\
& - k_{\rm SF} \frac{ {\biggr[}2 k_{\rm SF} \left(\sigma_{*}^{2}-\sigma_{g0}^{2} \left( \frac{k_{\rm SF}}{k_{0}}\right)^{-2b} \right)^{2}+4 b \sigma_{g0}^{2} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-2b} k_{\rm SF} \left(\sigma_{*}^{2}-\sigma_{g0}^{2} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-2b}\right){\biggr]}} {D}\nonumber\\
& + k_{\rm SF} \frac{ {\biggr[} 4\pi G \left(\Sigma_{g0}\left( \frac{k_{\rm SF}}{k_{0}}\right)^{-a}-\Sigma_{*}\right) \left(\sigma_{*}^{2}-\sigma_{g0}^{2}\left( \frac{k_{\rm SF}}{k_{0}}\right)^{-2b}\right) - 4\pi a G \Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a} \left(\sigma_{*}^{2}-\sigma_{g0}^{2} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-2b}\right) {\biggr]}} {D}\nonumber\\
& + k_{\rm SF} \frac{ {\biggr[} 8 a \pi^{2} G^{2} \Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a} k_{\rm SF}^{-1} \left(\Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a}+\Sigma_{*}\right) + 8 \pi b G \sigma_{g0}^{2} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-2b} \left(\Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a}-\Sigma_{*} \right) {\biggr]}} {D}\nonumber\\
& = 0
\label{eqa1}\end{aligned}$$
with
$$\begin{aligned}
D & = 4 {\biggr[}k_{\rm SF}^{2} \left(\sigma_{*}^{2}-\sigma_{g0}^{2} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-2b} \right)^{2}+4\pi G k_{\rm SF} \left(\Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a} - \Sigma_{*}\right) \left(\sigma_{*}^{2}-\sigma_{g0}^{2} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-2b}\right)+ \nonumber\\
& 4 \pi^{2} G^{2} \left(\Sigma_{g0} \left(\frac{k_{\rm SF}}{k_{0}}\right)^{-a}-\Sigma_{*}\right)^{2} {\biggr]^{1/2}}.
\label{eqa2}\end{aligned}$$
Given the values of $k_{0}=2\pi/\lambda_{0}$, where $\lambda_{0}$ is the physical size of the resolution element in the observations. For each resolution element of the NGC 628 galaxy, we solve Eq. \[eqa1\] numerically using a globally-convergent Broyden’s method (Press et al. 1992)
DO THE RESULTS DEPEND ON THE SPATIAL RESOLUTION OF THE OBSERVATIONS ? {#appb}
=====================================================================
The question may arise whether the solutions obtained for $\lambda_{\rm SF}$ using Eq. \[eq12\] (i.e., Eq. \[eqa1\] in its detailed form) depend on the spatial resolution of the observations (here $\lambda_{0}=750$ pc). It should be noted that the surface density and velocity dispersion of the gas have a scale dependance on the dimensionless number $k/k_{0}$ (and not merely on $k_{0}$). Nonetheless, we test this by performing the following simple test. We assume that the observations have been performed on a spatial resolution of $375$ pc (thus $k_{0}$ is now replaced by $2 k_{0}$, where $k_{0}$ is the wavenumber associated with the original spatial resolution of 750 pc). We do not possess observations that have been obtained self-consistently at a spatial resolution that is half of the spatial resolution of the observations at hand. However, we adapt the current observations to present those that could be obtained with an improved spatial resolution by a factor $2$. In the absence of a better guess, the stellar surface density and velocity dispersion for the resolution $\lambda_{0}/2$ are kept the same as on the scale $\lambda_{0}$. The velocity dispersion and surface density of the gas in Eqs. \[eq10\] and \[eq11\] have to be multiplied by the factors $2^{-\beta}$ and $2^{-\alpha}$, respectively. For $\alpha=\beta=1/3$, the gas velocity dispersion and surface density are both reduced by a factor $2^{-1/3}$. These assumptions generate only approximate conditions for the stellar and gas components in each constructed half-resolution element as one expects that there would be local fluctuations of the stellar velocity and surface density on smaller scales.
Fig. \[fig1app\] displays the distribution of the wavelengths of the most unstable mode ($\lambda_{\rm SF}$) with the new adopted spatial resolution. As expected, the choice of a different spatial resolution (here a higher resolution) does not affect the results and the distribution of $\lambda_{\rm SF}$ still peaks at $\approx 850-900$ pc. For this same adopted spatial resolution, Fig. \[fig2app\] displays the ratio of the theoretical to observed star formation rates while Fig. \[fig3app\] displays the surface density of the star formation rate as a function of the surface density of the gas (left panels) and of the stars (right panels) (for the model as a scatter plot in the top panels and as a closed $1-\sigma$ contour in the bottom panels). In this case, the efficiency of star formation per free-fall time $\epsilon_{ff}$ has been taken to include a correction for feedback (i.e., as in Fig. \[fig4\] and Fig. \[fig5\]). The existence of more outliers which result in a larger scatter is probably due to the approximations made in constructing the physical quantities (especially $\Sigma_{*}$ and $\sigma_{*}$) for the higher spatial resolution case.
\[lastpage\]
[^1]: E-mail: sami.dib@gmail.com
[^2]: Keeping with the terminology used in Shi et al. 2011, we also use the term “extended” to describe the dependence of the star formation rate on physical quantities pertaining to both gas and stars in galactic disks
[^3]: Kregel et al. (2002) argued that there is a constant ratio of the radial to vertical length scales in galactic disks of $l_{*}/h_{*} \approx,7.3 \pm 2.2$. With the measured value of $l_{*} \approx 2.3$ kpc in NGC 628 (Leroy et al. 2008), this yields a value of $h_{*} \approx 315$ pc, under the assumption that $h_{*}$ is independent of galactic radius. From an analysis of the line power spectrum, Dutta et al. (2008) argued for an upper limit on the gas vertical scales height of 800 pc.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In the thermal dark matter (DM) paradigm, primordial interactions between DM and Standard Model particles are responsible for the observed DM relic density. In [@boehm:2014MNRAS], we showed that weak-strength interactions between DM and radiation (photons or neutrinos) can erase small-scale density fluctuations, leading to a suppression of the matter power spectrum compared to the collisionless cold DM (CDM) model. This results in fewer DM subhaloes within Milky Way-like DM haloes, implying a reduction in the abundance of satellite galaxies. Here we use very high resolution $N$-body simulations to measure the dynamics of these subhaloes. We find that when interactions are included, the largest subhaloes are less concentrated than their counterparts in the collisionless CDM model and have rotation curves that match observational data, providing a new solution to the “too big to fail” problem.'
author:
- |
J. A. Schewtschenko,$^{1,2}$[^1] C. M. Baugh,$^{1}$ R. J. Wilkinson,$^{2}$ C. Bœhm,$^{2,3}$ S. Pascoli,$^{2}$ T. Sawala$^{4}$\
$^1$Institute for Computational Cosmology, Durham University, Durham DH1 3LE, UK\
$^2$Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK\
$^3$LAPTH, U. de Savoie, CNRS, BP 110, 74941 Annecy-Le-Vieux, France\
$^4$Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a, FI-00014 Helsinki, Finland
bibliography:
- 'IDM\_TBTF.bib'
title: 'Dark matter–radiation interactions: the structure of Milky Way satellite galaxies'
---
\[firstpage\]
astroparticle physics – dark matter – galaxies: haloes – large-scale structure of Universe.
Introduction {#sec:intro}
============
The cold dark matter (CDM) model has been remarkably successful at explaining measurements of the cosmic microwave background radiation and the large-scale structure of the Universe. However, in its simplest form, the model faces challenges on small scales; the most pressing of which are the “missing satellite” (@moore_dark_1999 [@Klypin:1999uc]) and “too big to fail” (@BoylanKolchin:2011de) problems. These discrepancies may indicate the need to consider a richer physics phenomenology in the dark sector, although they were first stated without the inclusion of baryonic physics.
The “missing satellite” problem refers to the overabundance of DM subhaloes in Milky Way (MW)-like DM haloes, compared to the observed number of MW satellite galaxies. This comparison between theory and observation requires a connection to be made between subhaloes and galaxies; in the absence of a good model for galaxy formation, this is most readily done using the halo circular velocity. Subsequent simulations that have taken into account baryonic physics suggest that a reduction in the efficiency of galaxy formation in low-mass DM haloes results in many of the excess subhaloes containing either no galaxy at all or a galaxy that is too faint to be observed (@Benson:2002 [@Somerville:2002; @Sawala:2014; @Sawala:2015]).
As the resolution of $N$-body simulations continued to improve, the “too big to fail” problem emerged (@BoylanKolchin:2011de). This concerns the largest subhaloes, which should be sufficiently massive that their ability to form a galaxy is not hampered by heating of the intergalactic medium by photo-ionising photons or heating of the interstellar medium by supernovae. Simulations of vanilla CDM showed that the largest subhaloes are more massive and denser than is inferred from measurements of the MW satellite rotation curves.
The severity of the small-scale problems can be reduced if one considers the mass of the MW, which impacts the selection of MW-like haloes in the simulations but remains difficult to determine (@Wang:2012 [@Cautun:2014dda; @Piffl:2014; @Wang:2015]). A range of alternatives to vanilla CDM have also been proposed e.g. warm DM (@schaeffer_silk), interacting DM (@Boehm:2000gq [@Boehm:2004th; @CyrRacine:2012fz; @Chu:2014lja]), self-interacting DM (@Spergel:1999mh [@Rocha:2012jg; @Vogelsberger:2014pda; @Buckley:2014PhRvD]), decaying DM (@Wang:2014ina) and late-forming DM (@Agarwal:2015). These “beyond CDM” models generally exhibit a cut-off in the linear matter power spectrum at small scales (high wavenumbers) that translates into a reduced number of low-mass DM haloes compared to collisionless CDM at late times.
Most numerical efforts so far to check whether such models could solve the small-scale problems have focussed on either warm DM or self-interacting DM. However, some works have studied the impact of DM scattering elastically with Standard Model particles in the early Universe; for example, with photons ($\boldsymbol{\gamma}$**CDM**) (@Boehm:2000gq [@boehm_interacting_2001; @Sigurdson:2004zp; @Boehm:2004th; @Dolgov:2013una; @Wilkinson:2013kia]), neutrinos ($\boldsymbol{\nu}$**CDM**) (@Boehm:2000gq [@boehm_interacting_2001; @Boehm:2004th; @Mangano:2006mp; @Serra:2009uu; @Wilkinson:2014ksa; @Escudero:2015yka]) and baryons (@Chen:2002yh [@Dvorkin:2013cea; @Aviles:2011ak]).
Such elastic scattering processes are intimately related to the DM annihilation mechanism in the early Universe and are thus directly connected to the DM relic abundance in scenarios where DM is a thermal weakly-interacting massive particle (WIMP). Therefore, rather than being viewed as exotica, interactions between DM and Standard Model particles should be considered as a more realistic realisation of the CDM model. Indeed, instead of assuming that CDM has no interactions beyond gravity, one can actually test this assumption by determining their impact on the linear matter power spectrum and ruling out values of the cross section that are in contradiction with observations. However, it should be noted that the strength of the scattering and annihilation cross sections can differ by several orders of magnitude, depending on the particle physics model.
The $\gamma$CDM and $\nu$CDM scenarios are characterised by the collisional damping of primordial fluctuations, which can lead to a suppression of small-scale power at late times. The collisional damping scale is determined by a single model-independent parameter: the ratio of the scattering cross section to the DM mass. The larger the ratio, the larger the suppression of the matter power spectrum. For simplicity, we assume that the scattering cross section is constant (i.e. temperature-independent), bearing in mind that temperature-dependence would give rise to the same effect but with a different value of the cross section today (@Wilkinson:2013kia [@Wilkinson:2014ksa]). In [@boehm:2014MNRAS], we confirmed that such models can provide an alternative solution to the missing satellite problem in the MW. Here we show that interacting DM could also solve the too big to fail problem[^2].
The paper is organised as follows. In Section \[sec:idmssp:sim\], we describe the setup of the $N$-body simulations that we use to study small structures. In Section \[sec:tbtf\], we investigate whether interacting DM can alleviate the too big to fail problem, using MW observations. Finally, we give our conclusions in Section \[sec:conc\].
Simulations {#sec:idmssp:sim}
===========
{width="90.00000%"}
While the CDM matter power spectrum predicts the existence of structures at all scales (down to earth mass haloes (@Diemand:2005Nature [@Springel:2008cc; @Angulo:2009hf])), interacting DM models predict a suppression of power below a characteristic damping scale that is determined by the ratio of the DM interaction cross section to the DM mass (@boehm_interacting_2001). For allowed $\gamma$CDM and $\nu$CDM models (@boehm:2014MNRAS), the suppression occurs for haloes with masses below $10^8-10^9~M_{\odot}$. Therefore, to study the distribution and properties of structures beyond the linear regime, it is essential to carry out high-resolution $N$-body simulations.
To reach the resolution required to model the dynamics of DM subhaloes within MW-mass DM haloes, we first identify Local Group (LG) candidates in an $N$-body simulation of a large cosmological volume, and then resimulate the region containing these haloes at much higher mass resolution in a “zoom” resimulation. We use the `DOVE` cosmological simulation to identify haloes for resimulation (the criteria used to select the haloes are listed below) [@Sawala:2014arXiv]. The `DOVE` simulation follows the hierarchical clustering of the mass within a periodic cube of side length $100~$Mpc, using particles of mass $8.8 \times 10^{6}~M_{\odot}$ and assuming a WMAP7 cosmology.
Following the `APOSTLE` project [@Fattahi:2015arXiv], which also uses the `DOVE` CDM simulation to identify LG candidates for study at higher resolution, we impose the following three criteria to select candidates for resimulation:
1. [**Mass:**]{} there should be a pair of MW and Andromeda mass host haloes, with masses in the range $(0.5 - 2.5) \times 10^{12}~{M}_\odot$.\
2. [**Environment:**]{} there should be no other large structures nearby, i.e. an environment with an unperturbed Hubble flow out to 4 Mpc.\
3. [**Dynamics:**]{} the separation between the two haloes should be $800 \pm 200$ kpc, with relative radial and tangential velocities below $250$ km $\mathrm{s}^{-1}$ and $100$ km $\mathrm{s}^{-1}$ respectively.
These criteria are more restrictive than those employed in our earlier work on the structure of haloes (@Schewtschenko:2015MNRAS) as they also take into account the internal kinematics of the LG. We obtain four LG candidates and therefore, eight MW-like haloes. If we assume that the gravitational interaction between the LG haloes is limited, the mass, environment and dynamics[^3] of the haloes would not be significantly different if we had run a $\gamma$CDM or $\nu$CDM version of the `DOVE` simulation.
------ ----------------------- ------------------------------------- -------------------------------------------- --
$M_{\rm vir}$ $V_{\max}$ $\sigma_{\mathrm{DM}-\gamma}$
$[10^{12}~M_{\odot}]$ \[$\mathrm{km}$ $\mathrm{s}^{-1}$\] $[\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})]$
AP-1 1.916 200.3
AP-2 1.273 151.5
AP-3 0.987 157.9
AP-4 0.991 163.0
AP-5 2.010 167.5
AP-6 1.934 165.1
AP-7 1.716 163.7 $0,~10^{-10},~10^{-9},$
AP-8 1.558 193.3 $2 \times 10^{-9},~10^{-8}$
------ ----------------------- ------------------------------------- -------------------------------------------- --
: Key properties of the MW-like haloes in the zoom resimulations (Section \[sec:idmssp:sim\]). The first column specifies the `APOSTLE` identifier (ID) for each MW-like halo, while the second and third columns list the virial mass, $M_{\rm vir}$, and maximum circular velocity, $V_{\rm max}$, respectively (for CDM). The fourth column lists the different DM–photon interaction cross sections, $\sigma_{\mathrm{DM}-\gamma}$, used in the zoom resimulations for each LG candidate, where $\sigma_{\rm Th}$ is the Thomson cross section and $\sigma_{\mathrm{DM}-\gamma} = 0$ corresponds to CDM.[]{data-label="tab:idmssp:LGs"}
We perform resimulations with the `P-Gadget3` $N$-body simulation code [@gadget2] assuming the $\gamma$CDM model, bearing in mind that the results for $\nu$CDM would be very similar (see @Schewtschenko:2015MNRAS). We use the same cosmology (WMAP7)[^4], random phases and second-order LPT method [@Jenkins:2010MNRAS] as [@Sawala:2014arXiv]. We resimulate the four LG candidates with a particle mass $m_{\rm part}=7.2\times10^5~M_\odot$ and a comoving softening length $l_{\rm soft}=216$ pc. This corresponds to a mass resolution that is intermediate between levels 4 and 5 in the Aquarius simulations of [@Springel:2008cc] (level 1 being the highest resolution). We also resimulate the two host haloes in one of our LG Candidates (AP-7/AP-8) at an even higher resolution ($m_{\rm part}=6\times10^4~M_\odot$, $l_{\rm soft}=94$ pc; which is comparable to Aquarius level 3). These simulations (denoted with the suffix `-HR`) are used to confirm that our results have converged[^5] and allow us to obtain more reliable predictions for the innermost parts of the halo. Substructures within the host haloes are located using the `AMIGA` halo finder [@ahf_refs].
We run resimulations for zero interaction cross section, which corresponds to collisionless CDM, and for a selection of DM–photon interaction cross sections, as listed in Tab. \[tab:idmssp:LGs\]. We note that the DM–photon interaction cross section value of $\sigma_{\mathrm{DM}-\gamma} = 2\times 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ was shown to solve the missing satellite problem in [@boehm:2014MNRAS], in the absence of baryonic physics effects.
Fig. \[fig:idmssp:apostles\] shows the projected matter density in the `DOVE` simulation box [@Sawala:2014arXiv] (central panel) along with renderings of the four LG candidates from the resimulations. The eight MW-like haloes are listed in Tab. \[tab:idmssp:LGs\] with their respective properties for CDM. Halo properties for $\gamma$CDM vary only slightly (within a few percent) from those in CDM for the cross sections considered here.
Results {#sec:tbtf}
=======
![Top: the circular velocity, $V_{\rm circ}$, versus radius, $r$, for the eleven most massive subhaloes in AP-7-HR for CDM (grey lines) and for $\gamma$CDM with $\sigma_{\mathrm{DM}-\gamma} = 2 \times 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ (red lines). The dashed lines indicate where $V_{\rm circ}$ can still be measured from the simulation but convergence cannot be guaranteed, according to the criteria set out by @power:2003MNRAS. The data points correspond to the observed MW satellites with 1$\sigma$ error bars [@Wolf:2010MNRAS]. Bottom: the $V_{\rm max}$ versus $R_{\rm max}$ results for all eight MW-like haloes, with the same scattering cross sections as in the top panel. The hatched region marks the 2$\sigma$ confidence interval for the observed MW satellites. $V_{\rm max}$ is derived from the observed stellar line-of-sight velocity dispersion, $\sigma_{\star}$, using the assumption that $V_{\rm max} = \sqrt{3} \sigma_{\star}$ [@Klypin:1999uc].[]{data-label="fig:idmssp:tbtf_gcdm"}](pics/tbtf_profile_combined "fig:"){width=".5\textwidth"} ![Top: the circular velocity, $V_{\rm circ}$, versus radius, $r$, for the eleven most massive subhaloes in AP-7-HR for CDM (grey lines) and for $\gamma$CDM with $\sigma_{\mathrm{DM}-\gamma} = 2 \times 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ (red lines). The dashed lines indicate where $V_{\rm circ}$ can still be measured from the simulation but convergence cannot be guaranteed, according to the criteria set out by @power:2003MNRAS. The data points correspond to the observed MW satellites with 1$\sigma$ error bars [@Wolf:2010MNRAS]. Bottom: the $V_{\rm max}$ versus $R_{\rm max}$ results for all eight MW-like haloes, with the same scattering cross sections as in the top panel. The hatched region marks the 2$\sigma$ confidence interval for the observed MW satellites. $V_{\rm max}$ is derived from the observed stellar line-of-sight velocity dispersion, $\sigma_{\star}$, using the assumption that $V_{\rm max} = \sqrt{3} \sigma_{\star}$ [@Klypin:1999uc].[]{data-label="fig:idmssp:tbtf_gcdm"}](pics/tbtf_allLGs "fig:"){width=".5\textwidth"}
We now explore the too big to fail problem and show how the theoretical predictions and observations can be reconciled by including DM interactions beyond gravity.
The too big to fail problem is illustrated in the top panel of Fig. \[fig:idmssp:tbtf\_gcdm\]. Here the rotation curves of the 11 most massive subhaloes[^6] in the CDM resimulation of the halo AP-7-HR clearly lie above the measurements for the “classical” dwarf spheroidal satellites in the MW taken from [@Wolf:2010MNRAS]. In general, one can see that the largest subhaloes in CDM simulations have a higher circular velocity, $V_{\rm circ}$, and therefore more enclosed (dark) matter, than is observed for a given radius.
In the case of $\gamma$CDM, the rotation curves of the most massive satellites are shifted to lower circular velocities, indicating that there is less (dark) matter enclosed within a given radius. Alternatively, one can interpret this as a lower central density or concentration for the haloes in $\gamma$CDM, as seen in [@Schewtschenko:2015MNRAS].
The circular velocity profiles shown in the top panel of Fig. \[fig:idmssp:tbtf\_gcdm\] are plotted using different line styles. The transition occurs at the scale determined by the convergence criteria devised by [@power:2003MNRAS]. At smaller radii (dashed lines), the velocity profiles are not guaranteed to have converged. However, the key point here is that the CDM and $\gamma$CDM resimulations have the same resolution and yet show a clear difference at all radii plotted, with a shift to lower circular velocities for the haloes in $\gamma$CDM.
The bottom panel of Fig. \[fig:idmssp:tbtf\_gcdm\] presents a related view of the too big to fail problem; this time showing the peak velocity in the rotation curve, $V_{\rm max}$, and the radius at which this occurs, $R_{\rm max}$. The hatched region indicates the $2\sigma$ range inferred for the observed MW satellites, assuming that these are DM-dominated and fitting NFW profiles (@Navarro:1997ApJ) to the rotation curve measurements. We allow the halo concentration parameter to vary, following the same technique and assumptions as described in [@Klypin:1999uc][^7].
Again, the collisionless CDM model predicts satellites that lie outside the $2\sigma$ range compatible with observations. Additionally, for CDM, there are many more subhaloes within the range of $V_{\rm max}$–$R_{\rm max}$ plotted than there are observed satellites. The abundance of massive, concentrated subhaloes varies depending on the mass and formation history of the host halo; however, for all the MW-like candidates studied, CDM exhibits a too big to fail problem, which is reduced in the case of $\gamma$CDM.
In Fig. \[fig:idmssp:tbtf\_gcdm2\], we present the results for AP-7 and AP-8 to show the impact of varying the DM–photon interaction cross section. As the cross section is increased, the predicted $V_{\rm max}$ values fall and shift to larger $R_{\rm max}$. This brings the model predictions well within the region compatible with the observational results and also reduces the number of satellites with such rotation curves. Therefore, one can clearly see that interacting DM can alleviate the too big to fail problem for a cross section $\sigma_{\mathrm{DM}-\gamma} \simeq 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ that also solves the missing satellite problem (@boehm:2014MNRAS).
![The $V_{\rm max}$ versus $R_{\rm max}$ results for a range of DM–photon interaction cross sections using the AP-7 and AP-8 haloes. As in Fig. \[fig:idmssp:tbtf\_gcdm\], the hatched region marks the 2$\sigma$ confidence interval for the observed MW satellites, following the methodology of @Klypin:1999uc.[]{data-label="fig:idmssp:tbtf_gcdm2"}](pics/tbtf_LG4){width=".5\textwidth"}
Conclusion {#sec:conc}
==========
There are a multitude of solutions proposed to overcome the small-scale “failures” of cold dark matter (CDM); namely, the “missing satellite” and “too big to fail” problems. Within the collisionless CDM model, these explanations fall into two camps: i) invoking baryonic physics to reduce the efficiency of galaxy formation in low-mass DM haloes [@Sawala:2014arXiv; @2015arXiv151101098S], and ii) exploiting the uncertainty in the mass of the Milky Way (MW) DM halo [@Wang:2012]. Both problems can be diminished using one or both of the above.
Solutions in which the properties of the DM are varied have also been explored. [@Lovell:2013ola] showed that replacing CDM by a warm DM particle of mass $1.5$ keV leads to a reduced abundance of subhaloes in MW-like haloes, and massive subhaloes that are less concentrated than their CDM counterparts, matching observations of the internal dynamics of the MW satellites. [@Vogelsberger:2014pda] investigated the impact of self-interacting DM on the properties of satellite galaxies, finding little change in the global properties of the galaxies but variation in their structure.
Here we have investigated the impact of interactions between DM and radiation on the structure of the MW satellites. Such interactions are well-motivated and may have helped to set the abundance of DM inferred in the Universe today [@Boehm:2003hm; @Peter:2012]; sometimes called the WIMP miracle. As well as its physical basis, this model has the attraction that it is as simple to simulate as CDM. The interactions took place in the early Universe when the densities of DM and radiation were much higher, and are negligible over the time period covered by the simulation. The DM particles are still cold, so there are no issues relating to particle velocity distributions, as would arise in high-resolution simulations of warm DM, particularly for lighter candidates. The only change compared to a CDM simulation is the modification to the matter power spectrum in linear perturbation theory; the DM–radiation interactions result in a damping of the matter power spectrum on small scales.
We have used high resolution $N$-body simulations of DM haloes, which have passed a set of Local Group selection criteria, to show the impact of DM–radiation interactions on the structure of massive subhaloes. Increasing the interaction cross section reduces the mass enclosed within a given radius in the subhaloes, compared to their CDM counterparts, as suggested by our results for a wider population of DM haloes (@Schewtschenko:2015MNRAS). When combined with our earlier work showing that stronger interactions also lead to a reduction in the number of MW subhaloes (@boehm:2014MNRAS), we find that a model with an elastic scattering cross section of $\simeq 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ can solve both of these small-scale problems of CDM. The next step will be to include baryonic physics. This will not alter the qualitative conclusions of our papers, but will relax the constraints on the DM–radiation scattering cross section.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank V. Springel for providing access to the `P-Gadget3` code and the `bPic` rendering code, A. Jenkins for sharing his IC generator code for the “zoom” resimulations with us and J. Halley for helpful discussions. JAS is supported by a Durham University Alumnus Scholarship and RJW is supported by the STFC Quota grant ST/K501979/1. This work was supported by the STFC (grant numbers ST/F001166/1, ST/G000905/1 and ST/L00075X/1) and the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). This work was additionally supported by the European Research Council under ERC Grant “NuMass” (FP7-IDEAS-ERC ERC-CG 617143). It made use of the DiRAC Data Centric system at Durham University, operated by the ICC on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. SP also thanks the Spanish MINECO (Centro de excelencia Severo Ochoa Program) under grant SEV-2012-0249. CMB acknowledges a research fellowship from the Leverhulme Trust.
Augmented content {#augmented-content .unnumbered}
=================
[l]{}[.14]{} {width=".16\textwidth"}
This paper provides additional multimedia and interactive content embedded in an augmented reality using the open DARO framework for mobile devices. In order to access the data, you need to use a DARO-compatible browser ([`http://icc.dur.ac.uk/\simdaro`](http://icc.dur.ac.uk/~daro)) and scan the DARO QR code printed here.\
\[lastpage\]
[^1]: E-mail: j.a.schewtschenko@dur.ac.uk
[^2]: Recently, it was also demonstrated that one can simultaneously alleviate the small-scale problems of CDM by including interactions between DM and dark radiation on the linear matter power spectrum and DM self-interactions during non-linear structure formation (@Cyr-Racine:2015ihg [@Vogelsberger:2015gpr]).
[^3]: The formation process of structures is slightly delayed by the presence of DM interactions. Therefore, both the separation and the relative velocities may actually lie outside the bound set by the “Dynamics” criterion as the haloes are at a different point in their orbit around each other for $\gamma$CDM. However, as long as this delay between CDM and $\gamma$CDM is not too large, we essentially have the same dynamical system in both cases and the substructures within the host haloes will be unaffected.
[^4]: The fact that we are using the older WMAP7 cosmology instead of the most recent data is not a concern since we are only interested in the effects of DM interactions on a selected local environment.
[^5]: While the properties of some small subhaloes may vary from one resolution level to another (due to minor changes in their formation and accretion histories), the overall scatter for all subhaloes remains the same and thus can be considered to have converged.
[^6]: We have included three more simulated subhaloes than the observed number of dwarf satellites as the most massive subhaloes are considered statistical outliers like the Magellanic clouds, which have been omitted in this study.
[^7]: A plot with the confidence bands for each of the MW satellites can be found in the provided online material and the augmented content of this paper.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the hydrodynamic forces acting on a finite-size impurity moving in a two-dimensional Bose-Einstein condensate at non-zero temperature. The condensate is modeled by the damped-Gross Pitaevskii (dGPE) equation and the impurity by a Gaussian repulsive potential giving the coupling to the condensate. The width of the Gaussian potential is equal to the coherence length, thus the impurity can only emit waves. Using linear perturbation analysis, we obtain analytical expressions corresponding to different hydrodynamic regimes which are then compared with direct numerical simulations of the dGPE equation and with the corresponding expressions for classical forces. For a non-steady flow, the impurity experiences a time-dependent force that, for small coupling, is dominated by the inertial effects from the condensate and can be expressed in terms of the local material derivative of the fluid velocity, in direct correspondence with the Maxey and Riley theory for the motion of a solid particle in a classical fluid. In the steady-state regime, the force is dominated by a self-induced drag. Unlike at zero temperature, where the drag force vanishes below a critical velocity, at finite temperatures, the drag force has a net contribution from the energy dissipated in the condensate through the thermal drag at all velocities of the impurity. At low velocities this term is similar to the Stokes’ drag in classical fluids. There is still a critical velocity above which the main drag pertains to energy dissipation by acoustic emissions. Above this speed, the drag behaves non-monotonically with impurity speed, reflecting the reorganization of fringes and wake around the particle.'
author:
- 'Jonas Rønning$^1$, Audun Skaugen$^2$, Emilio Hernández-García$^3$, Cristóbal López$^3$, Luiza Angheluta$^1$'
bibliography:
- 'ref-2.bib'
title: 'Classical analogies for the force acting on an impurity in a Bose-Einstein condensate'
---
Introduction
============
The motion of an impurity suspended in a quantum fluid depends on several key factors such as the superfluid nature and flow regime, as well as the size of the impurity and its interaction with the surrounding fluid [@winiecki2000motion; @wouters2010superfluidity; @astrakharchik2004motion; @shukla2016sticking; @pinsker2017gaussian]. Therefore, it is disputable whether the forces acting on an impurity in a quantum fluid should bear any resemblance to classical hydrodynamic forces. In the case of an impurity immersed in superfluid liquid helium, classical equations of motion and hydrodynamic forces are assumed a priori [@poole2005motion], since impurities are typically much larger than the coherence length and then quantum hydrodynamic effects like the quantum pressure can be neglected. For Bose-Einstein condensates (BEC) in dilute atomic gases, impurities can be neutral atoms [@chikkatur2000suppression], ion impurities [@zipkes2010trapped; @balewski2013coupling] or quasiparticles [@jorgensen2016observation]. The size of an impurity in a BEC is typically of the same order of magnitude or smaller than the coherence length, and quantum hydrodynamic effects cannot be ignored.
There are several theoretical and computational studies of the interaction force between an impurity and a BEC at zero absolute temperature, using different approaches depending on the nature of the particle and its interaction with the condensate. A microscopic approach is used to analyse the interaction of a rigid particle with a BEC by solving the Gross-Pitaevskii equation (GPE) for the condensate macroscopic wavefunction and using boundary conditions such that the condensate density vanishes at the particle boundary [@pham2005boundary]. This methodology allows to study complex phenomena such as vortex nucleation and flow instabilities, but it is more oriented to find the effects of an obstacle on the flow rather than the coupled particle-flow dynamics. In addition, the boundary condition introduces severe nonlinearities which can only be addressed numerically. At a more fundamental level of description, the impurity is treated as a quantum particle with its own wavefunction described by the Schrödinger equation and that is coupled with the GPE for the macroscopic wavefunction of the BEC [@berloff2000capture]. A more versatile model for the interaction of impurities with the BEC has been explored in several papers [@astrakharchik2004motion; @shukla2016sticking; @griffin2017vortex; @shukla2018particles; @pinsker2017gaussian]. Here, an additional repulsive interaction (a Gaussian or delta-function potential) is added to model scattering of the condensate particles with the impurity. The hydrodynamic force on the impurity is determined by this repulsion potential and the superfluid density through the Ehrenfest theorem. The strong-coupling limit of this repulsive potential would be equivalent to the rigid boundary-condition approach. Within this modeling approach, some works have studied the complex motion of particles interacting with vortices in the flow, and the indirect interactions between them arising from the presence of the fluid [@shukla2016sticking; @shukla2018particles]. Another line of research using this type of modeling focused mainly on the superfluidity criterion of a uniform BEC at zero temperature regime. Within the Bogoliubov perturbation analysis for a small impurity and weak interaction, analytical expressions can be derived for the steady-state force exerted by the superfluid as function of the constant velocity of the impurity [@astrakharchik2004motion; @roberts2006force; @sykes2009drag; @pinsker2017gaussian]. At zero temperature, this force vanishes below a critical velocity, the speed of long-wavelength sound waves, at least when we ignore the quantum fluctuations [@roberts2006force], and corresponds to the dissipationless motion. Above this velocity, there is a finite drag force and the motion of the impurity is damped by acoustic excitations. While this is a form of drag, in that the force opposes motion by dissipating energy, it is not the same as the classical Stokes’ drag in viscous fluids. Recent experiments probing superfluidity in a BEC are able to indirectly estimate the drag force by measuring the local heating rate in the vicinity of the moving laser beam and show that there is still a critical velocity even at non-zero temperatures and that the critical velocity is lower for a repulsive potential than for an attractive one [@singh2016probing].
In this paper, we study the forces exerted on an impurity moving in a two-dimensional BEC at finite temperature, using an approach similar to [@astrakharchik2004motion; @shukla2016sticking; @griffin2017vortex; @shukla2018particles; @pinsker2017gaussian], in which a repulsive Gaussian potential is used to describe the interaction of the particle with the BEC, but using a dissipative version of the GPE to model the fluid. Our aim is to bridge this microscopic approach with the phenomenological descriptions [@poole2005motion] that assume that the forces from the superfluid are the same as those from a classical fluid in the inviscid and irrotational case. As in the classical-fluid case, we find that the force is made of two contributions: One of them, dominant for very weak fluid-particle interaction, bears a rather complete analogy with the corresponding force in classical fluids (inertial or pressure-gradient force), which depends on local fluid acceleration and includes the so-called Faxén corrections arising from velocity inhomogeneities close to the particle position [@maxey1983equation]. The difference is that, in a classical fluid, these corrections arise from the finite size of the particle and vanish when the particle size becomes zero. In the BEC, Faxén-type corrections arise both from the particle size (modeled by the range of the particle repulsion potential) and from the BEC coherence length. As fluid-particle interaction becomes more important, a second contribution to the force becomes noticeable, which takes into account the drag on the particle arising from the perturbation of the flow produced by the presence of the particle. Thus it can be called a particle *self-induced* force. We are able to obtain explicit formulae for it in the case of constant-velocity motion of the particle in an otherwise homogeneous and steady BEC. This drag is a dissipative (damping) force due to viscous-like drag of the perturbed BEC with the thermal cloud. It occurs in addition to the drag due to acoustic excitations in the condensate that in the absence of dissipation occurs only above a critical velocity for the particle. Here, it can be compared with the corresponding force in classical fluids, namely the viscous Stokes drag. We find that, as the Stokes force, the self-induced dissipative drag is linear in the particle velocity for small velocities, and we obtain an expression for it also at arbitrary velocities.
The rest of the paper is structured as follows. In Sect. \[sec:model\], we discuss the general modeling setup and in Sect. \[sec:perturbation\] a perturbation analysis is used to derive the linearized equations for the perturbations in the wavefunction related to non-steady condensate flow and the particle repulsive potential. Subsections \[sec:inertial\] and \[sec:drag\] derive analytical expressions within perturbation theory for the two contributions to the force experienced by the particle. In Section \[sec:numerics\], we compare our theoretical predictions with numerical simulations of the dissipative GPE coupled to the impurity, and the final section summarizes our conclusions.
Modeling approach {#sec:model}
=================
We model the interaction between the impurity and a two-dimensional BEC through a Gaussian repulsive potential which can be reduced to a delta-function limit similar to previous studies [@astrakharchik2004motion; @pinsker2017gaussian]. The BEC itself, which is at a finite temperature, is described by a macroscopic wavefunction $\psi({\boldsymbol{r}},t)$ that evolves according to the damped Gross Pitaevskii equation (dGPE) [@Bradley_2012; @Reeves_2013; @skaugen2016vortex]: $$\begin{aligned}
\label{eq:GPe}
&& i\hbar\partial_t\psi = \nonumber \\
&& (1-i\gamma)\left(-\frac{\hbar^2}{2m}\nabla^2+g|\psi|^2-\mu+ V_{ext}+g_p{\cal U}_p\right)\psi,\end{aligned}$$ where $g$ is an effective scattering parameter between condensate atoms. $V_{ext}$ is any external potential used to confine the condensate atoms or to stir them. The damping coefficient $\gamma>0$ is related to finite-temperature effects due to thermal drag between the condensate and the stationary thermal reservoir of excited atoms at fixed chemical potential $\mu$. This damping $\gamma$ is very small at low temperatures and can be expressed as function of temperature $T$, chemical potential $\mu$ and the energy of the thermal cloud [@bradley2008bose]. The dGPE can be derived from the stochastic projected Gross-Pitaevskii equation in the low-temperature regime [@blakie2008dynamics] and has been used to study different quantum turbulence regimes and vortex dynamics [@Reeves_2014; @billam2015spectral; @Reeves_2013; @Bradley_2012] that are also observed in recent experiments [@neely2013characteristics]. An hydrodynamic description in terms of density and velocity of the BEC can be developed using the Madelung transformation of the wavefunction: $\psi=|\psi|e^{i\phi}$. The macroscopic number density is $\rho({\boldsymbol{r}},t)=|\psi({\boldsymbol{r}},t)|^2$ and the condensate velocity is ${\boldsymbol{v}}({\boldsymbol{r}},t)=(\hbar/m) \nabla
\phi({\boldsymbol{r}},t)$. This velocity can also be obtained from the superfluid current ${\boldsymbol{J}}({\boldsymbol{r}},t)$ as $${\boldsymbol{J}} = \frac{\hbar}{2 m i} \left(\psi^*\nabla\psi - \psi\nabla\psi^* \right) =\rho {\boldsymbol{v}} \ .
\label{eq:current}$$ $\psi^*$ denotes the complex conjugate of $\psi$. In addition to damping the BEC velocity, the presence of $\gamma\neq 0$ in the dGPE also singles out the value $\rho_h=|\psi|^2=g/\mu$ as the steady homogeneous density value when the phase is constant and $V_{ext}=0$.
The interaction potential ${\cal U}_p({\boldsymbol{r}}-{\boldsymbol{r}}_p)$ between the condensate and the impurity is modeled by a Gaussian potential ${\cal U}_p({\boldsymbol{r}}-{\boldsymbol{r}}_p)=
\mu/(2\pi \sigma^2) e^{-({\boldsymbol{r}}-{\boldsymbol{r}}_p)^2/(2\sigma^2)}$. The parameter $g_p>0$ is the weak coupling constant for repulsive impurity-condensate interaction, ${\boldsymbol{r}}_p={\boldsymbol{r}}_p(t)$ denotes the center-of-mass position of the impurity, and $\sigma$ its effective size. Here we consider an impurity of size $\sigma$ of the order the coherence length $\xi=\hbar/\sqrt{m\mu}$ of the condensate. The impurity is too small to nucleate vortices in its wake [@reeves2015identifying]. Instead, the impurity will create acoustic excitations which at supersonic speeds correspond to shock waves. Similar acoustic fringes in the condensate density have been reported numerically in [@wouters2010superfluidity] for a different realization of non-equilibrium Bose-Einstein condensates. In the limit of a point-like impurity, the Gaussian interaction potential converges to a two-body scattering potential ${\cal U}_p({\boldsymbol{r}}-{\boldsymbol{r}}_p,t)= \mu \delta({\boldsymbol{r}}-{\boldsymbol{r}}_p(t))$ that has been used in previous analytical studies [@astrakharchik2004motion; @shukla2016sticking; @griffin2017vortex; @shukla2018particles]. Note that we are modeling only the interaction of the particle with the BEC, so that the viscous-like drag we will obtain arises from the indirect coupling to the thermal bath via the BEC. Any direct interaction of the particle with the thermal cloud of normal atoms will lead to additional forces that we do not consider here.
In order to gain insight into the forces and their relationship with the classical case, we keep the set-up as simple as possible. We consider a strictly two-dimensional condensate, which can be obtained by plane optical traps. We assume that the size of the condensate is large enough so that we can neglect inhomogeneities in the confining part of $V_{ext}$ in the region of interest. Also, we consider a neutrally buoyant impurity so that effects of gravity can be neglected. This would imply $V_{ext}=0$ except if an external forcing is introduced to stir the system, in which case we assume the support of this external force is sufficiently far from the impurity.
The impurity and the condensate will exert an interaction force on each other that is determined by the Ehrenfest theorem for the evolution of the center-of-mass momentum of the particle. The potential force $-g_p\nabla {\cal U}_p({\boldsymbol{r}}-{\boldsymbol{r}}_p)$ is the force exerted by an impurity on a condensate particle at position ${\boldsymbol{r}}$. By space averaging over condensate density, we then determine the force exerted by the impurity on the condensate as $-g_p\int d{\boldsymbol{r}} |\psi({\boldsymbol{r}},t)|^2 \nabla {\cal
U}_p({\boldsymbol{r}}-{\boldsymbol{r}}_p)$ [@shukla2018particles]. Hence, the force acting on the impurity has the opposite sign and is equal to $$\begin{aligned}
\label{eq:fp0}
{\boldsymbol{F}}_p(t) &=& + g_p\int d^2{\boldsymbol{r}} |\psi({\boldsymbol{r}},t)|^2 \nabla {\cal U}_p({\boldsymbol{r}}-{\boldsymbol{r}}_p)\end{aligned}$$ which, through an integration by parts, is equivalent to $$\begin{aligned}
\label{eq:fp1}
{\boldsymbol{F}}_p(t) &=& -g_p\int d^2{\boldsymbol{r}} ~{\cal U}_p({\boldsymbol{r}}-{\boldsymbol{r}}_p ,t) \nabla |\psi({\boldsymbol{r}},t)|^2 .
\label{eq:forcegradrho}\end{aligned}$$ Note that this last expression can also be used, reversing the sign, to give the force exserted on the BEC by a laser of beam profile given by ${\cal U}_p$.
At zero temperature, i.e. $\gamma=0$, and neglecting the effect of quantum fluctuations [@roberts2006force; @sykes2009drag], the impurity moves without any drag through a uniform condensate below a critical velocity, which is the low-wavelength speed of sound $c=\sqrt{\mu/m}$, as determined by the condensate linear excitation spectrum, in agreement with Landau’s criterion of superfluidity [@astrakharchik2004motion]. Above the critical speed, the impurity will create excitations, and depending on the size of the impurity these excitations range from acoustic waves (Bogoliubov excitation spectrum) to vortex dipoles and to von-Karman street of vortex pairs [@reeves2015identifying]. Previous studies focused on the theoretical investigations of the self-induced drag force and energy dissipation rate in the presence of Bogoliubov excitations emitted by a pointwise [@astrakharchik2004motion; @roberts2006force; @sykes2009drag] or finite-size [@pinsker2017gaussian] particle, or numerical investigations of the drag force due to vortex emissions [@winiecki2000motion; @griffin2017vortex; @shukla2018particles]. The energy dissipation rate depends on whether the impurity is heavier, neutral or lighter with respect to the mass of the condensate particles [@shukla2018particles]. The dependence on the velocity of the self-induced drag force above the critical velocity changes with the spatial dimensions [@astrakharchik2004motion]. This means that the energy dissipation rate is also dependent on the spatial dimensions. If instead of a single impurity one considers many of them there will be, besides direct inter-particle interactions, additional forces between the impurities mediated by the flow, leading to a much more complex many-body dynamics even in an otherwise uniform condensate, as discussed in [@shukla2016sticking]. Here we neglect all these effects and consider a single impurity in a two-dimensional BEC.
We rewrite the dGPE in dimensionless units by using the characteristic units of space and time in terms of the long-wavelength speed of sound $c=\sqrt{\mu/m}$ in the homogeneous condensate and the coherence length $\xi=\hbar/(m
c)=\hbar/\sqrt{m\mu}$. Space is rescaled as ${\boldsymbol{r}}\rightarrow
\tilde{{\boldsymbol{r}}} \xi$ and time as $t\rightarrow \tilde t \xi/c$. In addition, the wavefunction is also rescaled $\psi
\rightarrow \tilde \psi\sqrt{\mu/g}$, where $g/\mu$ is the equilibrium particle-number density corresponding to the solution with constant phase if $V_{ext},\mathcal U_p=0$. The external potential, $V_{ext}=\mu \tilde V_{ext}$, and the interaction potential, $g_p \mathcal U_p = \mu \tilde g_p
\tilde{\mathcal U_p}$, are measured in units of the chemical potential $\mu$ with $\tilde{\mathcal U_p} = 1/(2\pi a^2)
e^{-({{\boldsymbol{\tilde}} r}-{{\boldsymbol{\tilde}} r}_p)^2/(2a^2)}$, and $a =
\sigma/\xi$, $\tilde{g}_p = g_p/(\xi^2\mu)$. Henceforth, the dimensionless form of the dGPE reads as $$\begin{aligned}
\label{eq:GPe_dimless}
\tilde \partial_t \tilde\psi =
(i+\gamma)\left(\frac{1}{2}\tilde\nabla^2+1-\tilde V_{ext}-\tilde g_p\tilde{\mathcal U_p}-|\tilde\psi|^2\right)\tilde\psi .\end{aligned}$$ We use these dimensionless units and express the force (\[eq:forcegradrho\]) exerted on an impurity as ${\boldsymbol{F}}_p=
(\mu^2\xi/g) \tilde{{\boldsymbol{F}}}_p$, where $$\begin{aligned}
\tilde{{\boldsymbol{F}}}_p(t) =
-\tilde g_p\int d^2\tilde{{\boldsymbol{r}}} \tilde{\mathcal U_p}({\boldsymbol{r}}-{\boldsymbol{r}}_p)\tilde\nabla |\tilde\psi(\tilde{{\boldsymbol{r}}},t)|^2. \label{eq:fp2}\end{aligned}$$ For the rest of the paper, we will now omit the tildes over the dimensionless quantities.
In the limit of a point-like particle, $\mathcal U_p=
\delta({\boldsymbol{r}}-{\boldsymbol{r}}_p)$, the force from Eq. (\[eq:fp2\]) becomes $$\begin{aligned}
{\boldsymbol{F}}_p(t) = -g_p\nabla|\psi({\boldsymbol{r}},t)|^2 |_{{\boldsymbol{r}}={\boldsymbol{r}}_p(t)}.
\label{eq:fp3}\end{aligned}$$
Perturbation analysis {#sec:perturbation}
=====================
For a weakly-interacting impurity, the condensate wavefunction $\psi$ can be decomposed into an unperturbed wavefunction $\psi_0({\boldsymbol{r}})$ describing the motion and density of the fluid in the absence of the particle and the perturbation $\delta\psi_1({\boldsymbol{r}})$ due to the impurity’s repulsive interaction with the condensate, hence $\psi = \psi_0+g_p
\delta\psi_1$. The unperturbed wavefunction $\psi_0({\boldsymbol{r}},t)$ can be spatially-dependent, if it is initialized in a nonequilibrium configuration, or if external forces characterized by $V_{ext}$ are at play. Here, we consider deviations with respect to the steady and uniform equilibrium state (which in our dimensionless units is $\psi_h=1$). As stated before, we do not consider large extended inhomogeneities produced by a trapping potential, and assume that any stirring force acting on the BEC is far from the particle. Thus, we treat inhomogeneities close to the particle as small perturbations to the uniform state $\psi_h=1$: $\psi_0({\boldsymbol{r}},t)=1+\delta\psi_0({\boldsymbol{r}},t)$. Combining the two types of perturbations, and using the relationships of the wavefunction to the density, velocity and current (Eq. (\[eq:current\]), which in dimensionless units reads $\rho{\boldsymbol{v}}=(\psi^*\nabla\psi-\psi\nabla\psi^*)/(2i)$) we find $$\begin{aligned}
\label{eq:density_perturb}
\psi &=& 1 +\delta \psi_0 +g_p \delta\psi_1 \\
\rho &=& 1 +\delta \rho_0+g_p \delta\rho_1, \\
{\boldsymbol{v}} &=& \delta{\boldsymbol{v}}^{(0)} + g_p \delta {\boldsymbol{v}}^{(1)},\end{aligned}$$ where $$\begin{aligned}
&&\delta \rho_0 =
\delta\psi_0+\delta\psi_0^*, \quad \delta\rho_1 =
\delta\psi_1+\delta\psi_1^* ,\label{eq:density_linear} \\
&&\delta {\boldsymbol{v}}^{(0)} = \frac{1}{2 i}\nabla \left(\delta \psi_0 - \delta \psi_0^*\right), \quad
\delta{\boldsymbol{v}}^{(1)} =\frac{1}{2 i}\nabla (\delta\psi_1 - \delta\psi_1^*).\nonumber\\
\label{eq:velocity_linear}\end{aligned}$$
Combining Eq. (\[eq:fp2\]) with the expressions for the density perturbations, we have that the total force can be split into the contribution from the density variations in the BEC by causes external to the particle (initial preparation, stirring forces in $V_{ext}$, ...), and the density perturbations due to the presence of the particle ${\boldsymbol{F}}_p=
{\boldsymbol{F}}^{(0)}+{\boldsymbol{F}}^{(1)}$: $$\begin{aligned}
{\boldsymbol{F}}^{(0)}(t) = - \frac{g_p}{2\pi a^2} \int d^2{\boldsymbol{r}} e^{- \frac{({\boldsymbol{r}}-{\boldsymbol{r}}_p(t))^2}{2a^2}}
\nabla \delta\rho_0({\boldsymbol{r}},t),
\label{eq:fp_unperturb}
\\
{\boldsymbol{F}}^{(1)}(t) = - \frac{g_p^2}{2\pi a^2} \int d^2{\boldsymbol{r}} e^{- \frac{({\boldsymbol{r}}-{\boldsymbol{r}}_p(t))^2}{2a^2}}
\nabla \delta\rho_1({\boldsymbol{r}},t).
\label{eq:fp_perturb}\end{aligned}$$ The perturbative splitting of the force in these two contributions is completely analogous to the corresponding classical-fluid case in the incompressible [@maxey1983equation] and in the compressible [@parmar2012equation] situations. The ${\boldsymbol{F}}^{(0)}$ contribution is the equivalent to the classical inertial or pressure-gradient force on a test particle, which does not disturb the fluid, in a inhomogeneous and unsteady flow. In the following it will be called the *inertial* force. The ${\boldsymbol{F}}^{(1)}$ contribution takes into account perturbatively the modifications on the flow induced by the presence of the particle, and it will be called the *self-induced drag* on the particle. To complete the comparison with the classical expressions [@maxey1983equation; @parmar2012equation], we need to express Eqs. (\[eq:fp\_unperturb\]) and (\[eq:fp\_perturb\]) in terms of the unperturbed velocity field ${\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)=\delta{\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)$ and of the particle speed ${\boldsymbol{V}}_p(t)={\boldsymbol{\dot r}}_p(t)$. We are able to do so in a general situation for the inertial force ${\boldsymbol{F}}^{(0)}$. For ${\boldsymbol{F}}^{(1)}$, we obtain analytical expressions in the simple case where the impurity is moving with a constant velocity in an otherwise uniform BEC.
The desired relationships between $\nabla \delta\rho_0$ and $\nabla \delta\rho_1$ in Eqs. (\[eq:fp\_unperturb\])-(\[eq:fp\_perturb\]), and $\delta{\boldsymbol{v}}^{(0)}$ and ${\boldsymbol{V}}_p$ will be obtained from the linearization of the dGPE Eq. (\[eq:GPe\_dimless\]) around the uniform steady state $\psi_h=1$: $$\begin{aligned}
\partial_t \delta\psi_0 &=& (i+\gamma)\left(\frac{1}{2}\nabla^2 -1\right)\delta\psi_0 \nonumber \\
&-& (i+\gamma)\delta\psi^*_0,
\label{eq:psi0lin} \\
\partial_t \delta\psi_1 &=&
(i+\gamma)\left(\frac{1}{2}\nabla^2-1\right)\delta\psi_1 \nonumber \\
&-&(i+\gamma)\delta\psi_1^* - (i+\gamma) \mathcal U_p({\boldsymbol{r}}-{\boldsymbol{r}}_p) \ .
\label{eq:psi1lin}\end{aligned}$$ Terms containing $V_{ext}$ are not included in Eq. (\[eq:psi0lin\]) because of our assumption of sufficient distance between possible stirring sources and the neighborhood of the particle position, the only region that–as we will see– will enter into the calculation of the forces. In the next sections we solve these linearized equations to relate density perturbations to undisturbed velocity field and particle velocity.
Inertial force {#sec:inertial}
--------------
To convert Eq. (\[eq:fp\_unperturb\]) for the inertial force into an expression suitable for comparison for the corresponding term in classical fluids, we need to express $\nabla\delta\rho_0$ in terms of the undisturbed velocity field ${\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)=\delta{\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)$. To this end, we substract Eq. (\[eq:psi0lin\]) from its complex conjugate, obtaining: $$\begin{aligned}
\left(\nabla^2-4\right)\nabla\delta\rho_0=
4 \left(\partial_t-\frac{\gamma}{2}\nabla^2\right)\delta{\boldsymbol{v}}^{(0)},
\label{eq:drhodv0}\end{aligned}$$ where we have used Eqs. (\[eq:density\_linear\]) and (\[eq:velocity\_linear\]). Since the force formulae require to obtain the condensate density in a neighborhood of the particle position, it is convenient to move to a coordinate frame with center always at the (possibly moving) particle location ${\boldsymbol{r}}={\boldsymbol{r}}_p(t)$. Thus we change variables from $({\boldsymbol{r}}, t)$ to $({\boldsymbol{z}}, t)$, with ${\boldsymbol{z}}={\boldsymbol{r}} -{\boldsymbol{r}}_p(t)$, and the velocity field will be now referred to the particle velocity ${\boldsymbol{V}}_p(t)={\boldsymbol{\dot r}}_p(t)$: $\delta {\boldsymbol{w}}^{(0)}({\boldsymbol{z}},t)=\delta{\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)- {\boldsymbol{V}}_p(t)$. Equation (\[eq:drhodv0\]) becomes: $$\begin{aligned}
\left(\nabla_z^2 - 4\right)&&\nabla_z\delta\rho_0 = \nonumber \\
&4& \left(\partial_t-{\boldsymbol{V}}_p\cdot\nabla_z-\frac{\gamma}{2}\nabla_z^2\right)\delta{\boldsymbol{w}}^{(0)} + {\boldsymbol{\dot V}}_p(t),
\label{eq:drhodv}\end{aligned}$$ which has the corresponding equation for its Green’s function given by $$\left(\nabla_z^2 - 4\right)G({\boldsymbol{z}}) = \delta({\boldsymbol{z}})$$ with the boundary condition $G(|{\boldsymbol{z}}|\rightarrow\infty)\rightarrow 0$ (corresponding to vanishing $\nabla_z \delta\rho_0({\boldsymbol{r}})$ at $|{\boldsymbol{r}}|=\infty$). The solution is given by the zeroth order modified Bessel function $G({\boldsymbol{z}})=-K_0(2|{\boldsymbol{z}}|)/(2\pi)$. Hence, the gradient of the density perturbation can be written as the convolution with the Green’s function:
$$\begin{aligned}
\nabla_z\delta\rho_0({\boldsymbol{z}},t)=
-\frac{2}{\pi}\int d{\boldsymbol{z}}' K_0(2|{\boldsymbol{z}}-{\boldsymbol{z}}'|)
\left[\left(\partial_t-{\boldsymbol{V}}_p\cdot\nabla_{{\boldsymbol{z}}'}-\frac{\gamma}{2}\nabla^2_{{\boldsymbol{z}}'}\right)\delta{\boldsymbol{w}}^{(0)}({\boldsymbol{z}}',t) + {\boldsymbol{\dot V}}_p(t) \right] ,
\label{eq:nablarho}\end{aligned}$$
and the expression for the force (\[eq:fp\_unperturb\]), using the comoving variables $({\boldsymbol{z}},t)$, becomes: $$\begin{aligned}
{\boldsymbol{F}}^{(0)}(t)=
-\frac{g_p}{\pi^2a^2}\int d{\boldsymbol{z}} e^{-\frac{z^2}{2 a^2}} \int d{\boldsymbol{z}}' K_0(2|{\boldsymbol{z}}-{\boldsymbol{z}}'|)
\left[\left(\partial_t-{\boldsymbol{V}}_p\cdot\nabla_{{\boldsymbol{z}}'}-\frac{\gamma}{2}\nabla^2_{{\boldsymbol{z}}'}\right)\delta{\boldsymbol{w}}^{(0)}({\boldsymbol{z}}',t) + {\boldsymbol{\dot V}}_p(t) \right] .
\label{eq:F0full}\end{aligned}$$
The above expression is a weighted average of contributions from properties of the fluid velocity in a neighborhood of the impurity center-of-mass position (${\boldsymbol{z}} = 0$ in the comoving frame). The size of this neighborhood is given by the combination of the range of the Bessel function kernel, which in dimensional units would be the correlation length $\xi$, and the range of the Gaussian potential, $a$, giving an effective particle size. In classical fluids, the analogous force on a spherical particle involves the average of properties of the undisturbed velocity field within the sphere size [@parmar2012equation], and there is no equivalent to the role of $\xi$.
As in the classical case [@maxey1983equation; @parmar2012equation], if fluid velocity variations are weak at scales below $a$ and $\xi$, we can approximate the condensate velocity by a Taylor expansion near the impurity, i.e.: $$\begin{aligned}
\delta w_i^{(0)}(\mathbf z',t)&\approx& \delta
w_i^{(0)}(t)+\sum_j
e_{ij}(t) z'_j \nonumber\\
&+&\frac{1}{2}\sum_{jk}e_{ijk}(t) z'_j z'_k + \ldots,\end{aligned}$$ where the indices $i,j,k=x,y$ denote the coordinate components. $e_{ij}(t)=\partial_j \delta w_i^{(0)}(\mathbf z,t)|_{{\boldsymbol{z}}=0}$ and $e_{ijk}(t)=\partial_j
\partial_k \delta w_i^{(0)}(\mathbf z,t)|_{{\boldsymbol{z}}=0}$ are gradients of the unperturbed condensate relative velocity. Inserting this expansion into Eq. (\[eq:F0full\]), and performing the integrals of the Gaussian and of the Bessel function (using for example $\int K_0(2|{\boldsymbol{z}}|)d{\boldsymbol{z}}=\pi/2$ and $\int z_i z_j K_0(2|{\boldsymbol{z}}|)d{\boldsymbol{z}}
=(\delta_{ij}/2)\int_0^\infty 2\pi z^3
K_0(2z)dz=\delta_{ij}\pi/4$), we obtain: $$\begin{aligned}
{\boldsymbol{F}}^{(0)}(t) &\approx& g_p {\boldsymbol{\dot V}}_p(t)
+ g_p\left[\partial_t
- {\boldsymbol{V}}_p(t)\cdot\nabla_z +\frac{a^2}{2}\partial_t\nabla^2_z \right. \nonumber \\
&-& \left. \frac{\gamma}{2}\nabla_z^2 +
\frac{1}{4}\partial_t\nabla^2_z \right] \delta{\boldsymbol{w}}^{(0)}({\boldsymbol{z}},t)|_{{\boldsymbol{z}}={\boldsymbol{0}}} \ .
\label{eq:Fi_th}\end{aligned}$$ The terms containing Laplacians are analogous to the Faxén corrections in classical fluids [@maxey1983equation] which arise for particles with finite size. Here, they arise from a combination of the finite effective size of the particle, $a$, and of the quantum coherence length, $\xi=1$. This last effect remains even in the limit of vanishing particle size $a\rightarrow 0$. Interestingly, one of the two terms in these quantum corrections depend on $\gamma$ hence indirectly on the presence of the thermal cloud.
As in the classical case, if flow inhomogeneities are unimportant below the scales $a$ and $\xi$, we can neglect the Laplacian terms in Eq. (\[eq:Fi\_th\]). Returning to the variables $({\boldsymbol{r}},t)$ in the lab frame of reference, the terms containing ${\boldsymbol{V}}_p$ cancel out, showing that the inertial force is mainly given by the local fluid acceleration: $$\begin{aligned}
\label{eq:pd1}
{\boldsymbol{F}}^{(0)}(t) = g_p \partial_t \delta{\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t) \big|_{{\boldsymbol{r}}={\boldsymbol{r}}_p(t)}.
\end{aligned}$$ We have assumed a small non-uniform unperturbed velocity field ${\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)= \delta{\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)$. To leading order in this small velocity, the partial derivative $\partial_t\delta{\boldsymbol{v}}^{(0)}$ and the material derivative $D\delta{\boldsymbol{v}}^{(0)}/Dt=\partial_t\delta{\boldsymbol{v}}^{(0)}+\delta{\boldsymbol{v}}^{(0)}\cdot \nabla \delta{\boldsymbol{v}}^{(0)}$ are identical. In classical fluids the same ambiguity occurs and it has been established, on physical grounds and by going beyond linearization, that using the material derivative is more correct [@maxey1983equation]. After all, using this material derivative in the equation of motion simply means that, under the above approximations and in places where stirring and other external forces are absent, the local acceleration on the impurity arises from the corresponding acceleration of the condensate. Since for $a\rightarrow 0$ the condensate-impurity interaction has a similar scattering potential (delta function) as that for the interaction between condensate particles, similar accelerations would be experienced by a condensate particle and by the impurity, just modulated by a different coupling constant. Thus, replacing $\partial_t$ by $D/Dt$ in (\[eq:pd1\]) the approximate inertial force becomes: $$\begin{aligned}
\label{eq:pd}
{\boldsymbol{F}}^{(0)}(t) &=&
g_p \left.\frac{D {\boldsymbol{v}}^{(0)}}{Dt}\right|_{{\boldsymbol{r}}={\boldsymbol{r}}_p(t)}\ ,
\label{eq:InertialSimpleDimless}
\end{aligned}$$ or, if we return back to dimensional variables: $$\begin{aligned}
\label{eq:pd_dim}
{\boldsymbol{F}}^{(0)}(t) &=&
\frac{g_p}{g} m \left.\frac{D {\boldsymbol{v}}^{(0)}}{Dt}\right|_{{\boldsymbol{r}}={\boldsymbol{r}}_p(t)}\ .
\label{eq:InertialSimpleDim}
\end{aligned}$$ This is equivalent to the equation for the inertial force in classical fluids [@maxey1983equation] except that the coefficient of the material derivative in the classical case is the mass of the fluid fitting in the size of the impurity. In the comoving frame, replacement of the partial by the material derivative amounts to replace $(\partial_t-{\boldsymbol{V}}_p\cdot
\nabla_{{\boldsymbol{z}}'})\delta{\boldsymbol{\omega}}^{(0)}$ in Eq. (\[eq:F0full\]) by $D\delta{\boldsymbol{\omega}}^{(0)}/Dt$. Eq. (\[eq:pd\]) is expected to be valid for small values of $g_p$ and in regions where fluid velocity and density inhomogeneities are both small and weakly varying. At this level of approximation neither compressibility nor dissipation effects appear explicitly in the inertial force, in analogy with classical compressible fluids [@parmar2012equation]. But these effects are indirectly present by determining the structure of the field ${\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)$.
Self-induced drag force {#sec:drag}
-----------------------
In classical fluids, consideration of the self-induced force on a particle moving at arbitrary time-dependent velocity, coming from the perturbation in produces in the flow, leads to different terms, namely [@maxey1983equation; @parmar2012equation] the viscous (Stokes) drag, the unsteady-inviscid term that in the incompressible case becomes the added-mass force, and the unsteady-viscous term that in the incompressible case becomes the Basset history force. They are expressed in terms of the undisturbed velocity flow ${\boldsymbol{v}}^{(0)}$ and the particle velocity ${\boldsymbol{V}}_p(t)$. Here, for the BEC case, we are able to obtain the self-induced force only for a particle moving at constant speed on the condensate. For the classical fluid case, in this situation the only non-vanishing force is the Stokes drag, so that this is the force we have to compare our quantum result with. We note that the condensate itself in the absence of the particle perturbation can be in any state of (weak) motion since in our perturbative approach summarized in Eqs (\[eq:psi0lin\])-(\[eq:psi1lin\]), the inhomogeneity $\delta\psi_0$ and the $g_p$-perturbation $\delta\psi_1$ are uncoupled.
It is convenient to transform the problem to the frame of reference moving with the particle $({\boldsymbol{r}},t)\rightarrow ({\boldsymbol{z}},t)$ with ${\boldsymbol{z}}={\boldsymbol{r}} - {\boldsymbol{r}}_p(t)$, so that Eq. (\[eq:psi1lin\]) becomes $$\begin{aligned}
\partial_t \delta\psi_1 - {\boldsymbol{V}}_p \cdot \nabla \delta\psi_1 &=&
(i+\gamma)\left(\frac{1}{2}\nabla^2-1\right)\delta\psi_1 \nonumber \\
&-&(i+\gamma)\delta\psi_1^* - (i+\gamma) \mathcal U({\boldsymbol{r}}) \ .
\label{eq:psi1linComoving}\end{aligned}$$ Note that such Galilean transformations of the GPE using a constant ${\boldsymbol{V}}_p$ are often accompanied by a multiplication of the transformed wavefunction by a phase factor $\exp(i{\boldsymbol{V}}_p \cdot {\boldsymbol{z}} + \frac i 2 V_p^2 t)$, in order to transform the condensate velocity (see below) to the new frame of reference, and account for the shift in kinetic energy. Indeed, such a combined transformation leaves the GPE unchanged at $\gamma = 0$ [@pismen1999] (but not for $\gamma>0$). The density perturbation $\delta\rho_1$ is already given correctly by $\delta\psi_1+\delta\psi_1^*$, where $\delta\psi_1({\boldsymbol{z}},t)$ is the solution of (\[eq:psi1linComoving\]), without the need of any additional phase factor. The velocity in the comoving frame would need to be corrected as $\delta\omega^{(1)}({\boldsymbol{z}},t)=\delta {\boldsymbol{v}}^{(1)}-{\boldsymbol{V}}_p$, with $\delta v^{(1)}$ given by expression (\[eq:velocity\_linear\]) in terms of he solution of (\[eq:psi1linComoving\]).
Eq. (\[eq:psi1linComoving\]) in the steady-state can be solved by using the Fourier transform $\delta\psi_1({\boldsymbol{z}})
=1/(2\pi)^2 \int d^2{\boldsymbol{k}} e^{i{\boldsymbol{k}}\cdot {\boldsymbol{z}}}
\delta\hat\psi_1({\boldsymbol{k}})$. It follows that the linear system of equations for $\delta\hat\psi_1({\boldsymbol{k}})$ and $\delta\hat\psi_1^*(-{\boldsymbol{k}})$ is given by $$\begin{aligned}
\left[-2i {\boldsymbol{k}}\cdot{\boldsymbol{V}}_p +(i+\gamma) (k^2+2)\right]&\delta\hat\psi_1& + 2(i+\gamma)\delta\hat\psi_1^* = \nonumber \\
&-& 2(i+\gamma) e^{-\frac{a^2k^2}{2}} ,\nonumber\\
\left[-2i {\boldsymbol{k}}\cdot{\boldsymbol{V}}_p +(-i+\gamma) (k^2+2)\right]&\delta\hat\psi_1^*& + 2(-i+\gamma)\delta\hat\psi_1 = \nonumber \\
&-& 2(-i+\gamma) e^{-\frac{a^2k^2}{2}} .\nonumber\\\end{aligned}$$ By solving these equations, we find $\delta\hat\psi_1({\boldsymbol{k}})$ and $\delta\hat\psi_1^*(-{\boldsymbol{k}})$, and the Fourier transform of the density perturbation $\delta\rho_1 =
\delta\psi_1^*+\delta\psi_1$ then follows as $$\delta\hat\rho_1 = \frac{e^{-\frac{k^2a^2}{2}} (4 k^2(1+\gamma^2)-8i\gamma{\boldsymbol{k}}\cdot {\boldsymbol{V}}_p)}{4{\boldsymbol{k}}\cdot{\boldsymbol{V}}_p({\boldsymbol{V}}_p \cdot {\boldsymbol{k}}+i\gamma k^2 +2i\gamma)- k^2(4+k^2)(1+\gamma^2)}.
\label{Rho1Delta}$$ Using the convolution theorem, we can express the self-induced force (\[eq:fp\_perturb\]) (in the co-moving frame, i.e. with $\mathbf r_p=0$) in terms of $\delta\hat\rho_1$ as $${\boldsymbol{F}}^{(1)}= - \frac{g_p^2}{(2\pi)^2}\int d^2{\boldsymbol{k}} i{\boldsymbol{k}}\delta\hat{\rho}_1({\boldsymbol{k}}) e^{-\frac{k^2a^2}{2}} .$$ This force can be decomposed into the normal and tangential components relative to the particle velocity ${\boldsymbol{V}}_p$: $ {\boldsymbol{F}}^{(1)} = F_{\|} {\boldsymbol{e}}_{\|}+F_{\perp} {\boldsymbol{e}}_{\perp}$. Due to symmetry, the normal component vanishes upon polar integration, and we are left with the tangential, or drag, force
$$\begin{aligned}
F_{\|}&=& -\frac{g_p^2}{(2\pi)^2} \int_0^\infty dk\int_0^{2\pi} d\theta e^{-k^2a^2}\frac{i k^2\cos(\theta) \left[4 k^2(1+\gamma^2) - 8i\gamma kV_p\cos(\theta)\right]}{4kV_p\cos(\theta)(kV_p\cos(\theta)+ i\gamma k^2 + 2i\gamma) - k^2(4 + k^2)(1+\gamma^2)}.
\label{eq:force_int}\end{aligned}$$
$V_p$ is the modulus of ${\boldsymbol{V}}_p$. At zero temperature, i.e. when $\gamma=0$, the drag force reduces to the one that has also been calculated for a point particle in Refs. [@astrakharchik2004motion] and in [@pinsker2017gaussian] for a finite-$a$ particle: $$F_\parallel = - \frac{g_p^2}{\pi^2}\int_0^\infty dk\int_0^{2\pi} d\theta\frac{i k^2 \cos{\theta}e^{-k^2a^2}}{4V_p^2 \cos^2{\theta}- (4+k^2)} \ ,$$ which is zero for particle speed smaller than the critical value given by the long-wavelength sound speed, $V_p < c=1$. Above the critical speed, the integral has poles and acquires a non-zero value given by $$F_\parallel = -\frac{g_p^2 k_{max}^2}{4V_p} e^{-\frac{a^2 k_{max}^2}{2}} \left[I_0\left(\frac{a^2 k_{max}^2}{2}\right) - I_1\left(\frac{a^2 k_{max}^2}{2}\right) \right]
\label{eq:forcegamma0}$$ in terms of the modified Bessel functions of the first kind $I_n(x)$ and where $k_{max} =2\sqrt{V_p^2 -1}$. For vanishing $a$ the dominant term is proportional to $(V_p^2-1)/V_p$ [@astrakharchik2004motion]. This drag is pertaining to energy dissipation by radiating sound waves in the condensate away from the impurity. We stress again that we assume $a$ small enough such that emission of other excitations, such as vortex pairs, does not occur. It is important to note [@astrakharchik2004motion; @pinsker2017gaussian] that in order to obtain a real value for the force in Eq. (\[eq:forcegamma0\]) one has to consider that it has been obtained from the limit $\gamma\rightarrow 0^+$ in (\[eq:force\_int\]), which implies that an infinitesimal positive imaginary part needs to be considered in the denominator to properly deal with the poles in the integral.
{width="\textwidth"}
In general, for a non-zero $\gamma$, Eq. (\[eq:force\_int\]) simplifies upon an expansion in powers of $V_p$ to the leading order. For the linear term in $V_p$, we can perform the polar integration and arrive at $$F_\parallel = -\frac{2}{\pi}V_p \frac{\gamma}{1+\gamma^2}g^2_p \int \frac{k^3 e^{-a^2k^2}}{(4 +k^2)^2}dk \ .$$ Substituting $u = a^2(k^2 + 4)$, we find $$\begin{aligned}
F_{\|} =-V_p \frac{\gamma}{1+\gamma^2}g_p^2 \frac{1}{\pi} \left[ e^{4a^2 }E_1(4a^2)(1+4a^2) - 1\right], \nonumber\\
\label{eq:drag_coeff_full}\end{aligned}$$ where $E_1(x)$ denotes the positive exponential integral. When $a \to 0$, the expression inside the bracket diverges as $-\gamma_E - 1 - \ln(4a^2)$ with $\gamma_E$ begin the Euler-Mascheroni constant. It is therefore necessary to keep a finite size $a$.
This drag force is analogous to the viscous Stokes drag force in classical fluids since it is due to an effective interaction of the impurity with the normal fluid through the thermal drag on the BEC. The effective drag coefficient depends on the thermal drag such that it vanishes at zero temperature. But it also depends non-trivially on the size of the impurity and it diverges in the limit of point-like particle. Faxén corrections involving derivatives of the unperturbed flow are not present here because of the decoupling between $\delta\psi_0$ and $\delta\psi_1$ arising in the perturbative approach leading to (\[eq:psi0lin\])-(\[eq:psi1lin\]).
Numerical results {#sec:numerics}
=================
{width="\textwidth"}
To test the analytical predictions of the inertial force and the self-induced drag deduced above from the total force expression Eq. (\[eq:fp2\]), we performed numerical simulations of the dGPE. Actually, our simulations are done in the co-moving frame of the impurity moving at constant velocity ${\boldsymbol{V}}_p$, so that the equation we solve is (see numerical details in the Appendix): $$\begin{aligned}
&\partial_t \psi - {\boldsymbol{V}}_p \cdot \nabla \psi =(i+\gamma)\left[\frac{1}{2}\nabla^2 \psi +\left(1-g_p\mathcal U_p - |\psi|^2\right)\psi\right], \nonumber \\
\label{eq:ComovingdGPE}\end{aligned}$$ where the impurity is described by the Gaussian potential of intensity $g_p=0.01$ and effective size $a=\xi=1$, and is situated in the middle of the domain with the coordinates $x/\xi= 128$ and $y/\xi= 64$. As an initial condition, we start with the condensate being at rest and in equilibrium with the impurity. This is done by imaginary time integration of Eq. (\[eq:ComovingdGPE\]) for $V_p=0$ and $\gamma=0$. Then, at $t=0$, we solve the full Eq. (\[eq:ComovingdGPE\]), and as a consequence, sound waves are emitted from the neighborhood of the impurity (the size of the impurity is below the critical size for vortex nucleation). Their speed is determined by the dispersion relation $\omega({\boldsymbol{k}})$ giving the frequency as a function of the wavenumber and can be obtained by looking for plane-wave solutions to Eq. (\[eq:psi0lin\]). If $\gamma=0$, $\omega({\boldsymbol{k}})$ is given by the Bogoliubov dispersion relation [@BEC-review] $\omega({\boldsymbol{k}}) =
k\sqrt{1+k^2/4}$ (with $c=\xi=1$). Note that the smallest velocity, $c=1$, is that of long waves, and that waves of smaller wavelength travel faster. For $\gamma>0$, the planar waves are dampened out and the dispersion relation becomes $\omega({\boldsymbol{k}})=-i\gamma (k^2/2+1)+\sqrt{k^2+k^4/4-\gamma^2}$. The damping rate is determined by $\gamma$ and increases quadratically with the wavenumber. Also, in this case all waves have a group velocity faster than a minimum one that for small $\gamma$ is close to $c=1$.
When $V_p<1$ all the waves escape the neighborhood of the impurity (see an example in Fig. \[fig:density\_gamma\](a)) and are dissipated in a boundary buffer region that has large $\gamma$ (see numerical details in the Appendix and Supplemental Material [@SupplMat]). After a transient the condensate achieves a steady state when seen in the frame comoving with the impurity. Fig. \[fig:density\_gamma\](b) shows a steady spatial configuration for $\gamma=0$ and $V_p=0.9$. Figs. \[fig:density\_gamma\](d-e) show different profiles of the condensate density along the $x$ direction across the impurity position for $V_p$. The condensate density is depleted near the impurity due to the repulsive interaction, and its general shape depends on the speed $V_p$ and thermal drag $\gamma$. If $\gamma=0$ and $V_p\leq 1$ the density of this steady state has a rear-front symmetry with respect to the particle position (see specially Fig. \[fig:density\_gamma\](d)), so that under integration in Eq. (\[eq:fp2\]) the net force is zero. The presence of dissipation ($\gamma>0$) breaks this symmetry even if $V_p<1$ so that a net drag will appear in agreement with the calculation of Sect. \[sec:drag\]. When $V_p>1$ there are waves that can not escape from the neighborhood of the impurity, forming fringes in front of it and a wake behind it. (see Fig. \[fig:density\_gamma\](c) and \[fig:density\_gamma\](f) and Supplemental material [@SupplMat]). Similar modulations in the condensate density around an obstacle in supersonic flows has also been observed experimentally [@carusotto2006bogoliubov]. As a consequence of this rear-front asymmetry, drag is nonvanishing even when $\gamma=0$.
Movies showing the transient and long-time density behavior for several values of $V_p$ and at $\gamma=0$ are included as Supplemental Material [@SupplMat]. They are presented in the comoving impurity frame, so that the impurity appears static. The fluid suddenly stars to move towards the negative $x$ direction, and density approaches a steady state after the transient. Note that during all the dynamics, the density deviation with respect to the equilibrium value $\rho=1$ is very small, justifying the perturbative approach of Sect. \[sec:perturbation\] for this situation. The time evolution for $\gamma>0$ is qualitatively similar to the $\gamma=0$ one shown in the movies, except that the waves become damped and that there is a front-rear asymmetry in the steady state.
Our numerical setup is well suited to measure the force produced by the perturbation of the impurity on the fluid, i.e. the self-induced drag. Nevertheless, in the absence of the impurity the unperturbed state is the trivial $\psi=1$, so that $\delta\psi_0=0$ and the inertial force is identically zero. In order to test the accuracy of our expressions for the inertial force without the need of additional simulations under a different set-up, we still use the computed condensate density and velocity dynamics, produced by the impurity introduced in the system at $t=0$, but we evaluate the inertial force exerted by this flow on another test particle located at a different position. In fact, there is no need to think on the flow as being produced by an impurity: it can be produced by a moving laser beam that can modeled by an external potential $V_{ext}$ and the only impurity present in the system is the test particle on which the force is evaluated. In the following we evaluate the inertial and the self-induced drag forces on the different particles from the general expressions Eqs. (\[eq:fp\_unperturb\])-(\[eq:fp\_perturb\]) and from the approximate expressions of Sects. \[sec:inertial\] and \[sec:drag\].
Numerical evaluation of the inertial force {#sec:numerical-inertial}
------------------------------------------
We consider a test particle traveling at the same speed $V_p$ as the impurity or laser beam producing the flow, but located at a distance of 10 coherence lengths in front of it, and 20 coherence lengths in the $y$ direction apart from it. This distance is sufficient to avoid inclusion of $\mathcal U_p$ or $V_{ext}$ in Eq. (\[eq:psi0lin\]) for the neighborhood of the test particle. Condensate and test particle interact via a coupling constant $g_p'$ sufficiently small so that the full force on the later, Eq. (\[eq:fp2\]), is well approximated by the inertial part Eq. (\[eq:fp\_unperturb\]), being the perturbation the particle induces on the flow, and thus the force (\[eq:fp\_perturb\]) completely negligible.
Figure (\[fig:inertial\_force\]) shows, for different values of $V_p = 0.1, 0.8, 1$ at $\gamma=0$, the $x$ component of the time-dependent force produced by the transient flow inhomogeneities hitting the test particle in the form of sound waves. The size of the test particle, taking several values, is called $a'$ to distinguish it from the size $a$ of the particle producing the flow perturbation. Blue lines are computed from the exact Eq. (\[eq:fp2\]) or equivalently from Eq. (\[eq:fp\_unperturb\]) to which it reduces for sufficiently small $g_p'$. Because of the rather explicit appearance of the interaction potential in this formula, we label the blue lines in Fig. (\[fig:inertial\_force\]) as ‘potential force’. High frequency waves arrive before low-frequency ones, because its larger sound speed. We also see how the frequencies become Doppler-shifted for increasing $V_p$. We have derived in Sect. \[sec:inertial\] several approximate expressions for the inertial force. First, Eq. (\[eq:F0full\]) is obtained with the sole assumption (besides $g_p$ sufficiently small) of smallness of the unsteady and/or inhomogeneous part $\delta\psi_0$ of the wavefunction, which allows linearization. Eq. (\[eq:Fi\_th\]) assumes in addition weak inhomogeneities below scales $a$ and $\xi$, and finally Eqs. (\[eq:InertialSimpleDimless\]) and (\[eq:InertialSimpleDim\]) (equivalent under the previous linearization approximation) completely neglects such inhomogeneities (or equivalently, they correspond to $a,\xi\rightarrow 0$). We show as black lines in Fig. (\[fig:inertial\_force\]) the prediction of this last approximation, similar to the most standard classical expressions. Since we have computed the wavefunction $\psi=1+\delta\psi_0$ in the comoving frame from Eq. (\[eq:ComovingdGPE\]), we actually use expression (\[eq:Fi\_th\]) without the Faxén Laplacian terms, with $\delta{\boldsymbol{\omega}}^{(0)}=\nabla(\delta \psi_0-\delta
\psi_0^*)/(2i) - {\boldsymbol{V}}_p$, and ${\boldsymbol{\dot V_p}}=0$. Fig. (\[eq:ComovingdGPE\]) shows that the full force computed from Eq. (\[eq:fp2\]) is well-captured by the approximate expression of the inertial force for small test-particle size $a'$. Accuracy progressively deteriorates for increasing $a'$, and also for increasing $V_p$, but this classical expression remains a reasonable approximation until $a'\approx 1$. The accuracy can be improved by considering higher-order Faxén corrections, Eq. (\[eq:Fi\_th\]), or even better, by considering the integral form in Eq. (\[eq:F0full\]). We have explicitly checked that keeping the full Gaussian integration in Eq. (\[eq:F0full\]) but approximating the integrand in the Bessel integral by its value at the particle position gives a very good approximation to the exact force even for $a'=1$.
Numerical evaluation of the drag force {#sec:numerical-drag}
--------------------------------------
![Plot of the drag force in the steady-state regime as function of the constant speed $V_p$. Dashed lines are the analytical predictions based on Eq. (\[eq:forcegamma0\]) (for $\gamma=0$) and Eq. (\[eq:force\_int\]) (for $\gamma>0$). The symbols correspond to the numerically computed force from Eq. (\[eq:fp2\]) based on direct simulations of the dGPE Eq. (\[eq:ComovingdGPE\]). The inset figure shows the small $V_p$ behavior, with solid straight lines giving the linear dependence of the drag force on the speed for $\gamma>0$, in the small-$V_p$ approximation given by Eq. (\[eq:drag\_coeff\_full\]). We use $a=1$ and $g_p=0.01$.[]{data-label="fig:force_drag"}](Figure_3.pdf){width="50.00000%"}
We now return to the situation in which there is a single impurity in the system, with size $a=\xi=1$ and $g_p=0.01$. It moves in the positive $x$ direction with speed $V_p$ producing a perturbation on the uniform and steady condensate state $\psi=1$. We compute it in the comoving frame, in which the particle is at rest and fluid moves with speed $-V_p$, by using Eq. (\[eq:ComovingdGPE\]). Since in the absence of the impurity there is no inhomogeneity nor time dependence, $\delta\psi_0=0$ and the exact force on the impurity, Eq. (\[eq:fp2\]), is also given by the self-induced drag expression given by Eq. (\[eq:density\_perturb\]). After a transient, that in analogy with the results for compressible classical fluids [@Longhorn1952; @parmar2012equation] we expect to be of the order of the time needed by the sound waves to cross a region of size $a$ or $\xi$, the condensate density near the particle achieves a steady state in the comoving frame, and we then measure the steady drag on the particle. Figure (\[fig:force\_drag\]) shows this force, for several values of $V_p$ and $\gamma$, as dots. The approximate value of the drag force that is obtained under the assumption of small perturbation (small $g_p$) that allows linearization is shown as dashed lines. It is computed from Eq. (\[eq:forcegamma0\]) for $\gamma=0$ and Eq. (\[eq:force\_int\]) for $\gamma>0$. the agreement is excellent. As shown in the inset figure, in the regime of small velocities, the self-induced drag is indeed linearly dependent on the speed with an effective drag coefficient that is well captured by Eq. (\[eq:drag\_coeff\_full\]). This Stokes-like drag at small speeds is due to the thermal drag of the condensate by the normal fluid, quantified by $\gamma$. We notice that the dependence of the drag force on $V_p$ is consistent with having a critical velocity for superfluidity even at $\gamma>0$, in the sense that there is still a relatively abrupt change in the force (sharper for smaller $\gamma$) around a particular impurity speed. The superfluidity of BECs at finite temperature is still an open question. Recent experiments [@sing2016probingsuperfluidity; @Weimer2015_criticavelocity] report superfluid below a critical velocity which is related to the onset of fringes [@wouters2010]. In the dGPE, the steady state drag is always nonzero. Nonetheless, there is a critical velocity above which acoustic emission significatively increases that can be associated to the breakdown of superfluidity. This is the regime where the drag force is dominated by the interaction of the impurity with the supersonic shock waves that are reminiscent of the Kelvin wake in classical fluids. This is seen in Fig. \[fig:density\_gamma\](c) and observed experimentally [@carusotto2006bogoliubov]. The maximum drag force occurs near the velocity for which the cusp lines forming the wake still retain an angle close to $\pi$. With increasing speed, this angle becomes more acute (Fig. \[fig:density\_gamma\](f)), and this lowers the density gradient around the impurity.
Conclusions
===========
We have studied, from analytic and numerical analysis of the dGPE, the hydrodynamic forces acting on a small moving impurity suspended in a 2D BEC at finite temperature. In the regime of small coupling constant $g_p$ and thermal drag $\gamma$, the force arising from the gradient of the condensate density can be decomposed onto the inertial force that is produced by the inhomogeneities and time-dependence of the condensate in the absence of the particle, and the self-induced force which is determined by the perturbation produced by the impurity on the condensate. When the unperturbed flow can be considered homogeneous on scales below the particle size and the condensate coherence length, the classical Maxey and Riley expression [@maxey1983equation], giving the inertial force in terms of the local or material fluid acceleration, is a good description of the force. When inhomogeneities become relevant below these scales, Faxén-type corrections arise, similar to the classical ones in the presence of a finite-size particle, but here the coherence length plays a role similar to the particle size. In addition, the condensate thermal drag enters into these expressions, at difference with the classical viscous case. We also determined the self-induced force in the steady-state regime and shown that it is non-zero at any velocity $V_p$ of the moving impurity if $\gamma>0$. For small $V_p$, this force is given as a Stokes drag which is linearly proportional to $V_p$ with a drag coefficient dependent on the thermal drag $\gamma$. With increasing velocities, there are corrections to the linear drag and above a critical speed of the order of $V_p=c=1$, the self-induced drag is dominated by the interactions of the impurity with the emitted shock waves. Following the recent experimental progress on testing the superfluidity in BEC at finite temperature [@singh2016probing], it would be interesting to test experimentally our prediction of the linear drag on the impurity due to the condensate thermal drag at small velocities by using measurements of the local heating rate.
We have checked our analytical expressions with numerical simulations in the situation in which the impurity moves at constant velocity, possibly driven by external forces different from the hydrodynamic ones analyzed here. When the coupling constant $g_p$ is sufficiently small so that only the inertial force is relevant, the equation of motion of the impurity under the sole action of the inertial force would be $m_p d{\boldsymbol{V}}_p(t)/dt={\boldsymbol{F}}^{(0)}(t)$, with $m_p$ the mass of the particle and ${\boldsymbol{F}}^{(0)}(t)$ one of the suitable approximations to the inertial force given in Sect. \[sec:inertial\]. For larger $g_p$, when the condensate becomes distorted by the impurity, we have computed the self-induced drag only in the steady case, so that can not write a general equation of motion for the impurity in interaction with a time-dependent perturbed flow. In analogy with classical compressible flows [@Longhorn1952; @parmar2012equation], we expect history-dependent forces in this unsteady situation. The dependence on the thermal drag, however, would be quite different from that of viscous classical fluids, because of the lack of viscous boundary layers in the BEC case.
In this study, we have focused on a small impurity that can only shed acoustic waves. Another interesting extension of this would be to further investigate the drag and inertial forces for larger impurity sizes, which can emit vortices, and study the effect of vortex-impurity interactions on the hydrodynamics forces.
Acknowledgements {#acknowledgements .unnumbered}
================
We are thankful to Vidar Skogvoll, Kristian Olsen, Zakarias Laberg Hejlesen and Per Arne Rikvold for stimulating discussions. This work was partly supported by the Research Council of Norway through its centers of Excellence funding scheme, Project No. 262644, and by Spanish MINECO/AEI/FEDER through the María de Maeztu Program for Units of Excellence in R&D (MDM- 2017-0711).
Appendix: Numerical integration of dGPE {#appendix-numerical-integration-of-dgpe .unnumbered}
=======================================
![Simulation domain showing the buffer region, outside the main simulation region, in which thermal drag is greatly enhanced to eliminate the emitted waves sufficiently far from the moving particle (which is at $x/\xi=128$, $y/\xi=64$). The density shown is the steady state (in the comoving frame, hence the direction of the arrows indicating the flow velocity in this frame) for $V_p=1.6$ and $\gamma=0$. []{data-label="fig:BufferRegion"}](Figure_4.pdf){width="45.00000%"}
Numerical simulations of dGPE Eq. (\[eq:ComovingdGPE\]) are run for a system size of $128\times 256$ (in units of $\xi$) corresponding to the grid size $dx=0.25\xi$, and $dt=0.01\xi/c$. To simulate an infinite domain where the density variations emitted by the impurity do not recirculate under periodic boundary conditions, we use the fringe method from [@reeves2015identifying]. This means that we define buffer (fringe) regions around the outer rim of the computational domain (see Fig. \[fig:BufferRegion\]) where the thermal drag $\gamma$ is much larger than its value inside the domain, such that any density perturbation far from the impurity is quickly damped out and a steady inflow is maintained. The thermal drag becomes thus spatially-dependent and given by $\gamma({\boldsymbol{r}}) = \max[\gamma(x),\gamma(y)]$, where $$\begin{aligned}
&\gamma(x)= \frac{1}{2}\big(2 + \tanh{[(x-x_p-w_x)/d]}\nonumber\\
&-\tanh{[(x-x_p+w_x)/d]}\big) + \gamma_0,\end{aligned}$$ and similarly for $\gamma(y)$. Here ${\boldsymbol{r}}_p = (x_p,y_p)=
(128\xi,64\xi)$ is the position of the impurity and $\gamma_0$ is the constant thermal drag inside the buffer regions (bulk region). We set the fringe domain as $w_x=100\xi$, $w_y=50\xi$ and $d=7\xi$ as illustrated in Figure \[fig:BufferRegion\].
By separating the linear and non-linear terms in Eq. (\[eq:ComovingdGPE\]), we can write the dGPE formally as [@audunsthesis] $$\partial_t \psi = \hat\omega(-i\nabla)\psi + N({\boldsymbol{r}},t),$$ where $\hat\omega(-i\nabla) = i[\frac 1 2 \nabla^2+1]+{\boldsymbol{V}}_p\cdot \nabla$ is the linear differential operator and $N({\boldsymbol{r}},t)=-(i+\gamma)(\mathcal U_p+|\psi|^2)\psi + \gamma
\psi +\frac{1}{2}\gamma\nabla^2\psi$ is the nonlinear function including the spatially-dependent $\gamma$ and $\mathcal U_p$. Taking the Fourier transform, we obtain ordinary differential equations for Fourier coefficients $\psi({\boldsymbol{k}},t)$ as $$\partial_t\hat\psi({\boldsymbol{k}},t) = \hat\omega({\boldsymbol{k}})\hat\psi({\boldsymbol{k}}, t) + \hat N({\boldsymbol{k}} ,t),
\label{eq:DecompositionGPE}$$ which can be solved by an operator-splitting and exponential-time differentiating method [@cox2002exponential]. It means that we exploit the fact that the linear part of Eq. (\[eq:DecompositionGPE\]) can be solved exactly by multiplying with the integrating factor $e^{-\hat\omega({\boldsymbol{k}})t}$. This leads to $$\partial_t \left(\hat\psi({\boldsymbol{k}},t) e^{-\hat\omega({\boldsymbol{k}})t}\right) = e^{-\hat\omega({\boldsymbol{k}})t}\hat N({\boldsymbol{k}}, t).
\label{eq:ETD}$$ The nonlinear term $\hat N({\boldsymbol{k}},t)$ is linearly approximated in time for a small time-interval $(t,t+\Delta t)$, i.e $$\hat N({\boldsymbol{k}}, t+\tau) = N_0 + \frac{N_1}{\Delta t}\tau$$ where $N_0 = \hat N(t)$ and $N_1 = \hat N(t+\Delta t) -N_0$. Inserting this into Eq. (\[eq:ETD\]) and integrating from $t$ to $t+\Delta t$ we get $$\begin{aligned}
&\hat{\psi}({\boldsymbol{k}},t+\Delta t) = \hat{\psi}({\boldsymbol{k}},t) e^{\hat{\omega}({\boldsymbol{k}})\Delta t} + \frac{N_0}{\hat\omega({\boldsymbol{k}})}\left(e^{\hat \omega({\boldsymbol{k}}) \Delta t} -1\right) \nonumber\\
&+ \frac{N_1}{\hat\omega({\boldsymbol{k}})} \left[\frac{1}{\hat\omega({\boldsymbol{k}}) \Delta t}(e^{\hat\omega({\boldsymbol{k}}) \Delta t} -1) -1 \right].\end{aligned}$$ Since computing the value of $N_1$ requires knowledge of the state at $t+\Delta t$ before we have computed it, we start by setting it to zero and find a value for the state at $t +\Delta
t$ given that $\hat N(t)$ is constant in the interval. We then use this state to calculate $N_1$, and add corrections to the value we got when assuming $N_1=0$.
| {
"pile_set_name": "ArXiv"
} |
---
address: |
Institute of Field Physics, Department of Physics and Astronomy,\
University of North Carolina, Chapel Hill, NC 27599-3255, USA\
E-mail: ng@physics.unc.edu
author:
- 'Y. JACK NG'
title: MAGNETIC CATALYSIS OF CHIRAL SYMMETRY BREAKING AND THE PAULI PROBLEM
---
=cmr8
1.5pt
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Let me begin with a joke which some of you may have heard before. One of Wolfgang Pauli’s life-long dreams was to understand why the fine structure constant in electrodynamics is 1/137 (in the infrared regime). Pauli was also known to be a difficult person, very hard to please. As the joke goes, the first thing Pauli asked God after his death was to explain why $\alpha$ = 1/137. As God went on with His explanation, Pauli grew more and more dissatisfied. After five minutes, Pauli was seen storming out of Heaven’s Gates mumbling, “Ridiculous!”
Like Pauli, I also would like to understand why $\alpha$ = 1/137. To dignify this problem, I will call it the Pauli problem. It is possible that chiral symmetry breaking by an external field in QED may provide some insight on this old problem by giving a critical value of $\alpha$ close to 1/137.[@JBW] Admittedly, nothing close to that magic value has arisen in the results we have obtained so far [@Lee], but our analysis is not yet complete.
My interest in chiral symmetry breaking by an external field dates back a dozen years ago, to the time when a multiple correlated and narrow-peak structures in electron and positron spectra[@GSI] was observed in heavy-ion experiments at GSI. Kikuchi and I [@newphase] interpreted the $e^+e^-$ peaks as decay products of a new type of positronium, which is formed in a new QED phase induced by the electromagnetic fields of the colliding heavy ions. The theoretical underpinning of this scenario was provided by earlier works [@Mir] which indicated that QED might have a non-perturbative strong-coupling phase, characterized by spontaneous chiral symmetry breaking, in addition to the familar weak-coupling phase. The negative results in recent heavy-ion collision experiments at Argonne [@Argonne] have rendered our interpretation moot. Nevertheless, the problem of chiral symmetry breaking by an external field is still interesting as it may shed light on the Pauli problem, and as it provides an example of vacuum engineering by manipulating external fields to alter the symmetry properties of the vacuum. But more concretely, our study provides a new non-perturbative phenomenon in (3+1)-dimensional quantum field theories and a new method to study it.
First, what kind of external fields can induce chiral symmetry breaking in gauge theories?[@Ng] Lessons gained from studying the Nambu-Jona-Lasinio model [@KL] lead us to believe that uniform magnetic fields are prime candidates. To put our problem in as general a setting as possible, we want an approach that treat both the coupling and the external field non-perturbatively. The former criterion is met by using the Schwinger-Dyson equations (or equivalently, the Nambu-Bethe-Salpeter equations [@GMS]); the latter condition is satisfied by applying the strong-field techniques introduced by Schwinger and others.
Let us start with the motion of a massless fermion of charge $e$ in an external electromagnetic field. It is described by the Green’s function that satisfies the modified Dirac equation proposed by Schwinger: $$\gamma \cdot \Pi(x) G_A(x,y) + \int d^4x' M(x,x') G_A(x',y) =
\delta^{(4)}(x-y),
\label{Greeneq}$$ where $\Pi_\mu(x) = - i \partial_\mu - e A_\mu(x)$, and $M(x,x')$ is the mass operator $M$ in the coordinate representation. For a constant magnetic field of strength $H$, we may take $A_2 = Hx_1$ to be the only nonzero component of $A_\mu$. We adopt the method due to Ritus[@Ritus], which is based on the use of the eigenfunctions of the mass operator and the diagonalization of the latter. As shown by Ritus, $M$ is diagonal in the representation of the eigenfunctions $E_p(x)$ of the operator $(\gamma \cdot \Pi)^2$: $$- (\gamma \cdot \Pi)^2 E_p(x) = p^2 E_p(x).
\label{eigeneq}$$ The advantage of using this representation is obvious: $M$ can now be put in terms of its eigenvalues, so the problems arising from its dependence on the operator $\Pi$ can be avoided. In the chiral representation in which $\sigma_3$ and $\gamma_5$ are diagonal with eigenvalues $\sigma = \pm 1$ and $\chi = \pm 1$, respectively, the eigenfunctions $E_{p\sigma\chi}(x)$ take the form $$E_{p\sigma\chi}(x) = N {\rm e}^{i (p_0x^0 + p_2x^2 + p_3x^3)} D_n(\rho)
\omega_{\sigma\chi} \equiv \tilde{E}_{p\sigma\chi} \omega_{\sigma\chi},
\label{eigenfcn}$$ where $D_n(\rho)$ are the parabolic cylinder functions with indices $$n = n(k,\sigma) \equiv k + \frac{e H \sigma}{2 |e H|} - \frac{1}{2},
~~~~k = 0, 1, 2, ...,
\label{index}$$ and argument $\rho = \sqrt{2 |e H|} (x_1 - \frac{p_2}{e H})$. Note that $n = 0,~1,~2,~...~$. The normalization factor is $N = (4 \pi |eH|)^{1/4}/\sqrt{n!}$; $p$ stands for the set $(p_0, p_2, p_3, k)$; and $\omega_{\sigma\chi}$ are the bispinors of $\sigma_3$ and $\gamma_5$.
Following Ritus, we form the orthonormal and complete eigenfunction-matrices $E_p = {\rm diag}(\tilde{E}_{p11},~
\tilde{E}_{p-11},~\tilde{E}_{p1-1},~\tilde{E}_{p-1-1})$. They satisfy $$\gamma \cdot \Pi~E_p(x) = E_p(x)~\gamma \cdot \bar{p}$$ and $$M(x,x') E_p(x') = E_p(x) \delta^{(4)}(x-x') {\Sigma}_A(\bar{p}),
\label{masseigeneq}$$ where ${\Sigma}_A(\bar{p})$ represents the eigenvalues of the mass operator, and $\bar{p}_0 = p_0,~\bar{p}_1 = 0,~\bar{p}_2
= - {\rm sgn}(eH) \sqrt{2|eH|k},~\bar{p}_3 = p_3$. These properties of the $E_p(x)$ allow us to express the Green’s function in the $E_p$-representation as $(\bar{E}_p \equiv \gamma^0 E_p^\dagger \gamma^0)$ $$G_A(x,y) = \Sigma \!\!\!\!\!\! \int \frac{d^4p}{(2 \pi)^4} E_p(x) \frac{1}
{\gamma \cdot \bar{p} + {\Sigma}_A(\bar{p})} \bar{E}_p(y),
~~~\Sigma \!\!\!\!\!\! \int d^4p \equiv \sum_{k} \int dp_0 dp_2 dp_3.
\label{Greenfcn}$$
We work in the ladder quenched approximation. In terms of the notations: $\bar{p"}_{\!\!\!\!_0} = p_0 - q_0$, $\bar{p"}_{\!\!\!\!_1} = 0$, $\bar{p"}_{\!\!\!\!_2} = -~{\rm sgn}(eH) \sqrt{2|eH|k"}$, $\bar{p"}_{\!\!\!\!_3} = p_3 - q_3$, the Schwinger-Dyson equation takes the form $$\Sigma_A(\bar{p}) \simeq \frac{i e^2}{(2 \pi)^3} |eH| \int dq_0 dq_3
\int_0^\infty dr^2 {\rm e}^{- r^2} \frac{-2}{q^2} \frac
{\Sigma_A(\bar{p"})}{\bar{p"}^2 + \Sigma_A(\bar{p"})}
\label{fermass}$$ where $q^2 = - q_0^2 + q_3^2 + 2 |eH| r^2$ and $\bar{p"}^2 =
- (p_0 - q_0)^2 + (p_3 - q_3)^2 + 2 |eH| k"$, and where summation over $k'' = k, ~k \pm 1$ is understood. We have assumed that, due to the factor e$^{- r^2}$ in the integrand, contributions from large values of $r$ are suppressed.
Let us make a Wick rotation to Euclidean space: $p_0 \rightarrow i p_4$, $q_0 \rightarrow i q_4$, and consider the case with $p = 0$, i.e., $p_0 = p_3
= k = 0$. We assume that the dominant contributions to the integral in Eq.(\[fermass\]) come from the infrared region of small $q_3$ and $q_4$, and that the $k'' = 0$ term dominates over the $k'' = 1$ term (this assumption has been checked to be self-consistent). Then, it is reasonable to replace $\Sigma_A(\bar{p"})$ in the integrand by $\Sigma_A(0) = m \times
{\bf 1}$. The SD equation yields the nonzero dynamical mass as $$m \simeq a~\sqrt{|eH|}~{\rm e}^{- b\sqrt{\frac{\pi}{\alpha}}},
\label{result}$$ where $a$ and $b$ are constants of order 1.
Eq.(\[result\]) clearly demonstrates the nonperturbative nature of the result. It also shows that our approximations break down when $\alpha > O(1)$. Our earlier assumption that effectively $r \ll 1$ is translated to the physical assumption that $m/\sqrt{|eH|} \ll 1$, which requires that $\alpha \ll O(1)$; in other words, the dynamical chiral symmetry breaking solution we have found applies to the weak-coupling regime of QED! We have checked that indeed the infrared region of $q_3$ and $q_4$ gives the dominant contributions to the integrals.
To establish that the above solution to the SD equation for the fermion self-energy does indeed correspond to a dynamical chiral symmetry-breaking solution, it is necessary to demonstrate the existence of the corresponding Nambu-Goldstone boson. One way to establish this is by studying the Nambu-Bethe-Salpeter equation of the bound-state NG boson, as was done by Gusynin [*et al.*]{} [@GMS], who found a solution consistent with our Eq.(\[result\]). As a consistency check on our approach, we [@Lee] have recovered the same result from the NBS equation, using the $E_p$-representation of the fermion propagator. The chiral condensate, the order parameter for dynamical symmetry breaking, can be easily computed, based on our formalism. We obtain ($\psi$ is the fermion field) $$\langle \bar{\psi} \psi \rangle
~\simeq~
-~\frac{|eH|}{2 \pi^2} ~m ~\ln\left(\frac{|eH|}{m^2}\right).
\label{psibpsi}$$
Thermal effects and the effects of a chemical potential can be readily incorporated into our study of chiral symmetry breaking in an external magnetic field. [@Lee] One finds that chiral symmetry is restored above a critical temperature which is of the order of the dynamical fermion mass (given by Eq.(\[result\])), and the corresponding phase transition is of second order. In contrast, the chiral symmetry restoration above a critical chemical potential (also of the order of the dynamical fermion mass) is a first order phase transition. The chiral condensates for both cases have the same form as given above in Eq.(\[psibpsi\]).
To summarize, we have shown that chiral symmetry is dynamically broken in quenched, ladder QED when an external magnetic field is present. So, the external magnetic field acts as a catalysis for chiral symmetry breaking. Furthermore, in the lowest Landau level approximation, this chiral symmetry breaking is generated in the infrared region where the QED gauge coupling is weak. As pointed out by Gusynin [*et al.*]{} [@GMS], the dynamics of LLL is (4 minus 2)-dimensional \[see the second propagator factor with $k'' = 0$ in Eq.(\[fermass\])\]. But this effective dimensional reduction is only for charged states. Propagators for neutral states (like the photon and NG boson) are still 4-dimensional \[see the first propagator factor in Eq.(\[fermass\])\]. Thus the Mermin-Wagner-Coleman theorem (which stipulates that there can be no spontaneous breakdown of continuous symmetries in dimensions less than three) is successfully evaded by our chiral symmetry breaking solution. In fact, it is the interplay between the 2-dimensional dynamics of the charged particles (in the LLL approxiamtion) and the 4-dimensional dynamics of the neutral particles that is partly responsible for some of the characteristics of our chiral symmetry-breaking solution. For the magnetic catalysis to take place, the Landau energy $(\sqrt{|eH|})$ must be much greater than the fermion mass. If the fermion mass is taken to be zero, the critical magnetic field is zero. Thus chiral symmetry is broken at an arbitrarily weak magnetic field. In the LLL approximation, the fermion pairing dynamics responsible for chiral condensates is (1 + 1)-dimensional, and is hence strong in the infrared region. Thus chiral symmetry breaking takes place even at the weakest attractive interactions between the fermions, yielding a zero critical gauge coupling.
As for applications, chiral symmetry breaking by an external magnetic field may be relevant to condensed matter systems (e.g., in quantum Hall effect and in superconductivity [@KM]). It may also find applications in astrophysics (e.g., in the cooling process of neutron stars). It has been suggested in the literature [@GMS] that the chiral symmetry-breaking solution may play a role in the electroweak phase transition during the early evolution of the Universe since a huge magnetic field (with the Landau energy of the order of electroweak breaking scale) was purportedly generated by the phase transition. Unfortunately, the electroweak phase transition took place at a temperature also of the order of the electroweak scale, which is much higher than the critical temperature for a magnetic field of such a magnitude. Therefore, it is very unlikely that such a magnetic field could change the character of the electroweak phase transition in any way. But, if (for reasons not yet understood) magnetic fields of such extraordinarily large magnitudes as to overwhelm the thermal effects were generated in the early epoch, then the chiral symmetry-breaking solution reported here could be very relevant to the early evolution of the Universe.
There remain several interesting questions which we intend to investigate. For instance, how do other background field configurations affect chiral symmetry breaking? (As an example, it is expected that an electric field would tend to break up the condensate and destabilize the vacuum, thus inhibiting chiral symmetry breaking [@DW]). Are there strong-coupling solutions of chiral symmetry breaking in an external magnetic field? If so, are the weak-coupling solutions we have found and the new strong-coupling solutions related by some kind of duality? How does one extend the chiral symmetry-breaking results to inhomogenous configurations? Can one understand the chiral symmetry breaking by a simple (perhaps topological) argument? [@Sem]
But, perhaps the most urgent task is to have a better understanding of this phenomenon of chiral symmetry breaking by a magnetic field. After all, if dynamical symmetry breaking occurs when particles interact strongly as in QCD, then how can [*weak*]{} electromagnetic interactions in an arbitrarily [*weak*]{} magnetic field lead to chiral symmetry breaking? Above, we have given a partial answer to this question by arguing that the dimensional reduction in the LLL approximation effectively enhances such weak interactions in the infrared region. But can the LLL approximation be the whole story?
For a better perspective, it is useful to recall that in strong-coupling QED [@Mir], where chiral symmetry is broken, the anomalous dimension of the fermion mass operator is one, thus four-fermion interactions are relevant operators. In fact, it is the dynamical running of the four-fermion couplings that leads to a finite dynamical fermion mass, and hence an infrared-meaningful theory in the continuum limit. Now, returning to the phenomenon of magnetic catalysis, one is naturally led to ask: what are the effects of four-fermion interactions in the chiral symmetry breaking triggered by an external magnetic field? A recent work by Hong [@Hong] suggests that quantum effects of fermions in higher Landau levels induce marginal four-fermion couplings (along with other interactions). In turn, the four-fermion interactions give rise to chiral symmetry breaking when the external magnetic field is super strong. Hong’s analysis, though incomplete, is intriguing. It hints at a non-zero critical magnetic field in the magnetic catalysis of chiral symmetry breaking. The next question to ask is this: Will a complete analysis of this phenomenon yield a non-zero critical gauge coupling as well as a non-zero critical magnetic field? Indications from the analysis of strong-coupling QED are encouraging: in strong-coupling QED, supplemented by the appropriate four-fermion interactions, both the critical gauge coupling and the critical four-fermion coupling are non-zero. This brings us back to the question posed at the beginning of this talk: Does chiral symmetry breaking by an external field shed light on the Pauli problem by giving a critical value of $\alpha$ close to 1/137?
Acknowledgements {#acknowledgements .unnumbered}
================
This talk is based on works done in collaboration with C.N. Leung, D.-S. Lee, and A. Ackley, supported in part by the U.S. Department of Energy under Grant No. DE-FG05-85ER-40219 Task A.
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Y.J. Ng and Y. Kikuchi, Phys. Rev. D [**36**]{}, 2880 (1987). See also D.G. Caldi and A. Chodos, Phys. Rev. D [**36**]{}, 2876 (1987); L.S. Celenza et. al., Phys. Rev. Lett. [**57**]{}, 55 (1986).
T. Maskawa and H. Nakajima, Prog. Theor. Phys. [**52**]{}, 1326 (1974); R. Fukuda and T. Kugo, Nucl. Phys. [**B117**]{}, 250 (1976); V.A. Miransky, Il. Nuovo Cim. [**90A**]{}, 149 (1985); W.A. Bardeen, C.N. Leung, and S.T. Love, Nucl. Phys. [**B273**]{}, 649 (1986); [*ibid*]{}., [**B323**]{}, 493 (1989); K. Yamawaki, M. Bando, and K. Matumoto, Phys. Rev. Lett. [**56**]{}, 1335 (1986); J.B. Kogut, E. Dagotto, and A. Kocic, Phys. Rev. Lett. [**60**]{}, 772 (1988).
I. Ahmad [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 2658 (1995).
See, e.g., contribution of Y.J. Ng and Y. Kikuchi in [*Vacuum Structure in Intense Fields*]{}, eds. H.M. Fried and B. Muller (Plenum, New York, 1991), and references therein.
S.P. Klevansky and R.H. Lemmer, Phys. Rev. D [**39**]{}, 3478 (1989).
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Sok Jérémy\
Ceremade, UMR 7534, Université Paris-Dauphine,\
Place du Maréchal de Lattre de Tassigny,\
75775 Paris Cedex 16, France.\
\
bibliography:
- 'bibliothese.bib'
title: '**The positronium and the dipositronium in a Hartee-Fock approximation of quantum electrodynamics**'
---
Introduction and main results
=============================
The Dirac operator
------------------
Relativistic quantum mechanics is based on the *Dirac operator* $D_0$, which is the Hamiltonian of the free electron. Its expression is [@Th]: $$\label{di_dirac_op}
D_0:=m_ec^2\beta-i\hbar c{\ensuremath{\displaystyle\sum}}_{j=1}^3\alpha_j \partial_{x_j}$$ where $m_e$ is the (bare) mass of the electron, $c$ the speed of light and $\hbar$ the reduced Planck constant and $\beta$ and the $\alpha_j$’s are $4\times 4$ matrices defined as follows: $$\label{di_beta_alpha}
\beta:=\begin{pmatrix}
\mathrm{Id}_{\mathbb{C}^2} & 0\\ 0 & -\mathrm{Id}_{\mathbb{C}^2}
\end{pmatrix},\ \alpha_j:= \begin{pmatrix}
0 & \sigma_j \\ \sigma_j & 0
\end{pmatrix},\ j\in\{1,2,3\}$$ $$\sigma_1:=\begin{pmatrix}
0 & 1\\ 1 & 0
\end{pmatrix},\ \sigma_2:=\begin{pmatrix}
0 & -i\\ i & 0
\end{pmatrix},\ \sigma_3\begin{pmatrix}
1 & 0 \\ -1 & 0
\end{pmatrix}.$$ The operator $D_0$ acts on the Hilbert space $ \mathfrak{H}$: $$\label{di_space_one_electron}
\mathfrak{H}:=L^2\big({\ensuremath{\mathbb{R}^3}},{\ensuremath{\mathbb{C}^4}}\big);$$it is self-adjoint on $\mathfrak{H}$ with domain $H^1({\ensuremath{\mathbb{R}^3}},{\ensuremath{\mathbb{C}^4}})$. Its spectrum is $\sigma(D_0)=(-\infty,m_ec^2]\cup[m_e c^2,+\infty)$, which leads to the existence of states with arbitrary small energy. Dirac postulated that all the negative energy states are already occupied by “virtual electrons”, with one electron in each state: by Pauli’s principle real electrons can only have a positive energy. In this interpretation the Dirac sea, composed by those negatively charged virtual electrons, constitutes a polarizable medium that reacts to the presence of an external field. This phenomenon is called the *vacuum polarization*.
After the transition of an electron of the Dirac sea from a negative energy state to a positive, there is a real electron with positive energy plus the absence of an electron in the Dirac sea. This hole can be interpreted as the addition of a particle with same mass, but opposite charge: the so-called positron. The existence of this particle was predicted by Dirac in 1931. Although firstly observed in 1929 independently by Skobeltsyn and Chung-Yao Chao, it was recognized in an experiment lead by Anderson in 1932.
Positronium and dipositronium
-----------------------------
The positronium is the bound state of an electron and a positron. This system was independently predicted by Anderson and Mohorovi$\check{\mathrm{c}}$ić in 1932 and 1934 and was experimentally observed for the first time in 1951 by Martin Deutsch.
It is unstable: depending on the relative spin states of the positron and electron, its average lifetime in vacuum is 125 ps (para-positronium) or 142 ns (ortho-positronium) [@karsh].
Here we are interested in positronium states in the Bogoliubov-Dirac-Fock (BDF) model.
In a previous paper we have proved the existence of a state that can be interpreted as the ortho-positronium. Our aim in this paper is to find another one that can be interpreted as the para-positronium and to find another state that can be interpreted as the dipositronium, the bound state of two electrons and two positrons. To find these states, we use symmetric properties of the Dirac operator.
Symmetries
----------
– Following Dirac’s ideas, the free vacuum is described by the negative part of the spectrum $\sigma(D_0)$: $$P^0_-=\chi_{(-\infty,0)}(D_0).$$ A correspondence between negative energy states and positron states is given by the *charge conjugation* ${\ensuremath{\mathrm{C}}}$ [@Th]. This is an antiunitary operator that maps $\mathrm{Ran}\,P^0_{-}$ onto $\mathrm{Ran}(1-P^0_{-})$. In our convention [@Th] it is defined by the formula: $$\label{di_chargeconj}
\forall\,\psi\in L^2({\ensuremath{\mathbb{R}^3}}),\ {\ensuremath{\mathrm{C}}}\psi(x)=i\beta\alpha_2\overline{\psi}(x),$$ where $\overline{\psi}$ denotes the usual complex conjugation. More precisely: $$\label{di_chargeconjprec}
{\ensuremath{\mathrm{C}}}\cdot \begin{pmatrix}\psi_1\\ \psi_2\\ \psi_2\\\psi_4\end{pmatrix}=\begin{pmatrix}\overline{\psi}_4\\ -\overline{\psi}_3\\ -\overline{\psi}_2\\\overline{\psi}_1\end{pmatrix}.$$ In our convention it is also an *involution*: ${\ensuremath{\mathrm{C}}}^2=\text{id}$. An important property is the following: $$\label{di_denspsi}
\forall\,\psi\in\,L^2,\forall\,x\in\mathbb{R}^3,\ |{\ensuremath{\mathrm{C}}}\psi(x)|^2=|\psi(x)|^2.$$ The Dirac operator anti-commutes with $D_0$, or equivalently there holds $$-{\ensuremath{\mathrm{C}}}D_0 {\ensuremath{\mathrm{C}}}^{-1}=-{\ensuremath{\mathrm{C}}}D_0{\ensuremath{\mathrm{C}}}=D_0.$$
– There exists another simple symmetry. We define $$\label{di_Isym}
{\ensuremath{\mathrm{I}_{\mathrm{s}}}}:=\begin{pmatrix}0 & -\mathrm{Id}_{\mathbb{C}^2}\\
\mathrm{Id}_{\mathbb{C}^2}& 0 \end{pmatrix}\in\mathbb{C}^{4\times 4}.$$ This operator is $-i$ the *time reversal operator* $\text{L}_T$ [@Th 2.5.7] in $\mathfrak{H}$, interpreted as a unitary reprsentation of the Poincar[é]{} group.
It acts on the spinor by simple multiplication, furthermore we have ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}^2=-\mathrm{Id}$ and $${\ensuremath{\mathrm{I}_{\mathrm{s}}}}:\begin{array}{rcl}
\mathrm{Ran}\,P^0_-&\overset{\simeq}{\longrightarrow}& \mathrm{Ran}\,(1-P^0_-)\\
\psi(x)&\mapsto& {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi(x)
\end{array}$$ Similarly we have $ -{\ensuremath{\mathrm{I}_{\mathrm{s}}}}D_0 {\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}D_0 {\ensuremath{\mathrm{I}_{\mathrm{s}}}}= D_0.$
– To end this part we recall that $\mathbf{SU}(2)$ acts on $\mathfrak{H}$ [@Th]. Writing $\boldsymbol{\alpha}:=(\alpha_j)_{j=1}^3$ and $$\label{di_L,S}
\mathbf{p}:=-i\hbar\nabla,\ {\ensuremath{\mathbf{L}}}:=\mathbf{x}\wedge \mathbf{p},\ {\ensuremath{\mathbf{S}}}:=-\frac{i}{4}\boldsymbol{\alpha}\wedge \boldsymbol{\alpha}=\frac{1}{2}\begin{pmatrix}\boldsymbol{\sigma}&0\\ 0&\boldsymbol{\sigma} \end{pmatrix},$$ we define $$\label{di_J_moment}
{\ensuremath{\mathbf{J}}}:={\ensuremath{\mathbf{L}}}+{\ensuremath{\mathbf{S}}}.$$ The operator $\mathbf{L}$ is the angular momentum operator and $\mathbf{J}$ is the total angular momentum. From a geometrical point of view, $-i\mathbf{J}$ gives rise to a unitary representation of $\mathbf{SU}(2)$ in $\mathfrak{H}$ by the following formula: $$\left\{\begin{array}{l}e^{-i\theta\mathbf{J}\cdot{\ensuremath{\omega}}}\psi(x)=e^{-i\mathbf{S}\cdot{\ensuremath{\omega}}}\psi\big( \mathbf{R}^{-1}_{{\ensuremath{\omega}},\theta}\big),\\
\forall\theta\in[0,4\pi),\forall\psi\in\mathfrak{H},\forall{\ensuremath{\omega}}\in\mathbb{S}^2,
\end{array}\right.$$where $\mathbf{R}_{{\ensuremath{\omega}},\theta}\in\mathrm{SO}(3)$ is the rotation with axis ${\ensuremath{\omega}}$ and angle $\theta$.
As each $\mathrm{S}_j$ is diagonal by block, it is clear that this group representation can be decomposed in two representations, the first acting on the upper spinors $\phi\in L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2)$ and the second on the lower spinors $\chi\in L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2)$: $$\psi=:\begin{pmatrix}\phi\\\chi\end{pmatrix}.$$ In [@Th pp. 122-129] it is proved that $D_0$ commutes with the action of $\mathbf{SU}(2)$, thus the representation can also be decomposed with respect to $\mathrm{Ran}\,P^0_-$ and $\mathrm{Ran}\,(1-P^0_-)$.
From an algebraic point of view, there exists a group morphism ${\ensuremath{\Phi_{\mathrm{SU}}}}:\mathbf{SU}(2)\to \mathbf{U}({\ensuremath{\mathfrak{H}_\Lambda}})$ where $\mathbf{U}(\mathfrak{H})$ is the set of unitary operator of $\mathfrak{H}$. We write $$\mathbf{S}:={\ensuremath{\Phi_{\mathrm{SU}}}}\big(\mathbf{SU}(2) \big).$$ The irreducible representations of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ are known and are expressed in terms of eigenspaces of ${\ensuremath{\mathbf{J}}}^2,{\ensuremath{\mathbf{S}}}$. The proofs of the following can be found in [@Th pp. 122-129].
The operators ${\ensuremath{\mathbf{J}}}^2,\mathrm{J}_3,{\ensuremath{\mathbf{K}}}$ all commute with each other, and ${\ensuremath{\mathbf{J}}}^2,{\ensuremath{\mathbf{K}}}$ with $D_0$. Moreover ${\ensuremath{\mathbf{K}}}$ commutes with the action ${\ensuremath{\Phi_{\mathrm{SU}}}}$. We have ${\ensuremath{\mathfrak{H}_\Lambda}}\subset L^2(\mathbb{R}^3)\simeq L^2((0,\infty),dr)\otimes L^2(\mathbb{S}^2)^4$, and ${\ensuremath{\mathbf{J}}}$, ${\ensuremath{\mathbf{L}}}$ only act on the part $L^2(\mathbb{S}^2)^4$.
Restricted to $L^2(\mathbb{S}^2)^4$, we have $$\sigma\,({\ensuremath{\mathbf{J}}}^2)=\big\{j(j+1),\ j\in\frac{1}{2}+\mathbb{Z}_+\big\},$$ and for each eigenvalue $j(j+1)\in \sigma\,{\ensuremath{\mathbf{J}}}^2$, the eigenspace $\mathrm{Ker}\big({\ensuremath{\mathbf{J}}}^2-j(j+1)\big)$ may be decomposed with respect to the eigenspaces of $\mathrm{J}_3$ and ${\ensuremath{\mathbf{S}}}$. The corresponding eigenvalues are
1. $m_j=-j,-j+1,\cdots,j-1,j$ for $\mathrm{J}_3$,
2. $\kappa_j=\pm\big(j+\frac{1}{2}\big)$ for ${\ensuremath{\mathbf{S}}}$.
The eigenspace $\mathfrak{k}_{m_j,\kappa_j}$ of a triplet $(j,m_j,\kappa_j)$ has dimension $2$ and is spanned by $\Phi^+_{m_j,\kappa_j}\perp\Phi^-_{m_j,\kappa_j}$, which have respectively a zero lower spinor and zero upper spinor.
\[di\_irreduc\] For each irreducible subrepresentation $\Phi'_{\mathrm{SU}}$ of ${\ensuremath{\Phi_{\mathrm{SU}}}}$, there exists $$(j,{\ensuremath{\varepsilon}},\mathbf{z}=[z_1:z_2], a_1(r),a_2(r))\in \big(\frac{1}{2}+\mathbb{Z}_+\big)\times\{+,-\}\times\mathbb{C}P^1\times \big(\mathbb{S}L^2((0,\infty),dr)\big)^2,$$ such that the representation $\Phi'_{\mathrm{SU}}$ is spanned by $\psi(x)$ defined as follows: $$\forall x=r{\ensuremath{\omega}}\in{\ensuremath{\mathbb{R}^3}}, \psi(x):=z_1 ra_1(r)\Phi^+_{j, {\ensuremath{\varepsilon}}(j+\tfrac{1}{2})}({\ensuremath{\omega}})+z_2 ra_2(r)\Phi^-_{j,{\ensuremath{\varepsilon}}(j+\tfrac{1}{2})}.$$
\[di\_def\_mathbb\_s\] We recall that for any Hilbert space $\mathfrak{h}$ and any subspace $V\subset \mathfrak{h}$, we define $\mathbb{S}V$ as the unitary vector in $V$: $$\mathbb{S}V:=\{x\in V,\ \lVert x\rVert_{\mathfrak{h}}=1\}.$$ We will use this notation throughout this paper.
We prove this Lemma in Section \[di\_proofmanif\].
\[di\_rep\_type\] An irreducible subrepresentation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ is characterized by the two numbers $(j,\kappa_j)$. Indeed, the irreducible representations of $\mathbf{SU}(2)$ are known: they can be described by homogeneous polynomials, and for any $n\in \mathbb{Z}_+$, there is but one irreducible representation of dimension $n+1$, up to isomorphism.
In the case of ${\ensuremath{\Phi_{\mathrm{SU}}}}$, the two cases $\kappa_j=\pm(j+\tfrac{1}{2})$ are different but *isomorphic*.
An irreducible subrepresentation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ spanned by an eigenvector of ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{K}}}$ with respective eigenvalues $j(j+1)$ and ${\ensuremath{\varepsilon}}(j+\tfrac{1}{2})$ will be refered as beeing of type $(j,{\ensuremath{\varepsilon}})$ (where ${\ensuremath{\varepsilon}}\in\{+,-\}$).
Throughout this paper we write $\text{Proj}\,E$ to mean the orthonormal projection onto the vector space $E$.
The BDF model
-------------
This model is a no-photon approximation of quantum electrodynamics (QED) which was introduced by Chaix and Iracane in 1989 [@CI], and studied in many papers [@stab; @ptf; @Sc; @mf; @at; @gs; @sok].
It allows to take into account the Dirac vacuum together an electronic system in the presence of an external field. This is a Hartree-Fock type approximation in which a state of the system “vacuum plus real electrons” is given by an infinite Slater determinant $\psi_1\wedge\psi_2\wedge \cdots$. Such a state is represented by the projector onto the space spanned by the $\psi_j$’s: its so-called one-body density matrix. For instance $P^0_-$ represents the free Dirac vacuum.
We do not recall the derivation of the BDF model from QED: we refer the reader to [@CI; @ptf; @mf] for full details.
To simplify the notations, we choose relativistic units in which, the mass of the electron $m_e$, the speed of light $c$ and $\hbar$ are set to $1$.
Let us say that there is an external density $\nu$, *e.g.* that of some nucleus. We write $\alpha>0$ the so-called *fine structure constant* (physically $e^2/(4\pi{\ensuremath{\varepsilon}}_0\hbar c)$, where $e$ is the elementary charge and ${\ensuremath{\varepsilon}}_0$ the permittivity of free space).
The relative energy of a Hartree-Fock state represented by its 1pdm $P$ with respect to a state of reference ($P^0_-$ in [@CI; @ptf]) turns out to be a function of $Q=P-P^0_-$, the so-called reduced one-body density matrix. A projector $P$ is the one-body density matrix of a Hartree-Fock state in $\mathcal{F}_{\text{elec}}$ *iff* $P-P^0_-$ is Hilbert-Schmidt, that is compact such that its singular values form a sequence in $\ell^2$ [@ptf Appendix].
An ultraviolet cut-off $\Lambda>0$ is needed: we only consider electronic states in $${\ensuremath{\mathfrak{H}_\Lambda}}:=\big\{ f\in\mathfrak{H},\ \text{supp}\,{\ensuremath{\widehat{f}}}\subset B(0,{\ensuremath{\Lambda}})\big\},$$ where ${\ensuremath{\widehat{f}}}$ is the Fourier transform of $f$.
This procedure gives the BDF energy introduced in [@CI] and studied in [@ptf; @Sc].
Our convention for the Fourier transform $\mathscr{F}$ is the following $$\forall\,f\in L^1({\ensuremath{\mathbb{R}^3}}),\ {\ensuremath{\widehat{f}}}(p):=\frac{1}{(2\pi)^{3/2}}{\ensuremath{\displaystyle\int}}f(x)e^{-ixp}dx.$$
Let us notice that ${\ensuremath{\mathfrak{H}_\Lambda}}$ is invariant under $D_0$ and so under $P^0_-$.
We write $\Pi_{\ensuremath{\Lambda}}$ for the orthogonal projection onto ${\ensuremath{\mathfrak{H}_\Lambda}}$: $\Pi_{\ensuremath{\Lambda}}$ is the Fourier multiplier $\mathscr{F}^{-1}\chi_{B(0,\Lambda)}\mathscr{F}$. By means of a thermodynamical limit, Hainzl *et al.* showed that the formal minimizer and hence the reference state should not be given by $\Pi_{\ensuremath{\Lambda}}P^0_-$ but by another projector ${\ensuremath{\mathcal{P}^0_-}}$ in ${\ensuremath{\mathfrak{H}_\Lambda}}$ that satisfies the self-consistent equation [@mf]: $$\label{di_PP_self}
\left\{ \begin{array}{ccl}
{\ensuremath{\mathcal{P}^0_-}}-\tfrac{1}{2}&=&-\text{sign}\big({\ensuremath{\mathcal{D}^0}}\big),\\
{\ensuremath{\mathcal{D}^0}}&=&D_0\Pi_{\ensuremath{\Lambda}}-\dfrac{\alpha}{2}\dfrac{({\ensuremath{\mathcal{P}^0_-}}-\tfrac{1}{2})(x-y)}{|x-y|}
\end{array}
\right.$$ We have ${\ensuremath{\mathcal{P}^0_-}}=\chi_{(-\infty,0)}({\ensuremath{\mathcal{D}^0}})$. This operator ${\ensuremath{\mathcal{D}^0}}$ was previously introduced by Lieb *et al.* in [@ls]. In $\mathfrak{H}$, the operator ${\ensuremath{\mathcal{D}^0}}$ coincides with a bounded, matrix-valued Fourier multiplier whose kernel is ${\ensuremath{\mathfrak{H}_\Lambda}}^{\perp}\subset \mathfrak{H}$.
Throughout this paper we write $$m=\inf \sigma \big(|{\ensuremath{\mathcal{D}^0}}|\big)\ge 1,$$ and $${\ensuremath{\mathcal{P}^0_+}}:=\Pi_{\ensuremath{\Lambda}}-{\ensuremath{\mathcal{P}^0_-}}=\chi_{(0,+\infty)}({\ensuremath{\mathcal{D}^0}}).$$
The resulting BDF energy $\mathcal{E}^\nu_{\text{BDF}}$ is defined on Hartree-Fock states represented by their one-body density matrix $P$: $$\mathscr{N}:=\big\{P\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ P^*=P^2=P,\ P-{\ensuremath{\mathcal{P}^0_-}}\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})\big\}.$$
We recall that $\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})$ is the set of bounded operators and that for $p\ge1$, $\mathfrak{S}_p({\ensuremath{\mathfrak{H}_\Lambda}})$ is the set of compact operators $A$ such that ${\ensuremath{\mathrm{Tr}}}\big(|A|^p\big)<+\infty$ [@ReedSim; @Sim]. In particular $\mathfrak{S}_{\infty}({\ensuremath{\mathfrak{H}_\Lambda}})$ is the set $\text{Comp}({\ensuremath{\mathfrak{H}_\Lambda}})$ of compact operators.
This energy depends on three parameters: the fine structure constant $\alpha>0$, the cut-off ${\ensuremath{\Lambda}}>0$ and the external density $\nu$. We assume that $\nu$ has finite *Coulomb energy*, that is $${\ensuremath{\widehat{\nu}}}\ \text{measurable\ and\ }D(\nu,\nu):=4\pi\underset{{\ensuremath{\mathbb{R}^3}}}{{\ensuremath{\displaystyle\int}}}\frac{|{\ensuremath{\widehat{\nu}}}(k)|^2}{|k|^2}dk<+\infty.$$ The above integral coincides with $\underset{{\ensuremath{\mathbb{R}^3}}\times{\ensuremath{\mathbb{R}^3}}}{\iint}\frac{\nu(x)^*\nu(y)}{|x-y|}dxdy$ whenever this last one is well-defined.
The same symmetries holds for ${\ensuremath{\mathcal{P}^0_-}}$ and ${\ensuremath{\mathcal{P}^0_+}}$: the charge conjugation ${\ensuremath{\mathrm{C}}}$ and the operator ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ maps $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_-}}$ onto $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$. Moreover thanks to [@Th pp. 122-129] we can easily check that ${\ensuremath{\mathcal{D}^0}}$ also commutes with the action of $\mathbf{SU}(2)$ and with the operators ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{K}}}$.
Minimizers and critical points
------------------------------
For $P\in\mathscr{N}$, we have the identity $$\label{di_eqq}
(P-{\ensuremath{\mathcal{P}^0_-}})^2={\ensuremath{\mathcal{P}^0_+}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_+}}-{\ensuremath{\mathcal{P}^0_-}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_-}}\in \mathfrak{S}_1.$$ The charge of a state $P$ is given by the ${\ensuremath{\mathcal{P}^0_-}}$-trace of $P-{\ensuremath{\mathcal{P}^0_-}}$, defined by the formula: $$\begin{aligned}
{\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big(P-{\ensuremath{\mathcal{P}^0_-}}\big)&:={\ensuremath{\mathrm{Tr}}}\big({\ensuremath{\mathcal{P}^0_-}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_-}}+{\ensuremath{\mathcal{P}^0_+}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_+}}\big),\\
&=\text{Dim}\mathrm{Ran}({\ensuremath{\mathcal{P}^0_+}})\cap \mathrm{Ran} (P)-\text{Dim}\mathrm{Ran}({\ensuremath{\mathcal{P}^0_-}})\cap \mathrm{Ran} (1-P).\end{aligned}$$ A minimizer over states with charge $N\in\mathbb{N}$ is interpreted as a ground state of a system with $N$ electrons, in the presence of an external density $\nu$
The existence problem was studied in several papers [@at; @sok; @sokd]: by [@at Theorem 1], it is sufficient to check binding inequalities.
The following results hold under technical assumptions on $\alpha$ and ${\ensuremath{\Lambda}}$ (different for each result).
In [@at], Hainzl *et al.* proved existence of minimizers for the system of $N$ electrons with $\nu\ge 0$, provided that $N-1<\int \nu$ .
In [@sok], we proved the existence of a ground state for $N=1$ and $\nu=0$: an electron can bind alone in the vacuum. This surprising result holds due to the vacuum polarization.
In [@sokd], we studied the charge screening effect: due to vacuum polarization, the observed charge of a minimizer $P\neq {\ensuremath{\mathcal{P}^0_-}}$ is different from its real charge ${\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}(P-{\ensuremath{\mathcal{P}^0_-}})$. We also proved it is possible to keep track of this effect in the non-relativistic limit $\alpha\to 0$: the resulting limit is an altered Hartree-Fock energy.
Here we are looking for states with an equal number of electrons and positrons, that is we study $\mathcal{E}^0_{\text{BDF}}$ on $$\mathscr{M}:=\Big\{P\in\mathscr{N},\ {\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big(P-{\ensuremath{\mathcal{P}^0_-}}\big)=0\Big\}.$$ From a geometrical point of view $\mathscr{M}$ is a Hilbert manifold and $\mathcal{E}^0_{\text{BDF}}$ is a differentiable map on $\mathscr{M}$ (Propositions \[di\_manim\] and \[di\_gragra\]).
We thus seek a critical point on $\mathscr{M}$, that is some $P\in\mathscr{M},\ P\neq{\ensuremath{\mathcal{P}^0_-}}$ such that $\nabla \mathcal{E}^0_{\text{BDF}}(P)=0$. In [@pos_sok], we have found the ortho-positronium by studying the BDF energy restricted to states with the ${\ensuremath{\mathrm{C}}}$-symmetry: $$\label{di_mm_cc}
P\in\mathscr{M}\text{\ s.t.\ }P+{\ensuremath{\mathrm{C}}}P{\ensuremath{\mathrm{C}}}=\mathrm{Id}_{{\ensuremath{\mathfrak{H}_\Lambda}}}.$$ We write $\mathscr{M}_{\mathscr{C}}$ the set of such states. We will seek the para-positronium in the set $\mathscr{M}_{\mathscr{I}}$ of states having the ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-symmetry.
$$\label{di_ii_ss}
\mathscr{M}_{\mathscr{I}}:=\{P\in\mathscr{M}\text{\ s.t.\ }P+{\ensuremath{\mathrm{I}_{\mathrm{s}}}}P{\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}=P-{\ensuremath{\mathrm{I}_{\mathrm{s}}}}P{\ensuremath{\mathrm{I}_{\mathrm{s}}}}=\mathrm{Id}_{{\ensuremath{\mathfrak{H}_\Lambda}}}\}.$$
Equivalently $P\in\mathscr{M}_{\mathscr{I}}$ if and only if $Q:=P-{\ensuremath{\mathcal{P}^0_-}}$ is Hilbert-Schmidt and satisfies $$-{\ensuremath{\mathrm{I}_{\mathrm{s}}}}Q {\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}Q {\ensuremath{\mathrm{I}_{\mathrm{s}}}}=Q.$$
We seek a projector $P$ “close” to a state $P_0$ that can be written as:$$\label{di_imagine}
P_0={\ensuremath{\mathcal{P}^0_-}}+{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_-\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_-|}\xspace}-{\ensuremath{|\psi_-\rangle}\xspace}{\ensuremath{\langle \psi_-|}\xspace},\ {\ensuremath{\mathcal{P}^0_+}}\psi_-=0.$$ To deal with the dipositronium, we impose an additional symmetry: we define $\mathscr{W}\subset \mathscr{M}_{\mathscr{C}}$ as follows.
$$\label{di_def_w}
\mathscr{W}:=\big\{P\in \mathscr{M}_{\mathscr{C}},\ \forall U\in\mathbf{S},\ UP U^{-1}=P \big\}.$$
Equivalently $$P\in\mathscr{W}\,\iff\,Q:=P-{\ensuremath{\mathcal{P}^0_-}}\mathrm{\ satisfies\ }-{\ensuremath{\mathrm{C}}}Q{\ensuremath{\mathrm{C}}}=Q\mathrm{\ and\ }UQU^{-1}=Q,\ \forall\,U\in \mathbf{S}.$$
Those sets $\mathscr{M}_{\mathscr{C}},\mathscr{M}_{\mathscr{I}},\mathscr{W}$ have fine properties: they are all submanifolds of $\mathscr{M}$, invariant under the gradient flow of $\mathcal{E}^0_{\text{BDF}}$ (Proposition \[di\_mani\_ci\_sym\]).
However while $\mathscr{M}_{\mathscr{C}}$ has two connected components, $\mathscr{M}_{\mathscr{I}}$ has only one connected component and $\mathscr{W}$ has countable connected components. So we may find critical points by searching a minimizer of the BDF energy over the different connected components of $\mathscr{W}$. For the para-positronium, a critical point is found by an argument of mountain pass.
\[di\_conn\_comp\] There is a one-to-one correspondence between the connected components of $\mathscr{W}$ and the set $\mathbb{Z}_2^2[X]$ of polynomials with coefficients in the ring $\mathbb{Z}_2\times \mathbb{Z}_2$.
Let $P$ be in $\mathscr{W}$. The vector space $E_1:=\mathrm{Ran}\,P\cap \mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$ has finite dimension and is invariant under ${\ensuremath{\Phi_{\mathrm{SU}}}}$. We decompose it into irreducible representations.
The projector is associated to $\sum_{\ell=1}^{\ell_0}t_\ell X^\ell$ with $t_\ell=(t_{\ell,1};t_{\ell,-1})$ if and only if for any $j\in\tfrac{1}{2}+\mathbb{Z}_+$:
1. The number $b_{j-\tfrac{1}{2},1}$ of irreducible representations of $E_1$ of type $(j,+)$ satisfies $b_{j-\tfrac{1}{2},1}\equiv t_{j-\tfrac{1}{2},1}[2]$.
2. The number $b_{j-\tfrac{1}{2},-1}$ of irreducible representations of $E_1$ of type $(j,-)$ satisfies $b_{j-\tfrac{1}{2},-1}\equiv t_{j-\tfrac{1}{2},-1}[2]$.
\[di\_a\_c\_i\] The symbols $\mathscr{Y}$ and ${\ensuremath{\mathrm{Y}}}$ denotes respectively $\mathscr{C}$ and ${\ensuremath{\mathrm{C}}}$ or $\mathscr{I}$ and ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$. Furthermore the different connected components of $\mathscr{W}$ are written $\mathscr{W}_{p(X)}$ with $p(X)\in\mathbb{Z}_2^2[X]$.
To state our main Theorems, we need to introduce the mean-field operator.
An operator $Q\in \mathscr{V}$ is Hilbert-Schmidt and we write $Q(x,y)$ its integral kernel. Its density $\rho_Q$ is defined by the formula $$\label{di_dens_def}
\forall x\in{\ensuremath{\mathbb{R}^3}},\ \rho_Q(x):={\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathbb{C}^4}}}\big(Q(x,x)\big),$$ we prove in the next Section that it is well-defined. The mean-field operator $D^{({\ensuremath{\Lambda}})}_{Q}$ associated to $Q$ *in the vacuum* is : $$\label{di_mean_field}
D^{({\ensuremath{\Lambda}})}_{Q}:=\Pi_{\ensuremath{\Lambda}}\Big({\ensuremath{\mathcal{D}^0}}+\alpha \big(\rho_Q*\frac{1}{|\cdot|}-\frac{Q(x,y)}{|x-y|}\big)\Big).$$
\[di\_main\] There exist $\alpha_0,L_0,{\ensuremath{\Lambda}}_0>0$ such that if $$\alpha\le \alpha_0;\ \alpha{\ensuremath{\log(\Lambda)}}:=L\le L_0\text{\ and\ }{\ensuremath{\Lambda}}^{-1}\le {\ensuremath{\Lambda}}_0^{-1},$$ then there exists a critical point ${\ensuremath{\overline{P}}}={\ensuremath{\overline{Q}}}+{\ensuremath{\mathcal{P}^0_-}}$ of $\mathcal{E}^0_{\text{BDF}}$ in $\mathscr{M}_{\mathscr{I}}$ that satisfies the following equation. $$\exists 0<\mu<m,\ \exists \psi_a\in \mathrm{Ker}\big(D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})}-\mu\big),\ {\ensuremath{\overline{P}}}=\chi_{(-\infty,0)}\big(D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})}\big)+{\ensuremath{|\psi_a\rangle}\xspace}{\ensuremath{\langle \psi_a|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a|}\xspace}.$$
As $\alpha$ tends to $0$, the upper spinor of $U_{{\ensuremath{\lambda}}}\psi_a:={\ensuremath{\lambda}}^{3/2}\psi_a({\ensuremath{\lambda}}(\cdot))$ with ${\ensuremath{\lambda}}:=\tfrac{g'_1(0)^2}{\alpha m}$ tends to a Pekar minimizer.
– *We recall that the Pekar energy is defined as follows* $$\forall\,\psi\in H^1,\ \mathcal{E}_{\text{PT}}(\psi):={\ensuremath{\lVert\nabla\psi\rVert_{L^{2}}}}^2-D\big(|\psi|^2,|\psi|^2\big).$$ *The infimum over $\mathbb{S}L^2\cap H^1$ is written $E_{\text{PT}}(1)$.*
\[di\_main\_1\] There exist $L_0,{\ensuremath{\Lambda}}_0>0$, and for any $j\in\tfrac{1}{2}+\mathbb{Z}_+$, there exists $\alpha_j$ such that if $$\alpha\le \alpha_j;\ \alpha{\ensuremath{\log(\Lambda)}}:=L\le L_0\text{\ and\ }{\ensuremath{\Lambda}}^{-1}\le {\ensuremath{\Lambda}}_0^{-1},$$ then there exists a minimizer $P_{\mathbf{t}X^{\ell_0}}=Q+{\ensuremath{\mathcal{P}^0_-}}$ of $\mathcal{E}^0_{\text{BDF}}$ over the connected component of $\mathscr{W}_{\mathbf{t}X^{\ell_0}}$ with $\mathbf{t}\in\{(1,0),(0,1)\}$.
Moreover there exists $0<\mu_{\ell_0,\mathbf{t}}<1$ and $\psi\in \mathrm{Ker}\big(D_Q^{({\ensuremath{\Lambda}})}-\mu_{\ell_0,\mathbf{t}}\big)$ such that $$P_{\mathbf{t}X^{\ell_0}}=\chi_{(-\infty,0)}(D_Q^{({\ensuremath{\Lambda}})})+\mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)-\mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi).$$ Any upper spinor ${\ensuremath{\widetilde{{\ensuremath{\varphi}}}}}$ of ${\ensuremath{\widetilde{\psi}}}\in {\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)$ can be written as $$\forall\,x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ {\ensuremath{\widetilde{{\ensuremath{\varphi}}}}}=:ra(r)\sum_{m=-j}^j c_m({\ensuremath{\widetilde{{\ensuremath{\varphi}}}}})\Phi^+_{m,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})},\ c_m({\ensuremath{\widetilde{{\ensuremath{\varphi}}}}})\in\mathbb{C}.$$
Furthermore, as $\alpha$ tends to $0$, the function $ U_{{\ensuremath{\lambda}}} a(r)={\ensuremath{\lambda}}^{3/2}a({\ensuremath{\lambda}}r)$ tends to a minimizer of the energy $\mathcal{E}_{\mathbf{t}X^{\ell_0}}$ over $\mathbb{S}L^2(\mathbb{R}_+,r^2dr)\cap H^1(\mathbb{R}_+,r^2dr):$
$$\label{di_non_rel_w}
\mathcal{E}_{\mathbf{t}X^{\ell_0}}\big(f(r)\big):= {\ensuremath{\mathrm{Tr}}}\big(-\Delta\, \mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(rf(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}) \big)-{\ensuremath{\lVert\mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,(rf(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})})\rVert_{\text{Ex}}}}^2.$$
In particular, the dipositronium corresponds to the case $\ell_0=j_0-\tfrac{1}{2}=0$.
The minimum is written $E_{\mathbf{t}X^{\ell_0}}^{nr}$ for the non-relativistic energy and $E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}$ for the BDF energy over $\mathscr{W}_{\mathbf{t}X^{j_0-1/2}}$.
\[di\_eps\_t\] For $\mathbf{t}X^{\ell_0}\in \mathbb{Z}_2^2[X]$ as in Theorem \[di\_main\_1\], ${\ensuremath{\varepsilon}}(\mathbf{t})\in\{+,-\}$ denotes $+$ if $\mathbf{t}=(1,0)$ or $-$ if $\mathbf{t}=(0,1)$.
We expect the existence of minimizers over any connected components of $\mathscr{W}$ (associated to $p(X)\in \mathbb{Z}_2^2[X]$), provided that $\alpha$ is smaller than some $\alpha_{p(X)}$.
\[di\_prec\_non\_rel\] The non-relativistic energy can be computed: $$\left\{\begin{array}{rcl}
\mathcal{E}_{\mathbf{t}X^{\ell_0}}\big(f(r)\big)&:=&(2j_0+1)\underset{0}{\overset{+\infty}{{\ensuremath{\displaystyle\int}}}} \Big[r^2|f'(r)|^2+(j_0+{\ensuremath{\varepsilon}}\tfrac{1}{2})(j_0+1+{\ensuremath{\varepsilon}}\tfrac{1}{2})|f(r)|^2\Big]dr\\
&& \ \ \ -\underset{\mathbb{R}_+^2}{{\ensuremath{\displaystyle\iint}}}r_1^2r_2^2|f(r_1)|^2|f(r_2)|^2w_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}(r_1,r_2),\\
w_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}(r_1,r_2)&:=&\underset{(\mathbb{S}^2)^2}{{\ensuremath{\displaystyle\iint}}}\frac{dn_1 dn_2}{|r_1n_1-r_2n_2|}\Big({\ensuremath{\displaystyle\sum}}_{m_1,m_2}((\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})^*\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})(n_1) \Big)\\
&&\ \ \ \times\Big({\ensuremath{\displaystyle\sum}}_{m_1,m_2}((\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})^*\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})(n_2) \Big).
\end{array}\right.$$
It corresponds to the energy $$\mathcal{E}_{nr}\big({\ensuremath{\Gamma}}\big):= {\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\Gamma}}\big)-{\ensuremath{\lVert{\ensuremath{\Gamma}}\rVert_{\text{Ex}}}}^2,\ 0\le {\ensuremath{\Gamma}}\le 1,\ {\ensuremath{\Gamma}}\in\mathfrak{S}_1(H^1({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2))$$ restricted to the subspace $$\mathscr{S}_{(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))}:=\big\{{\ensuremath{\Gamma}},\ {\ensuremath{\Gamma}}^*={\ensuremath{\Gamma}}^2={\ensuremath{\Gamma}},\ \mathrm{Ran}\,({\ensuremath{\Phi_{\mathrm{SU}}}})_{\big|_{{\ensuremath{\Gamma}}}}\ \text{irreducible\ of\ type\ }(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))\big\}.
$$ This subspace is invariant under the action of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ and it is easy to see that it is a submanifold of $ \big\{ {\ensuremath{\Gamma}},\ {\ensuremath{\Gamma}}^*={\ensuremath{\Gamma}}^2={\ensuremath{\Gamma}},\ {\ensuremath{\mathrm{Tr}}}\,{\ensuremath{\Gamma}}=2j_0+1\big\}$.
The subspace $\mathscr{S}_{(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))}$ is invariant under the flow of $\mathcal{E}_{nr}$.
The energies can be estimated.
\[di\_est\] In the same regime as in Theorem \[di\_main\], the following holds. The critical point ${\ensuremath{\overline{P}}}$ of the BDF functional over $\mathscr{M}_{\mathscr{I}}$ satisfies $$\label{di_en_para}
\mathcal{E}^0_{\text{BDF}}({\ensuremath{\overline{P}}})=2m +\frac{\alpha^2m}{g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3).$$ Furthermore the minimizer ${\ensuremath{\overline{P}}}_{\ell_0}$ over $\mathscr{W}_{\mathbf{t}X^{\ell_0}}$ satisfies: $$\label{di_est_mult}
\mathcal{E}^0_{\text{BDF}}({\ensuremath{\overline{P}}}_{\ell_0})=2(2j_0+1)+\frac{\alpha^2 m}{g'_1(0)^2} E_{\mathbf{t}X^{\ell_0}}^{nr}+\mathcal{O}(\alpha^3K(j_0)).$$
The Pekar model describes an electron trapped in its own hole in a polarizable medium. Thus it is not surprising to find it here. We recall that there is a unique minimizer of the Pekar energy up to translation and a phase in $\mathbb{S}^7$ (in $\mathbb{C}^4$).
The asymptotic expansion coincides with that of the ortho-positronium [@pos_sok]. In fact, it can be proved that the first difference between the energies occurs at order $\alpha^4$.
Throughout this paper we write $K$ to mean a constant independent of $\alpha,{\ensuremath{\Lambda}}$. Its value may differ from one line to the other. When we write $K(a)$, we mean a constant that depends solely on $a$. We also use the symbol $\apprle$: $0\le a\apprle b$ means there exists $K>0$ such that $a\le Kb$.
We also recall the reader our use of the notation $\mathbb{S}V$ for any subspace $V$ of some Hilbert space that denotes the set of unitary vector in $V$.
Remarks and notations about ${\ensuremath{\mathcal{D}^0}}$
----------------------------------------------------------
${\ensuremath{\mathcal{D}^0}}$ has the following form [@mf]: $$\label{di_D_form}
{\ensuremath{\mathcal{D}^0}}=g_0(-i\nabla)\beta -i\boldsymbol{\alpha}\cdot \frac{\nabla}{|\nabla|}g_1(-i\nabla)$$ where $g_0$ and $g_1$ are smooth radial functions on $B(0,{\ensuremath{\Lambda}})$. Moreover we have: $$\forall\,p\in B(0,{\ensuremath{\Lambda}}),\ 1\le g_0(p),\text{\ and\ }|p|\le g_1(p)\le |p|g_0(p).$$
For $\alpha{\ensuremath{\log(\Lambda)}}$ sufficiently small, we have $m=g_0(0)$ [@LL; @sok].
The smallness of $\alpha$ is needed to get estimates that hold close to the non-relativistic limit.
The smallness of $\alpha{\ensuremath{\log(\Lambda)}}$ is needed to get estimates of ${\ensuremath{\mathcal{D}^0}}$: in this case ${\ensuremath{\mathcal{D}^0}}$ can be obtained by a fixed point scheme [@mf; @LL], and we have [@sok Appendix A]: $$\label{di_estim_g}
\begin{array}{c}
g'_0(0)=0,\ \text{and}\ {\ensuremath{\lVertg'_0\rVert_{L^{\infty}}}},{\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}\le K\alpha\\{\ensuremath{\lVertg'_1-1\rVert_{L^{\infty}}}}\le K\alpha{\ensuremath{\log(\Lambda)}}\le \tfrac{1}{2}\ \text{and}\ {\ensuremath{\lVertg_1''\rVert_{L^{\infty}}}}\apprle 1.
\end{array}$$
Description of the model
========================
The BDF energy
--------------
For any ${\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}}'\in\{+,-\}$ and $A\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})$, we write $$A^{{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}}'}:=\mathcal{P}^0_{{\ensuremath{\varepsilon}}}A\mathcal{P}^0_{{\ensuremath{\varepsilon}}'}.$$
For an operator $Q\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$, we write $R_Q$ the operator given by the integral kernel: $$R_Q(x,y):=\frac{Q(x,y)}{|x-y|}.$$
Let $\alpha>0,{\ensuremath{\Lambda}}>0$ and $\nu\in\mathcal{S}'({\ensuremath{\mathbb{R}^3}})$ a generalized function with $D(\nu,\nu)<+\infty$. For $P\in\mathscr{N}$ we write $Q:=P-{\ensuremath{\mathcal{P}^0_-}}$ and $$\label{di_formule_bdf}
\left\{\begin{array}{l}
\mathcal{E}^0_{\text{BDF}}(Q)={\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big({\ensuremath{\mathcal{D}^0}}Q \big)-\alpha D(\rho_Q,\nu)+\dfrac{\alpha}{2}\Big(D(\rho_Q,\rho_Q)-{\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2\Big),\\
\forall\,x,y\in{\ensuremath{\mathbb{R}^3}},\ \rho_Q(x):={\ensuremath{\mathrm{Tr}}}_{\mathbb{C}^4}\big(Q(x,x)\big),\ {\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2:={\ensuremath{\displaystyle\iint}}\frac{|Q(x,y)|^2}{|x-y|}dxdy,
\end{array}\right.$$ where $Q(x,y)$ is the integral kernel of $Q$.
The term ${\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big({\ensuremath{\mathcal{D}^0}}Q \big)$ is the kinetic energy, $-\alpha D(\rho_Q,\nu)$ is the interaction energy with $\nu$. The term $\dfrac{\alpha}{2}D(\rho_Q,\rho_Q)$ is the so-called *diract term* and $-\dfrac{\alpha}{2}{\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2$ is the *exchange term*.
Let us see that formula is well-defined whenever $Q$ is ${\ensuremath{\mathcal{P}^0_-}}$-trace-class [@ptf; @at].
#### $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ and the variational set $\mathcal{K}$
The set $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ of ${\ensuremath{\mathcal{P}^0_-}}$-trace class operator is the following Banach space: $$\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}=\big\{Q\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}),\ Q^{++},Q^{--}\in\mathfrak{S}_1({\ensuremath{\mathfrak{H}_\Lambda}})\big\},$$ with the norm $$\lVert Q\rVert_{\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}}:={\ensuremath{\lVertQ^{+-}\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVertQ^{-+}\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVertQ^{++}\rVert_{\mathfrak{S}_{1}}}}+{\ensuremath{\lVertQ^{--}\rVert_{\mathfrak{S}_{1}}}}.$$
We have $\mathscr{N}\subset {\ensuremath{\mathcal{P}^0_-}}+\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ thanks to . The closed convex hull of $\mathscr{N}-{\ensuremath{\mathcal{P}^0_-}}$ under $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ is $$\mathcal{K}:=\big\{Q\in\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}}),\ Q^*=Q,\ -{\ensuremath{\mathcal{P}^0_-}}\le Q\le {\ensuremath{\mathcal{P}^0_+}}\big\}$$ and we have [@ptf; @Sc] $$\forall\,Q\in \mathcal{K},\ Q^2\le Q^{++}-Q^{--}.$$
#### The BDF energy for $Q\in \mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$
We have $${\ensuremath{\mathcal{P}^0_-}}({\ensuremath{\mathcal{D}^0}}Q){\ensuremath{\mathcal{P}^0_-}}=-|{\ensuremath{\mathcal{D}^0}}|Q^{--}\in\,\mathfrak{S}_1({\ensuremath{\mathfrak{H}_\Lambda}}),\ \text{because}\, |{\ensuremath{\mathcal{D}^0}}|\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),$$ this proves that the kinetic energy is defined.
By the Kato-Seiler-Simon (KSS) inequality [@Sim], $Q$ is locally trace-class: $$\forall\,\phi\in \mathbf{C}^\infty_0({\ensuremath{\mathbb{R}^3}}),\ \phi \Pi_{\ensuremath{\Lambda}}\in\mathfrak{S}_2\text{\ so\ }\phi Q \phi=\phi\Pi_{\ensuremath{\Lambda}}Q\phi\in\mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}})).$$ We recall this inequality states that for all $2\le p\le \infty$ and $d\in\mathbb{N}$, we have $$\forall\,f,g\in L^p(\mathbb{R}^d),\ f(x)g(-i\nabla)\in\mathfrak{S}_{p}({\ensuremath{\mathfrak{H}_\Lambda}})\text{\ and\ }{\ensuremath{\lVertf(x)g(-i\nabla)\rVert_{\mathfrak{S}_{p}}}}\le (2\pi)^{-d/p}{\ensuremath{\lVertf\rVert_{L^{p}}}}{\ensuremath{\lVertg\rVert_{L^{p}}}}.$$ It follows that the *density* $\rho_Q$ of $Q$, defined in is well-defined. By the KSS inequality, we can also prove that ${\ensuremath{\lVert\rho_Q\rVert_{\mathcal{C}}}}\apprle K({\ensuremath{\Lambda}})\lVert Q \rVert_{\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}}$ [@gs Proposition 2]. By Kato’s inequality: $$\label{di_kato}
\dfrac{1}{|\cdot|}\le \dfrac{\pi}{2}|\nabla|,$$ the exchange term is well-defined.
Moreover the following holds: if $\alpha < \tfrac{4}{\pi}$, then the BDF energy is bounded from below on $\mathcal{K}$ [@stab; @Sc; @at]. We have $$\label{di_below}
\forall\,Q_0\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}),\ \mathcal{E}^0_{\text{BDF}}(Q_0)\ge \big(1-\alpha\frac{\pi}{4}\big){\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}||Q_0|^2\big).$$
Here we assume it is the case. This result will be often used throughout this paper.
#### Minimizers
For $Q\in\mathcal{K}$, its charge is its ${\ensuremath{\mathcal{P}^0_-}}$-trace: $q={\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}(Q)$. We define the Charge sector sets: $$\forall\,q\in{\ensuremath{\mathbb{R}^3}},\ \mathcal{K}^q:=\big\{Q\in\mathcal{K},\ {\ensuremath{\mathrm{Tr}}}(Q)=q\big\}.$$ A minimizer of $\mathcal{E}^\nu_{\text{BDF}}$ over $\mathcal{K}$ is interpreted as the polarized vacuum in the presence of $\nu$ while a minimizer over charge sector $N\in\mathbb{N}$ is interpreted as the ground state of $N$ electrons in the presence of $\nu$, by Lieb’s principle [@at Proposition 3], such a minimizer is in $\mathscr{N}-{\ensuremath{\mathcal{P}^0_-}}$.
We define the energy functional $E^\nu_{\text{BDF}}$: $$\forall\,q\in{\ensuremath{\mathbb{R}^3}},\ E^\nu_{\text{BDF}}(q):=\inf\big\{\mathcal{E}^\nu_{\text{BDF}}(Q),\ Q\in\mathcal{K}^q\big\}.$$
We also write: $$\label{di_koci}
\mathcal{K}^0_{\mathscr{Y}}:=\{ Q\in\mathcal{K},\ \text{Tr}_{{\ensuremath{\mathcal{P}^0_-}}}(Q)=0,\ -{\ensuremath{\mathrm{Y}}}Q {\ensuremath{\mathrm{Y}}}^{-1} =Q\}.$$ Proposition \[di\_weakclosed\] states that this set is sequentially weakly-$*$ closed in $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}})$.
Structure of manifold
---------------------
We consider $$\mathscr{V}=\big\{P-{\ensuremath{\mathcal{P}^0_-}},\ P^*=P^2=P\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ {\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big( P-{\ensuremath{\mathcal{P}^0_-}}\big)=0\big\}\subset \mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}).$$ and write: $\mathscr{M}:={\ensuremath{\mathcal{P}^0_-}}+\mathscr{V}=\big\{P,\ P^*=P^2=P,\ {\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big( P-{\ensuremath{\mathcal{P}^0_-}}\big)=0\big\}.$
We recall the following proposition, proved in [@pos_sok].
\[di\_manim\] The set $\mathscr{M}$ is a Hilbert manifold and for all $P\in\mathscr{M}$, $$\mathrm{T}_P \mathscr{M}=\{ [A,P],\,A\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ A^*=-A\text{\ and\ }PA(1-P)\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})\}.$$ Writing $$\mathfrak{m}_P:=\{ A\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ A^*=-A,\ PAP=(1-P)A(1-P)=0\text{\ and\ }PA(1-P)\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})\},$$ any $P_1\in\mathscr{M}$ can be written as $P_1=e^A P e^{-A}$ where $A\in\mathfrak{m}_P$.
The BDF energy $\mathcal{E}_{\text{BDF}}^\nu$ is a differentiable function in $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}})$ with: $$\label{di_eqdebdf}
\left\{ \begin{array}{l}
\forall\, Q,\delta Q\in\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}}),\ \text{d}\mathcal{E}_{\text{BDF}}^\nu(Q)\cdot \delta Q=\text{Tr}_{{\ensuremath{\mathcal{P}^0_-}}}\big(D_{Q,\nu}\delta Q\big).\\
D_{Q,\nu}:={\ensuremath{\mathcal{D}^0}}+\alpha \big((\rho_Q-\nu)*\frac{1}{|\cdot|}-R_Q\big).
\end{array}\right.$$ We may rewrite as follows: $$\forall\, Q,\delta Q\in\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}}),\ \text{d}\mathcal{E}_{\text{BDF}}^\nu(Q)\cdot \delta Q=\text{Tr}_{{\ensuremath{\mathcal{P}^0_-}}}\big(\Pi_\Lambda D_{Q,\nu}\Pi_\Lambda \delta Q\big)$$ We recall the mean-field operator $ D_Q^{({\ensuremath{\Lambda}})}$ is defined in Notation \[di\_mean\_field\].
\[di\_gragra\] Let $(P,v)$ be in the tangent bundle $\mathrm{T}\mathscr{M}$ and $Q=P-{\ensuremath{\mathcal{P}^0_-}}$. Then we have $[[\Pi_\Lambda D_Q \Pi_\Lambda,P],P]\in\mathrm{T}_P\mathscr{M}$ and: $$\label{di_difftan}
\mathrm{d}\mathcal{E}_{\text{BDF}}^0(P)\cdot v=\text{Tr}\Big(\big[\big[ D_{Q}^{({\ensuremath{\Lambda}})} ,P\big],P\big]v\Big).$$ In other words: $$\label{di_defgradient}
\forall\,P\in\mathscr{M},\ \nabla \mathcal{E}_{\text{BDF}}^0(P)=\big[\big[\Pi_{\ensuremath{\Lambda}}D_Q \Pi_{\ensuremath{\Lambda}},P\big],P\big].$$
The operator $[[\Pi_{\ensuremath{\Lambda}}D_Q\Pi_{\ensuremath{\Lambda}},P],P]$ is the “projection” of $\Pi_{\ensuremath{\Lambda}}D_Q \Pi_{\ensuremath{\Lambda}}$ onto $\text{T}_P\mathscr{M}$.
In [@pos_sok], we proved that $\mathscr{M}_{\mathscr{C}}$ is a submanifold of $\mathscr{M}$. We recall that the notations $\mathscr{Y}$, ${\ensuremath{\mathrm{Y}}}$ are specified in Notation \[di\_a\_c\_i\].
\[di\_mani\_ci\_sym\] The sets $\mathscr{M}_{\mathscr{I}}$ and $\mathscr{W}$ are *submanifolds* of $\mathscr{M}$, which are *invariant* under the flow of $\mathcal{E}_{\text{BDF}}^0$. The following holds: for any $P\in \mathscr{M}_{\mathscr{Y}}$, writing $$\mathfrak{m}^{\mathscr{Y}}_P=\{a\in \mathfrak{m}_P,\ {\ensuremath{\mathrm{Y}}}a {\ensuremath{\mathrm{Y}}}^{-1}=a\},$$ we have $$\label{di_tangentc}
\mathrm{T}_P \mathscr{M}_{\mathscr{Y}}=\{[a,P],\ a\in \mathfrak{m}_P^{\mathscr{Y}}\}=\{v\in\mathrm{T}_P \mathscr{M},\ -{\ensuremath{\mathrm{Y}}}v {\ensuremath{\mathrm{Y}}}^{-1}=v\}.$$ Furthermore, for any $P\in\mathscr{M}_{\mathscr{Y}}$ we have $\rho_{P-{\ensuremath{\mathcal{P}^0_-}}}=0.$
For $P\in\mathscr{W}$, the same holds with $$\left\{ \begin{array}{rcl}
\mathfrak{m}^{\mathscr{W}}_{P}&:=&\big\{a\in \mathfrak{m}^{\mathscr{C}}_{P},\ \forall\,U\in\mathbf{S},\ U a U^{-1}=a\big\},\\
\mathrm{T}_P\mathscr{W}&:=&\big\{[a,P],\ a\in\mathfrak{m}_P^{\mathscr{W}}\big\}.
\end{array}
\right.$$
\[Lagrangians\] The operator ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ induced a symplectic structure on the *real* Hilbert space $({\ensuremath{\mathfrak{H}_\Lambda}},\mathfrak{Re}{\ensuremath{\langle \cdot\,,\,\cdot\rangle}\xspace}_{\mathfrak{H}})$: $$\forall\,f,g\in{\ensuremath{\mathfrak{H}_\Lambda}},\ {\ensuremath{\omega}}_{\mathrm{I}}(f,g):=\mathfrak{Re}{\ensuremath{\langle f\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}g\rangle}\xspace}.$$ The manifold $\mathscr{M}_{\mathscr{I}}$ is constituted by *Lagrangians* of ${\ensuremath{\omega}}_{\mathrm{I}}$ that are in $\mathscr{M}$.
We end this section by stating technical results.
Form of trial states
--------------------
The following Theorem is stated in [@at Appendix] and proved in [@pos_sok].
\[di\_structure\] Let $P_1,P_0$ be in $\mathscr{N}$ and $Q=P_1-P_0$. Then there exist $M_+,M_-\in\mathbb{Z}_+$ such that there exist two orthonormal families $$\begin{array}{ll}
(a_1,\ldots,a_{M_+})\cup(e_i)_{i\in\mathbb{N}}& \mathrm{in}\ \mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}},\\
(a_{-1},\ldots,a_{-M_+})\cup(e_{-i})_{i\in\mathbb{N}}&\mathrm{in}\ \mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_-}},
\end{array}$$ and a nonincreasing sequence $({\ensuremath{\lambda}}_i)_{i\in\mathbb{N}}\in\ell^2$ satisfying the following properties:
1. The $a_i$’s are eigenvectors for $Q$ with eigenvalue $1$ (resp. $-1$) if $i>0$ (resp. $i<0$).
2. For each $i\in\mathbb{N}$ the plane $\Pi_i:=\text{Span}(e_i,e_{-i})$ is spanned by two eigenvectors $f_i$ and $f_{-i}$ for $Q$ with eigenvalues ${\ensuremath{\lambda}}_i$ and $-{\ensuremath{\lambda}}_i$.
3. The plane $\Pi_i$ is also spanned by two orthogonal vectors $v_i$ in $\mathrm{Ran}(1-P)$ and $v_{-i}$ in $\mathrm{Ran}(P)$. Moreover ${\ensuremath{\lambda}}_i=\sin(\theta_i)$ where $\theta_i\in (0,\tfrac{\pi}{2})$ is the angle between the two lines $\mathbb{C}v_i$ and $\mathbb{C}e_i$.
4. There holds: $$Q={\ensuremath{\displaystyle\sum}}_i^{M_+}{\ensuremath{|a_i\rangle}\xspace}{\ensuremath{\langle a_i|}\xspace}-{\ensuremath{\displaystyle\sum}}_i^{M_-}{\ensuremath{|a_{-i}\rangle}\xspace}{\ensuremath{\langle a_{-i}|}\xspace}+{\ensuremath{\displaystyle\sum}}_{j\in \mathbb{N}}{\ensuremath{\lambda}}_j({\ensuremath{|f_j\rangle}\xspace}{\ensuremath{\langle f_j|}\xspace}-{\ensuremath{|f_{-j}\rangle}\xspace}{\ensuremath{\langle f_{-j}|}\xspace}).$$
We have $$\label{di_++--}
\begin{array}{l}
Q^{++}={\ensuremath{\displaystyle\sum}}_i^{M_+}{\ensuremath{|a_i\rangle}\xspace}{\ensuremath{\langle a_i|}\xspace}+{\ensuremath{\displaystyle\sum}}_{j\in\mathbb{N}}\sin(\theta_j)^2{\ensuremath{|e_j\rangle}\xspace}{\ensuremath{\langle e_j|}\xspace},\\
Q^{--}=-{\ensuremath{\displaystyle\sum}}_i^{M_-}{\ensuremath{|a_{-i}\rangle}\xspace}{\ensuremath{\langle a_{-i}|}\xspace}-{\ensuremath{\displaystyle\sum}}_{j\in\mathbb{N}}\sin(\theta_j)^2{\ensuremath{|e_{-j}\rangle}\xspace}{\ensuremath{\langle e_{-j}|}\xspace}.
\end{array}$$
Thanks to Theorem \[di\_structure\], it is possible to characterize states in $\mathscr{M}_{\mathscr{Y}}$ and $\mathscr{W}$. We restate a proposition of [@pos_sok] and add the case of ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$.
\[di\_chasym\] Let ${\ensuremath{\gamma}}=P-{\ensuremath{\mathcal{P}^0_-}}$ be in $\mathscr{M}_{\mathscr{Y}}$. For $-1\le \mu\le 1$ and $X\in\{{\ensuremath{\gamma}},{\ensuremath{\gamma}}^2\}$, we write $$E^X_\mu=\mathrm{Ker}(X-\mu).$$ Then for any $\mu\in\sigma({\ensuremath{\gamma}})$, ${\ensuremath{\mathrm{Y}}}E^{\ensuremath{\gamma}}_\mu=E^{\ensuremath{\gamma}}_{-\mu}$. Moreover for $|\mu|<1$ if we decompose $E^{\ensuremath{\gamma}}_{\mu}\oplus E^{\ensuremath{\gamma}}_{-\mu}$ into a sum of planes $\Pi$ as in Theorem \[di\_structure\], then
1. If ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}$, then we can choose the $\Pi$’s to be ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-invariant.
2. If ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{C}}}$, then each $\Pi$ is *not* ${\ensuremath{\mathrm{C}}}$-invariant and $\mathrm{Dim}\,E^{\ensuremath{\gamma}}_{\mu}$ is even.
Equivalently $\text{Dim}\,E^{{\ensuremath{\gamma}}^2}_{\mu^2}$ is divisible by $4$. Moreover there exists a decomposition $$E^{{\ensuremath{\gamma}}^2}_{\mu^2}=\underset{1\le j\le \tfrac{N}{2}}{\overset{\perp}{\oplus}}V_{\mu,j}\text{\ and\ }V_{\mu,j}=\Pi^a_{\mu,j}\overset{\perp}{\oplus}{\ensuremath{\mathrm{C}}}\Pi^a_{\mu,j}$$ where the $\Pi^a_{\mu,j}$’s and ${\ensuremath{\mathrm{C}}}\Pi^a_{\mu,j}$’s are spectral planes described in Theorem \[di\_structure\].
The Cauchy expansion
--------------------
In this part, we introduce a useful trick in the model. The Cauchy expansion is an application of functional calculus: we refer the reader to [@ptf; @sok] for further details.
We assume $Q_0\in \mathfrak{S}_2$ with $$\label{di_supp}
\alpha {\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}Q_0\rVert_{\mathfrak{S}_{2}}}}\ll 1.$$
We recall the following inequality, proved in [@sok] $$\label{di_cauchy_est0}
\forall\,Q_0\in\mathfrak{S}_2,\ {\ensuremath{\lVertR_{Q_0}\tfrac{1}{|\nabla|^{1/2}}\rVert_{\mathfrak{S}_{2}}}}^2\apprle {\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2\apprle {\ensuremath{\displaystyle\iint}}|p+q||{\ensuremath{\widehat{Q}}}(p,q)|^2dpdq,$$
From now on, we only deal with $Q_0$ whose density vanishes: $\rho_{Q_0}=0$. The mean-field operator $D_{Q_0}^{({\ensuremath{\Lambda}})}$ is away from $0$ thanks to . Indeed, there holds $$\begin{aligned}
|\Pi_{\ensuremath{\Lambda}}R_{Q_0}\Pi_{\ensuremath{\Lambda}}|^2&\le |\nabla|^{1/2}\,\frac{\Pi_{\ensuremath{\Lambda}}}{|\nabla|^{1/2}}R_{Q_0}^* R_{Q_0}\frac{\Pi_{\ensuremath{\Lambda}}}{|\nabla|^{1/2}}\,|\nabla|^{1/2}\\
&\le \Pi_{\ensuremath{\Lambda}}|\nabla|{\ensuremath{\lVert\tfrac{1}{|\nabla|^{1/2}} R_{Q_0}\rVert_{\mathcal{B}}}}^2\\
&\apprle \Pi_{\ensuremath{\Lambda}}|\nabla|{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}\apprle |{\ensuremath{\mathcal{D}^0}}|^2{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}^2,\end{aligned}$$ thus $$|D_{Q_0}^{({\ensuremath{\Lambda}})}|\apprge |{\ensuremath{\mathcal{D}^0}}|\big(1-\alpha K{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}\big).$$
The Cauchy expansion gives an expression of $${\ensuremath{\gamma}}_0:=\chi_{(-\infty,0)}\big(D_{Q_0}^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}:={\ensuremath{\overline{\boldsymbol{\pi}}}}_0.$$
We have [@ptf] $$\label{di_cauchy0}
\chi_{(-\infty,0)}\big(D_{Q_0}^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}=\frac{1}{2\pi}{\ensuremath{\displaystyle\int}}_{-\infty}^{+\infty}\frac{d {\ensuremath{\omega}}}{{\ensuremath{\mathcal{D}^0}}+i\omega}\big(\alpha\Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}\big)\dfrac{1}{D_{Q_0}+i{\ensuremath{\omega}}}\Pi_{\ensuremath{\Lambda}}.$$
We also expand in power of $Y[Q_0]:=-\alpha \Pi_{\ensuremath{\Lambda}}R_{Q_0}\Pi_{\ensuremath{\Lambda}}$: $$\label{di_cauchy20}
\left\{
\begin{array}{rcl}
{\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}&=&{\ensuremath{\displaystyle\sum}}_{j\ge 1}\alpha^j M_j[Y[Q_0]],\\
M_j[Y_n]&=&-\dfrac{1}{2\pi}{\ensuremath{\displaystyle\int}}_{-\infty}^{+\infty}\frac{ d{\ensuremath{\omega}}}{{\ensuremath{\mathcal{D}^0}}+i{\ensuremath{\omega}}}\Big(Y_n\frac{1}{{\ensuremath{\mathcal{D}^0}}+i{\ensuremath{\omega}}} \Big)^{j}.
\end{array}
\right.$$ Each $M_j[Y[Q_0]]$ is polynomial in $\Pi_{\ensuremath{\Lambda}}R_{Q_0} \Pi_{\ensuremath{\Lambda}}$ of degree $j$.
By using , the decomposition is well-defined in several Banach space, provided that $\alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}$ is small enough.
– First, integrating the norm of bounded operator in , we obtain $${\ensuremath{\lVert{\ensuremath{\overline{\boldsymbol{\pi}}}}_0-{\ensuremath{\mathcal{P}^0_-}}\rVert_{\mathcal{B}}}}\apprle \alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}<1.$$
– We take the Hilbert-Schmidt norm [@ptf; @sok]: we get $$\label{di_estim_gn}
{\ensuremath{\lVert{\ensuremath{\gamma}}_{0}\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}.$$ – We take the norm ${\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}(\cdot)\rVert_{\mathfrak{S}_{2}}}}$ we get the rough estimate $$\label{di_estim_kin}
{\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{0}\rVert_{\mathfrak{S}_{2}}}}\apprle \min(\sqrt{L\alpha}{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}},\alpha {\ensuremath{\lVertR_{Q_0}\rVert_{\mathfrak{S}_{2}}}}\big)+\alpha^2{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}^2.$$
\[di\_diff\_ch\] The same estimates holds for the differential of $Q_0\mapsto {\ensuremath{\gamma}}_0$, for sufficiently small $\alpha$. As shown in [@sok], the upper bound of each norm is a power series of kind $$\lVert{\ensuremath{\gamma}}_0\rVert\le \alpha \lVert M_1[Y[Q_0]]\rVert+{\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}\sqrt{j}\alpha^j\big(K{\ensuremath{\lVert Q_0\rVert_{\text{Ex}}}} \big)^j.$$ In the case of the differential, we get an upper bound of kind $$\lVert \text{d}{\ensuremath{\gamma}}_0\rVert\le \alpha \lVert M_1[Y[Q_0]]\rVert+{\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}j^{3/2}\alpha^j\big(K{\ensuremath{\lVert Q_0\rVert_{\text{Ex}}}} \big)^j.$$ The power series converge for sufficiently small $\alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}$.
– It is also possible to consider other norms, using from the fact that a (scalar) Fourier multiplier $F(\mathbf{p}-\mathbf{q})=F(-i\nabla_x+i\nabla_y)$ commutes with the operator $R[\cdot]:Q(x,y)\mapsto \tfrac{Q(x,y)}{|x-y|}$. We can also consider the norm $$\lVert Q_0\rVert_{w}^2:={\ensuremath{\displaystyle\iint}}w(p-q)({\ensuremath{\widetilde{E}\left(p\right)}}+{\ensuremath{\widetilde{E}\left(q\right)}})|{\ensuremath{\widehat{Q}}}_0(p,q)|^2dpdq,$$ where $w(\cdot)\ge 0$ is any weight satisfying a subadditive condition [@sok]: $$\forall\,p,q\in{\ensuremath{\mathbb{R}^3}},\ \sqrt{w(p+q)}\le K(w)\big(\sqrt{w(p)}+\sqrt{w(q)}\big).$$
Proof of Theorems \[di\_main\] and \[di\_main\_1\]
==================================================
Strategy and tools of the proof: the dipositronium
--------------------------------------------------
### Topologies
The existence of a minimizer over $\mathscr{W}_{\mathbf{t}X^\ell}$ (with $\mathbf{t}\in\mathbb{Z}_2^2$) is proved with the same method used in [@pos_sok].
We use a lemma of Borwein and Preiss [@borw; @at], a smooth generalization of Ekeland’s Lemma [@ek]: we study the behaviour of a specific minimizing sequence $(P_n)_n$ or equivalently $(P_n-{\ensuremath{\mathcal{P}^0_-}}=:Q_n)_n$.
This sequence satisfies an equation close to the one satisfied by a real minimizer and we show this equation remains in some weak limit.
\[di\_topo\] We recall different topologies over bounded operator, besides the norm topology ${\ensuremath{\lVert\cdot\rVert_{\mathcal{B}}}}$ [@ReedSim].
1. The so-called *strong topology*, the weakest topology $\mathcal{T}_s$ such that for any $f\in{\ensuremath{\mathfrak{H}_\Lambda}}$, the map $$\begin{array}{rcl}
\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})&\longrightarrow&{\ensuremath{\mathfrak{H}_\Lambda}}\\
A&\mapsto& Af
\end{array}$$ is continuous.
2. The so-called *weak operator topology*, the weakest topology $\mathcal{T}_{w.o.}$ such that for any $f,g\in{\ensuremath{\mathfrak{H}_\Lambda}}$, the map $$\begin{array}{rcl}
\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})&\longrightarrow&\mathbb{C}\\
A&\mapsto& {\ensuremath{\langle A f\,,\,g\rangle}\xspace}
\end{array}$$ is continuous.
We can also endow $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ with its weak-$*$ topology, the weakest topology such that the following maps are continuous: $$\begin{array}{|l}
\begin{array}{rcl}
\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}&\longrightarrow&\mathbb{C}\\
Q&\mapsto& {\ensuremath{\mathrm{Tr}}}\big(A_0(Q^{++}+Q^{--})+A_2(Q^{+-}+Q^{-+})\big)
\end{array}\\
\forall\,(A_0,A_2)\in\mathrm{Comp}({\ensuremath{\mathfrak{H}_\Lambda}})\times \mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}).
\end{array}$$
\[di\_weakclosed\] The set $\mathcal{K}^0_{\mathscr{Y}}$, defined in , is weakly-$*$ sequentially closed in $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}})$.
This Lemma was stated for $\mathscr{Y}=\mathscr{C}$ in [@pos_sok]. For $\mathscr{Y}=\mathscr{I}$ the proof is the same and we refer the reader to this paper.
### The Borwein and Preiss Lemma
We recall this Theorem as stated in [@at]:
\[di\_bp\_lemma\] Let $\mathcal{M}$ be a closed subset of a Hilbert space $\mathcal{H}$, and $F:\mathcal{M}\to (-\infty,+\infty]$ be a lower semi-continuous function that is bounded from below and not identical to $+\infty$. For all ${\ensuremath{\varepsilon}}>0$ and all $u\in \mathcal{M}$ such that $F(u)<\inf_{\mathcal{M}}+{\ensuremath{\varepsilon}}^2$, there exist $v\in\mathcal{M}$ and $w\in{\ensuremath{\overline{\mathrm{Conv}(\mathcal{M})}}}$ such that
1. $F(v)< \inf_{\mathcal{M}}+{\ensuremath{\varepsilon}}^2$,
2. $\lVert u-v\rVert_{\mathcal{H}}<\sqrt{{\ensuremath{\varepsilon}}}$ and $\lVert v-w\rVert_{\mathcal{H}}<\sqrt{{\ensuremath{\varepsilon}}}$,
3. $F(v)+{\ensuremath{\varepsilon}}\lVert v-w\rVert_{\mathcal{H}}^2=\min\big\{F(z)+{\ensuremath{\varepsilon}}\lVert z-w\rVert_{\mathcal{H}}^2,\ z\in\mathcal{M}\big\}.$
– Here we apply this Theorem with $\mathcal{H}=\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$, $\mathcal{M}=\mathscr{W}_{p(X)}-{\ensuremath{\mathcal{P}^0_-}}$ and $F=\mathcal{E}^0_{\mathrm{BDF}}$.
The BDF energy is continuous in the $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$-norm topology, thus its restriction over $\mathscr{V}$ is continuous in the $\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$-norm topology.
This subspace $\mathcal{H}$ is closed in the Hilbert-Schmidt norm topology because $\mathscr{V}=\mathscr{M}_{\mathscr{C}}$ is closed in $\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$ and $\mathscr{E}_{-1}-{\ensuremath{\mathcal{P}^0_-}}$ is closed in $\mathscr{V}$.
Moreover, we have $${\ensuremath{\overline{\text{Conv}(\mathscr{W}_{p(X)}-{\ensuremath{\mathcal{P}^0_-}})}}}^{\mathfrak{S}_2}\subset \mathcal{K}_{\mathscr{C}}^0.$$
– For every $\eta>0$, we get a projector $P_\eta\in\mathscr{W}_{p(X)}$ and $A_\eta\in \mathcal{K}_{\mathscr{C}}^0$ such that $P$ that minimizes the functional $F_\eta: P\in\mathscr{E}_{-1}\mapsto \mathcal{E}_{\text{BDF}}^0(P-{\ensuremath{\mathcal{P}^0_-}})+{\ensuremath{\varepsilon}}{\ensuremath{\lVertP-{\ensuremath{\mathcal{P}^0_-}}-A_\eta\rVert_{\mathfrak{S}_{2}}}}^2.$
We write $$\label{di_almost}
Q_\eta:= P_\eta -{\ensuremath{\mathcal{P}^0_-}},\ {\ensuremath{\Gamma}}_\eta:=Q_\eta -A_\eta,\ {\ensuremath{\widetilde{D}}}_{Q_\eta}:=\Pi_{\ensuremath{\Lambda}}\big({\ensuremath{\mathcal{D}^0}}-\alpha R_{Q_\eta}+2\eta {\ensuremath{\Gamma}}_\eta\big)\Pi_{\ensuremath{\Lambda}}.$$ Studying its differential on $\text{T}_{P_\eta} \mathscr{W}$, we get: $$\label{di_eq_almost}
\big[{\ensuremath{\widetilde{D}}}_{Q_\eta}, P_\eta\big]=0.$$ In particular, by functional calculus, we have: $$\label{di_pimoins}
\big[\boldsymbol{\pi}_-^{\eta},P_\eta\big]=0,\ \boldsymbol{\pi}_{\eta}^-:=\chi_{(-\infty,0)}({\ensuremath{\widetilde{D}}}_{Q_\eta}).$$ We also write $$\label{di_piplus}
\boldsymbol{\pi}_{\eta}^+:=\chi_{(0,+\infty)}({\ensuremath{\widetilde{D}}}_{Q_\eta})=\Pi_{\ensuremath{\Lambda}}-\boldsymbol{\pi}_{\eta}^-.$$ We decompose ${\ensuremath{\mathfrak{H}_\Lambda}}$ as follows (here R means $\mathrm{Ran}$): $$\label{di_decomp_hl}
{\ensuremath{\mathfrak{H}_\Lambda}}=\text{R}(P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^-)\overset{\perp}{\oplus}\text{R}(P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^+)\overset{\perp}{\oplus}\text{R}(\Pi_{\ensuremath{\Lambda}}-P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^-)\overset{\perp}{\oplus}\text{R}(\Pi_{\ensuremath{\Lambda}}-P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^+).$$
We will prove
1. $\mathrm{Ran}\,P\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}_{\eta}^+$ has dimension $2j+1$ and is invariant under ${\ensuremath{\Phi_{\mathrm{SU}}}}$, spanned by a unitary $\psi_\eta\in{\ensuremath{\mathfrak{H}_\Lambda}}$.
2. As $\eta$ tends to $0$, up to translation and a subsequence, $\psi_\eta\rightharpoonup \psi_a\neq 0$, $Q_\eta\rightharpoonup {\ensuremath{\overline{Q}}}$. There holds ${\ensuremath{\overline{P}}}_{j_0}={\ensuremath{\overline{Q}}}+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{W}_{p(X)}$, $\psi_a$ is a unitary eigenvector of $ D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})} $ and $$\label{di_eq_min}
{\ensuremath{\overline{Q}}}+{\ensuremath{\mathcal{P}^0_-}}=\chi_{(-\infty,0)}\big( D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})} \big)+\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi_a)-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi_a),$$ where $\text{Proj}\,E$ means the orthonormal projection onto the vector space $E$.
In the following part we write the spectral decomposition of trial states and prove Lemma \[di\_weakclosed\].
### Spectral decomposition
Let $(Q_n)_n$ be any minimizing sequence for $E_{\mathbf{t}X^{(j_0-1/2)}}^{nr}$ for $j_0\in \tfrac{1}{2}+\mathbb{Z}_+$.
Thanks to the upper bound, $\text{Dim}\,\mathrm{Ker}(Q_n-1)=1$, as shown in Subsection \[di\_subscritic\].
There exist a *non-increasing* sequence $({\ensuremath{\lambda}}_{j;n})_{j\in\mathbb{N}}\in\ell^2$ of eigenvalues and an orthonormal family $\mathbf{B}_n$ of $\mathrm{Ran}\, Q_n$: $$\label{di_basen}
\mathbf{B}_n:=(\psi_n,{\ensuremath{\mathrm{C}}}\psi_n)\cup (e_{j;n}^a,e_{j;n}^b,{\ensuremath{\mathrm{C}}}e_{j;n}^a,{\ensuremath{\mathrm{C}}}e_{j;n}^b),\ {\ensuremath{\mathcal{P}^0_-}}\psi_n={\ensuremath{\mathcal{P}^0_-}}e_{j;n}^{\star}=0,\ \star\in\{a,b\},$$ such that the following holds. We omit the index $n$.
1\. For any $j$, the vector spaces $V_{j;n}^\star:={\ensuremath{\Phi_{\mathrm{SU}}}}(e_{j;n}^\star)$ are irreducible, and so is $V_{0;n}:={\ensuremath{\Phi_{\mathrm{SU}}}}(\psi_n)$.
2\. That last one is of type $(\ell_0,{\ensuremath{\varepsilon}}(\mathbf{t}))$ (see Notation \[di\_eps\_t\]).
3\. Moreover for any $j\in \mathbb{N}$ we write:
\[di\_formtrial\] $$\label{di_formtrial1}
\begin{array}{| l}
e_{-j}^a:=-{\ensuremath{\mathrm{C}}}e_{j}^b\text{\ and\ } e_{-j}^b:={\ensuremath{\mathrm{C}}}e_j^a,\\
V_{-j}^a:={\ensuremath{\Phi_{\mathrm{SU}}}}\,e_{-j}^a\text{\ and\ }V_{-j}^b:={\ensuremath{\Phi_{\mathrm{SU}}}}\,e_{-j}^b.
\end{array}$$
$$\label{di_formtrial2}
\begin{array}{rll}
f_{j}^\star&:=& \sqrt{\tfrac{1-{\ensuremath{\lambda}}_j}{2}} e_{-j}^\star+\sqrt{\tfrac{1+{\ensuremath{\lambda}}_j}{2}}e_{j}^\star,\\
f_{-j}^\star&:=& -\sqrt{\tfrac{1+{\ensuremath{\lambda}}_j}{2}}e_{-j}^\star+\sqrt{\tfrac{1+{\ensuremath{\lambda}}_j}{2}} e_{j}^\star,
\end{array}$$
and $$\label{di_formtrial22}
\forall\,j\in \mathbb{Z}^*,\ F_j^\star:={\ensuremath{\Phi_{\mathrm{SU}}}}(f_j^\star).$$ The trial state $Q_n$ has the following form. $$\label{di_formtrial3}
\left\{\begin{array}{rll}
Q_n&=&\text{Proj}\,V_{0,n}-\text{Proj}\,{\ensuremath{\mathrm{C}}}V_{0,n}+{\ensuremath{\displaystyle\sum}}_{j\ge 1}{\ensuremath{\lambda}}_jq_{j;n} \\
q_{j;n}&=&\text{Proj}\,F_{j}^a-\text{Proj}\,F_{-j}^a+\text{Proj}\,F_{j}^b-\text{Proj}\,F_{-j}^b.
\end{array}
\right.$$
\[di\_diag\_extrac\] Thanks to the cut-off the sequences $(\psi_n)_n$ and $(e_{j;n})_n$ are $H^1$-bounded. Up to translation and extraction ($(n_k)_k\in\mathbb{N}^{\mathbb{N}}$ and $(x_{n_k})_k\in(\mathbb{R}^3)^{\mathbb{N}}$), we can assume that the weak limit of $(\psi_n)_n$ is non-zero (if it were then there would hold $E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}=2m(2j_0+1)$).
We can consider the weak limit of each $(e_n)$: by means of a diagonal extraction, we assume that all the $(e_{j,n_{k}}(\cdot -x_{n_k}))_k$ and $(\psi_{j,n_k}(\cdot-x_{n_k}))_k$, converge along the same subsequence $(n_k)_k$. We also assume that $$\label{di_spec_conv}
\forall\,j\in\mathbb{N},\ {\ensuremath{\lambda}}_{j,n_k}\to\mu_j,\ (\mu_j)_j\in\ell^2,\ (\mu_j)_j\text{\ non-increasing},$$ and that the above convergences also hold in $L^2_{\text{loc}}$ and almost everywhere.
Upper bound and rough lower bound of $E_{j_0,\pm}$ {#di_subscritic}
--------------------------------------------------
We aim to prove the upper bound of Proposition \[di\_est\]. The method will also give a rough lower bound of $E_{j_0,\pm}$.
We write: $$C(j_0):=j_0^2\underset{-j_0\le m\le j_0}{\sup}{\ensuremath{\lVert\Psi_{m,j_0\pm \tfrac{1}{2}}\rVert_{L^{\infty}}}}^4,$$ where the functions $\Psi_{m,j_0\pm\tfrac{1}{2}}$ are defined in [@Th p. 125]: they are the upper or lower spinors of the $\Phi^{\pm}_{m,\kappa_{j_0}}$’s.
For $E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t}})$, we only consider $\mathbf{t}\in\{(1,0);(0,1)\}$ and ${\ensuremath{\varepsilon}}(\mathbf{t})$ is defined in Notation \[di\_eps\_t\].
– We consider trial state of the following form: $$Q=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi),$$ where ${\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)$ is of type $(\ell_0+\tfrac{1}{2},{\ensuremath{\varepsilon}}(\mathbf{t}))$ and ${\ensuremath{\mathcal{P}^0_-}}\psi=0$. For short, we write $$N_\psi:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)\text{\ and\ }N_{{\ensuremath{\mathrm{C}}}\psi}:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi).$$
The set of these states is written $\mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$. We will prove that the energy of a particular $Q$ gives the upper bound. The BDF energy of $Q\in \mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$ is: $$\label{di_form_no_pol}
2{\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|N_{\psi}\big)-\alpha{\ensuremath{\lVertN_\psi\rVert_{\text{Ex}}}}^2-\alpha \mathfrak{Re}\,{\ensuremath{\mathrm{Tr}}}\big(N_\psi R[N_{{\ensuremath{\mathrm{C}}}\psi}]\big).$$
– We will study the non-relativistic limit $\alpha\to 0$.
– To get an upper bound, we choose a specific trial state in $\mathscr{W}_{\mathbf{t}X^{\ell_0}}$, the idea is the same as in [@sok; @pos_sok]: the trial state is written in . Before that, we precise the structure of elements in $\mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$.
#### Minimizer for $E_{\mathbf{t}X^{\ell_0}}^{nr}$
By an easy scaling argument, there exists a minimizer for the non-relativistic energy $E_{\mathbf{t}X^{\ell_0}}^{nr}$ . The scaling argument enables us to say that this energy is negative. Then it is clear that a minimizing sequence converges to a minimizer ${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}$, up to extraction. Writing $$H_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}:=-\Delta-R_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}},$$ this minimizer satisfies the self-consistent equation $$\big[H_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}, {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big]=0.$$ This comes from Remark \[di\_prec\_non\_rel\]. In particular, $H_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}$ restricted to $\mathrm{Ran}\,{\ensuremath{\Gamma}}$ is a homothety by some $-e^2<0$, so $$\forall\,\psi\in\mathrm{Ran}\,{\ensuremath{\overline{{\ensuremath{\Gamma}}}}},\ {\ensuremath{\lVert\psi\rVert_{L^{2}}}}=1,\ {\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}\le {\ensuremath{\lVertR_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}\psi\rVert_{L^{2}}}}\apprle {\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi\rVert_{L^{2}}}},$$ and we get $${\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}^{3/4}\apprle {\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}\ i.e.\ {\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}\apprle {\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}^{4/3}\apprle (2j_0+1)^{2/3}.$$ The last estimate comes from a simple study of a minimizer for $E_{\mathbf{t}X^{\ell_0}}^{nr}$: we have $${\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)-\frac{\pi}{2}{\ensuremath{\mathrm{Tr}}}\big(|\nabla|{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\le \mathcal{E}_{nr}({\ensuremath{\overline{{\ensuremath{\Gamma}}}}})<0,$$ thus ${\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\apprle j_0^2$ and $ {\ensuremath{\mathrm{Tr}}}\big((-\Delta)^2{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\apprle j_0^{5/2}.$
We end this bootstrap argument at ${\ensuremath{\lVert|\nabla|^{3}\psi\rVert_{L^{2}}}}$ for $\psi\in\mathrm{Ran}\,\psi$: we have $$\begin{aligned}
|\nabla|^3\psi&=\frac{-\Delta}{e^2-\Delta}\Big([|\nabla|,R_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}]\psi+R_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}\psi\Big),\\
{\ensuremath{\lVert\,|\nabla|^3\psi\rVert_{L^{2}}}}&\apprle {\ensuremath{\lVert\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVert\nabla {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\mathfrak{S}_{2}}}}\apprle j_0^{5/2}.\end{aligned}$$
#### Trial state
We take the following trial state. First, let ${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}=\text{Proj}\,ra_0(r)\Psi_{j_0,j_0+{\ensuremath{\varepsilon}}(\mathbf{t})\tfrac{1}{2}}$ be a minimizer for $E_{\mathbf{t}X^{\ell_0}}^{nr}$. We form $$\label{di_trial_non_rel_1}
{\ensuremath{\overline{N}}}_+:= \text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\, {\ensuremath{\mathcal{P}^0_+}}U_{{\ensuremath{\lambda}}^{-1}}(ra_0(r)\Phi^{+}_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})(j_0+\tfrac{1}{2})})$$ where we recall that $${\ensuremath{\lambda}}:=\frac{g'_1(0)^2}{\alpha m}\text{\ and\ }U_a \phi(x):=a^{3/2} \phi(ax),\ a>0.$$ This corresponds to dilating ${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}$ by ${\ensuremath{\lambda}}^{-1}$ and projecting the range of the dilation onto $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$. Of course $ {\ensuremath{\Gamma}}\in \mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2))$ is embedded in $\mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2\times\mathbb{C}^2))$ as follows: $${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\mapsto \begin{pmatrix} {\ensuremath{\overline{{\ensuremath{\Gamma}}}}} & 0 \\ 0 & 0\end{pmatrix}\in \mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2\times\mathbb{C}^2)).$$
Then we define $$\label{di_trial_non_rel_2}
{\ensuremath{\overline{N}}}_-:={\ensuremath{\mathrm{C}}}{\ensuremath{\overline{N}}}_-{\ensuremath{\mathrm{C}}}^{-1}={\ensuremath{\mathrm{C}}}{\ensuremath{\overline{N}}}_-{\ensuremath{\mathrm{C}}}.$$ Our trial state is $$\label{di_trial_non_rel_end}
{\ensuremath{\overline{N}}}:={\ensuremath{\overline{N}}}_+-{\ensuremath{\overline{N}}}_-.$$
#### Upper bound for $E_{j_0,\pm}$
We compute $\mathcal{E}^0_{\text{BDF}}({\ensuremath{\overline{N}}})$.
Before that, we study a general projector $\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi$ where ${\ensuremath{\mathcal{P}^0_-}}\psi=0$ and ${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi$ irreducible of type $(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))$.
As an element of $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$, the wave function $\psi$ can be written $$\psi={\ensuremath{\mathcal{P}^0_+}}\begin{pmatrix}{\ensuremath{\varphi}}\\ 0\end{pmatrix}.$$ As it spans an irreducible representation of type $(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))$, we can choose $$\forall\, x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ {\ensuremath{\varphi}}(x):= ia(r)\Psi_{j_0+{\ensuremath{\varepsilon}}(\mathbf{t})\tfrac{1}{2}}^{j_0}({\ensuremath{\omega}}_x),\ a(r)\in L^2\big((0,\infty),r^2dr\big),$$ where we used notations of [@Th p. 126]. This corresponds to taking $$\psi:={\ensuremath{\mathcal{P}^0_+}}ra(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})},\ {\ensuremath{\varepsilon}}={\ensuremath{\varepsilon}}(\mathbf{t}).$$
We recall the following formulae of [@Th pp. 125-127] (with $\boldsymbol{{\ensuremath{\omega}}}:x\mapsto\tfrac{x}{|x|}$) $$\label{di_form_th_125}
\begin{array}{l}
-i\boldsymbol{\alpha}\cdot \nabla=-i(\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}})\partial_r+\frac{i}{r}(\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}})(2{\ensuremath{\mathbf{S}}}\cdot {\ensuremath{\mathbf{L}}}),\\
\big\{{\ensuremath{\mathbf{S}}}\cdot {\ensuremath{\mathbf{L}}},\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}}\big\}=-\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}}\text{\ and\ }i\boldsymbol{\sigma}\cdot \boldsymbol{{\ensuremath{\omega}}}\Psi^{m_j}_{j\pm \tfrac{1}{2}}=\Psi^{m_j}_{j\mp\tfrac{1}{2}}.
\end{array}$$ This gives $$\label{di_form_en_trial1}
\begin{array}{rcl}
{\ensuremath{\mathcal{P}^0_+}}a(r)\Phi^+_{m,{\ensuremath{\varepsilon}}(\mathbf{t})(j_0+\tfrac{1}{2})}&=&\dfrac{1}{2}\begin{pmatrix}i\big(1+\frac{g_0(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}|} \big)a(r) \Psi^m_{j_0+{\ensuremath{\varepsilon}}\tfrac{1}{2}}\\ \frac{g_1(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}||\nabla|}\big(\partial_r (a(r))+{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})\tfrac{a(r)}{r}\big)\Psi^m_{j_0-{\ensuremath{\varepsilon}}\tfrac{1}{2}}\end{pmatrix},\\
&=:&\begin{pmatrix} ia_{\uparrow}(r)\Psi^m_{j_0+{\ensuremath{\varepsilon}}\tfrac{1}{2}}\\ a_{\downarrow}({\ensuremath{\varepsilon}},j_0;r)\Psi^m_{j_0-{\ensuremath{\varepsilon}}\tfrac{1}{2}}\end{pmatrix}.
\end{array}$$ We write $ \mathrm{Op}:=\frac{g_1(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}||\nabla|}:$ the following holds. $$\label{di_form_en_trial2}
\begin{array}{rcl}
\Big| {\ensuremath{\mathrm{Tr}}}\big(N_\psi R[N_{{\ensuremath{\mathrm{C}}}\psi}]\big)\Big|&\apprle &j_0^2\sup_{m}{\ensuremath{\lVert\Psi^m_{j_0\pm\tfrac{1}{2}}\rVert_{L^{\infty}}}}^2{\ensuremath{\lVert\,|a_{\uparrow}a_{\downarrow}({\ensuremath{\varepsilon}},j_0,\cdot)|\rVert_{\mathcal{C}}}}^2\\
&\apprle& C(j_0)D\Big(|a_{\uparrow}|^2; |\mathrm{Op}\cdot\partial_r (a(r))|^2+j_0^2|\mathrm{Op}\cdot r^{-1}a(r)|^2\Big),\\
&\apprle& C(j_0) {\ensuremath{\langle |\nabla|\psi\,,\,\psi\rangle}\xspace}{\ensuremath{\lVert\nabla\psi\rVert_{L^{2}}}}^2=: \mathcal{R}em_0(j_0,\psi).
\end{array}$$
In fact, we have ${\ensuremath{\mathrm{Tr}}}\big(N_\psi R[N_{{\ensuremath{\mathrm{C}}}\psi}] \big)\ge 0$ by direct computation.
Let us deal with ${\ensuremath{\lVertN_\psi\rVert_{\text{Ex}}}}^2$.
We write ${\ensuremath{P_{\uparrow}}}$ the projection onto the upper part of $\mathbb{C}^2\times\mathbb{C}^2$ and ${\ensuremath{P_{\downarrow}}}$ the projection onto the lower part. That is: ${\ensuremath{P_{\uparrow}}}\psi$ has no lower spinor and the same upper spinor as $\psi$.
Similarly, $$\begin{aligned}
{\ensuremath{\lVertN_\psi\rVert_{\text{Ex}}}}^2-{\ensuremath{\lVert{\ensuremath{P_{\uparrow}}}N_\psi {\ensuremath{P_{\uparrow}}}\rVert_{\text{Ex}}}}^2&={\ensuremath{\mathrm{Tr}}}\big({\ensuremath{P_{\uparrow}}}N_\psi {\ensuremath{P_{\downarrow}}}R_{N_\psi}\big)+{\ensuremath{\mathrm{Tr}}}\big({\ensuremath{P_{\downarrow}}}N_\psi {\ensuremath{P_{\uparrow}}}R_{N_\psi}\big)\\
&\ \ +{\ensuremath{\lVert{\ensuremath{P_{\downarrow}}}N_\psi {\ensuremath{P_{\downarrow}}}\rVert_{\text{Ex}}}},\\
&\apprle \mathcal{R}em(j_0,\psi)+C(j_0){\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2{\ensuremath{\lVert\tfrac{|\nabla|^{3/2}}{|D_0|} \psi\rVert_{L^{2}}}}^2,\\
&=:\mathcal{R}em_1(j_0,\psi).\end{aligned}$$ For the trial state , this gives: $$\begin{aligned}
{\ensuremath{\lVert{\ensuremath{\overline{N}}}_+\rVert_{\text{Ex}}}}^2&={\ensuremath{\lVert{\ensuremath{P_{\uparrow}}}N_\psi {\ensuremath{P_{\uparrow}}}\rVert_{\text{Ex}}}}^2+\mathcal{O}\Big( C(j_0)\big( \alpha^3j_0+\alpha^5j_0^{5/3}\big)\Big)\\
&=\frac{\alpha m}{g'_1(0)^2}{\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}^2(1+\mathcal{O}({\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2))\\
&\ \ \ +\mathcal{O}\big( {\ensuremath{\lVert\tfrac{\Delta}{1-\Delta} \psi\rVert_{L^{2}}}}^2({\ensuremath{\lVert\tfrac{|\nabla|^{5/2}}{1-\Delta} \psi\rVert_{L^{2}}}}+{\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2)\big),\\
&=\frac{\alpha m}{g'_1(0)^2}{\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}^2\\
&+\mathcal{O}\Big[C(j_0)\Big(\alpha^3j_0^{5/3}+ \underset{0\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big(\alpha^2j_0^{2/3}+\underset{2^{-1}\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big) \Big)\Big].\end{aligned}$$
We compute the kinetic energy as in [@sok; @pos_sok]: we get $$\begin{aligned}
{\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}| {\ensuremath{\overline{N}}}_+\big)&=\frac{\alpha^2m}{g'_1(0)^2}{\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\big(1+K\alpha\big)+\mathcal{O}\big(\alpha^4{\ensuremath{\mathrm{Tr}}}\big((\Delta)^2{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\big),\\
&=\frac{\alpha^2m}{g'_1(0)^2}{\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)+\mathcal{O}\big(\alpha^3 j_0+\alpha^4j_0^{5/2}\big).\end{aligned}$$ This proves $$\begin{array}{|l}
E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}\le 2m(2j_0+1)+\frac{\alpha^2m}{g'_1(0)^2}E_{\mathbf{t}X^{\ell_0}}^{nr}+\mathcal{O}\big(\varrho(\alpha,j_0)\big)\\
\varrho(\alpha,j_0):=\alpha^3 j_0+\alpha^4j_0^{5/2}+C(j_0)\Big(\alpha^3j_0^{5/3}+ \underset{0\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big(\alpha^2j_0^{2/3}+\underset{2^{-1}\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big) \Big).
\end{array}$$
First, by Kato’s inequality , we have $${\ensuremath{\lVertN_\psi-N_{{\ensuremath{\mathrm{C}}}\psi}\rVert_{\text{Ex}}}}^2\le \frac{\pi}{2}{\ensuremath{\mathrm{Tr}}}\big(|\nabla| (N_\psi+N_{{\ensuremath{\mathrm{C}}}\psi})\big)=\pi{\ensuremath{\mathrm{Tr}}}\big(|\nabla|N_{\psi}\big).$$ So $$\mathcal{E}^0_{\text{BDF}}(Q)\ge 2\Big({\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|N_{\psi}\big)-\alpha\frac{\pi}{2}{\ensuremath{\mathrm{Tr}}}\big(|\nabla|N_{\psi}\big)\Big)=:2\big((2j_0+1)m+\mathcal{F}(N_\psi)\big).$$ As $\alpha$ tends to $0$, a minimizer over $\mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$ should be localized in Fourier space around $0$. Indeed, for $\alpha,L$ sufficiently small, we have $$\forall\,p\in B(0,{\ensuremath{\Lambda}}),\ {\ensuremath{\widetilde{E}\left(p\right)}}-m=\frac{g_0(p)^2-m^2+g_1(p)^2}{{\ensuremath{\widetilde{E}\left(p\right)}}+m}\ge \frac{p^2}{2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}|D_0|},$$ and for any $0<s\le 2$: $$\frac{p^2}{2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}|D_0|}\ge s\frac{\alpha\pi}{2}|p|\iff |p|\ge \frac{\alpha s\pi {\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}}{\sqrt{1-(\alpha s\pi{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}})^2}}=:\vartheta_{s}.$$ We get $$2\mathcal{F}\big( \Pi_{\vartheta_1} N_{\psi} \Pi_{\vartheta_1}\big)\le \mathcal{E}^0_{\text{BDF}}(Q)-2(2j_0+1)m.$$ By Cauchy-Schwartz inequality, we get a rough lower bound $${\ensuremath{\mathrm{Tr}}}\big(-\Delta \Pi_{\vartheta_1} N_{\psi} \Pi_{\vartheta_1} \big)\apprle \alpha^2(2j_0+1)\text{\ and\ }\mathcal{E}^0_{\text{BDF}}(Q)-2(2j_0+1)m\apprge -\alpha^2(2j_0+1).$$ For an almost minimizer $Q$, the same argument shows that $$\label{di_alm_min}
{\ensuremath{\mathrm{Tr}}}\big(\frac{-\Delta}{|{\ensuremath{\mathcal{D}^0}}|}Q^2\big)\apprle \alpha^2 (2j_0+1).$$
A precise lower bound is obtained once we know that there exists a minimizer ${\ensuremath{\overline{P}}}_{j_0}$. This state satisfies the self-consistent equation : see Subsection \[di\_low\_bound\].
The same method can be used to get an upper bound of $E_{p(X)}^{nr}$ for any $p(X)=\sum_{\ell=0}^{\ell_0}\mathbf{t}_{\ell}X^\ell$. By scaling we have $E_{p(X)}^{nr}<0.$
Strategy of the proof: the para-positronium
-------------------------------------------
The method is more subtle because $\mathscr{M}_{\mathscr{I}}$ has only one connected component. We first consider the subset $\mathscr{M}_{\mathscr{I}}^{1}$ defined by: $$\label{di_trial_isym}
\mathscr{M}_{\mathscr{I}}^{1}=\big\{P_\psi:={\ensuremath{\mathcal{P}^0_-}}+{\ensuremath{|\psi\rangle}\xspace}{\ensuremath{\langle \psi|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace},\ \psi\in\mathbb{S}\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}\big\}.$$
\[di\_infimum\_1\] Let $F_{\mathscr{I}}$ be the infimum of the BDF energy over $\mathscr{M}_{\mathscr{I}}^{1}$. Then we have $$F_{\mathscr{I}}\ge 2m-\alpha^2\frac{E_{\mathrm{PT}}(1) m}{g'_1(0)^2}+\mathcal{O}(\alpha^3).$$
We will prove the existence of a critical point in the neighbourhood of $\mathscr{M}_{\mathscr{I}}^{1}$ *via* a mountain pass argument. In this part, we aim to prove the following Proposition.
\[di\_para\_method\] 1. In the regime of Theorem \[di\_main\], there exists a bounded sequence in $\mathscr{M}_{\mathscr{I}}-{\ensuremath{\mathcal{P}^0_-}}$ of almost critical points: $(Q_n=P_n-{\ensuremath{\mathcal{P}^0_-}})_n$ such that $$\underset{n\to+\infty}{\lim}{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\rVert_{\mathfrak{S}_{2}}}}=0\mathrm{\ with\ }\mathcal{E}^0_{\text{BDF}}(Q_n)= 2m-\frac{\alpha^2 m}{g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3).$$ Furthermore, for sufficiently big $n$, there exists $\psi_{a;n}$ such that $$\mathbb{C}\psi_{a;n}=\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,\chi_{(0,+\infty)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\big)$$ and $P_n=\chi_{(-\infty,0)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\big)+{\ensuremath{|\psi_{a;n}\rangle}\xspace}{\ensuremath{\langle \psi_{a;n}|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_{a;n}\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_{a;n}|}\xspace}.$
2\. Up to a subsequence and up to translation the sequence tends to a critical point $Q_{\infty}$ of $\mathcal{E}^0_{\text{BDF}}$ in $\mathscr{M}_{\mathscr{I}}-{\ensuremath{\mathcal{P}^0_-}}$.
Moreover, writing ${\ensuremath{\overline{P}}}=Q_\infty+{\ensuremath{\mathcal{P}^0_-}}$, there exists $0<\mu<m$ and $\psi_a\in \mathbb{S}\,{\ensuremath{\mathfrak{H}_\Lambda}}$ such that $$\left\{\begin{array}{ccl}
{\ensuremath{\overline{P}}}&=&\chi_{(-\infty,0)}(D_{Q_\infty}^{({\ensuremath{\Lambda}})})+{\ensuremath{|\psi_a\rangle}\xspace}{\ensuremath{\langle \psi_a|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a|}\xspace},\\
\mathbb{C}\psi_a&=&\mathrm{Ker}\big(D_{Q_\infty}^{({\ensuremath{\Lambda}})}-\mu\big),\\
\inf\sigma(|D_{Q_\infty}^{({\ensuremath{\Lambda}})}|)&=&\mu.
\end{array}
\right.$$
#### Proof of Proposition \[di\_para\_method\]: first part
For any $\psi\in \mathbb{S}\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$, we define: $$\label{di_def_c(t)}
c_{\psi}:\begin{array}{rcl}
[0,1] &\longrightarrow& \mathscr{M}_{\mathscr{I}}-{\ensuremath{\mathcal{P}^0_-}}\\
s&\mapsto& {\ensuremath{|\sin(\pi s)\psi+\cos(\pi s){\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle \sin(\pi s)\psi+\cos(\pi s){\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace}.
\end{array}$$
\[di\_remark\_cross\] The loop $c_\psi+{\ensuremath{\mathcal{P}^0_-}}$ crosses $\mathscr{M}_{\mathscr{I}}^{1}$ at $t_0=\tfrac{1}{2}$ where the BDF energy is maximal: $$\underset{s\in[0,1]}{\sup}\mathcal{E}^0_{\text{BDF}}(c(s)).$$ Indeed, there holds $$\mathcal{E}^0_{\text{BDF}}(c(s))=2\sin(\pi s)^2{\ensuremath{\langle |{\ensuremath{\mathcal{D}^0}}|\psi\,,\,\psi\rangle}\xspace}-\alpha\sin(\pi s)^2\big[D\big(|\psi|^2,|\psi|^2\big)+\cos(2\pi s)D\big(\psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi, \psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\big)\big],$$ and the derivative with respect to $s$ is: $$\begin{array}{l}
\frac{d}{d s} \mathcal{E}^0_{\text{BDF}}(c(s_0))=2\pi\sin(2\pi s_0)\Big({\ensuremath{\langle |{\ensuremath{\mathcal{D}^0}}|\psi\,,\,\psi\rangle}\xspace}-\frac{\alpha}{2}\big[D\big(|\psi|^2,|\psi|^2\big)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +(\sin(\pi s_0)^2-\tfrac{1}{2}\cos(2\pi s_0)) \alpha D\big(\psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi, \psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\big)\big]\Big).
\end{array}$$ For sufficiently small $\alpha$, this quantity vanishes only at $2\pi s_0\equiv 0[\pi]$.
What happens when we apply the gradient flow $\Phi_{\text{BDF},t}$ of the BDF energy ? The loop $c_{\psi}$ is transformed into $c_{t}:=\Phi_{\text{BDF},t}(c_\psi)$ and we still have $$c_t(s=0)=c_t(s=1)=0.$$ This follows from the fact that ${\ensuremath{\mathcal{P}^0_-}}$ is the global minimizer of $\mathcal{E}^0_{\text{BDF}}$.
We recall that for all $s\in[0,1]$, the function $c_t(s)$ satisfies the equation $$\forall\,t_0\in\mathbb{R}_+,\ \frac{d}{dt}(c_{t_0}(s))=-\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_0}(s))\in \text{T}_{c_{t_0}(s)+{\ensuremath{\mathcal{P}^0_-}}}\mathscr{M}_{\mathscr{I}}.$$
The non-trivial result holds.
\[di\_non\_triv\] Let $P_\psi\in\mathscr{M}_{\mathscr{I}}^1$ be a state whose energy is close to the infimum $F_{\mathscr{I}}$: $$\mathcal{E}^0_{\text{BDF}}\big(P_\psi\big)<F_{\mathscr{I}}+\alpha^3.$$ Let $c_\psi$ be the loop associated to $\psi$ (see ) and $c_t:=\Phi_{\text{BDF},t}(c_\psi)$. Then for all $t\in\mathbb{R}_+$, the loop $c_t$ crosses the set $\mathscr{M}_{\mathscr{I}}^{1}$ at some ${\ensuremath{\widetilde{s}}}(t)\in(0,1)$.
\[di\_ex\_critic\] Let $(c_t)_{t\ge 0}$ be the family of loops defined in Lemma \[di\_non\_triv\] and let $(s(t))_{t\ge 0}$ be a family of reals in $(0,1)$ such that $$\forall\,t\ge 0,\ \mathcal{E}^0_{\text{BDF}}\big(c_t(s(t))\big)=\underset{s\in[0,1]}{\sup}\mathcal{E}^0_{\text{BDF}}(c_t(s)).$$ Then there exists an increasing sequence $(t_n)_{n\in\mathbb{N}}$ the sequence $(c_{t_n}(s(t_n)))_{n\ge 0}$ satisfies the first point of Proposition \[di\_para\_method\]
We prove Lemmas \[di\_infimum\_1\] and \[di\_non\_triv\] in Subsection \[di\_fait\_ch\]. We assume they are true to prove Lemma \[di\_ex\_critic\] and Proposition \[di\_para\_method\].
The proof of Lemma \[di\_non\_triv\] uses an index argument. We kept it elementary but it is possible to rephrase it in terms of the Maslov index [@LagGrass] once we notice that ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ induces a symplectic structure and that the projectors in $\mathscr{M}_{\mathscr{I}}$ are Lagrangians (see Remark \[Lagrangians\]).
##### Spectral decomposition of $P_n$
We define $$F_1:=\liminf_{t\to +\infty}\mathcal{E}^0_{\text{BDF}}(c_t(s(t)))=\liminf_{t\to+\infty}\underset{s\in[0,1]}{\sup}\mathcal{E}^0_{\text{BDF}}(c_t(s)).$$ We assume $(t_n)_{n\ge 0}$ is a minimizing sequence for $F_1$.
We may assume that $\lim_{n\to+\infty}t_n=+\infty$. – First we prove that along the path $c_t$ the gradient $\nabla \mathcal{E}^0_{\text{BDF}}$ (see ) is bounded in $\mathfrak{S}_2$. Indeed, for all $P=Q+{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{M}$, we write $${\ensuremath{\widetilde{Q}}}:=P-\chi_{(-\infty,0)}\big(\Pi_{\ensuremath{\Lambda}}D_Q\Pi_{\ensuremath{\Lambda}}\big),$$ We recall that $D_Q^{({\ensuremath{\Lambda}})}:=\Pi_{\ensuremath{\Lambda}}D_Q \Pi_{\ensuremath{\Lambda}}$: $$\label{di_form_grad}
\begin{array}{rcl}
\nabla \mathcal{E}^0_{\text{BDF}}(P)&=&\big[\big[D_Q^{{\ensuremath{\Lambda}}},P\big],P\big]=\big\{|D_Q^{({\ensuremath{\Lambda}})}|;{\ensuremath{\widetilde{Q}}}\big\}-2{\ensuremath{\widetilde{Q}}} D_{Q}^{({\ensuremath{\Lambda}})} {\ensuremath{\widetilde{Q}}},\\
\lVert \nabla \mathcal{E}^0_{\text{BDF}}(P)\rVert_{\mathfrak{S}_2}&\apprle&{\ensuremath{\lVert{\ensuremath{\widetilde{Q}}}\rVert_{\mathfrak{S}_{2}}}}{\ensuremath{\widetilde{E}\left({\ensuremath{\Lambda}}\right)}}\Big[(1+{\ensuremath{\lVertQ\rVert_{\mathfrak{S}_{2}}}})(1+{\ensuremath{\lVert{\ensuremath{\widetilde{Q}}}\rVert_{\mathfrak{S}_{2}}}})\Big]\\
&\apprle& K({\ensuremath{\Lambda}},F_1+\alpha^3).
\end{array}$$ We have used the Cauchy expansion to get an expression $$\chi_{(-\infty,0)}\big(D_Q^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}={\ensuremath{\displaystyle\sum}}_{k=1}^{+\infty}\alpha^k M_k[Y[Q]]$$ where $M_k[Y[Q]]$ is a polynomial function of $\pi_{\ensuremath{\Lambda}}R_{Q}\Pi_{\ensuremath{\Lambda}}$ of degree $k$. We refer the reader to these papers or to - above for more details. From formula and Remark \[di\_diff\_ch\] we see that the gradient, as a function of $Q$ is *locally Lipschitz*, at least in some ball $\{Q_0:\,{\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2} Q_0\rVert_{\mathfrak{S}_{2}}}}\le C_0\}$ in which there holds $$\inf \sigma\big(|D_{Q_0}^{({\ensuremath{\Lambda}})}|\big)\ge K(C_0),$$ where $C_0$ is some constant. The Lipschitz constant depends on the constant $C_0$ and in the present case, we can take $C_0\apprle 1$.
Let us prove that $$\label{di_grad_zero}
\lim_{n\to+\infty}{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_n}(s(t_n))) \rVert_{\mathfrak{S}_{2}}}}=0.$$ If not, the $\limsup$ is bigger than some $\eta>0$ and then we get a contradiction when we consider $n_0$ large enough such that $$|F_1-\mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}} (s(t_{n_0})))|\ll \eta\text{\ and\ }{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}}(s(t_{n_0})))\rVert_{\mathfrak{S}_{2}}}}\ge \frac{\eta}{2},$$ because $$\forall\,\tau>0,\ \mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}+\tau}(s(t_{n_0})))-\mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}}(s(t_{n_0})))=-{\ensuremath{\displaystyle\int}}_0^{\tau}{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}+u})(s_{t_{n_0}}) \rVert_{\mathfrak{S}_{2}}}}^2du.$$
– We recall that the gradient at $P\in\mathscr{M}$ is the “projection” of the mean-field operator onto the tangent plane $\text{T}_{P}\mathscr{M}$, in the sens that $$\begin{array}{l}
\forall\,v\in \text{T}_{P}\mathscr{M},P D_Q (1-P)\in\mathfrak{S}_1\text{\ and\ }\\
\ \ \ \ \ \ \ \ \ \ \ \ {\ensuremath{\mathrm{Tr}}}\big(P D_Q (1-P) v+(1-P)D_Q P v\big)={\ensuremath{\mathrm{Tr}}}\big( \nabla \mathcal{E}^0_{\text{BDF}} \big)
\end{array}$$
For short, we write $$Q_n:=c_{t_n}\big(s(t_n)\big)\text{\ and\ }P_n:=Q_n\text{\ and\ }v_n:=\nabla \mathcal{E}^0_{\text{BDF}}(Q_n).$$ Moreover, we write $${\ensuremath{\widetilde{D}}}_{Q_n}:=D_{Q_n}-v_n\text{\ and\ } {\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}:=\chi_{(-\infty,0)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-v_n\big).$$
We have shown that $\lim_{n\to+\infty}{\ensuremath{\lVertv_n\rVert_{\mathfrak{S}_{2}}}}=0.$
But as $v_n$ is an element of the tangent plane $\text{T}_{P_n}\mathscr{M}$, we have $$\big[ \big[v_n,P_n\big] ,P_n\big]=P_n v_n(1-P_n)+(1-P_n)v_nP_n=v_n$$ thus $$\big[ \big[ D_{Q_n}^{({\ensuremath{\Lambda}})}-v_n,P_n\big] ,P_n\big]=0.$$ Equivalently, we have $$\label{di_comm_alm}
\big[{\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})},P_n \big]=(1-P_n){\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}P_n-P_n{\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}(1-P_n)=0.$$
Thus the projector $P_n$ commutes with the distorted mean-field operator ${\ensuremath{\widetilde{D}}}_{Q_n}$. We recall that $$\lim_n {\ensuremath{\lVert{\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}-D_{Q_n}^{({\ensuremath{\Lambda}})}\rVert_{\mathfrak{S}_{2}}}}=0,$$ and thus up to taking $n$ big enough, we can neglect the distortion $v_n$: all its Sobolev norms tend to zero as $n$ tends to infinity *thanks to the cut-off*.
– Thanks to Lemma \[di\_infimum\_1\] we have the following energy condition: $$2m+\mathcal{O}(\alpha^2)\le F_1\le \mathcal{E}^0_{\text{BDF}}(Q_n)\le F_1+\alpha^3=2m+\mathcal{O}(\alpha^2).$$ Using the Cauchy expansion -, we have $${\ensuremath{\lVert\,|{\ensuremath{\mathcal{D}^0}}|^{1/2}({\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}-{\ensuremath{\mathcal{P}^0_-}})\rVert_{\mathfrak{S}_{2}}}}\apprle \sqrt{L\alpha}{\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}\apprle \sqrt{L\alpha}.$$
Thus we get $$\big|{\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}-{\ensuremath{\lVertP_n-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}\rVert_{\mathfrak{S}_{2}}}}\big|\le {\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_-}}-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}\rVert_{\mathfrak{S}_{2}}}}\apprle \sqrt{L\alpha}.$$ As ${\ensuremath{\widetilde{D}}}_{Q_n}$ and $P_n$ commutes, then necessarily ${\ensuremath{\lVertP_n-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}\rVert_{\mathfrak{S}_{2}}}}^2$ is an integer equal to twice the dimension of $\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,(1-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n})$.
But we know that $$m{\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}^2\le {\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|Q_n^2\big)\le \frac{1}{1-\alpha \tfrac{\pi}{4}}\mathcal{E}^0_{\text{BDF}}(Q_n)\le \frac{2m}{1-\alpha\frac{\pi}{4}}=2m+\mathcal{O}(\alpha).$$ Then the above dimension is lesser than $1$ and it cannot be $0$ because of the energy condition $$\mathcal{E}^0_{\text{BDF}}(Q_n)\ge F_{\mathscr{I}}\ge 2m-K\alpha^2\gg \sqrt{L\alpha}.$$ This proves the first part of Proposition \[di\_para\_method\]. We have $\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,(1-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n})=\mathbb{C}\psi_{a;n}$ where $\psi_{a;n}$ is unitary. It is an eigenvector for ${\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}$ with eigenvalue $\mu_n$. From the equality: $$\mathcal{E}^0_{\text{BDF}}(Q_n)=\mathcal{E}^0_{\text{BDF}}({\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}-{\ensuremath{\mathcal{P}^0_-}})+2\mu_n-\frac{\alpha}{2}{\ensuremath{\displaystyle\iint}}\frac{|\psi_{a;n}\wedge {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_{a;n}(x,y)|^2}{|x-y|}dxdy,$$ we get $0<\mu_n<m$. We end the proof as follows.
#### Proof of Proposition \[di\_para\_method\]: second part
We follow the method of [@pos_sok]. We recall the main steps and refer the reader to this paper for further details.
– The idea is simple: we must ensure that there exists a non-vanishing weak-limit and that this weak-limit is in fact a critical point.
Let us say that $\psi_{a;n}$ is associated to the eigenvalue $\mu_n$.
– The condition of the energy ensures that the sequence $(\psi_{a;n})_n$ does not vanish in the sense that we *do not* have the following: $$\forall\,A>0,\ \limsup_n \sup_{x\in{\ensuremath{\mathbb{R}^3}}}{\ensuremath{\displaystyle\int}}_{B(x,A)}|\psi_{a;n}|^2=0.$$ Up to translation and extraction of a subsequence, we may suppose that $(Q_n)$ (resp. $(\psi_{a;n})$) converges in the weak topology of $H^1$ to $Q_\infty\neq 0$ (resp. $\psi_{a}\neq 0$). In particular these sequences also converge in $L^2_{loc}$ and *a.e.* We recall that thanks to the cut-off and Kato’s inequality , we have $Q_n\in H^1({\ensuremath{\mathbb{R}^3}}\times {\ensuremath{\mathbb{R}^3}})$ with $${\ensuremath{\lVert|D_0|Q_n\rVert_{\mathfrak{S}_{2}}}}^2\le {\ensuremath{\widetilde{E}\left({\ensuremath{\Lambda}}\right)}}{\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}Q_n\rVert_{\mathfrak{S}_{2}}}}^2\le \frac{{\ensuremath{\widetilde{E}\left({\ensuremath{\Lambda}}\right)}}}{1-\alpha \pi/4}\sup_n \mathcal{E}^0_{\text{BDF}}(Q_n).$$ A similar estimate hold for $(\psi_{a;n})$. We also suppose that $\lim_n\mu_n=\mu_{\infty}$.
– As shown in [@pos_sok], the operator $R_{Q_n}$ converges in the strong operator topology to $R_{Q_\infty}$. Thanks to the Cauchy expansion , we also have $$\text{s}.\,\lim_n\Big[\chi_{(-\infty,0)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\big)-{\ensuremath{\mathcal{P}^0_-}}\Big]=\chi_{(-\infty,0)}\big(D_{Q_\infty}^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}.$$ By that strong convergence, we also have the weak-convergence of ${\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}\psi_{a;n}$ to $D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_a$ in $L^2$ and it follows that: $$D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_a=\mu_{\infty}\psi_a\neq 0.$$
– The condition of the energy ensures that for $\alpha$ sufficiently small, the $\psi_{a;n}$’s are close to a scaled Pekar minimizer: for any $n$, there exists a Pekar minimizer ${\ensuremath{\widetilde{\phi}}}_n$ such that $$\lVert\psi_{a;n}-{\ensuremath{\lambda}}^{-3/2}{\ensuremath{\widetilde{\phi}}}_n({\ensuremath{\lambda}}^{-1}(\cdot))\rVert_{H^1}^2\le \alpha K\text{\ where\ }{\ensuremath{\lambda}}:=\frac{g'_1(0)^2}{\alpha m}.$$ The constant $K$ depends on the energy estimate of Proposition \[di\_para\_method\].
– Thanks to that, for all $n$, $\mu_n$ is an isolated eigenvalue of ${\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}$, uniformly in $n$: we have $$\mathbb{C}\psi_{a;n}=\mathrm{Ker}\big({\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}-\mu_n\big),$$ and $$\text{dist}\Big(\mu_n;\sigma\big({\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}\big)\backslash \{ \mu_n\}\Big)>K\alpha^2.$$ By functional calculus, we finally get the norm convergence of $(\psi_{a;n})_n$ to $\psi_a$ in $L^2$.
– This proves that $$\text{s}.\,\lim_n P_n=\chi_{(-\infty,0)}\big(D_{Q_\infty}^{({\ensuremath{\Lambda}})}\big)+{\ensuremath{|\psi_a\rangle}\xspace}{\ensuremath{\langle \psi_a|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a|}\xspace}\in \mathscr{M}_{\mathscr{I}},$$ and ends the proof.
Existence of a minimizer for $E_{j_0,\pm}$
------------------------------------------
We consider a family of almost minimizers $(P_{\eta_n})_n$ of type where $(\eta_n)_n$ is any decreasing sequence. We also consider the spectral decomposition of any
$Q_n:=P_{\eta_n}-{\ensuremath{\mathcal{P}^0_-}}$.
For short we write $P_n:=P_{\eta_n}$ and we replace the subscript $\eta_n$ by $n$ (for instance $\psi_n:=\psi_{\eta_n}$). Moreover, we will often write ${\ensuremath{\varepsilon}}$ instead of ${\ensuremath{\varepsilon}}(\mathbf{t})$.
We study weak limits of $(Q_{n})_n$. We recall that $Q_n$ can be written as follows:$$\label{di_spec_no}
\left\{ \begin{array}{l}
N_{+;n}={\ensuremath{\mathcal{P}^0_+}}N_{+;n}=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{\eta_n}\text{\ and\ }N_{-;n}={\ensuremath{\mathrm{C}}}N_{+;n}{\ensuremath{\mathrm{C}}},\\
Q_n=N_{+;n}-N_{-;n}+{\ensuremath{\gamma}}_n,\ \mathrm{Ran}\,N_{\pm;n}\cap\mathrm{Ker}\,{\ensuremath{\gamma}}_n=\{ 0\}.
\end{array}
\right.$$ We can suppose $$\psi_n={\ensuremath{\mathcal{P}^0_+}}a_n(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}{\mathbf{t}}},\ a_n(r)\in \mathbb{S}L^2(\mathbb{R}_+,r^2dr).$$
\[di\_newton\_rem\] The functions $\psi\in\mathrm{Ran}\,N_{\pm;n}$ are “almost” radial. We recall , giving $$\label{di_radial}
\begin{array}{| l}
\forall\,x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ |\psi(x)|\le {\ensuremath{\lVert\psi\rVert_{L^{2}}}}|s_n(r)|{\ensuremath{\lVert\Phi^{\pm}_{j_0,\pm(j_0+\tfrac{1}{2})}\rVert_{L^{\infty}}}},\\
4|s_n(r_0)|^2:=\big|(1+\frac{g_0(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}|})a_n\big|(r_0)^2+\big|\frac{g_1(|\nabla|)}{|\nabla||{\ensuremath{\mathcal{D}^0}}|}(\partial_r a_n+{\ensuremath{\varepsilon}}\tfrac{a_n}{r})\big|(r_0)^2.
\end{array}$$ In particular by Newton’s Theorem for radial function we have: $$\label{di_newton}
\forall\,\psi\in \mathrm{Ran}\,N_{\pm;n},\ |\psi|^2*\frac{1}{|\cdot|}(x_0)\le K(j_0)\frac{{\ensuremath{\lVert\psi\rVert_{L^{2}}}}^2}{|x_0|}.$$
– We first prove that there is no vanishing, that is $$\exists A>0,\ \limsup_n \sup_{z\in{\ensuremath{\mathbb{R}^3}}}\underset{B(z,A)}{{\ensuremath{\displaystyle\int}}}|\psi_{n}(x)|^2dx>0.$$ Indeed, let assume this is false. Then using , it is clear that $${\ensuremath{\lVertN_{\pm;n}\rVert_{\text{Ex}}}}^2\to 0,$$ and we get $\liminf \mathcal{E}^0_{\text{BDF}}\ge 2(2j_0+1)m+\liminf \mathcal{E}^0_{\text{BDF}}({\ensuremath{\gamma}}_n)\ge 2(2j_0+1)m,$
an inequality that is false as shown in the previous section.
**Thus, we have: $Q_n\rightharpoonup Q_{\infty}\neq 0$.**
– As the BDF energy is sequential weakly lower continuous [@Sc], we have $$E_{j_0,{\ensuremath{\varepsilon}}}\ge \mathcal{E}_{\text{BDF}}^0(Q_{\infty}).$$ Our aim is to prove that $Q_{\infty}+{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{W}_{\mathbf{t}X^{\ell_0}}$: in other words that $Q_{\infty}$ is a minimizer for $E_{j_0,{\ensuremath{\varepsilon}}}$.
The spectral decomposition is not the relevant one: let us prove we can describe $P_n$ in function of the spectral spaces of the “mean-field operator” ${\ensuremath{\widetilde{D}}}_{Q_n}$: the first step is to prove below.
We recall that $Q_n$ satisfies Eq. , that we have the decomposition .
Using , we have for all $\psi$ in $\mathbb{S}\mathrm{Ran}\,N_{+;n}$: $$\begin{aligned}
{\ensuremath{\langle {\ensuremath{\widetilde{D}}}_{Q_n} \psi\,,\,\psi\rangle}\xspace}-m&={\ensuremath{\langle (|{\ensuremath{\mathcal{D}^0}}|-m)\psi\,,\,\psi\rangle}\xspace}- {\ensuremath{\langle (\alpha R_{Q_n}+2\eta_n {\ensuremath{\Gamma}}_n )\psi\,,\,\psi\rangle}\xspace},\\
&\apprge -\alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi\rVert_{L^{2}}}}-\\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\apprge -\alpha^2(2j_0+1).\end{aligned}$$ Thus $\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_+\neq \{0\}.$
– Let us prove this subspace has dimension $2j_0+1$: we use the minimizing property of $Q_n$. The condition on the first derivative gives . The estimation of the energy (from above and below) obtained in the previous section gives this result. Indeed, using the Cauchy expansion and the method of [@sok], we have $$\label{di_kin_no_proof}
\begin{array}{|l}
\sqrt{{\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|{\ensuremath{\gamma}}_{vac;n}^2\big)}\apprle \alpha({\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}})\apprle \sqrt{L\alpha}\sqrt{\alpha j_0} ,\\
{\ensuremath{\gamma}}_{vac;n}:=\chi_{(-\infty,0)}\big({\ensuremath{\widetilde{D}}}_{Q_n}\big)-{\ensuremath{\mathcal{P}^0_-}}.
\end{array}$$ The Cauchy expansion is explained in - below, we assume the above estimate for the moment (see ).
We write $Q_n=N_n+{\ensuremath{\overline{{\ensuremath{\gamma}}}}}_{n}$: there holds $$\big|{\ensuremath{\lVertN_n\rVert_{\mathfrak{S}_{2}}}}^2-{\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}^2\big|\apprle L^{1/2}\alpha(2j_0+1).$$ As $2(2j_0+1)\le {\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}^2\le 2(2j_0+1)\big(1-\alpha \pi/4\big)^{-1}$, then necessarily $$\label{di_arg_en}
\big|{\ensuremath{\lVertN_n\rVert_{\mathfrak{S}_{2}}}}^2-2(2j_0+1)\big|\apprle \alpha (2j_0+1),$$ and for $\alpha$ sufficiently small, the upper bound is smaller than $4$. This proves $$\text{Dim}\,\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_+=2j_0+1.$$
\[di\_form\_nn\] There exists a unitary $\psi_{a;n}$ such that $${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}=\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_+.$$ We can assume that $\psi_{a;n}\in\mathrm{Ker}\big( \mathrm{J}_3-j_0\big)$. Then we have $$N_n:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,{\ensuremath{\mathrm{C}}}\psi_{a;n}.$$ Equivalently writing $\psi_{w;n}:={\ensuremath{\mathrm{C}}}\psi_{a;n}$ there holds ${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{w;n}=\mathrm{Ran}\,(1-P_n)\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_-$.
– We have: $$\label{di_spec_yes}
P_n=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{w;n}+{\ensuremath{\boldsymbol{\pi}}}_-^n.$$ We thus write $$Q_n=N_n+{\ensuremath{\gamma}}_{vac;n}.$$
As $ \mathrm{Ran}\,P_n$ is ${\ensuremath{\widetilde{D}}}_{Q_n}$ invariant and that ${\ensuremath{\widetilde{D}}}_{Q_n}$ is bounded (with a bound that depends on ${\ensuremath{\Lambda}}$), necessarily $${\ensuremath{\widetilde{D}}}_{Q_n}\psi_{a;n}=\mu_n\psi_{a;n},\ \mu_n\in \mathbb{R}_+.$$ As in [@pos_sok], studying the Hessian we have$$m-\mu_n+2\eta_n\ge 0.$$
– As for $\psi_n$, there is no vanishing for $(\psi_{a,n})_n$ for $\alpha$ sufficiently small: decomposing $\psi_+\in\mathrm{Ran}\,P_n$: $$\psi_+=a\psi_{a;n}+\phi,\ \phi\in \mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}_-^n,$$ we have $$|a|^2\ge\frac{1}{\mu}\big(m+{\ensuremath{\langle |{\ensuremath{\widetilde{D}}}_{Q_n}|\phi\,,\,\phi\rangle}\xspace}-K(\alpha^2j_0+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}) \big).$$ Provided that $\mu_n$ is close to $m$, the absence of vanishing for $\psi_n$ implies that of $\psi_{a;n}$.
By Kato’s inequality : $$\begin{aligned}
{\ensuremath{\widetilde{D}}}_{Q_n}^2&\ge |{\ensuremath{\mathcal{D}^0}}|\big(1-2\alpha{\ensuremath{\lVertR_{Q_n}|{\ensuremath{\mathcal{D}^0}}|^{-1}\rVert_{\mathcal{B}}}}-4\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathcal{B}}}}\big)|{\ensuremath{\mathcal{D}^0}}|\\
&\ge |{\ensuremath{\mathcal{D}^0}}|^2\big(1-\alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}-4\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\big)\end{aligned}$$ Thus $$\big| {\ensuremath{\widetilde{D}}}_{Q_n}\big|\ge |{\ensuremath{\mathcal{D}^0}}|\big(1-\alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}-2\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\big)\text{\ and\ }\mu_n\ge 1-K(\alpha^2 j_0 +\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}).$$ In the same way we can prove that $$|\mu_n-m|\apprle \alpha^2j_0+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}$$ So $$\psi_{a,n}\rightharpoonup\psi_{a}\neq 0.$$
– We decompose ${\ensuremath{\gamma}}_{vac;n}={\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{W}_{0}-{\ensuremath{\mathcal{P}^0_-}}$ as in : using Cauchy’s expansion -, we have $$\label{di_cauchy}
{\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}=\frac{1}{2\pi}{\ensuremath{\displaystyle\int}}_{-\infty}^{+\infty}\frac{d {\ensuremath{\omega}}}{{\ensuremath{\mathcal{D}^0}}+i\omega}\big(2\eta_n {\ensuremath{\Gamma}}_n-\alpha\Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}+2\eta_n {\ensuremath{\Gamma}}_n \big)\dfrac{1}{{\ensuremath{\widetilde{D}}}_{Q_n}+i{\ensuremath{\omega}}}\Pi_{\ensuremath{\Lambda}}.$$ To justify this equality, we remark that $|{\ensuremath{\widetilde{D}}}_{Q_n}|$ is uniformly bounded from below, it follows that the r.h.s. of is well-defined provided that $\alpha\le \alpha_{j_0}$: $$\begin{aligned}
\Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}^2&\apprle |\nabla|{\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}^2\apprle \alpha(2j_0+1)|\nabla|\le \alpha(2j_0+1)|{\ensuremath{\mathcal{D}^0}}|^2.\end{aligned}$$ We must ensure that $\alpha \sqrt{\alpha(2j_0+1)}$ is sufficiently small.
Integrating the norm of bounded operator in , we obtain $${\ensuremath{\lVert{\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}\rVert_{\mathcal{B}}}}\apprle \alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}<1.$$
We also expand in power of $Y_n:=-\alpha \Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}+2\eta_n {\ensuremath{\Gamma}}_n$ as in $$\label{di_cauchy2}
\begin{array}{rcl}
{\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}&=&{\ensuremath{\displaystyle\sum}}_{j\ge 1}\alpha^j M_j[Y_n].
\end{array}$$ We have $$\label{di_estim_gn1}
{\ensuremath{\lVert{\ensuremath{\gamma}}_{vac;n}\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha^2.$$ We take the norm ${\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}(\cdot)\rVert_{\mathfrak{S}_{2}}}}$: $$\label{di_estim_kin1}
{\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{vac;n}\rVert_{\mathfrak{S}_{2}}}}\apprle \sqrt{L\alpha}{\ensuremath{\lVertQ_N\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\apprle L^{1/2}\alpha j_0.$$
– We thus write $$\begin{array}{rcl}
{\ensuremath{\gamma}}_{vac;n}&=&{\ensuremath{\displaystyle\sum}}_{j\ge 1}{\ensuremath{\lambda}}_{j;n}q_{j;n},
\end{array}$$ where $q_{j;n}$ has the same form as the one in .
Up to a subsequence, we may assume all weak convergence as in Remark : the sequence of eigenvalues $({\ensuremath{\lambda}}_{j;n})_n$ tends to $(\mu_j)_j\in\ell^2$ and each $(e_{j;n}^\star)_n$ (with $\star\in\{a,b\}$) tends to $e_{j;\infty}^\star$, $(\psi_{e;n})_n$ tends to $\psi_{e}$. We can also assume that the sequence $(\mu_n)_n$ tends to $\mu$ with $0\le \mu\le m$.
For shot we write $\psi_v:={\ensuremath{\mathrm{C}}}\psi_e$.
Furthermore, we write ${\ensuremath{\overline{P}}}:=Q_{\infty}+{\ensuremath{\mathcal{P}^0_-}}$ and ${\ensuremath{\overline{\boldsymbol{\pi}}}}:=\chi_{(-\infty,0)}(D_{Q_\infty}^{({\ensuremath{\Lambda}})})$.
– We will prove that
1. $\big[D^{({\ensuremath{\Lambda}})}_{Q_\infty} ,{\ensuremath{\overline{P}}}\big]=0$,
2. $D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_{a}=\mu\psi_a$ and so ${\ensuremath{\overline{\boldsymbol{\pi}}}}\psi_a=0$.
Moreover $D_{Q_\infty}^{({\ensuremath{\Lambda}})}{\ensuremath{\mathrm{C}}}\psi_{a}=-\mu{\ensuremath{\mathrm{C}}}\psi_a$ and ${\ensuremath{\langle {\ensuremath{\mathrm{C}}}\psi_a\,,\,\psi_a\rangle}\xspace}=0$.
3. $$\label{di_marre_form}
{\ensuremath{\overline{\boldsymbol{\pi}}}}={\ensuremath{\overline{P}}}-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi_a)+\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi_a)=:{\ensuremath{\overline{P}}}-N.$$
These results follow from the strong convergence $$\label{di_strong}
\text{s}.\,\lim_n R_{Q_n}=R_{Q_\infty}.$$ This fact enables us to show $$\begin{array}{|l}
\lim_n R_{Q_n}\psi_{a;n}=R_{Q_\infty}\psi_a\text{\ in\ }L^2,\\
\text{s.\,op.}\ \lim_n\big({\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}\big)={\ensuremath{\overline{\boldsymbol{\pi}}}}-{\ensuremath{\mathcal{P}^0_-}}\text{\ in\ }\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\\
\text{w.\,op.}\ \lim_n P_n={\ensuremath{\overline{\boldsymbol{\pi}}}}-{\ensuremath{\mathcal{P}^0_-}}+\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_a-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\psi_w\text{\ in\ }\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\\
\lim_n\psi_{a;n}=\psi_a\text{\ in\ }L^2.
\end{array}$$
\[di\_assume\] We only write in this paper the proof of $$R_{Q_n}\psi_{a;n}\underset{n\to+\infty}{\overset{L^2}{\longrightarrow}}R_{Q_\infty}\psi_a\text{\ and\ }\psi_{a;n}\underset{n\to+\infty}{\overset{L^2}{\longrightarrow}}\psi_a.$$ The convergence in the weak-topology can be proved using the same method as in [@pos_sok]. For the first limit this follows from the convergence of $R_{Q_n}$ in the strong topology. For the proof of this fact and of the strong convergence of ${\ensuremath{\gamma}}_{vac;n}={\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}$, we refer the reader to [@pos_sok].
For $R_{Q_n}$, it suffices to remark that $Q_n(x,y)$ converges in $L^2_{loc}$ and $a.e.$. To estimate the mass at infinity, we simply use the term $\tfrac{1}{|x-y|}$ in $\tfrac{Q_n(x,y)}{|x-y|}$.
The strong convergence of ${\ensuremath{\gamma}}_{vac;n}$ follows from that of $R_{Q_n}$ and the Cauchy expansion .
Then, assuming all these convergences, the convergence of $Q_n$ resp. $\big[ {\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}; P_n\big]$ in the weak operator topology to $Q_\infty$ resp. $\big[ D_{Q_\infty}^{({\ensuremath{\Lambda}})},{\ensuremath{\overline{P}}}\big]$ are straightforward.
Similarly, using , it is clear that $${\ensuremath{\widetilde{D}}}_{Q_n}\psi_{a;n}\underset{n\to+\infty}{\rightharpoonup}D_{Q_\infty}\psi_a,$$ and that $$D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_a=\mu\psi_a.$$ To get the existence of minimizer, it suffices to prove that ${\ensuremath{\lVert\psi_a\rVert_{L^{2}}}}=1$ or equivalently $\lim_n\psi_{a;n}=\psi_a$ in $L^2$.
– To prove the norm convergence of $\psi_{a;n}$ to $\psi_a$, we need a uniform upper bound of $\mu_n$, or precisely, we need the following: $$\label{di_need}
\limsup_n(m-\mu_n)>0.$$ Indeed, we then get $$\label{di_lim_l2}
({\ensuremath{\mathcal{D}^0}}-\mu_n)\psi_{a;n}=\alpha R_{Q_n}\psi_{a;n}-2\eta_n {\ensuremath{\Gamma}}_n\psi_{a;n}\text{\ and\ }\psi_{a;n}=\frac{\alpha}{{\ensuremath{\mathcal{D}^0}}-\mu_n}\big(R_{Q_n}\psi_{a;n}-2\eta_n {\ensuremath{\Gamma}}_n\psi_{a;n}\big).$$ Provided that holds and that we have norm convergence of $R_{Q_n}\psi_{a;n}$ we obtain the norm convergence of $\psi_{a;n}$.
– To prove the norm convergence of $R_{Q_n}\psi_{a;n}$ to $R_{Q_\infty}\psi_a$, we use the fact that the element of ${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}$ are “almost radial” (see in Remark \[di\_newton\_rem\]). We recall holds. In the following, we write $\delta Q_n:=Q_n-Q_\infty$ and $\delta \psi_n:=\psi_{a;n}-\psi_a$ and use Cauchy-Schwartz inequality: for any $A>0$ there hold $$\begin{aligned}
{\ensuremath{\displaystyle\int}}_{|x|\ge A}\Big|{\ensuremath{\displaystyle\int}}\frac{\delta Q_n(x,y)}{|x-y|}\psi_{a;n}(y)dy\Big|^2dx&\le {\ensuremath{\lVert\delta Q_n\rVert_{\text{Ex}}}}^2 \frac{K(j_0)}{A},\\
{\ensuremath{\displaystyle\int}}_{|x|\le A}\Big|{\ensuremath{\displaystyle\int}}\frac{\delta Q_n(x,y)}{|x-y|}\psi_{a;n}(y)dy\Big|^2dx&\le \frac{2\pi}{2}{\ensuremath{\langle |\nabla| \psi_{a;n}\,,\,\psi_{a;n}\rangle}\xspace}\underset{B(0,A)\times B(0,2A)}{{\ensuremath{\displaystyle\iint}}}\frac{|\delta Q_n(x,y)|^2}{|x-y|}dxdy\\
&\ \ \ +\frac{2}{A^2}{\ensuremath{\lVert\delta Q_n\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\psi_{a;n}\rVert_{L^{2}}}}^2.\end{aligned}$$ Thus $$\limsup_n {\ensuremath{\lVertR[Q_n-Q_\infty]\psi_{a;n}\rVert_{L^{2}}}}=0.$$ Similarly $$\begin{aligned}
{\ensuremath{\displaystyle\int}}_{|x|\ge A}\Big|\frac{Q_\infty(x,y)}{|x-y|}\delta \psi_n(y)dy\Big|^2dx&\le \frac{2}{A-\tfrac{A}{2}}{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\delta\psi_n\rVert_{L^{2}}}}^2+2{\ensuremath{\lVert\delta \psi_n\rVert_{L^{2}}}}^2\frac{2}{A}{\ensuremath{\lVertQ_\infty\rVert_{\text{Ex}}}}^2,\\
{\ensuremath{\displaystyle\int}}_{|x|\le A}\Big|\frac{Q_\infty(x,y)}{|x-y|}\delta \psi_n(y)dy\Big|^2dx&\le\frac{2\pi}{2}{\ensuremath{\langle |\nabla|\delta \psi_n\,,\,\delta\psi_n\rangle}\xspace}\underset{B(0,A)\times B(0,2A)}{{\ensuremath{\displaystyle\iint}}}\frac{|\delta Q_n(x,y)|^2}{|x-y|}dxdy\\
&\ \ \ +\frac{2}{A^2}{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\delta\psi_n\rVert_{L^{2}}}}^2,\end{aligned}$$ and $$\limsup_n{\ensuremath{\lVertR_{Q_\infty}(\psi_{a;n}-\psi_a)\rVert_{L^{2}}}}=0.$$ This proves that $$\lim_{n\to+\infty}{\ensuremath{\lVertR_{Q_n}\psi_{a;n}-R_{Q_\infty}\psi_a\rVert_{L^{2}}}}=0.$$ – Let us prove . We have: $$\label{di_need_to_proof}
\begin{array}{rcl}
2\mu_n(2j_0+1)&=&{\ensuremath{\mathrm{Tr}}}\Big({\ensuremath{\widetilde{D}}}_{Q_n}N_n\Big),\\
&=&{\ensuremath{\mathrm{Tr}}}\Big({\ensuremath{\widetilde{D}}}_{{\ensuremath{\gamma}}_{vac;n}}N_n\Big)-\alpha {\ensuremath{\lVertN_n\rVert_{\text{Ex}}}}^2,\\
&=&\mathcal{E}^0_{\text{BDF}}(Q_n)-\mathcal{E}^0_{\text{BDF}}({\ensuremath{\gamma}}_{vac;n})-\frac{\alpha}{2}{\ensuremath{\lVertN_n\rVert_{\text{Ex}}}}^2,\\
&<&2m(2j_0+1)-K(j_0)\alpha^2.
\end{array}$$ This upper bound holds provided that $\alpha\le \alpha_{j_0}$ thanks to the upper bound of $E_{j_0,{\ensuremath{\varepsilon}}}$ obtained in the previous section.
Lower bound of $E_{j_0,\pm}$ {#di_low_bound}
----------------------------
Our aim is to prove the estimate of Proposition \[di\_est\]. We consider the minimizer $Q_{\infty}=N+{\ensuremath{\gamma}}_{vac}$ found in the previous subsection. It satisfies Eq. where $$\label{di_marre_form_re}
{\ensuremath{\overline{P}}}={\ensuremath{\mathcal{P}^0_-}}+Q_\infty\text{\ and\ }{\ensuremath{\gamma}}_{vac}=\chi_{(-\infty,0)}(D_{Q_infty}^{({\ensuremath{\Lambda}})})-{\ensuremath{\mathcal{P}^0_-}}.$$
– The proof is the same as that in [@sok; @pos_sok] and relies on estimates on the Sobolev norms ${\ensuremath{\lVert\,|\nabla|^s N_+\rVert_{\mathfrak{S}_{2}}}}$ where we write $$\label{di_eq_recall}
N_+:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_a=\mathrm{Ker}\,(D_{Q_\infty}^{({\ensuremath{\Lambda}})}-\mu).$$ Using , we get $$\begin{aligned}
{\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|^2N_+\big)&= 2(2j_0+1)\mu^2+2\alpha \mu{\ensuremath{\mathrm{Tr}}}\big(R_{Q_\infty}N_+\big)+\alpha^2{\ensuremath{\mathrm{Tr}}}\big(R_{Q_\infty}^2N_+\big),\\
&\le 2(2j_0+1)\mu^2+4\alpha \mu{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}{\ensuremath{\lVert\nabla N_+\rVert_{\mathfrak{S}_{2}}}}+4\alpha^2{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\nabla N_+\rVert_{\mathfrak{S}_{2}}}}^2\end{aligned}$$ and provided that $\alpha\le \alpha_{j_0}$, we get $${\ensuremath{\mathrm{Tr}}}\big((-\Delta)N_+\big)\apprle \frac{\alpha^2(2j_0+1)}{1-4\alpha^2(2j_0+1)-2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}{\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}}.$$ We have used Hardy’s inequality: $$\label{di_hardy}
\dfrac{1}{4|\cdot|^2}\le -\Delta\text{\ in\ }{\ensuremath{\mathbb{R}^3}}.$$ We recall that $$0\le {\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}-1\apprle \alpha{\ensuremath{\log(\Lambda)}}\text{\ and\ } {\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}\apprle \alpha.$$ See (or [@sok Appendix A] for more details).
Thus for sufficiently small $\alpha$, we have $$\label{di_wf_to_scale}
\forall\,\psi\in\mathbb{S}\mathrm{Ran}\,N_+,\ {\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2\apprle \frac{\alpha^2}{1-4\alpha^2(2j_0+1)-2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}{\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}}\apprle \alpha^2.$$
– By *bootstrap argument*, we can estimate ${\ensuremath{\lVert\,\Delta N_+\rVert_{\mathfrak{S}_{2}}}}$. We have: $$\label{di_marre_boot}
\forall\,\psi\in\mathbb{S}\mathrm{Ran}\,N_+,\ {\ensuremath{\lVert\,|\nabla|^{3/2}\psi\rVert_{L^{2}}}}^2\apprle \alpha^{3}\sqrt{2j_0+1}\text{\ and\ }{\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}\apprle \alpha^4(2j_0+1)^{3/2}.$$ We prove this result below.
Furthermore, using the Cauchy expansion and , we get $$\begin{array}{| rcl}
{\ensuremath{\lVert\,|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{vac}\rVert_{\mathfrak{S}_{2}}}}&\apprle&\alpha {\ensuremath{\lVert\nabla N\rVert_{\mathfrak{S}_{2}}}}+\sqrt{L\alpha}{\ensuremath{\lVert{\ensuremath{\gamma}}_{vac}\rVert_{\text{Ex}}}}+\alpha^2{\ensuremath{\lVertQ_\infty\rVert_{\text{Ex}}}}^2\big({\ensuremath{\lVert\nabla N\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVert{\ensuremath{\gamma}}_{vac}\rVert_{\text{Ex}}}} \big),
\end{array}$$ hence $$\label{di_marre_gvac}
{\ensuremath{\lVert\,|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{vac}\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha^2\sqrt{2j_0+1}.$$
Now, if we assume -, then we get $$\text{For\ } \alpha\le \alpha_{j_0},\ \mathcal{E}^0_{\text{BDF}}\big(Q_\infty\big)=2m(2j_0+1)+\frac{\alpha^2m}{g'_1(0)^2}E_{\mathbf{t}X^{\ell_0}}^{nr}+\mathcal{O}\big(\alpha^3 K(j_0)\big).$$ We do not prove this fact: the method is the same as in [@sok; @pos_sok] (in the proof of the lower bound of $E^0_{\text{BDF}}(1)$ resp. $E_{1,1}$).
We just recall how we get .
##### Proof of
We scale the wave functions of by ${\ensuremath{\lambda}}:=\frac{g'_1(0)^2}{\alpha m}$: $$\forall\,x\in {\ensuremath{\mathbb{R}^3}},\ U_{{\ensuremath{\lambda}}}\psi(x)={\ensuremath{\underline{\psi}}}(x):={\ensuremath{\lambda}}^{3/2}\psi({\ensuremath{\lambda}}x),$$ and we split $\psi$ (resp. ${\ensuremath{\underline{\psi}}}$) into the upper spinor ${\ensuremath{\varphi}}$ (resp. ${\ensuremath{\underline{{\ensuremath{\varphi}}}}}$) and the lower spinor $\chi$ (resp. ${\ensuremath{\underline{\chi}}}$). Thanks to , we have $$\alpha^{-2}(m-\mu)=:\alpha^{-2}\delta m\ge K(j_0)>0$$ provided that $\alpha$ is sufficiently small $(\alpha\le \alpha_{j_0})$.
We write $$\forall\,Q_0\in\mathfrak{S}_2,\ {\ensuremath{\underline{Q_0}}}:=U_{{\ensuremath{\lambda}}} Q_{0}U_{{\ensuremath{\lambda}}}^{-1}=U_{{\ensuremath{\lambda}}} Q_{0}U_{{\ensuremath{\lambda}}^{-1}}.$$ For all $\psi$ in $\mathbb{S}\mathrm{Ran}{\ensuremath{\underline{N_+}}}$ we have $$\label{di_eq_scale}
\left\{\begin{array}{rcl}
{\ensuremath{\lambda}}^2\delta m{\ensuremath{\underline{{\ensuremath{\varphi}}}}}&=&i{\ensuremath{\lambda}}\boldsymbol{\sigma}\cdot \nabla {\ensuremath{\underline{\chi}}}+\alpha {\ensuremath{\lambda}}\big(R_{{\ensuremath{\underline{Q}}}_\infty} {\ensuremath{\underline{\psi}}}\big)_{\uparrow},\\
{\ensuremath{\underline{\chi}}}&=&\frac{-i{\ensuremath{\lambda}}\boldsymbol{\sigma}\cdot \nabla {\ensuremath{\underline{{\ensuremath{\varphi}}}}}}{{\ensuremath{\lambda}}(m+\mu)}-\tfrac{\alpha}{{\ensuremath{\lambda}}} \big(R_{{\ensuremath{\underline{Q}}}_\infty} {\ensuremath{\underline{\psi}}}\big)_{\downarrow}.
\end{array}
\right.$$ – We recall $$\label{di_marre_Rtrois}
\forall\,Q_0\in\mathfrak{S}_2,\ \lVert \big[\nabla,R_{Q_0}\big]\tfrac{1}{|\nabla|^{1/2}} \rVert_{\mathcal{B}}^2\apprle {\ensuremath{\displaystyle\iint}}|p-q|^2|p+q||{\ensuremath{\widehat{Q_0}}}(p,q)|^2dpdq.$$ This result was previously proved in [@pos_sok] and follows from the fact that a (scalar) Fourier multiplier $F(\mathbf{p}-\mathbf{q})=F(-i\nabla_x+i\nabla_y)$ commutes with the operator $R[\cdot]:Q(x,y)\mapsto \tfrac{Q(x,y)}{|x-y|}$. Then it suffices to use Hardy’s inequality : $${\ensuremath{\lVert\big[\nabla,R_{{\ensuremath{\underline{Q_\infty}}}}\big]{\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}^2\apprle {\ensuremath{\lambda}}^2{\ensuremath{\displaystyle\iint}}|p-q|^2|{\ensuremath{\widehat{Q}}}_{\infty}(p,q)|^2dpdq\times {\ensuremath{\lVert\nabla {\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}^2.$$
By Hardy’s inequality and , the following holds: $$\label{di_est_un_scale}
\begin{array}{| rcl}
{\ensuremath{\lVert{\ensuremath{\underline{\chi}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le& \frac{2}{4{\ensuremath{\lambda}}^2m^2}{\ensuremath{\lVert\nabla {\ensuremath{\underline{{\ensuremath{\varphi}}}}}\rVert_{\mathfrak{S}_{2}}}}^2+2\alpha^2{\ensuremath{\lVertR_{{\ensuremath{\underline{Q_\infty}}}}{\ensuremath{\underline{\psi}}}\rVert_{\mathfrak{S}_{2}}}}^2\apprle \alpha^2,\\
{\ensuremath{\lVert\nabla {\ensuremath{\underline{\chi}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le&2({\ensuremath{\lambda}}\delta m)^2+2\alpha^2{\ensuremath{\lVertR_{{\ensuremath{\underline{Q_\infty}}}} {\ensuremath{\underline{\psi}}}\rVert_{\mathfrak{S}_{2}}}}^2\apprle \frac{(\delta m)^2}{\alpha^2}+\alpha^2(2j_0+1),\\
{\ensuremath{\lVert\Delta {\ensuremath{\underline{{\ensuremath{\varphi}}}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le&2{\ensuremath{\lambda}}^2 m{\ensuremath{\lVert\nabla {\ensuremath{\underline{\chi}}}\rVert_{L^{2}}}}^2+2\alpha^2({\ensuremath{\lVert\big[\nabla,R_{{\ensuremath{\underline{Q_\infty}}}}\big]{\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}+{\ensuremath{\lVertR_{Q_\infty}\rVert_{L^{2}}}}\nabla {\ensuremath{\underline{\psi}}})^2 \\
&\apprle& \frac{(\delta m)^2}{\alpha^4}+(2j_0+1)+\alpha^2(2j_0+1)^{3/2}, \\
{\ensuremath{\lVert\Delta {\ensuremath{\underline{\chi}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le& 2{\ensuremath{\lambda}}^2(\delta m)^2{\ensuremath{\lVert\nabla {\ensuremath{\underline{{\ensuremath{\varphi}}}}}\rVert_{L^{2}}}}+2\alpha^2({\ensuremath{\lVert\big[\nabla,R_{{\ensuremath{\underline{Q_\infty}}}}\big]{\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}+{\ensuremath{\lVertR_{Q_\infty}\rVert_{L^{2}}}}\nabla {\ensuremath{\underline{\psi}}})^2 \\
&\apprle& \frac{(\delta m)^2}{\alpha^2}+(2j_0+1)+\alpha^2(2j_0+1)^{3/2}.
\end{array}$$
– There remains to estimate $${\ensuremath{\displaystyle\iint}}|p-q|^2|{\ensuremath{\widehat{Q_0}}}(p,q)|^2dpdq,\ \text{for\ }Q_0=N\text{\ and\ }{\ensuremath{\gamma}}_{vac}.$$ For $Q_0=N$, we just have to estimate ${\ensuremath{\mathrm{Tr}}}\big(|\nabla|^2 N_+\big)$.
The case $Q_0={\ensuremath{\gamma}}_{vac}$ is dealt with as in [@sok; @sokd]: by a *fixed-point* argument (valid for $\alpha\le \alpha_{j_0}$), we prove that $$\left\{{\ensuremath{\displaystyle\iint}}|p-q|^2|{\ensuremath{\widehat{{\ensuremath{\gamma}}_{vac}}}}(p,q)|^2dpdq\right\}^{1/2}\apprle \alpha\min\big({\ensuremath{\lVert\Delta N\rVert_{\mathfrak{S}_{2}}}},{\ensuremath{\lVert\,|\nabla|^{3/2}N\rVert_{\mathfrak{S}_{2}}}}\big).$$
Now, we can prove that $${\ensuremath{\mathrm{Tr}}}\big(|\nabla|^{3}N_+\big)\apprle \alpha^{5/2}(2j_0+1)^{3/2}.$$
For a unitary $\psi$ in $\mathrm{Ran}\,N_+$, there holds $$\label{di_sobtrois}
\begin{array}{rcl}
{\ensuremath{\lVert\,|\nabla|^{1/2}{\ensuremath{\mathcal{D}^0}}\psi\rVert_{L^{2}}}}^2&\le& \mu^2{\ensuremath{\langle |\nabla|\psi\,,\,\psi\rangle}\xspace}+\alpha K {\ensuremath{\lVert\,|\nabla|^{1/2}\psi\rVert_{L^{2}}}}{\ensuremath{\lVertR_{Q_\infty}\psi\rVert_{L^{2}}}}\\
&&\ \ \ +\alpha^2\big({\ensuremath{\lVert[R_{Q_\infty},|\nabla|^{1/2}] \psi\rVert_{L^{2}}}}+2{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}{\ensuremath{\lVert\,|\nabla|^{3/2}\rVert_{L^{2}}}}\big)^2.
\end{array}$$ Similarly, in Fourier space we have: $$\Big|\mathscr{F}\big([R_{Q_\infty},|\nabla|^{1/2}];p,q\big) \Big|\apprle |p-q|^{1/2}|{\ensuremath{\widehat{R}}}_{Q_\infty}(p,q)|,$$ and by Hardy’s inequality $${\ensuremath{\lVert[R_{Q_\infty},|\nabla|^{1/2}] \psi\rVert_{L^{2}}}}^2\apprle {\ensuremath{\displaystyle\iint}}|p-q| |{\ensuremath{\widehat{Q_\infty}}}(p,q)|^2dpdq{\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2\apprle {\ensuremath{\mathrm{Tr}}}\big(|\nabla| Q_\infty^2\big){\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2.$$ Substituting in , we get $${\ensuremath{\langle |\nabla|^3\psi\,,\,\psi\rangle}\xspace}\apprle \alpha^{5/2}\sqrt{2j_0+1},\text{\ hence\ }{\ensuremath{\mathrm{Tr}}}\big(|\nabla|^3N_+ \big)\apprle \alpha^{5/2}(2j_0+1)^{3/2}.$$
Proof of Lemmas \[di\_infimum\_1\] and \[di\_non\_triv\] {#di_fait_ch}
--------------------------------------------------------
### Proof of Lemma \[di\_infimum\_1\]
We consider a trial state $P_\psi\in\mathscr{M}_{\mathscr{I}}^1$: $$Q_\psi:=P_\psi-{\ensuremath{\mathcal{P}^0_-}}={\ensuremath{|\psi\rangle}\xspace}{\ensuremath{\langle \psi|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace},\ {\ensuremath{\mathcal{P}^0_+}}\psi=\psi\in\mathbb{S}\,{\ensuremath{\mathfrak{H}_\Lambda}}.$$ Its BDF energy is $$\begin{aligned}
\mathcal{E}^0_{\text{BDF}}(Q_\psi)&=2{\ensuremath{\langle |{\ensuremath{\mathcal{D}^0}}|\psi\,,\,\psi\rangle}\xspace}-\frac{\alpha}{2}{\ensuremath{\displaystyle\iint}}\frac{|\psi\wedge {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi(x,y)|^2}{|x-y|}dxdy\\
&\ge 2m+2{\ensuremath{\langle \big(|{\ensuremath{\mathcal{D}^0}}|-m\big)\psi\,,\,\psi\rangle}\xspace}-\alpha D\big(|\psi|^2,\psi^2\big)=:2m+\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi).\end{aligned}$$ We recall the following $$\begin{aligned}
|{\ensuremath{\mathcal{D}^0}}|-m&=\frac{1}{|{\ensuremath{\mathcal{D}^0}}|+m}\big((g_0(-i\nabla)-m)(g_0(-i\nabla)+m)+g_1(-i\nabla)^2\big).\end{aligned}$$
Thanks to Estimates and Kato’s inequality , we have$$\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi)\le (1-K\alpha){\ensuremath{\langle \tfrac{-\Delta}{2|{\ensuremath{\mathcal{D}^0}}|} \psi\,,\,\psi\rangle}\xspace}-\alpha\frac{\pi}{4}{\ensuremath{\langle |\nabla|\psi\,,\,\psi\rangle}\xspace}$$ We split $\psi$ into two with respect to the frequency cut-off $\Pi_{\alpha K_0}$: we get $$\psi=\Pi_{\alpha K_0}\psi+\psi_2=\psi_1+\psi_2.$$ The constant $K_0$ is chosen such that $$\frac{\alpha^2 K_0^2}{2{\ensuremath{\widetilde{E}\left(\alpha K_0\right)}}}\apprge \alpha \pi \alpha K_0.$$ Then we have $$\begin{aligned}
D\big(|\psi|^2,|\psi|^2\big)&=D\big(|\psi_1|^2,|\psi_1|^2\big)+\mathcal{O}\big({\ensuremath{\langle |\nabla|\psi_2\,,\,\psi_2\rangle}\xspace}+{\ensuremath{\lVert|\psi_1|^2\rVert_{\mathcal{C}}}}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi_2\rVert_{L^{2}}}}\big)\\
&=D\big(|\psi_1|^2,|\psi_1|^2\big)+\mathcal{O}\big({\ensuremath{\langle |\nabla|\psi_2\,,\,\psi_2\rangle}\xspace}+\sqrt{\alpha}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi_2\rVert_{L^{2}}}}\big),\end{aligned}$$ where we recall that ${\ensuremath{\lVert\rho\rVert_{\mathcal{C}}}}^2=D(\rho,\rho)$. This gives $$\begin{array}{rcl}
\tfrac{1}{2}\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi)&=&{\ensuremath{\langle \frac{g_1(-i\nabla)^2}{|{\ensuremath{\mathcal{D}^0}}|+m}\psi_1\,,\,\psi_1\rangle}\xspace}-\alpha\frac{\pi}{2}D\big(|\psi_1|^2,|\psi_1|^2\big)\\
&&\ \ \ +K{\ensuremath{\langle \frac{g_1^2(-i\nabla)}{|{\ensuremath{\mathcal{D}^0}}|} \psi_2\,,\,\psi_2\rangle}\xspace}+\mathcal{O}(\alpha^3),\\
&\ge&\frac{\alpha^2 g'_1(0)^2}{2m}{\ensuremath{\lVert\nabla\psi_1\rVert_{L^{2}}}}^2-\frac{\alpha}{2}D\big(|\psi_1|^2,|\psi_1|^2\big)+\mathcal{O}(\alpha^3),\\
&\ge&\frac{\alpha^2 m}{2g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3).
\end{array}$$
We have obtained a lower bound. Let us prove that it is attained up to an error $\mathcal{O}(\alpha^3)$. That is let us prove there exists a unitary $\psi_0\in\mathrm{Ran}{\ensuremath{\mathcal{P}^0_+}}$ such that $$\label{di_di_test}
\begin{array}{rcl}
\mathcal{E}^0_{\text{BDF}}(Q_{\psi_0})-2m&=&\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi_0)+\mathcal{O}(\alpha^3)\\
&=& \frac{\alpha^2 m}{g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3).
\end{array}$$
As in [@pos_sok], we consider the unique positive radially symetric Pekar minimizer $\phi_{\text{PT}}$ in $L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C})$. We form $$\label{di_test0}
\phi_1:=\begin{pmatrix}\phi_{\text{PT}}\\ 0\\0\\0 \end{pmatrix}\in L^2({\ensuremath{\mathbb{R}^3}},{\ensuremath{\mathbb{C}^4}}),$$ which is a Pekar minimizer in the space of spinors. We scale this wave function by ${\ensuremath{\lambda}}^{-1}:=\frac{\alpha m}{g'_1(0)^2}$: $$\label{di_test1}
\forall\,x\in{\ensuremath{\mathbb{R}^3}},\ \phi_{{\ensuremath{\lambda}}^{-1}}(x):={\ensuremath{\lambda}}^{-3/2}\phi_1({\ensuremath{\lambda}}^{-1}x).$$ To get a proper $\psi_0\in\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$, we form $$\label{di_test2}
\psi_0:=\frac{1}{{\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_+}}\phi_{{\ensuremath{\lambda}}^{-1}}\rVert_{L^{2}}}}}{\ensuremath{\mathcal{P}^0_+}}\phi_{{\ensuremath{\lambda}}^{-1}}.$$ Our trial state is:
$$\label{di_test3}
Q_0:={\ensuremath{|\psi_0\rangle}\xspace}{\ensuremath{\langle \psi_0|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_0\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_0|}\xspace}.$$
We do not compute its energy: the method is as in [@pos_sok] (except that instead of ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$, the operator ${\ensuremath{\mathrm{C}}}$ is considered in [@pos_sok], but that does not change anything). Eventually we refer the reader to the proof of the upper bound of $E_{\mathbf{t}X^{\ell_0}}$ above in Section \[di\_subscritic\] for the ideas.
### Proof of Lemma \[di\_non\_triv\]
We remark the following fact.
\[di\_lem\_orientation\] Let $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\subset {\ensuremath{\mathfrak{H}_\Lambda}}$ be the set $$\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}=\big\{ f\in {\ensuremath{\mathfrak{H}_\Lambda}},\ {\ensuremath{\lVertf\rVert_{L^{2}}}}=1,\ {\ensuremath{\langle f\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\rangle}\xspace}=0\big\}=\big\{ f\in {\ensuremath{\mathfrak{H}_\Lambda}},\ {\ensuremath{\lVertf\rVert_{L^{2}}}}=1,\ \mathfrak{Im}{\ensuremath{\langle {\ensuremath{\mathcal{P}^0_-}}f\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}{\ensuremath{\mathcal{P}^0_+}}f\rangle}\xspace}=0\big\}.$$
There exists a smooth angle operator $\mathcal{A}:\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\to \mathbb{R}/ \pi \mathbb{Z}$.
For two $\mathbb{C}$-colinear wave functions $f_1,f_2$ in $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ we have $\mathcal{A}(f_1)=\mathcal{A}(f_2)$.
Furthermore we have $\mathcal{A}^{-1}(0)=\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_-}}$ and $\mathcal{A}^{-1}(\tfrac{\pi}{2})=\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$.
#### Proof:
Let $f$ be in $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$: the space $\text{Span}_{\mathbb{C}}(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)$ is spanned by the eigenvectors $g_{-}:=\tfrac{f+i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f}{{\ensuremath{\lVertf+i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\rVert_{L^{2}}}}}$ and $g_+:=\tfrac{f-i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f}{{\ensuremath{\lVertf-i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\rVert_{L^{2}}}}}$. We have $$\text{Span}_{\mathbb{C}}(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)=\text{Span}({\ensuremath{\mathcal{P}^0_-}}g_{\pm},{\ensuremath{\mathcal{P}^0_+}}g_{\pm}).$$ It follows that $\mathcal{P}^0_{\pm} f\parallel \mathcal{P}^0_{\pm} g_+$ and ${\ensuremath{\mathcal{P}^0_-}}f \parallel {\ensuremath{\mathrm{I}_{\mathrm{s}}}}{\ensuremath{\mathcal{P}^0_+}}f$. As $f\in \mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$, for ${\ensuremath{\varepsilon}}\in\{ +,-\}$ with $\mathcal{P}^0_{{\ensuremath{\varepsilon}}} f\neq 0$, we have $$\mathcal{P}^0_{-{\ensuremath{\varepsilon}}} f\in \text{Span}_{\mathbb{R}}(\mathcal{P}^0_{{\ensuremath{\varepsilon}}} f).$$ Thus we have with $$\label{di_di_cond}
\mathrm{Span}_{\mathbb{R}}(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)=\mathrm{Span}_{\mathbb{R}}(e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-),\ e_-\in\text{Ran}\,{\ensuremath{\mathcal{P}^0_-}}\text{\ and\ }{\ensuremath{\lVerte_-\rVert_{L^{2}}}}=1.$$ Indeed if ${\ensuremath{\mathcal{P}^0_-}}f\neq 0$ we can choose $e_-:=\tfrac{{\ensuremath{\mathcal{P}^0_-}}f}{{\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_-}}f\rVert_{L^{2}}}}}$, else we can choose $e_-:={\ensuremath{\mathrm{I}_{\mathrm{s}}}}\tfrac{{\ensuremath{\mathcal{P}^0_+}}f}{{\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_+}}f\rVert_{L^{2}}}}}$.
Then we decompose $f$ w.r.t. the basis $(e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ and there exists $\theta\in \mathbb{R}/(2\pi \mathbb{Z})$ with $f=\cos(\theta) e_-+\sin(\theta) {\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-$. In fact the function $f\mapsto (e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ that maps $f$ to a basis is bi-valued: if $(e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ is a possibility, then $(-e_-,-{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ is another possibility. It follows that the angle $\theta$ is defined up to $\pi$: we thus obtain a function $$\mathcal{A}:\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\to \mathbb{R}/ \pi \mathbb{Z}.$$ The smoothness of $\mathcal{A}$ is straightforward. The end of the proof is also clear.
We use the angle operator to get a mountain pass argument: see Lemma \[di\_mount\] below.
We use and Theorem \[di\_structure\] and Proposition \[di\_chasym\].
Let $\mathscr{U}\subset \mathscr{M}_{\mathscr{I}}$ be the *open* subset $$\mathscr{U}\subset \mathscr{M}_{\mathscr{I}}:=\Big\{P=Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{M}_{\mathscr{I}},\ \text{dim}\,\mathrm{Ker}(Q-{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}})=1 \Big\}.$$ For all $P=Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{U}$, the eigenspace $\mathrm{Ker}(Q-{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}})$ is spanned by a unitary vector $f_0$. By ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-symmetry, we have $${\ensuremath{\mathrm{I}_{\mathrm{s}}}}\mathrm{Ker}(Q-{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}})=\mathrm{Ker}(Q+{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}}),$$ and we have ${\ensuremath{\langle f_0\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f_0\rangle}\xspace}=0.$ By Proposition \[di\_chasym\], the plane $\text{Span}_{\mathbb{C}}\,(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)$ is spanned by $f_-\in\text{Ran}\,P$ and $f_+\in\text{Ran}\,(1-P)$.
By ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-symmetry, we have ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}f_-\in\mathbb{R}f_+$. In other words:
**the wave function $f_-$ is in $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$**.
Let $Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{U}\subset \mathscr{M}_{\mathscr{I}}$ and $f_-$ as above. We define the smooth function $\mathcal{A}_U$ as follows: $$\mathcal{A}_U:Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{U}\subset \mathscr{M}_{\mathscr{I}}\mapsto \mathcal{A}(f_-).$$ It is clear it does not depend on the choice of $f_-$ but is a function of $\mathbb{C}f_-$. Furthermore, we have $$\forall\,P\in \mathscr{U},\ \nabla \mathcal{A}_U(P)\neq 0$$
The following Lemma is an application of classical results in geometry.
\[di\_mount\] Let $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ be the subset $$\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}:=\big\{Q+{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{U},\ {\ensuremath{\lVertQ\rVert_{\mathcal{B}}}}=1\big\}=\mathcal{A}_U^{-1}\big(\big\{\frac{\pi}{2}\big\}\big),$$ in other words the set of projectors in $\mathscr{U}$ whose range intersects nontrivially $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$. For any differentiable function $c:(-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})\to \mathscr{M}_{\mathscr{I}}$ such that ${\ensuremath{\varepsilon}}>0$, $c(0)\in \mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ and $${\ensuremath{\mathrm{Tr}}}\big(\nabla \mathcal{A}_U(c(0))^* \frac{d}{ds}c(0)\big)\neq 0,$$ the following holds: any sufficiently small smooth perturbation $$c+\delta c:(-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})\to \mathscr{M}_{\mathscr{I}},$$ in the norm $$\lVert \widetilde{c}\rVert:=\sup_{s\in (-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})}{\ensuremath{\lVert\widetilde{c}(s)-{\ensuremath{\mathcal{P}^0_-}}\rVert_{\mathfrak{S}_{2}}}}+\sup_{s\in (-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})}{\ensuremath{\lVert\tfrac{d}{ds}\widetilde{c}(s)\rVert_{\mathfrak{S}_{2}}}}$$ still intersects $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ at some $s(\delta c)$.
– Let us now prove Lemma \[di\_non\_triv\]. We recall that we have defined a loop $c_\psi=c_0$ that crosses $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ at $s=\tfrac{1}{2}$ and we can easily check that ${\ensuremath{\mathrm{Tr}}}\big(\mathcal{A}_U(c(2^{-1}))^*\frac{d}{ds}c(2^{-1})\big)=1\neq 0.$
Furthermore we have defined the family $(c_t)_{t\ge 0}$ by $c_t:=\Phi_{\text{BDF};t}(c_\psi)$ where $\Phi_{\text{BDF};t}$ is the gradient flow of the BDF energy.
– By Lemma \[di\_mount\], the loop $c_t$ still intersects $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ for sufficiently small $t$. We must ensure that this fact holds for all $t\ge 0$ to end the proof.
We use a continuation principle and set $$t_{\infty}:=\sup\Big\{t\ge 0,\ \forall\,0\le \tau\le t,\exists s_0\in[0,1] c_\tau\text{\ crosses\ }\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\text{\ at\ }s=s_0\Big\}.$$ We also define for all $0\le \tau<t_{\infty}$: $$\begin{array}{| rcl}
s_{-}(\tau)&=&\sup\{ s\in[0,1],\ \forall\,s'\le s,\ {\ensuremath{\lVertc_\tau(s')\rVert_{\mathcal{B}}}}<1\}>0,\\
s_{+}(\tau)&=&\inf\{ s\in[0,1],\ \forall\,s'\ge s,\ {\ensuremath{\lVertc_\tau(s')\rVert_{\mathcal{B}}}}<1\}<1.
\end{array}$$ – We assume that $t_\infty<+\infty$ and prove this implies a contradiction.
The initial loop $c_0$ induces $$\mathcal{L}_0:s\in [0,1]\mapsto \mathcal{A}_U(c_0(s))=\pi s\in \mathbb{T},$$ and we notice that $\mathcal{L}_0$ has a non-trivial homotopy.
Thus, at least for $\tau$ close to $0$, the following holds.
1. There exist $0<\eta_\tau,\eta_\tau'\ll 1$ such that $$\label{di_cont_0}
\mathcal{A}_U\big[c_\tau\big((s_-(\tau)-\eta_\tau,s_-(\tau)) \big)\big]\cap (\tfrac{\pi}{2},\tfrac{\pi}{2}+\eta_\tau')=\varnothing.$$
2. There exist $0<\eta_\tau,\eta_\tau'\ll 1$ such that $$\label{di_cont_1}
\mathcal{A}_U\big[c_\tau\big((s_+(\tau),s_+(\tau)+\eta_\tau) \big)\big]\cap (\tfrac{\pi}{2}-\eta_\tau',\tfrac{\pi}{2})=\varnothing.$$
The functions $\tau\ge 0\mapsto s_{\pm}(\tau)$ are well-defined and continuous in a neighbourhood of $0$ with $ s_-(0)=s_+(0)=\tfrac{1}{2}.$
– We prove that by continuity in $\tau$ we have $$\label{di_contz}
\forall\,s\in[0,1],\ {\ensuremath{\lVertc_{\tau}(s)\rVert_{\mathcal{B}}}}=1\Rightarrow c_{\tau}(s)+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$$ and in particular $$\label{di_cont}
c_\tau(s_\pm(\tau))\in \mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}-{\ensuremath{\mathcal{P}^0_-}}.$$ If not, this implies that as $\tau$ increases, the second highest eigenvalue of $c_\tau(s_0)$ also increases to reach $1$ where becomes false, at some $(\tau_0,s_0)$.
This cannot occurs because of the energy condition: if this was true, we would have by Kato’s inequality $$\mathcal{E}^0_{\text{BDF}}\big(c_{\tau_0}(s_0)\big)\ge (1-\alpha \tfrac{\pi}{4}){\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|c_{\tau_0}(s_0)^2 \big)\ge 4m(1-\alpha \tfrac{\pi}{4})>2m.$$
Thus - hold for all $0\le \tau<t_\infty$.
– Thanks to this fact, by continuity for all $0\le \tau<t_\infty$, - hold: if we follow the point $s_{\pm}(\tau)$ from $\tau=0$, we see that there cannot exist $\tau_0$ such that or becomes false, because the set $\{t\ge 0,\ \forall\,0\le \tau< t,$ (resp. ) holds for $\tau \}$ is non-empty and open. – Up to an isomorphism of $[0,1]$, we can suppose that for all $0\le \tau\le t_\infty$, $$\forall\, s\in[0,1],\ {\ensuremath{\lVert\partial_s c_\tau(s_0)\rVert_{\mathfrak{S}_{2}}}}\apprle 1.$$
In $\mathfrak{S}_2$, the function $\partial_sc_t(s_0)$ satsifies the following equation: $$\frac{d}{dt}\partial_sc_t(s_0)=\partial_s \nabla \mathcal{E}^0_{\text{BDF}}(c_t(s_0))\in\mathfrak{S}_2.$$
These new loops are written ${\ensuremath{\widetilde{c}}}_\tau$ and have the same range as the $c_\tau$’s and define the same arc length.
Studying the limit of ${\ensuremath{\widetilde{c}}}_\tau$ as $\tau$ tends to $t_\infty$, we get that at $t=t_\infty$, - still holds for the loop ${\ensuremath{\widetilde{c}}}_{t_\infty}$ at some $0< s_{-}(t_\infty)\le s_+(t_\infty)< 1$.
Then necessarily, the loop ${\ensuremath{\widetilde{c}}}_{t_\infty}$ crosses $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ at some $s\in [s_-(t_\infty),s_+(t_\infty)]$. Going back to $c_{t_\infty}$, this proves that the same holds for $c_{t_\infty}$, which contradicts the definition of $t_\infty$.
Proofs on results on the variational set {#di_proofmanif}
========================================
Proof of Lemma \[di\_irreduc\] {#di_irr_proof}
------------------------------
Let $${\ensuremath{\Phi_{\mathrm{SU}}}}':\mathbf{SU}(2)\to \mathbf{U}(E),\ E\subset{\ensuremath{\mathfrak{H}_\Lambda}}$$ be an irreducible representation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$. As ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{S}}}$ commutes with the action of $\mathbf{SU}(2)$, then necessarily $E$ is an eigenspace for ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{S}}}$, associated to $j(j+1)$ and $\kappa_j={\ensuremath{\varepsilon}}(j+\tfrac{1}{2})$ where $j\in \tfrac{1}{2}+\mathbb{Z}_+$ and ${\ensuremath{\varepsilon}}=\pm$. The eigenspaces are known [@Th p. 126]: they are spanned by wave functions of type $$\label{di_irr_0}
\forall\,x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ \psi(x):=a(r)\Phi^{\pm}_{m,\kappa_j},\ m=-j,-j+1,\ldots,j,$$ where
\[di\_courage\] $$\label{di_irr_1}
a(r)\in L^2(\mathbb{R}_+,r^2dr),$$ $$\label{di_irr_2}
\Phi^+_{m,\pm(j+\tfrac{1}{2})}:=\begin{pmatrix}i\Psi^{m}_{j\pm\tfrac{1}{2}}\\ 0\end{pmatrix}\text{\ and\ }\Phi^-_{m,\pm(j+\tfrac{1}{2})}:=\begin{pmatrix}0 \\ \Psi^{m}_{j\mp \tfrac{1}{2}} \end{pmatrix}$$ $$\Psi^{m}_{j-\tfrac{1}{2}}=\frac{1}{\sqrt{2j}}\begin{pmatrix}\sqrt{j+m}Y^{m-\tfrac{1}{2}}_{j-\tfrac{1}{2}}\\ \sqrt{j-m}Y^{m+\tfrac{1}{2}}_{j-\tfrac{1}{2}}\end{pmatrix}\text{\ and\ }\Psi^{m}_{j+\tfrac{1}{2}}=\frac{1}{\sqrt{2j+2}}\begin{pmatrix} \sqrt{j+1-m}Y^{m-\tfrac{1}{2}}_{j+\tfrac{1}{2}}\\ -\sqrt{j+1+m}Y^{m+\tfrac{1}{2}}_{j+\tfrac{1}{2}}\end{pmatrix}.$$
We recall that the $Y^m_{\ell}$ are the spherical harmonics (eigenvectors of ${\ensuremath{\mathbf{L}}}^2$).
Hence $E$ is spanned by a wave function which is a linear combination of that of type . We recall that for any integer $n\ge 1$ there is but one irreducible representation of $\mathbf{SU}(2)$ of dimension $n$ up to isomorphism. They can be found by the number of eigenvalues of $J_3'$, the infinitesimal “rotation” around the $z$ axis which induces a representation of $\mathbf{SO}(3)$.. Here $J'_3$ corresponds to $J_3$.
Thus we get that for ${\ensuremath{\varepsilon}}\in\{+,-\}$ $$E_{{\ensuremath{\varepsilon}}}:={\ensuremath{\Phi_{\mathrm{SU}}}}\,a(r)\Phi^{\ensuremath{\varepsilon}}_{j,\kappa_j}$$ is irreducible with respect to ${\ensuremath{\Phi_{\mathrm{SU}}}}$. By unicity of the irreducible representation of dimension $2j+1$, there exists an isomorphism from $E_-$ to $E_+$. As there must be a correspondence between the eigenspace of $J_3(E_-)$ and that of $J_3(E_+)$, necessarily $\mathbb{C}a\Phi^-_{m,\kappa_j}$ is sent to $\mathbb{C}a\Phi^+_{m,\kappa_j}$.
In particular as ${\ensuremath{P_{\uparrow}}}E$ and ${\ensuremath{P_{\downarrow}}}E$ are also representation of $\mathbf{SU}(2)$ with same eigenvalues of ${\ensuremath{\mathbf{J}}}^2,{\ensuremath{\mathbf{S}}}$ (or $=\{0\}$). If one of them is zero then $E$ is of type $E_\pm$. If both are non-zero, then there exists $a_\uparrow(r),a_{\downarrow}(r)$ such that $${\ensuremath{P_{\uparrow}}}E={\ensuremath{\Phi_{\mathrm{SU}}}}a_\uparrow(r)\Phi^{+}_{j,\kappa_j}\text{\ and\ }{\ensuremath{P_{\downarrow}}}E={\ensuremath{\Phi_{\mathrm{SU}}}}a_{\downarrow}(r)\Phi^{-}_{j,\kappa_j}.$$ Both ${\ensuremath{P_{\uparrow}}}E$ and ${\ensuremath{P_{\downarrow}}}E$ are irreducible. We can suppose that there exists $f\in E$ with $${\ensuremath{P_{\uparrow}}}f= a_\uparrow(r)\Phi^{+}_{j,\kappa_j}\text{\ and\ }{\ensuremath{P_{\downarrow}}}f=a_{\downarrow}(r)\Phi^{-}_{j,\kappa_j}.$$
The isomorphism between the two representations implies that $$E={\ensuremath{\Phi_{\mathrm{SU}}}}\big( a_\uparrow(r)\Phi^{+}_{j,\kappa_j}+a_{\downarrow}(r)\Phi^{-}_{j,\kappa_j}\big).$$
Proof of Proposition \[di\_mani\_ci\_sym\]
------------------------------------------
We have to prove that $\mathscr{M}_{\mathscr{I}}$ and $\mathscr{W}$ are submanifold of $\mathscr{M}$. The method is similar to the one used in [@pos_sok] to prove that $\mathscr{M}_{\mathscr{C}}$ is a submanifold of $\mathscr{M}$.
Let $P_0=Q_0+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{M}$. We will prove that in a neighbourhood of $P_0$ in ${\ensuremath{\mathcal{P}^0_-}}+\mathfrak{S}_2$, the projectors $P_1$ in $\mathscr{M}_{\mathscr{I}}$ (resp. $\mathscr{W}$) can be written as $$P_1=e^{A}P_0e^{-A},$$ where $A\in\mathfrak{m}^{\mathscr{I}}_{P_0}$ (resp. $\mathfrak{m}^{\mathscr{W}}_{P_0}$). – If we assume this point, then it is clear that the two sets are submanifolds of $\mathscr{M}$. Indeed $e^A$ is a global linear isometry of ${\ensuremath{\mathfrak{H}_\Lambda}}$, whose restriction to the $\mathfrak{m}_P^{\cdot}$’s maps $\mathfrak{m}_{P_0}^{\cdot}$ onto $\mathfrak{m}_{P_1}^{\cdot}$.
Equivalently it maps the first tangent plane onto the other: $$\{ [a,P_0],\ a\in\mathfrak{m}_{P_0}^{\cdot}\}\underset{\simeq}{\to} \{ [a,P_1],\ a\in\mathfrak{m}_{P_1}^{\cdot}\}.$$
– We use Theorem \[di\_structure\] to write $$\label{di_di_chiant}
Q_0={\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}{\ensuremath{\lambda}}_j\big({\ensuremath{|f_j\rangle}\xspace}{\ensuremath{\langle f_j|}\xspace}-{\ensuremath{|f_{-j}\rangle}\xspace}{\ensuremath{\langle f_{-j}|}\xspace}\big)$$ where $({\ensuremath{\lambda}}_i)_i\in\ell^2$ is non-increasing and the $f_i$’s form an orthonormal basis of $\mathrm{Ran}\,Q$. Provided that $${\ensuremath{\lVertP_1-P_0\rVert_{\mathfrak{S}_{2}}}}<1,$$ then ${\ensuremath{\lambda}}_1<1$ and there is no $j$ such that $f_j$ or $f_{-j}$ is in the range of ${\ensuremath{\mathcal{P}^0_+}}$ or ${\ensuremath{\mathcal{P}^0_-}}$.
We decompose with respect with the eigenvalues $\mu_1>\mu_2>\cdots>0$ as follows: $$Q_0={\ensuremath{\displaystyle\sum}}_{k=1}^{+\infty}\mu_k\big(\text{Proj}\ \mathrm{Ker}(Q_0-\mu_k)-\text{Proj}\ \mathrm{Ker}(Q_0+\mu_k)\big).$$ For short we write $\mu_{-k}:=-\mu_{k}$, and $$M_k:=\text{Proj}\ \mathrm{Ker}(Q_0-\mu_k)\text{\ and\ }E_{\mu_k}^{Q_0}:=\mathrm{Ker}(Q_0-\mu_k).$$
As any ${\ensuremath{\mathrm{Y}}}\in\{{\ensuremath{\mathrm{C}}},{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\}$ is an isometry (linear or antilinear) and as the eigenvalues are the sine of the angles between vectors in $P_0$ and ${\ensuremath{\mathcal{P}^0_-}}$, for any $k$ we have $$\label{di_invar_y}
{\ensuremath{\mathrm{Y}}}E_{\mu_k}^{Q_0}=E_{-\mu_k}^{Q_0}$$ and the eigenspaces $E_{\mu_k}^{Q_0}\oplus E_{-\mu_k}^{Q_0}=\mathrm{Ker}(Q_0^2-\mu_k^2)$ are invariant under ${\ensuremath{\mathrm{Y}}}$.
#### Case of $\mathscr{W}$
– In the case ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{C}}}$ and $P_0\in \mathscr{W}$, each eigenspace
$\mathrm{Ker}(Q_0^2-\mu_k^2)$ is also invariant under the action of ${\ensuremath{\Phi_{\mathrm{SU}}}}$. In other words, $\mathrm{Ker}(Q_0^2-\mu_k^2)$ is a finite dimensional representation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$, and we can decompose it into irreducible representations $E_{\mu_k}^{(\ell)}$, where $0\le \ell\le \ell_k$.
By ${\ensuremath{\mathrm{C}}}$-symmetry, we have $${\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell_1)}=E_{-\mu_k}^{(\ell_1')},$$ there is a one-to-one correspondence between irreducible representations of type $E_{\mu_k}^{(\ell)}$ and that of type $E_{-\mu_k}^{(\ell)}$. Up to changing indices $\ell'_j$, we can suppose that $${\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}=E_{-\mu_k}^{(\ell)},\ 0\le \ell\le \ell_k.$$ Decomposing $E_{\mu_k}^{(\ell)}$ with respect with ${\ensuremath{\mathcal{P}^0_-}}$ and ${\ensuremath{\mathcal{P}^0_+}}$, we see that $$\mathcal{P}^0_{\pm} E_{\mu_k}^{(\ell)}\ \text{is\ irreducible},$$ and from the spectral decomposition of $Q_0$ $${\ensuremath{\mathcal{P}^0_-}}E_{\mu_k}^{(\ell)}\oplus {\ensuremath{\mathcal{P}^0_+}}E_{\mu_k}^{(\ell)}=E_{\mu_k}^{(\ell)}\oplus F_{-\mu_k},$$ where $F_{-\mu_k}$ is an irreducible subset of $\mathrm{Ker}(Q_0+\mu_k)$.
– Let us show that $$\label{di_inters}
F_{-\mu_k}\cap {\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}=\{ 0\}.$$ Indeed, from Lemma \[di\_irreduc\] and the expression of the $\Phi^{\pm}_{m,\kappa}$, we see that $${\ensuremath{\mathrm{C}}}\mathrm{Ker}\big(J_3-m\big)=\mathrm{Ker}\big(J_3+m\big).$$ Thus if the intersection is non-zero, then we have by ${\ensuremath{\mathrm{C}}}$-symmetry and ${\ensuremath{\Phi_{\mathrm{SU}}}}$-symmetry: $$F_{-\mu_k}={\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}.$$ But as shown in [@pos_sok], this cannot happen: let us say that $E_{\mu_k}^{(\ell)}$ is associated to the eigenvalues $j_0(j_0+1),\kappa$ of ${\ensuremath{\mathbf{J}}}^2$ resp. ${\ensuremath{\mathbf{S}}}$. We consider: $$\mathrm{Ker}(J_3-m)\cap \mathcal{P}^0{\pm} E_{\mu_k}^{(\ell)}=\mathbb{C}e_{\pm;m},\ -j_0\le m\le j_0,\ {\ensuremath{\lVerte_{\pm;m}\rVert_{L^{2}}}}=1.$$ We would have $${\ensuremath{\mathrm{C}}}e_{\pm;m}=\text{exp}{i\theta(\pm;m)}e_{\mp;-m}.$$ The constant $\theta(\pm;m)$ does not depend on $m$ by ${\ensuremath{\Phi_{\mathrm{SU}}}}$-symmetry. Moreover, if $$\mathrm{Ker}(J_3-m)\cap E_{\mu_k}^{(\ell)}=\mathbb{C}f_{m},$$ then $$\mathcal{P}^0_{\pm} f_m\parallel e_{\pm;m}.$$ As in [@pos_sok] for $\mathscr{M}_{\mathscr{C}}$, the condition ${\ensuremath{\mathrm{C}}}^2=1$ implies $\theta_+-\theta_-\equiv 0[2\pi]$ while $$-{\ensuremath{\mathrm{C}}}Q_0{\ensuremath{\mathrm{C}}}=Q_0$$ implies $\theta_+-\theta_-\equiv \pi[2\pi]$, which cannot occur.
Similarly, we can prove that holds and that in fact $F_{-\mu_k}$ is orthogonal to $ {\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}$.
As a consequence, the number of $E_{\mu_k}^{(\ell)}$’s is even, or equivalently, the number of ${\ensuremath{\mathcal{P}^0_-}}E_{\mu_k}^{(\ell)}$ is even.
– The fact that $$\label{di_check}
P_1=e^A P_0 e^{-A},\ \text{with}\ {\ensuremath{\Phi_{\mathrm{SU}}}}A=A,\ {\ensuremath{\mathrm{C}}}A {\ensuremath{\mathrm{C}}}=A,\ {\ensuremath{\lVertA\rVert_{\mathfrak{S}_{2}}}}<+\infty,$$ follows from Theorem \[di\_structure\] and the different symmetries.
The $f_j$’s in can be written as (${\ensuremath{\lambda}}_j=\sin(\theta_j)$) $$f_j=\sqrt{\frac{1-{\ensuremath{\lambda}}_j}{2}}e_{-;j}+\sqrt{\frac{1+{\ensuremath{\lambda}}_j}{2}}e_{+;j},\ \mathcal{P}^0_{\pm}e_{\pm;j}=e_{\pm;j}.$$ We also have $$f_{-j}=-\sqrt{\frac{1+{\ensuremath{\lambda}}_j}{2}}e_{-;j}+\sqrt{\frac{1-{\ensuremath{\lambda}}_j}{2}}e_{+;j}.$$ Then we define $$\label{di_aaa}
A={\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}\theta_j\big({\ensuremath{|e_{+;j}\rangle}\xspace}{\ensuremath{\langle e_{-;j}|}\xspace}-{\ensuremath{|e_{-;j}\rangle}\xspace}{\ensuremath{\langle e_{+;j}|}\xspace}\big).$$ It is easy to check that $A$ satisfies . In fact, we can assume that $f_j$ spans an irreducible representation of $\mathbf{SU}(2)$, and in this case the same holds for $e_{+;j}$ and $e_{-;j}$.
As in Section \[di\_irr\_proof\], the correspondence $e_{-;j}\mapsto e_{+;j}$ induces an isomorphism between ${\ensuremath{\Phi_{\mathrm{SU}}}}e_{-;j}$ and ${\ensuremath{\Phi_{\mathrm{SU}}}}e_{+;j}$. This fact together with the ${\ensuremath{\Phi_{\mathrm{SU}}}}$-symmetry implies that $$\forall\,U\in\mathrm{Ran}\,{\ensuremath{\Phi_{\mathrm{SU}}}},\ UAU^{-1}=A.$$ The fact that ${\ensuremath{\mathrm{C}}}A{\ensuremath{\mathrm{C}}}=A$ was proved in [@pos_sok] in the case $P_0,P_1\in\mathscr{M}_{\mathscr{C}}$. Here this remains true because $$\mathscr{W}\subset \mathscr{M}_{\mathscr{C}}.$$
– We can now determine the connected component of $\mathscr{W}$. Let $P_0,P_1$ be in $\mathscr{W}$ and let $Q=P_1-P_0$.
We consider $$E_1^{Q}:=\mathrm{Ker}(Q-1).$$ If $E_1^Q=\{0\}$, then we can write $P_1=e^A P_0 e^{-A}$ as in . And we see that the path in $\ell^2$: $$t\in[0,1]\mapsto (t\theta_j)_j\in \ell^2$$ induces a path connecting $P_0$ and $P_1$.
If $E_1^Q\neq \{ 0\}$, we count the number of irreducible representation in $E_1^Q$: let $b_{j,\kappa_j}$ be the number of irr. rep. in $$\mathrm{Ker}\big({\ensuremath{\mathbf{J}}}^2-j(j+1)\big)\cap\mathrm{Ker}\big({\ensuremath{\mathbf{S}}}-\kappa_j\big).$$ If all the $b_{j,\kappa_j}$’s are even, we can still write $P_1$ as $P_1=e^A P_0 e^{-A}$ with $A$ as in with the first $\theta_j$ equal to $\tfrac{\pi}{2}$. In particular the two projectors can be connected by a path in $\mathscr{W}$.
Let us say that $b_{j_0,\kappa_{0}}\equiv 1[2]$ for some $j_0,\kappa_0$. We have shown that for $P\in\mathscr{W}$ with ${\ensuremath{\lVertP-P_0\rVert_{\mathcal{B}}}}<1$, the number of planes $\Pi_j$’s in the decomposition of Theorem \[di\_structure\] is even. Precisely, due to the ${\ensuremath{\mathrm{C}}}$-symmetry, there exists a sequence $(\ell_\mu(j,\kappa))_j$ in $\mathbb{N}$, with $$\begin{array}{l}
\mathrm{Ker}\big((P-P_0)-\mu\big)\cap\mathrm{Ker}\big({\ensuremath{\mathbf{J}}}^2-j(j+1)\big)\cap \mathrm{Ker}\big({\ensuremath{\mathbf{S}}}-\kappa\big)\\
\ \ \ =\underset{1\le \ell\le \ell_\mu(j,\kappa)}{\bigoplus}E^{(\ell)}_{\mu},
\end{array}$$ where each $E^{(\ell)}_{\mu}$ is irreducible as a representation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ and $\ell_\mu(j,\kappa)$ is *even*.
We show that there cannot exist a continuous path linking $P_0$ and $P_1$ by a contradiction argument.
Let us say that ${\ensuremath{\gamma}}:t\in [0,1]\to \mathscr{W}$ is a continuous path with ${\ensuremath{\gamma}}(0)=P_0$ and ${\ensuremath{\lVert{\ensuremath{\gamma}}(1)-P_0\rVert_{\mathcal{B}}}}=1$.
Then by the previous remarks, we have by continuity: $$\begin{array}{l}
\forall t\in[0,1], \forall\,j\in\frac{1}{2}+\mathbb{Z}_+,\ \forall\kappa\in\big\{\pm\big(j+\frac{1}{2}\big)\big\},\\
\ \ \ \ \ell_1(Q_t={\ensuremath{\gamma}}(t)-P_0;j,\kappa)\equiv 0[2].
\end{array}$$ In particular it is not possible to have ${\ensuremath{\gamma}}(1)=P_1$.
#### Case of $\mathscr{M}_{\mathscr{I}}$
For ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ and $P_0\in \mathscr{M}_{\mathscr{I}}$, we use . For each $f\in E_\mu^Q$, we have ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}\in E_{-\mu}^Q$ where $\mu\in \sigma(Q)$. We may assume that $\mu>0$.
Thus the plane $$\Pi:=\text{Span}\big(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\big)$$ is invariant under $Q$ and ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$. We decompose $f$ and ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}f$ with respect to $P_0$ and $1-P_0$. By a dimension argument:
1. either $\mu=1$, $P_0 f=0$ and $(1-P_0){\ensuremath{\mathrm{I}_{\mathrm{s}}}}f=0$,
2. or $0<\mu<1$ and $$\mathbb{C}P_0 f=\mathbb{C} P_0 {\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\text{\ and\ }\mathbb{C}(1-P_0) f=\mathbb{C} (1-P_0) {\ensuremath{\mathrm{I}_{\mathrm{s}}}}f.$$
In each case, we write $e_{-}$ a unitary vector in $\mathrm{Ran}\,P_0\cap \Pi$ and $e_+={\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-$.
If we consider the sequence $(\mu_i)_i$ of positive eigenvalues of $Q$ (counted with multiplicities), we get the correspondent sequences $(e_{-;j})_j$ and $(e_{+;j})$. Moreover by Theorem \[di\_structure\], we know that $\mu_j=\sin(\theta_j)$ where $\theta_j\in[0,\tfrac{\pi}{2}]$ is the angle between the two lines $\mathbb{C}e_{-;j}$ and $\mathbb{C}f_j$.
Provided that we take $-\theta_j$ instead of $\theta_j$ and up to a phase, we can suppose that $$f_j=\cos(\theta_j)e_{-;j}+\sin(\theta_j){\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_{-;j}.$$
In particular we have $$P_1=e^A P_0 e^{-A},$$ with $$A={\ensuremath{\displaystyle\sum}}\theta_j\big({\ensuremath{|e_{+;j}\rangle}\xspace}{\ensuremath{\langle e_{-;j}|}\xspace}-{\ensuremath{|e_{-;j}\rangle}\xspace}{\ensuremath{\langle e_{+;j}|}\xspace}\big).$$ It is straightforward to check that ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}A {\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}=A$.
*Acknowledgment* The author wishes to thank Éric séré for useful discussions and helpful comments. This work was partially supported by the Grant ANR-10-BLAN 0101 of the French Ministry of research.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we utilize coupled mode theory (CMT) to model the coupling of surface plasmon-polaritons (SPPs) between tri-layered corrugated thin films (CTF) structure coupler in the terahertz region. Employing the stimulated raman adiabatic passage (STIRAP) quantum control technique, we propose a novel directional coupler based on SPPs evolution in tri-layered CTF in some curved configuration. Our calculated results show that the SPPs can be completely transferred from the input to the output CTF waveguides, and even we consider SPPs propagation loss, the transfer rate is still above $70 \%$. The performance of our coupler is also robust that it is not sensitive to the geometry of device and wavelength of SPPs. As a result, our device can tolerate defect induced by fabrication and manipulate THz wave at broadband.'
author:
- Wei Huang
- Shan Yin
- Wentao Zhang
- Kaili Wang
- Yuting Zhang
- Jiaguang Han
title: 'Adiabatic following of terahertz surface plasmon-polaritons based on tri-layered corrugated thin film coupler'
---
Introduction
============
Terahertz (THz) radiation has drawn enormous attentions these years. Since many material responses are located at THz frequency, THz technologies can obtain unique spectral characteristics and abundant information about matters, which is widely used in spectroscopy [@Lee2009] and imaging [@Chan2007]. Naturally, the THz applications in information processing and transmission [@Ozbay2006; @Lee2010] are vital. On the other hand, with the rapid development of the network and popularization of portable terminals, the miniaturization of the integrated devices is an irresistible trend. THz technologies are promising to accelerate the next generation of communications [@Naeem2018; @Withayachumnankul2018; @Yu2016] due to the capabilities of high capacity and micro-size [@Koenig2013; @Ostmann2011]. To realize further integration, how to manipulate the electromagnetic (EM) waves in subwavelength scale is a key issue. Surface plasmon polaritons (SPPs) are the EM waves propagating along metal-dielectric interfaces with exponential decay in the direction perpendicular to the interfaces. The recent emerged SPPs-based elements, such as antennas [@Schnell2009; @Maguid2016], waveguides [@Sorger2011; @Ebbesen2008] and logic circuitry [@Ebbesen2008; @Cohen2013], demonstrated their potential application on the microscale and nanoscale chips since the wavelength of SPPs can be scaled down below diffraction limit [@Maier2007; @Gramotnev2010; @Kawata2009].
At terahertz regime, SPPs-based waveguides [@Zhang2017], couplers [@Ma2017] and coders [@Yin2018] have been investigated recently, which will make great contributions to the THz applications. Due to these advantages of SPPs at terahertz regime, completely transfer energies and information of THz SPPs is significant to implement compact device in THz regime. Two recent researches studying on the coupling of THz SPPs waveguides [@Liu2014; @Zhang2018] involved in coupled mode theory (CMT), which is a widely used theory in describing coupling between two optical waveguides, through the overlap of their evanescent electromagnetic fields [@Yariv1973; @Huang2014]. Base on this concept, if two thin films are close enough, the two evanescent fields of SPPs in each thin film have overlapping and SPPs can transfer from one thin film to another [@Liu2014; @Zhang2018]. In our paper, we employ and derive the CMT to describe the SPPs coupling between courrgated thin films structure.
However, the present stuctures of two parallel THz SPPs waveguides (e.g. ref. [@Liu2014; @Zhang2018]) require rigorous fabrication precision and only operate at specific excited frequency of THz waves, otherwise, the fidelity of device will drop rapidly. Most recently, to overcome this shortcoming, a remarkable paper applied coherent quantum control (stimulated raman adiabatic passage, short for STIRAP) into transferring the SPPs on the graphene sheets [@Huang20181]. STIRAP is the well-known three-level coherent quantum control, which provides completely transfer population from first state to third state, without any population remaining In intermediate state [@Vitanov20011; @Vitanov20012; @Vitanov2017]. Furthermore, it is shown that STIRAP is exceedingly robust against controlling parameters under perturbations. The SITRAP has already widely used in various domains, such as atomic molecular and optical physics [@Yale2016; @Huang2017], waveguide coupler [@Mrejen2015; @Longhi2007], graphene electronic and optical effect [@Huang20181; @Huang20182]. In this paper, we firstly introduce the STIRAP technique into the SPPs waveguide coupler at terahertz regime, to achieve very robust device against varying frequency of input THz waves and disturbances on the geometry parameters. We propose the tri-layered corrugated thin film coupler structure with some curved configuration and we substantiate that the performance of our coupler is also robust to the geometry of device and wavelength of SPPs. As a result, our device can tolerate defect induced by fabrication and manipulate THz wave at broadband, which is meaningful in developing THz functional devices.
Model
=====
We first consider terahertz radiation to excited surface plasmon-polaritons on the surface of the courrgated thin films structure. Assuming a slab courrgated thin films locates at $z=0$ at $x z$ plane, we illuminate the terahertz waves on the surface of the thin film to excited SPPs propagating along $x$ direction. In order to SPPs extend the propagation distance, it is remarkable to utilize courrgated structure cutting on the thin film [@Zhang2017; @Liu2014; @Zhang2018], as shown in Fig. 1, with cutting depth $h$, width $a$, period $d$ and thickness of thin film $t$. If we contemplate the mode profile of SPPs, electric field of SPPs has exponentially decay along with $y$ and $z$ directions outside the SPPs waveguide, as the evanescent field of electric field.
In this paper, we only study the coupling mechanism along $z$ direction. Therefore, SPPs’ electric fields of $x$ direction (SPPs propagation) and $z$ direction (SPPs’ evanescent field) are observed and we ignore the impacts of $y$ direction. Assume that we place two corrugated thin films at $z=g/2$ and $-g/2$ and these two films are parallel to $x y$ plane, where $g$ is the gap distance between two parallel thin films (see Fig. 1). The TM polarized SPPs modes are excited on one corrugated thin films. The electric fields can be described by $E_1 = (E_{1x}, 0, E_{1z}) e^{iqx} e^{-k_m |z-g/2|}$ and $E_2 = (E_{2x}, 0, E_{2z}) e^{iqx} e^{-k_m |z+g/2|}$. Here $k_m$ is the decay rate of evanescent field in the surrounding dielectric mediums, given by $k_m = \sqrt{ (\omega^2 \epsilon_m - q^2) /c^2}$ [@Saleh1991]. $\epsilon_m$ is the permittivity of medium material (we use silicon as surrounding mediums) and $\omega$ is the frequency of incident light in air. In addition, $q$ is the propagation constant of SPPs and it well depends on the geometry structure and frequency of incident terahertz light $\omega$ [@Zhang2017; @Ma2017; @Maier2006]. We can numerically solve it by dispersion equation, given by $q=\frac{\omega}{c}\sqrt{1+\frac{a^2}{d^2} \tan^2 \frac{\omega h}{c}}$ [@Maier2006].
{width="50.00000%"}
In our parallel coupling model, we take the notations $\Psi_1(x,z)$ ($\Psi_2(x,z)$) as the electric field of SPPs on the first (second) thin film, written as
$$\begin{aligned}
\Psi_1(x,z)= a_{1}(x) u_{1}(z) \exp(-i q x), \\
\Psi_2(x,z)= a_{2}(x) u_{2}(z) \exp(-i q x),
\end{aligned}$$
where $a_{1}(x)$ and $a_{2}(x)$ are the amplitudes of the modes with respect to SPPs on two thin film. Due to the extremely thickness of film ($t = 100$ nm) comparing to other geometry parameters, we can obtain the mode profiles of SPPs as $u_{1} = E_{1z} exp(- k_m |z-g/2|)$ and $u_{2} = E_{2z} exp(- k_m |z+g/2|)$. The electric filed have to normalize by the normalization factor as $N_{1,2} = \sqrt{ \int^{+\infty}_{-\infty} |u_{1,2}(z)|^2 dz}$ for the thin film 1 and 2, respectively. We take the notation as $\psi_{1}$=$u_{1}(z) \exp(-i q x)$ and $\psi_{2}$=$u_{2}(z) \exp(-i q x)$, where $\psi_{1}$ and $\psi_{2}$ must be satisfied Helmholtz equations in the $x$ direction.
Based on the CMT model, we can manipulate the Helmholtz equations with the source terms to obtain $$\begin{aligned}
\dfrac{\partial^2}{\partial x^2} \Psi_{1}(x,z) + q^2 \Psi_{1}(x,z) = -(k_{2}^2 - k_0^2)\Psi_{2}(x,z), \\
\dfrac{\partial^2}{\partial x^2} \Psi_{2}(x,z) + q^2 \Psi_{2}(x,z) = -(k_{1}^2 - k_0^2)\Psi_{1}(x,z),
\end{aligned}$$ where $k_0=\sqrt{(\omega^2 \epsilon_{g} - q^2 )/ c^2}$ with thin film (gold) permittivity $\epsilon_{g}$. These equations are consistent with the conventional optical waveguide coupled equations [@Saleh1991].
By substituting the wave functions of the SPPs on two thin films into the given Helmholtz equations, we simplify the formation by using the slowly varying envelope approximation [@Saleh1991], namely $\frac{d^2 a_1}{dx^2} \ll \frac{d a_1}{dx}$ and $\frac{d^2 a_2}{dx^2} \ll \frac{d a_2}{dx}$. Under this approximation, the coupling equations can be rewritten as a Schrödinger-like equation of a two-level system, given by $$i\dfrac{d}{d x}
\begin{bmatrix}
a_{1} \\
a_{2}
\end{bmatrix}
= \begin{bmatrix}
0 & C_{12} \\
C_{21} & 0
\end{bmatrix} \begin{bmatrix}
a_{1} \\
a_{2}
\end{bmatrix}.$$ Here, $C_{12}$ and $C_{21}$ are the coupling coefficients: $C_{12} = \frac{1}{2} \frac{k_{2}^2 - k_0^2}{q} \int^{+\infty}_{-\infty} u_{1}(z) u_{2}(z) dx$ and $C_{21} = \frac{1}{2} \frac{k_{1}^2 - k_0^2}{q} \int^{+\infty}_{-\infty} u_{1}(z) u_{2}(z) dx$.
{width="50.00000%"} {width="50.00000%"}
As an example, it is straight forward to obtain the coupling strength $C_{12} = C_{21} = 1255$ $m^{-1}$, with setting up $a = 40$ $\mu m$, $d = 50$ $\mu m$, $g = 4$ $\mu m$ and input frequency of terahertz waves at 1 THz. Beneficial to illustrate trends of the coupling strength against to different parameters, we demonstrate coupling strength against varying gap distance and height of corrugated structure (see Fig 2a). Furthermore, Fig. 2b shows the coupling strength as the function of changing frequency of input terahertz radiation and gap distance. From results of Fig. 2a, the coupling strength will exponentially decrease either by increasing gap distance or by increasing the depth courrgated structure $d$. It is remarkable to attain that escalating the frequency of input terahertz radiation will enhance the coupling strength from 0.2 THz to 1 THz, shown in Fig. 2b. These results are noticeably the same as the trends of the SPPs’ coupling strength on two parallel thin film structure, Ref. [@Liu2014].
In our first example, we only consider SPPs coupling between two corrugated thin films. Based on the coupling equation of bi-layered SPPs coupling (Eq. 3), it is obviously to extend the SPPs coupling between multi-layered thin films by using analogy of multi-level Schrödinger equation, written as
$$i\dfrac{d}{d x}
\begin{bmatrix}
a_{1} \\
\vdots \\
a_{n}
\end{bmatrix}
= \begin{bmatrix}
0 & C_{12}(x) & \ddots \\
C_{21}(x) & \ddots & C_{n-1,n}(x) \\
\ddots & C_{n, n-1}(x) & 0 \\
\end{bmatrix} \begin{bmatrix}
a_{1} \\
\vdots \\
a_{n}
\end{bmatrix}.$$
Here, $C_{12}(x)$ ($C_{21}(x)$) is the coupling SPPs between first and second layer courrgated thin film and $C_{n-1,n}(x)$ ($C_{n,n-1}(x)$) is the coupling SPPs between $(n-1)^{th}$ and $n^{th}$ layer thin film. Similarly, $a_1$ and $a_n$ are the SPPs amplitudes of first and $n^{th}$ courrgated thin film.
Adiabatic following SPPs on tri-layered thin film coupling
==========================================================
In our novel design of SPPs’ adiabatic following, we employ tri-layered SPPs on the corrugated thin film coupler. The geometry scheme of our structure is shown in Fig 3. We have tri-layered of corrugated thin film, which are placed at $x z$ plane to obtain the $z$ directional coupling and $x$ direction SPPs propagation. The middle layer is flat courrgated thin film, located at $x = 0$. The input (first) layer of courrgated thin film has slight curve with radius $R$ and its center is above $x$ axis. The minimum distance between input and middle layer takes the notation as $d_{min}$. After that, we slightly bend the output (third) layer courrgated thin film with radius $R$, whereas its center is below $x$ axis. $d_{min}$ is also the minimum distance between middle and output layer. Notice that the offset between two centers of circles (input and output layers) in the $x$ axis is $\delta$ and center of output layer is in front of input layer in the $x$ direction. The spatial dependence of the spacing $d_1(x)$ and $d_2(x)$ of the input and output courrgated thin film with respective to the middle layer is given by $d_1(x)=\sqrt{R^2-(x-\delta/2)^2}+(d_{min}+R)$ and $d_2(x)=\sqrt{R^2-(x+\delta/2)^2}-(d_{min}+R)$. Therefore, at the beginning, we excited the SPPs on the input layer of corrugated thin film and the all power of SPPs will completely transfer from input to output layer of corrugated thin film via coupling mechanism by our designing structure.
{width="50.00000%"}
Thus, based on the multi-layered SPPs coupling (Eq. 4), the coupling equation for tri-layered SPPs coupler is described as, $$i\dfrac{d}{d x}
\begin{bmatrix}
a_{1} \\
a_{2} \\
a_{3}
\end{bmatrix}
= \begin{bmatrix}
0 & C_{12} & 0 \\
C_{21} & 0 & C_{23} \\
0 & C_{32} & 0
\end{bmatrix} \begin{bmatrix}
a_{1} \\
a_{2} \\
a_{3}
\end{bmatrix},$$ where $a_1$ ($a_2$, $a_3$) is the power amplitude of the first (second, third) SPPs waveguide. $C_{12}$ ($C_{23}$) is the coupling strength between first and middle layer (middle and third layer), where $ C_{12} = C_{21}$ and $ C_{23} = C_{32}$.
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
With the geometry structure of our adiabatic following design (see Fig. 3), we set the geometry parameters as $R = 45$ mm, device length as $L = 4$ mm and distance between two maximum coupling point $\delta = 1.5$ mm. Based on these geometry parameters, we can obtain the coupling strength between input and middle layer $C_{12} = C_{21}$ (middle and output layer $C_{23} = C_{32}$) SPPs waveguide, as shown in Fig. 4a. At the beginning of the transition, the coupling strength of input and middle layer $C_{12} = C_{21}$ is much larger than coupling strength of output and middle layer $C_{23} = C_{32}$. Eventually, the coupling strength $C_{12} = C_{21}$ is much smaller than $C_{23} = C_{32}$. Thus, the whole transition is the quintessential STIRAP transition. The evolution intensities of SPPs within input, middle and output waveguides are shown in Fig. 4b. From the result of Fig. 4b, it is conspicuously to acquire that the intensity of input waveguide ($P_1$) completely transfer to output SPPs waveguide ($P_3$) in the ideal case (without lossy, shown with solid line). However, there is some lossy during SPPs propagation within the corrugated thin film coupler, which is settled as 8 dB/cm in this paper [@Zhang2017]. Therefore, the results of SPPs intensity transition with lossy are shown with dashed line in Fig. 4b. We still can achieve efficient transfer of intensity above $70 \%$. The corresponding visualized results with population transfer and geometry structure is demonstrated in Fig. 4c. This intuitionistic outcome illustrates the the SPPs propagation within our adiabatic device, with geometry structure of input, middle, output SPPs waveguides.
{width="50.00000%"} {width="50.00000%"}
To authenticate the robustness (by varying against input frequency of THz light and geometry parameters) of our adiabatic following design, we lay out the contour plot of final population at output SPPs waveguide, by scanning the frequency of frequency of input THz light (from 0.5 THz to 1.3 THz) and device length (from 2 mm to 5 mm), shown as Fig. 5a. We conclude that our design can suffer broadband frequency of input THz light (roughly from 0.9 THz to 1.3 THz) and larger perturbation of device length do not deteriorate our performance, which can endure length from 2 mm to 4 mm (energy transfer rate larger than 0.6, even the lossy in consideration). Furthermore, we set the device length L = 4 mm and frequency of THz wave with 1 THz. We plot the final intensity of SPPs in output SPPs waveguide with varying the offset between two centers of curve $\delta$ (from 1 mm to 2 mm) and curve with radius $R$ (from 40 mm to 50 mm), shown as Fig. 5b. It is very easy to observe that even though our device has relative large errors on the geometry structure parameters ($\delta$ and $R$), intensity of SPPs of output SPPs waveguide still relatively maintain at good performance. Therefore, our adiabatic device is also robust device against geometry structure parameters and fabrication of our device do not require high precision processing of manufacture to achieve low-cost device and high fidelity device.
At the last of this section, we propose a possible fabrication processing to manufacture our designed device. The fabrication techniques of multi-layered or 3D metamaterials have been widely reported. The two key issues in fabricating the coupler we designed are i) how to transfer of metallic corrugated patterns and ii) how to stack the curved waveguides. The optional methods for the former process are prevalent shadow mask lithography, soft lithography or nanoimprint lithography [@Walia2015; @Moser2012], and all techniques can guarantee the high-resolution in sub-microscale. As for the assembly, we can choose the suitable flexible polymer as the substrate, and peel off the structure after every metallic layer is transferred. Since our coupler can work in wide band, its outstanding superiority is the high tolerance for the structural imperfection induced by the fabrication.
Conclusion
==========
Based on stimulated raman adiabatic passage (STIRAP) quantum control technique, we have proposed a novel coupler using a tri-layered surface plasmon-polaritons (SPPs) waveguide curved configuration, in which SPPs can be completely transferred from input corrugated thin film to output corrugated thin film in terahertz (THz) region. We demonstrate that our design realizes highly efficient transfer with strong robustness against the perturbations of geometry parameters, and also illustrate that our device has good performance at broadband excited THz waves. This finding will make contribute to develop compact and robust integrated THz devices, which will promote the future applications in all-optical network and THz communications.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is acknowledged for funding National Science and Technology Major Project (grant no. 2017ZX02101007-003); National Natural Science Foundation of China (grant no. 61565004); National Natural Science Foundation of China (grant no. 61665001); Natural Science Foundation of Guangxi Province (Nos. 2017GXNSFBA198116, 2018GXNSFAA281163).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The $U(1)$ Calogero-Sutherland Model with anti-periodic boundary condition is studied. This model is obtained by applying a vertical magnetic field perpendicular to the plane of one dimensional ring of particles. The trigonometric form of the Hamiltonian is recast by using a suitable similarity transformation. The transformed Hamiltonian is shown to be integrable by constructing a set of momentum operators which commutes with the Hamiltonian and amongst themselves. The function space of monomials of several variables remains invariant under the action of these operators. The above properties imply the quasi-solvability of the Hamiltonian under consideration.'
author:
- Arindam Chakraborty
- Subhankar Ray
- 'J. Shamanna'
date: 30 December 2006
title: 'Quasi-solvability of Calogero-Sutherland model with Anti-periodic Boundary Condition'
---
Introduction
============
The study of Calogero-Sutherland system has inspired significant research activity since the pioneering work of Calogero and Sutherland [@cal62; @suth71]. The integrability of the model has been studied for different root systems over the past few decades [@ols83]. A few of the classical and spin varieties of the model are found to be exactly solvable and the solutions in terms of their eigenvalues and eigenfunctions have been used extensively to describe physical properties of several condensed matter systems. The study of Calogero systems is also related to various other research areas in physics and mathematics, e.g., Yang-Mills theories [@gor94; @mina94], soliton theory [@poly95], random matrix model [@dyson62], multivariable orthogonal polynomials [@jack69], Selberg integral formula [@forr93], $W^{\infty}$ algebra [@hika93] etc.
This article investigates the Calogero-Sutherland Model (CSM) with anti-periodic boundary condition. The anti-periodic boundary condition is a special case of the general twisted boundary condition which arises when a one dimensional chain of particles is placed in a transverse magnetic field. A one dimensional chain of particles with a periodic boundary condition is topologically equivalent to a one dimensional ring. A particle transported adiabatically around this ring an integral number of times returns to the same point. In absence of a magnetic field this implies that the particle returns to the same quantum state. However, in the presence of a transverse magnetic field, one adiabatic transportation around the ring introduces a phase factor $\exp(i \phi)$. This is called a twisted boundary condition. When the phase factor is $\exp(i \phi) = -1$, it is called an anti-periodic boundary condition. Though the introduction of a magnetic field is physically important in this context, the model becomes mathematically more involved; and the CSM with anti-periodic boundary condition remains less extensively investigated.
The original version of the Calogero system incorporates long-range interaction by considering a two-body inverse square potential. The integrability of such systems was initially studied by Calogero and Perelomov [@cal75; @perel77] by means of Lax pair formulation. The integrability of CSM has since been investigated in a variety of ways [@ols83; @mina93; @berm97; @poly99].
The general form of CSM Hamiltonian is often represented by the following equation: \[hamilton\] H\_N=\_[j=1]{}\^N[\_j]{}\^2-(-1) \_ U(x\_[jk]{}\^-) The two-body potential, represented by $U(x_{jk}^-)$ is a long-range interaction in a chain of spinless nonrelativistic particles in one dimension. Here, $\lambda$ is a dimensionless interaction parameter, $x_j$ and $x_k$ denote the coordinates of the $j$-th and $k$-th particle respectively and $x_{jk}^-=x_j-x_k$. While studying the solvability of $A_{N-1}$-type Calogero model, the Hamiltonian is operated on a partially ordered state space of all symmetric polynomials of several variables. This results in an upper triangular representation of the Hamiltonian. The diagonal terms of this matrix are the eigenvalues of the Hamiltonian. The orthonormal eigenfunctions are expressed in terms of Jack symmetric polynomials [@jack69] which are very useful in determining the various physical properties of many particle systems with long-range interactions [@habook].
The search for an exact form of eigenfunction sometimes leads to partial diagonalization of the Hamiltonian [@tana05; @fin01]. Among the one dimensional systems with periodic boundary condition, several such quasi-solvable models exist. The eigenvalues and eigenfunctions for many of them have been obtained[@tur87; @ushbook]. The model with $sl(2)$ structure was first discovered by Turbiner and Ushveridze [@tur88]. It was also observed that the well known $N$ body Calogero-Sutherland models [@cal71; @suth71; @ruhl95] have similar Lie algebraic structure of $sl(N+1)$.
It may be noted that these models are in fact different generalizations of the classically integrable Inozemtsev model [@tana04; @ino83]. The common feature of these models with some underlying Lie algebraic structure is the existence of an invariant finite dimensional module of the associated Lie algebra. Post and Turbiner [@post95] studied a classification of linear differential operators of a single variable which have a finite dimensional invariant subspace spanned by monomials. One of the basic advantages of quasi-solvability is that, one can restrict the study to a finite dimensional submanifold of the full Hamiltonian. The finite dimensional matrix elements can be calculated by allowing the Hamiltonian to act on finite-dimensional subspaces of a Hilbert space on which it is originally defined. When the Hamiltonian operator preserves an infinite number of subsequences of such finite dimensional subspaces [@tana05] it becomes solvable. The exact solvability of a model is ensured when the closure property is imposed on the space on which the Hamiltonian is allowed to act.
In this article we study the integrability and solvability of a spinless non-relativistic Calogero-Sutherland model (CSM) with anti-periodic boundary condition. The two-body long-range interaction incorporating the anti-periodic boundary condition is derived. The Hamiltonian so obtained is reduced to a more apparent integrable form, using a similarity transformation. The integrability is then verified by constructing a set of mutually commuting momentum-like differential operators which further commute with the Hamiltonian. Finally, the concept of quasi-solvability is discussed for a model of many particle system. For CSM with anti-periodic boundary condition the quasi-solvability is studied by operating the Hamiltonian on a multivariable polynomial space [@fin01]. The momentum operators in the anti-periodic model remind us of the well known Dunkl operator [@dunk89; @che91] which resembles the Laplace-Beltrami-type operator acting on a symmetric Riemannian space. These operators are extensively used in the study of integrability and solvability of Calogero-Sutherland models. It is shown that these commuting momentum operators preserve the space spanned by all monomials of degree $n$, i.e., $\{\prod_i z_i^{\ell_i}\}$, where $\ell_i\geq 0$ and $\sum \ell_i = n$, $n$ being a non-negative integer. This property ensures the quasi-solvability of the Hamiltonian under study.
Trigonometric version of CSM Hamiltonian
========================================
Let us first consider the periodic CSM with inverse square long-range interaction in the absence of a magnetic field. The topological representation of a one dimensional chain of particles with periodic boundary condition is simply a circular ring. A particle when transported adiabatically around the ring an integral number of times, does not take up any phase factor, and so the eigenfunctions retain their initial form. Then the pairwise interaction summed around a unitarily equivalent circle of circumference $L$, an infinite number of times is given as, \[pair\_int\] \_[n=-]{}\^[+ ]{} = where, as shown in the figure, $x$ is the interparticle distance along the ring and $d(x)$ is the chord length. It is easy to verify that $d(x) = L/\pi \sin(\pi x/L)$.
Therefore, the potential $U(x)=(\pi^2/L^2)\sin^{-2}{x}$ is an inverse trigonometric function of the inter-particle distance $x$. The Hamiltonian with the above potential is given by, \[htrig1\] H\_N=\_[j=1]{}\^[N]{}[\_j]{}\^2-(-1) \_ where $x_{jk}^-= x_j-x_k$. Using standard trigonometric identity, making a change of variable $(\pi/2L)x_j \rightarrow x_j$ and rescaling the Hamiltonian $(4L^2/\pi^2) H_N \rightarrow H_N $, Eq. (\[htrig1\]) may be written as, \[htrig2\] H\_N=\_[j=1]{}\^[N]{}[\_j]{}\^2-(-1) \_ (+ ) . Let us now consider the anti-periodic case. When a magnetic field is introduced transverse to the one dimensional ring, a general twisted boundary condition arises. A particle transported adiabatically around the entire system $n$ number of times picks up a net phase $\exp{(in\phi)}$. The pairwise interaction summed around a unitarily equivalent circle of circumference $L$, an infinite number of times, is now given as, \[antipair\_int\] \_[n=-]{}\^[+ ]{} . The above summation can be performed by making the choice, $\phi=2\pi p/q$, with $p, \,q$ relative primes, and $n=jq+k $, with $j, \, k$ integers ($-\infty < j < +\infty$ and $0 \le k \le q-1$). The interaction term becomes, \_[n=- ]{}\^[+]{} = \_[k=0]{}\^[q-1]{} \_[j=- ]{}\^[+]{} =\_[k=0]{}\^[q-1]{} . The last expression represents an interaction with a general twisted boundary condition. The model can be viewed as a system of interacting particles residing on a circle with circumference $qL$. For $p/q =1/2$ the sum becomes, \_[k=0]{}\^[q-1]{} \_[k=0]{}\^[1]{} =\_[k=0]{}\^[1]{} . This corresponds to an anti-periodic boundary condition [@habook]. The potential then takes the following form \[antiu\] U(x)= . Thus the Hamiltonian with anti-periodic boundary condition, using standard trigonometric identity, and scale changes $(\pi/2L)x_j \rightarrow x_j$ and $(4L^2/\pi^2) H_N \rightarrow H_N $, becomes \[htrig3\] H\_N\^[\^[ap]{}]{}=\_[j=1]{}\^[N]{}[\_j]{}\^2-(-1) \_ (- ) .
Integrability of the model Hamiltonian
======================================
The integrability of such types of Hamiltonian is established by constructing a complete set of commuting momentum operators that also commute with the model Hamiltonian. These operators were initially introduced in the study of Calogero-Sutherland model with periodic boundary conditions (both spin and classical cases) and are known as Dunkl operators. Similar operators have been used to study Calogero-Sutherland-type models derived from different root systems and are called Dunkl-type operators [@fin01; @buch94].
To construct commutative Dunkl-type operators we introduce variables, $z_j = \exp(2ix_j)$. Using this substitution, the anti-periodic Hamiltonian becomes, \[globe1\] H\_N\^[\^[ap]{}]{}=\_[j=1]{}\^N(z\_j\_j)\^2-(-1) \_ -(-1) \_ , z\_[jk]{}\^ = z\_j z\_k . Let us apply the following similarity transformation, \_N=\^[-1]{}H\_N\^[\^[ap]{}]{}where, =\_ , \_1()=,1-, \_2() = \[1\] . Thus, the anti-periodic Hamiltonian becomes \[sim2\] \_N=\_[j=1]{}\^N(z\_j\_j)\^2 + \_ (z\_j\_j-z\_k\_k) + \_ (z\_j\_j-z\_k\_k) The term $\mu_1(\lambda)$ is real for all $\lambda$, however, $\mu_2(\lambda)$ is real only for $1+4\lambda-4\lambda^2 \geq 0$, i.e., $\vert\lambda-\frac{1}{2}\vert \leq \frac{1}{\sqrt{2}}$. Under this restriction, the Hamiltonian $\widetilde{H}_N$ becomes hermitian. In the following, the integrability of the Hamiltonian is studied for different allowed values of $\lambda$.
Let us introduce the coordinate exchange operators $\{\Lambda_{jk}\vert j,k = 1,..N; j \neq k\}$ and the sign reversing operators $\{\Lambda_j\vert j,k = 1,..N\}$. The coordinate exchange operator acting on the coordinates of $j$-th and $k$-th particle may be defined by the operation $\Lambda_{jk}f(z_1,..,z_j,..,z_k,.., z_N)=
f(z_1,..,z_k,..,z_j,.., z_N)$. This operator is (i) self-adjoint, (ii) unitary, and satisfies (iii) $\Lambda_{ij}\Lambda_{jk}=\Lambda_{ik}\Lambda_{ij}
=\Lambda_{jk}\Lambda_{ik}$ , (iv) $\Lambda_{ij}\Lambda_{kl}=\Lambda_{kl}\Lambda_{ij}$ , (v) $\Lambda_{jk} z_k \partial_k = z_j \partial_j$ .
The sign reversing operator $\Lambda_j$ may be defined by its action on the coordinates of the $j$-th particle as $\Lambda_j f(z_1,..,z_j,.., z_N)=
f(z_1,..,-z_j,.., z_N)$. This operator is (i) self-inverse $\Lambda_{j}^{-1} = \Lambda_{j}$, (ii) mutually commuting $[\Lambda_j, \Lambda_k] = 0$, and satisfies (iii) $[\Lambda_{ij},\Lambda_k]= 0$ , $i\neq j\neq k$, (iv) $\Lambda_{ij}\Lambda_j=\Lambda_i\Lambda_{ij}$.
In terms of $\Lambda_{jk}$ and $\Lambda_j$, we introduce operator $\widetilde{\Lambda}_{jk} = \Lambda_j \Lambda_k
\Lambda_{jk}$, which is (i) self-adjoint and (ii) unitary. In addition it satisfies (iii) $\widetilde{\Lambda}_{ij}\widetilde{\Lambda}_{jk}
=\widetilde{\Lambda}_{ik}\widetilde{\Lambda}_{ij}
=\widetilde{\Lambda}_{jk}\widetilde{\Lambda}_{ik}$, (iv) $\widetilde{\Lambda}_{ij}\widetilde{\Lambda}_{kl}
=\widetilde{\Lambda}_{kl}\widetilde{\Lambda}_{ij}$, (v) $\widetilde{\Lambda}_{jk} z_k \partial_k = z_j \partial_j $. In terms of the above mentioned operators, the Dunkl-type momentum operators $\{ D_j \vert j=1,\dots, N \}$ may be represented by the following equation, \[mom11\] D\_j = z\_j\_j +\_[k(j)]{} (1-\_[jk]{}) . +\_[k(j)]{} (1-\_[jk]{}) $ \widetilde{H}_N = \sum_j D_j^2 $. These Dunkl-type operators commute with the coordinate exchange operators and the operators $\widetilde{\Lambda}_{jk}$, i.e; $[D_j, \Lambda_{jk}] = 0$, $[D_j, \widetilde{\Lambda}_{jk}] = 0$. They also commute among themselves and because of the very nature of their construction, commute with the Hamiltonian; $[D_j, D_k] = 0$, $[D_j, \widetilde{H}_N ] = 0$. The existence of such an operator establishes the integrability of the system.
Quasi-solvability of the model Hamiltonian
==========================================
The integrability does not necessarily imply the existence of a function space involving the variables $\{z_j \vert j = 1,\dots, N \}$ such that $ \widetilde{H}_N $ can be represented in a diagonal form. However, sometimes it may so happen that operators like $\widetilde{H}_N $, acting on a suitably chosen function subspace can preserve the space partially. In such cases we introduce the term quasi-solvability. A linear differential operator $H_N$ of several variables $\{z_j \vert j=1,\dots,N\}$, is said to be quasi-solvable if it preserves a finite dimensional function space $V_{\nu}$ whose basis admits an analytic expression in a closed form i.e., H\_N V\_ V\_, V\_ = n() < , 0.5cm 0.5cm V\_ = v\_1(z),…, v\_[n()]{}(z).
One of the advantages of quasi-solvability is that one can explicitly evaluate finite dimensional matrix elements $A_{kl}$ defined by H\_N v\_k = \_[l=1]{}\^[n()]{}A\_[kl]{}v\_l, (k = 1,…, n()). The finite dimensional submatrices $A_{kl}$ may be diagonalizable even when the entire $H_N$ is not. If the space $V_{\nu}$ is the subspace of a Hilbert space on which the operator $H_N$ is defined, the spectrum of $H_N$ can be computed algebraically, so as to obtain the exact eigenvectors of $H_N$ that belong entirely to $V_{\nu}$. This is the typical nature of quasi-solvability.
A quasi-solvable operator is said to be solvable if the quasi-solvability condition holds for an infinite number of sequences of finite dimensional proper subspaces each containing its previous descendant. V\_1 V\_2 …V\_ … Moreover, if the closure of $V_{\nu}$, as $\nu \rightarrow \infty $, is the Hilbert space on which $H_N$ acts, we call $H_N$ to be exactly solvable.
Now, we shall show that the Dunkl-type momentum operators obtained in Eq.(\[mom11\]) preserve the space $\mathcal{R}_n$ i.e., the space spanned by all monomials of the form $\prod_i z_i^{\ell_i}$, where $\ell_i \geq 0$ and $\sum_i \ell_i = n$, $n$ being a non negative integer.
It is easy to verify that the operator $(z_j\partial_j)$ preserves the space $\mathcal{R}_n$. (z\_j\_j) \_i z\_i\^[\_i]{} = \_j \_i z\_i\^[\_i]{}
We shall show that the second and third operators in Eq. (\[mom11\]) preserve $\mathcal{R}_n$, i.e., $(z_j+z_k)/(z_j-z_k)(1-\Lambda_{jk})\prod_i z_i^{\ell_i}
\in \mathcal{R}_n $ and $(z_j-z_k)/(z_j+z_k)(1-\widetilde{\Lambda})_{jk}\prod_i z_i^{\ell_i}
\in \mathcal{R}_n $. They can be rewritten as \[ratio1\] (1-\_[jk]{})\_i z\_i\^[\_i]{} =(\_[i(j,k)]{} z\_i\^[\_i]{})(z\_j+z\_k)(z\_jz\_k) \^[(\_j,\_k)]{}(\_j-\_k) \_[r=0]{}\^[\_j-\_k-1]{}z\_j\^[\_j -\_k-1-r]{}z\_k\^r and \[ratio2\] (1-)\_[jk]{}\_i z\_i\^[\_i]{} =(\_[i ()]{} z\_i\^[\_i]{})(z\_j-z\_k)(z\_jz\_k) \^[(\_j,\_k)]{}(\_j,\_k) \_[r=0]{}\^[\_j-\_k-1]{}(-z\_j)\^[\_j -\_k-1-r]{}z\_k\^r where $\textrm{sign}(0) = 0$ and $\textrm{sign}(\alpha) = \alpha / \vert \alpha \vert$ for $\alpha \neq 0 $. And (,) = {
[cc]{} 1 &
. The right hand side of Eq.(\[ratio1\]) can be expressed as a sum of the following two terms, \[term1\] (\_[i]{} z\_i\^[\_i]{})z\_j (z\_jz\_k) \^[(\_j,\_k)]{}(\_j-\_k) \_[r=0]{}\^[\_j-\_k-1]{}z\_j\^[\_j -\_k-1-r]{}z\_k\^r and \[term2\] (\_[i]{} z\_i\^[\_i]{})z\_k(z\_jz\_k) \^[(\_j,\_k)]{}(\_j-\_k) \_[r=0]{}\^[\_j-\_k-1]{}z\_j\^[\_j -\_k-1-r]{}z\_k\^r . Let $p_r^j$ and $p_r^k$ denote the powers of $z_j$ and $z_k$ in the $r$-th summand of Eq. (\[term1\]). Then $p_r^j = \max(\ell_j, \ell_k) - r $ and $p_r^k = \min(\ell_j, \ell_k) + r$ . Thus, $p_r^j + p_r^k = \ell_j + \ell_k$. Therefore, the sum of powers of $z_j$ and $z_k$, in the $r$-th summand is $(\sum_{i\neq j, k} \ell_i ) + \ell_j+ \ell_k = \sum_i \ell_i$.
Hence, the expression (\[term1\]) is a member of $\mathcal{R}_n$. Similar calculation shows that the expression (\[term2\]) also belongs to a space spanned by monomials of degree $n$. Thus, the second operator in Eq. (\[mom11\]) preserves $\mathcal{R}_n$. In a similar manner it can be verified that the third operator in Eq. (\[mom11\]) also preserves $\mathcal{R}_n$. As the operators $\{D_j \vert j=1,\dots,N \}$ are linear and preserve the space $\mathcal{R}_n$, the Hamiltonian $\widetilde{H}_N(=\sum D_j^2)$ also preserves $\mathcal{R}_n$, and hence is quasi-solvable.
Conclusion
==========
In this article we have studied the behaviour of one dimensional chain of particles in a magnetic field interacting through an inverse square potential. The anti-periodic boundary condition allows one to analyze a special form of the above situation. The extension of the model to anti-periodic case reduces the algebraic symmetry of the root systems. This makes this model mathematically more challenging. Here, we recast the Hamiltonian to a new form by using a suitable similarity transformation. The transformed Hamiltonian is shown to be integrable in the sense that there exists a complete set of commuting momentum operators which also commute with the Hamiltonian. It is observed that the momentum operators are hermitian for a certain range of the interaction parameter.
The new form of the Hamiltonian and its constituent momentum operators indicate the existence of a multivariable polynomial space which is invariant under the action of the Hamiltonian. Indeed, it is observed that the momentum operators constructed in this article keep the monomial space $\mathcal{R}_n$ invariant. This invariance demonstrates the quasi-solvability of the model.
Acknowledgment {#acknowledgment .unnumbered}
==============
AC wishes to acknowledge the Council of Scientific and Industrial Research, India (CSIR) for fellowship support.
[99]{} F. Calogero, J. Math. Phys. 10 (1969) 2191; 10 (1969) 2197. B. Sutherland, J. Math. Phys. 12 (1971) 246; 12 (1971) 251. M. A. Olshanetsky, A. M. Perelomov, Phys. Rep. 94 (1983) 313. A. Gorsky and N. Nekrasov, Nucl. Phys. B 414 (1994) 213. J. A. Minahan and A. P. Polychronakos, Phys. Lett. B 326 (1994) 288. A. P. Polychronakos, Phys. Rev. Lett. 74 (1995) 5153. F. J. Dyson, J. Math. Phys. 3 (1962) 140, 157, 166. H. Jack, Proc. Roy. Soc. (Edinburgh) A 69 (1970) 1. P. J. Forrester, Phys. Lett. A 179 (1993) 127. K. Hikami, M. Wadati, J. Phys. Soc. Japan 62 (1993) 469. F. Calogero, Lett. Nuovo Cim. 13 (1975) 411. A. M. Perelomov,Lett. Math. Phys. 1 (1977) 531. J. A. Minahan, A. P. Polychronakos, Phys. Lett. B 302 (1993) 265. D. Bernard, M. Gaudin, F. D. M. Haldane, V. Pasquier, J. Phys. A 26 (1993) 5219; hep-th/9301084 v1. A. P. Polychronakos, Nucl. Phys. B 543 (1999) 485. Z. N. C. Ha, Quantum Many-Body Systems in One Dimension, World Scientific, Singapore, 1996. T. Tanaka, ann. Phys. 320 (2005) 199; hep-th/0502019 v2. F. Finkel, D. Gomez-Ullate, A. Gonzalez-Lopez, M. A. Rodriguez, R. Zhdanov, Nucl. Phys. B 613 (2001) 472; hep-th/0103190 v2. A. V. Turbiner, A. G. Ushveridze, Phys. Lett. A 126 (1987) 181. A. G. Ushveridze, Quasi exactly solvable models in Quantum Mechanics, IOP publishing, Bristol, 1994. A. V. Turbiner, Commun. Math. Phys. 118 (1988) 467. F. Calogero, J. Math. Phys. 12 (1971) 419; J. Math. Phys. 37 (1996) 3646. W. Ruhl, A. Turbiner, Mod. Phys. Lett. A 10 (1995) 2213; hep-th/9506105. T. Tanaka, Ann. Phys. 309 (2004) 239. V. I. Inozemtsev, Phys. Lett. A 98 (1983) 316. G. Post, A. Turbiner, Russ. J. Math. Phys. 3 (1995) 113. C. F. Dunkl, Trans. Amer. Math. Soc. 311 (1989) 167. I. V. Cherednik, Invent. Math. 106 (1991) 411; Adv. Math. 106 (1994) 65. V. M. Buchstaber, G. Felder, A. P. Vaselov, Duke Math. J. 76 (1994) 885; hep-th/9403178 v1.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, the exact dynamics of open quantum systems in the presence of initial system-reservoir correlations is investigated for a photonic cavity system coupled to a general non-Markovian reservoir. The exact time-convolutionless master equation incorporating with initial system-reservoir correlations is obtained. The non-Markovian dynamics of the reservoir and the effects of the initial correlations are embedded into the time-dependent coefficients in the master equation. We show that the effects induced by the initial correlations play an important role in the non-Markovian dynamics of the cavity but they are washed out in the steady-state limit in the Markovian regime. Moreover, the initial two-photon correlation between the cavity and the reservoir can induce nontrivial squeezing dynamics to the cavity field.'
author:
- 'Hua-Tang Tan'
- 'Wei-Min Zhang'
title: 'Dynamics of open quantum systems with initial system-reservoir correlations'
---
= 10000
Introduction
============
The study of dynamics of open quantum systems continuously receives attentions because of its fundamental importance in quantum physics and also because of the rapid development of quantum technologies. Previous studies on the dynamics of open quantum systems mainly lie on the Lindblad-type master equation [@bm1; @bm2; @bm3], where the characteristic time of the environment is sufficiently shorter than that of the system such that the non-Markovian memory effect is negligible, and so does for initial system-reservoir correlations. However, the new development in ultrafast photonics, ultracold atomic physics, nanoscience and technology as well as quantum information science strongly suggests that the non-Markovian dynamics in ultrafast and ultrasmall open systems should play an important role, and the associated effects (including the initial system-reservoir correlations) should be fully taken into account. To this end, the more rigorous approach is demanded for the study of non-Markovian dynamics of open quantum systems incorporating with the initial system-reservoir correlations.
The exact description of open quantum systems has indeed been explored extensively in the literature, mainly focusing on quantum Brown motion based on Feynman-Vernon influence functional [@Fey63118; @Cal83587; @Haa852462; @Hu922843; @Hal962012; @Food01105020] and stochastic diffusion Schrödinger equation [@Str981699; @Str994909; @Yu04062107]. Extending the Feynman-Vernon influence functional to other open quantum systems has also achieved a great success recently, including the exact master equation for electron systems and the nonequilibrium quantum transport theory in various nanostructures [@Tu08235311; @Tu09631; @Jin10083013] and the exact master equation for micro- or nanocavities in photonic crystals and the quantum transport theory for photonic crystals [@Xio10012105; @Wu1018407; @Lei104570]. However, in most of these investigations, the system and the reservoir are often assumed to be initially uncorrelated with each other [@Leg871]. Realistically, it is possible and often unavoidable in experiments that the system and its environment are correlated closely at the beginning, especially for the cases of the system strongly coupled to the reservoir [@ee]. Various initial-correlation induced effects have been investigated in different open quantum systems [@src0; @src1; @src2; @src3; @src4; @src5; @src7; @src8; @src6; @src9; @src10]. For example, it has been recently shown that the initial correlations between a qubit and its environment can lead to the distance growth of two quantum states over its initial value [@src7; @src8]. It has also been demonstrated that the initial correlations have nontrivial differences in quantum tomography process [@src6]. Besides, it has been found that the initial system-reservoir correlations have significant effects on the entanglement in a two-qubit system [@src9; @src10].
In this paper, the dynamics of open quantum systems in the presence of initial system-reservoir correlations is investigated with a photonic cavity system coupled to a non-Markovian reservoir as a specific example. By solving the exact dynamics of the cavity system, the effects of the initial correlations are explicitly built into the equations of motion for the intensity and the two-photon correlation function of the cavity field. We then obtain the exact master equation incorporating with the initial correlations which induce new terms and also modify the time-dependent dissipation and fluctuation coefficients in the master equation. Taking a nanocavity coupled to a coupled resonator optical waveguide (serving as a structured reservoir) as an experimentally realizable system, we find that the effects of the initial correlations are fragile for a Markovian reservoir but play an important role in the non-Markovian regime. In fact, in the strong non-Markovian regime, the initial two-photon correlation between the cavity and the reservoir can induce oscillating squeezing dynamics in the cavity. But in the Markovian regime, the initial correlations will be washed out in the steady-state limit.
The rest of the paper is organized as follows. In Sec. II, the dynamics of open quantum systems with initial system-reservoir correlations is formulated for a photonic cavity system coupled to a general non-Markovian reservoir. In Sec. III, we construct the exact time-convolutionless master equation incorporating with the initial correlations, where the effects from the initial correlations are explicitly embedded into the time-dependent coefficients in the master equation. In Sec. IV, an experimentally realizable example is considered to analytically and numerically examine the influence of the initial correlations on the dynamics of open quantum systems. At last, a summary is given in Sec. V.
Non-Markovian dynamics with initial system-reservoir correlations
=================================================================
To be specific, we consider here a single-mode photonic cavity system coupled to a general non-Markovian reservoir, where the single-mode cavity system could be a nanocavity in nanostructures or photonic crystals, and the non-Markovian environment may be a structured photonic reservoir [@stru-reservoir]. The Hamiltonian of the system can be expressed as a Fano-type model of a localized state coupled with a continuum [@Fano611866]: $$\begin{aligned}
H=\omega_c a^\dag a+\sum_{k}\omega_k b_k^\dag b_k\ +\sum_kV_k (a
b_k^\dag +b_k a^\dag),\label{H1}\end{aligned}$$ where the first term is the Hamiltonian of the cavity field with frequency $\omega_c$, and $a^\dag$ and $a$ are the creation and annihilation operators of the cavity field; the second term describes a general non-Markovian reservoir which is modeled as a collection of infinite photonic modes, where $b_k^\dag$ and $b_k$ are the corresponding creation and annihilation operators of the $k$-th photonic mode with frequency $\omega_k$. The third term characterizes the system-reservoir coupling with the coupling strength $V_k$ between the cavity field and the $k$-th photonic mode. For convenience, we take $\hbar=1$ throughout the paper.
We shall use the equation of motion approach to solve the dynamics of the cavity system and the reservoir, from which the general initial correlations between the cavity and the reservoir can be fully taken into account. The time evolution of the cavity field operator $a(t)=e^{iHt}ae^{-iHt}$ and the reservoir field operators $b_k(t)=e^{iHt}b_ke^{-iHt}$ in the Heisenberg picture obey the equations of motion
$$\begin{aligned}
&\frac{d}{dt}a(t)=-i[a(t), H]=-i\omega_c a(t)-i\sum_k V_k b_k(t),\\
&\frac{d}{dt}b_k(t)=-i[b_k(t),H]=-i\omega_k b_k(t)-iV_k a(t).
\label{bk}\end{aligned}$$
Solving Eq. (\[bk\]) for $b_k(t)$ $$\begin{aligned}
b_k(t)=b_k(0)e^{-i\omega_k t}-iV_k\int_0^t d\tau
a(\tau)e^{-i\omega_k (t-\tau)},\end{aligned}$$ we obtain $$\begin{aligned}
\frac{d}{dt}a(t)=-i\omega_c a(t) -\int_0^t d\tau g(t-\tau)a(\tau)
\nonumber\\-i\sum_k V_k b_k(0)e^{-i\omega_k t}. \label{lat}\end{aligned}$$ Here, the memory kernel $g(\tau)=\sum_{k}|V_k|^2e^{-i\omega_k\tau}$ characterizes the non-Markovian dynamics of the reservoir. For a continuous reservoir spectrum, we have $g(\tau)=\int_{0}^\infty
\frac{d\omega}{2\pi}J(\omega)e^{-i\omega\tau}$, where $J(\omega)=2\pi
\varrho(\omega)|V(\omega)|^2$ is the spectral density of the reservoir, with $\varrho(\omega)$ being the density of states and $V(\omega)$ the coupling between the cavity and the reservoir in the frequency domain.
Because of the linearity of Eq.(\[lat\]), the cavity field operator $a(t)$ can be expressed, in terms of the initial field operators $a(0)$ and $b_k(0)$ of the cavity and the reservoir, as $$\begin{aligned}
a(t)=u(t)a(0)+f(t) , \label{at}\end{aligned}$$ where the time-dependent coefficient $u(t)$ and $f(t)$ are determined from Eq.(\[lat\]) and given by
$$\begin{aligned}
\frac{d}{dt}u(t)=-i\omega_c u(t)-\int_0^t d\tau
g(t-\tau)u(\tau),\label{ut}
\\
\frac{d}{dt}f(t)= -i\omega_c f(t)-\int_0^t d\tau g(t-\tau)f(\tau) \label{ft} \nonumber\\
-i\sum_kV_k b_k(0)e^{-i\omega_k \tau},\end{aligned}$$
subjected to the initial conditions $u(0)=1$ and $f(0)=0$. The integrodifferential equation (\[ut\]) shows that $u(t)$ is just the propagating function of the cavity field (the retarded Green function in nonequilibrium Green function theory [@green]). In addition, $f(t)$ is in fact an operator coefficient and its solution can be obtained analytically from the inhomogeneous equation of Eq. (\[ft\]): $$\begin{aligned}
f(t)=-i\sum_kV_k b_k(0)\int_0^t d\tau e^{-i\omega_k \tau}u(t-\tau).
\label{ress}\end{aligned}$$
From Eqs. (\[at\])-(\[ress\]) we can determine the exact non-Markovian dynamics of the cavity field coupled to a general reservoir with arbitrary initial system-reservoir correlations, upon a given initial state $\rho_{\rm tot}(0)$ of the whole system. In the Heisenberg picture, quantum states are time-independent. Once $\rho_{\rm
tot}(0)=\rho_{\rm tot}$ is given, the time evolution of any physical observable can be obtained directly from Eqs. (\[at\])-(\[ress\]) through the relation $$\begin{aligned}
\langle f(a^\dag(t),a(t))\rangle = {\rm
tr}[f(a^\dag(t),a(t))\rho_{\rm tot}].\end{aligned}$$ For example, the time evolution of the expectation values $\langle
a(t) \rangle$, $n(t) \equiv \langle a^\dag(t) a(t) \rangle$, and $s(t) \equiv \langle a(t) a(t)\rangle$, which respectively describe the cavity amplitude, the cavity intensity, and the two-photon correlation of the cavity field, can be expressed explicitly by the following solution
\[e\] $$\begin{aligned}
&\langle a (t) \rangle=u(t)\langle a (0)\rangle + \upsilon_0(t),\label{e1}\\
& n(t)=|u(t)|^2 n(0)
+2{\rm Re}[u^*(t)\nu_1(t)]+\upsilon_1(t),\label{e2}\\
& s(t)=u^2(t) s (0) +2u(t)\nu_2(t)+\upsilon_2(t) , \label{e3}\end{aligned}$$
where $\langle a (0)\rangle, n(0)$ and $ s (0)$ are the corresponding initial conditions. Other time-dependent functions in Eq. (\[e\]) are given by
\[core\]
$$\begin{aligned}
\nu_1(t)&=\langle a^\dag(0)f(t)\rangle
=-i\int_0^t\sum_k V_k \langle a^\dag(0)b_k(0)\rangle e^{-i\omega_k \tau}u(t-\tau)d\tau,\label{u1}\\
\nu_2(t)&=\langle a(0)f(t)\rangle
=-i\int_0^t\sum_k V_k \langle a(0) b_k(0)\rangle e^{-i\omega_k \tau}u(t-\tau)d\tau,\label{u2}\\
\upsilon_0(t)&=-i\int_0^t\sum_k V_k \langle b_k(0)\rangle
e^{-i\omega_k\tau}u(t-\tau)d\tau,\label{u0}\\
\upsilon_1(t)&=\langle f^\dag(t)f(t)\rangle =\int_0^t d\tau\int_0^t
d\tau'\sum_{kk'} V^*_k V_{k'}\langle b^\dag_k(0)
b_{k'}(0)\rangle e^{-i(\omega_{k'}\tau'-\omega_k\tau)}u^*(t-\tau)u(t-\tau'),\label{v1}\\
\upsilon_2(t)&=\langle f(t)f(t)\rangle =-\int_0^t d\tau\int_0^t
d\tau'\sum_{kk'} V_k V_{k'}\langle b_k(0) b_{k'}(0)\rangle
e^{-i(\omega_k\tau+\omega_{k'}\tau')}u(t-\tau)u(t-\tau')\label{v2}.\end{aligned}$$
In these solutions, $\upsilon_j(t)$ ($j=0,1,2$) characterize respectively the contributions from the initial field amplitudes $\langle b_k(0)\rangle$, the initial photon scattering amplitudes $\langle b^\dag_k(0) b_{k'}(0)\rangle$ and the initial two-photon correlations $\langle b_k(0) b_{k'}(0)\rangle$ of all the photonic modes in the reservoir. While $\nu_1(t)$ and $\nu_2(t)$ correspond to the contributions from the different initial system-reservoir correlations $\langle a(0)b_k^\dag(0)\rangle$ and $\langle
a(0)b_k(0)\rangle$, respectively.
If the initial state of the total system is uncorrelated, and the reservoir is in a thermal equilibrium state, i.e., $$\begin{aligned}
\rho_{\rm tot}(0)=\rho(0) \times \rho_R(0), ~~
\rho_R(0)=\frac{e^{-\beta H_R}}{tr e^{-\beta H_R}},\end{aligned}$$ with $H_R=\sum_k\omega_kb_k^\dag b_k$ and $\beta=1/k_BT$ being the initial temperature of the reservoir, it is easy to check that $\nu_i(t)=0, \upsilon_i(t)=0$ except for $v_1(t)$ which is given by $$\begin{aligned}
\upsilon_1(t) =\int_0^t d\tau\int_0^t d\tau'
u(t-\tau')\widetilde{g}(\tau'-\tau) u^*(t-\tau). \label{vts}\end{aligned}$$ Here, $\widetilde{g}(\tau)=\sum_k |V_k|^2 \bar{n}_k e^{-i\omega_k
\tau}$ and $\bar{n}_k=\langle b^\dag_k(0) b_{k}(0)\rangle=
1/(e^{\beta \omega_k} -1)$ is the initial photonic distribution function of the reservoir. Then Eq. (\[e\]) reproduces the same solution solved from the exact master equation without initial system-reservoir correlations [@Xio10012105]. However, as we see, the exact non-Markovian dynamics in Eq.(\[e\]) derived via the equation of motion approach has explicitly included the effects induced by the initial correlations between the system and the reservoir.
Exact master equation with initial system-reservoir correlations
================================================================
To further understand the effects of the initial system-reservoir correlations on the dynamics of open quantum systems, we shall attempt to derive the exact master equation for the reduced density matrix of the cavity system $\rho(t)$. In the literature, exact master equations for open systems are mostly derived without initial correlations, such as the systems associated with quantum Brown motions [@Hu922843; @Hal962012; @Food01105020], quantum dot systems in various nanostructures [@Tu08235311; @Tu09631] and cavity systems coupled to structured reservoirs as well as general non-Markovian reservoirs [@An07042127; @Xio10012105; @Wu1018407]. Here, we concentrate the exact master equation for the photonic system in the presence of initial Gaussian correlated states. Based on the bilinear operator structure of the system as well as the techniques in deriving exact master equation for the cavity system described by Eq. (\[H1\]) [@Xio10012105; @Wu1018407], the master equation with the initial system-reservoir correlations would have a general time-convolutionless form as follows: $$\begin{aligned}
\dot{\rho}(t)=& -i\Delta(t)[a^\dag a, \rho] \notag \\
& +\gamma_1(t)(2a\rho a^\dag -a^\dag a\rho-\rho_a a^\dag a)\nonumber\\
&+\gamma_2(t)(a\rho a^\dag+a^\dag\rho a -a^\dag a\rho-\rho a a^\dag)\nonumber\\
&+\gamma_3^*(t)(2a\rho a-aa\rho-\rho aa)\nonumber\\
&+\gamma_3(t)(2a^\dag \rho a^\dag -a^\dag a^\dag \rho-\rho a^\dag
a^\dag),\label{me}\end{aligned}$$ where the coefficient $\Delta(t)$ is the renormalized cavity frequency, $\gamma_1(t)$ and $\gamma_2(t)$ usually denote respectively the dissipation (damping) and fluctuation (noise) due to the back-reactions between the system and the reservoir, and $\gamma_3(t)$ is related to a two-photon decoherence process. As we see, the first three terms have the standard form as the exact master equation for the Hamiltonian in Eq. (\[H1\]) without the initial correlations [@Xio10012105; @Wu1018407], but with the coefficients modified by the initial correlation $\langle
a(0)b^\dag_k(0) \rangle$. The last two terms are contributed from the two-photon correlation $\langle b_k(0)b_{k'}(0) \rangle$ in the reservoir as well as by the initial system-reservoir two-photon correlation $\langle a(0)b_k(0) \rangle$.
To figure out the time-convolutionless coefficients in Eq. (\[me\]), we compute the physical observables in Eq. (\[e\]) from the above master equation. From Eq. (\[me\]), it is easy to find that
\[opem\] $$\begin{aligned}
&\frac{d}{dt}\langle a(t)\rangle=-[\gamma_1(t)+i\Delta(t)]\langle a(t) \rangle, \\
&\frac{d}{dt}n(t)=-2\gamma_1(t)n(t)+2\gamma_2(t), \\
&\frac{d}{dt}s(t)=-2[\gamma_1(t)+i\Delta(t)]s(t)-2\gamma_3(t).\label{meq}\end{aligned}$$
On the other hand, with Eq.(\[at\]) we obtain $$\begin{aligned}
\frac{d}{dt}a(t)=\frac{\dot{u}(t)}{u(t)}a(t)-\frac{\dot{u}(t)}{u(t)}f(t)
+\dot{f}(t).\label{at3}\end{aligned}$$ Note that the photonic modes in the reservoir usually cannot be a coherent state so that $\langle b_k(0)\rangle=0$. Then using Eq. (\[at3\]), we find that
\[opem1\] $$\begin{aligned}
&\frac{d}{dt}\langle a(t)\rangle=\frac{\dot{u}(t)}{u(t)}\langle a(t)\rangle, \\
&\frac{d}{dt}n(t)=2{\rm
Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big]n(t) +\dot{\upsilon}_1(t) -2{\rm Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big]\upsilon_1(t)\nonumber\\
&~~~~~~~~~~ ~~~~~~ +2{\rm Re}\Big[u^*(t)\dot{\nu}_1(t)-\frac{\dot{u}(t)u^*(t)}{u(t)}\nu_1(t)\Big] ,\\
&\frac{d}{dt}s(t)=2\frac{\dot{u}(t)}{u(t)}s(t) +\dot{\upsilon}_2(t) -2\frac{\dot{u}(t)}{u(t)}\upsilon_2(t)
\nonumber\\
&~~~~~~~~~~ ~~~~~~~~~~~~~~~ +2u(t)\dot{\nu}_2(t)
-2\dot{u}(t)\nu_2(t).\label{ap1}\end{aligned}$$
By comparing Eq. (\[opem\]) with Eq. (\[opem1\]), the coefficients $\Delta(t)$ and $\gamma_j(t)$ in the master equation can be uniquely determined and given by
\[ecoff\] $$\begin{aligned}
&\Delta(t)=-{\rm Im}\Big[\frac{\dot{u}(t)}{u(t)}\Big],~~
\gamma_1(t)=-{\rm Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big],\\
&\gamma_2(t)=\dot{\upsilon}_1(t)+2{\rm Re}\Big[ u(t)\dot{\nu}_1^*(t)
-\frac{\dot{u}(t)}{u(t)}[ \upsilon_1(t)+u^*(t)\nu_1(t)]\Big],\\
&\gamma_3(t)=-\frac{1}{2}\dot{\upsilon}_2(t)
+\frac{\dot{u}(t)}{u(t)}\upsilon_2(t)-
u(t)\dot{\nu}_2(t)+\dot{u}(t)\nu_2(t),\end{aligned}$$
which shows that the coefficients $\gamma_2(t)$ and $\gamma_3(t)$ in the master equation depend explicitly on the initial correlations $\langle a(0)b_k^\dag(0)\rangle$ and $\langle a(0)b_k(0)\rangle$ in the presence of the initial Guassian correlated states of the whole system.
If the reservoir is initially in a thermal state uncorrelated to the system, we have $\langle a(0)b^\dag_k(0) \rangle =\langle a(0)b_k(0)
\rangle=\langle b_k(0)b_{k'}(0) \rangle = 0$ except for $\langle
b^\dag_k(0)b_{k'}(0)=\bar{n}_k$. Accordingly, from Eq.(\[core\]) we have $\upsilon_2(t)=0=\nu_1(t)=\nu_2(t)$ so that $\gamma_3(t)=0$ and $$\begin{aligned}
\gamma_2(t)=\dot{\upsilon}_1(t)-2{\rm
Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big]\upsilon_1(t),\end{aligned}$$ where $\upsilon_1(t)$ is then given by Eq. (\[vts\]). Consequently, the master equation (\[me\]) in this situation is reduced to the exact master equation for the cavity system coupled with a general non-Markovian reservoir presented recently in Ref. [@Xio10012105; @Wu1018407], which is obtained originally using the Feynman-Vernon influence functional. In addition, if there are no initial correlations but the reservoir involves initially two-photon correlation, namely, $\langle a(0)b^\dag_k(0) \rangle
=\langle a(0)b_k(0) \rangle=0$ but $\langle b_k(0)b_{k'}(0) \rangle
\neq 0$, then we have $\nu_1(t)=0=\nu_2(t)$ but $\upsilon_2(t)\neq0$. As a result, the coefficient $\gamma_3(t) \neq
0$, which induces a two-photon decoherence process in the cavity [@TMSS1]. However, if the initial states of the whole system only contains the two-photon correlation $\langle a(0)b_k(0)\rangle$ but the reservoir itself stays in an initial thermal state, then we have $\nu_1(t)=0=\upsilon_2(t)$ but $\nu_2(t) \neq 0$. This situation also leads to a non-zero $\gamma_3(t)$ which is essentially equivalent to the situation in which the reservoir involves initially two-photon correlation but without the initial system-reservoir correlations.
Therefore, the master equation, Eq. (\[me\]) with the time-dependent coefficients in Eq. (\[ecoff\]), describes the exact non-Markovian dynamics of a cavity system coupled with a general reservoir involving two-photon correlation in the presence of the quadratic-type initial correlations between the system and reservoir. It shows explicitly that the initial correlation $\langle
a(0)b_k^\dag(0)\rangle$ modifies the fluctuation coefficient $\gamma_2(t)$ but without altering the damping (dissipation) rate $\gamma_1(t)$, which in turn changes the cavity field intensity given by Eq. (\[e2\]) without changing the cavity field amplitude of Eq. (\[e1\]). The initial correlation $\langle
a(0)b_k(0)\rangle$ affects on the two-photon decoherence process which leads to a two-photon process $s(t)=\langle a(t)a(t) \rangle$ of the cavity field. It should be pointed out that if the system and the reservoir are initially in non-Gaussian correlated states, the form of Eq. (\[me\]) may need to be modified further. Nevertheless, the master equation of Eq. (\[me\]) is exact for the initial Gaussian correlated states of the whole system, and it remains in a time-convolutionless form in which the non-Markovain memory dynamics is fully embedded into the time-dependent coefficients. As we see, all these time-dependent coefficients are determined by a unique function, the cavity field propagating function $u(t)$, through the relations given by Eqs. (\[ecoff\]) and (\[core\]). While the propagating function $u(t)$ is determined by Eq. (\[ut\]) in which the integral kernel contains all the non-Markovian memory effects characterizing the back-reactions between the system and the reservoir.
Examples with initial system-reservoir correlations
===================================================
In this section, we shall take two different initial correlated states to examine the effects of the initial correlation on the non-Markovian dynamics in such an open system. To be more specific, we consider an experimentally realizable nanocavity system. Fig. 1 is a schematic plot for a single-mode nanocavity coupled to a coupled resonator optical waveguide (CROW) structure. The nanocavity could be a point defect created in photonic crystals and the waveguide consists of a linear defects in which light propagates due to the coupling of the adjacent defects. The CROW is called as a structured reservoir which possess strong non-Markovian effects [@FanoAnderson-2; @Wu1018407]. The Hamiltonian of the whole system is given by $$\begin{aligned}
H&=&\omega_ca^\dag a+\sum_{n}\omega_0b_n^\dag b_n
+\lambda (ab_1^\dag+b_1a^\dag)\nonumber\\
&&-\sum_{n}\lambda_0(b_nb_{n+1}^\dag+b_{n+1}b_n^\dag),\label{H2}\end{aligned}$$ where $a$ and $a^\dag$ are the annihilation and creation operators of the nanocavity field with frequency $\omega_c$, and the annihilation and creation operators $b_n$ and $b_n^\dag$ describe the resonators at site $n$ in the waveguide with identical frequency $\omega_0$. The frequencies $\omega_c$ and $\omega_0$ are tunable by changing the size of the relevant defects. The third terms describes the coupling of the nanocavity field to the resonator at the first site in the waveguide with the coupling strength $\lambda$ which is also controllable experimentally by adjusting the distance between defects. The last term characterizes the photon hopping between two consecutive resonators in the waveguide structure with the controllable hopping amplitude $\lambda_0$.
Consider the waveguide is semi-infinite long, then performing the following Fourier transformation $b_k=\sqrt{2/\pi}\sum_{n=1}^\infty\sin(nk)b_n$, where the operators $b_k$ and $b_k^\dag$ correspond to the Bloch modes of the waveguide, the Hamiltonian of Eq. (\[H2\]) becomes $$\begin{aligned}
H&=&\omega_ca^\dag a+\sum_{k}\omega_kb_k^\dag b_k
+\sum_{k}g_k(ab_k^\dag+b_ka^\dag),\label{H3}\end{aligned}$$ where $$\begin{aligned}
\omega_k=\omega_0-2\lambda_0\cos k~,
~~g_k=\sqrt{\frac{2}{\pi}}\lambda\sin k. \label{strength}\end{aligned}$$ with $0\le k\le \pi$. As we see, Eq. (\[H3\]) is reduced to the same form of Eq. (\[H1\]) for the system considered in Secs. II-III.
Initially system-reservoir correlated squeezed state
----------------------------------------------------
For the above specific physical system, we shall first consider an initial system-reservoir correlated state with two-photon correlation $\langle a(0)b_{k}(0)\rangle \neq 0$. We assume that the cavity field is correlated initially with the first resonator mode of the CROW in terms of a two-mode entangled squeezed vacuum state [@TMSS1] as $$\begin{aligned}
|\psi_{ab_1}(0)\rangle=\exp(-r_se^{-i\theta_s}ab_1+r_se^{i\theta_s}a^\dag
b_1^\dag)|0_a0_{b_1}\rangle,\label{cstate1}
%&=&\sec r_s\sum_{n=0}^\infty[-\exp(i\theta_s)\tanh %r_s]^n|n_an_{b_1}\rangle.\nonumber\\\end{aligned}$$ and the other resonators in the CROW are in vacuum, with $r_s$ and $\theta_s$ being the squeezing parameter and the reference phase, respectively. The strength of the nonclassical correlations (entanglement) contained in the above state increases with the increasing of the squeezing parameter $r_s$ [@TMSS2]. The reduced density matrices of the cavity field and the resonator mode from Eq. (\[cstate1\]) is a mixed state which can be expressed as $$\begin{aligned}
\rho_{a/b_1} (0)=\sum_{n=0}^\infty\frac{\sinh^{2n}r_s}
{(\sinh^2r_s+1)^{n+1}}|n_{a/b_1}\rangle\langle
n_{a/b_1}|,\label{rstate}\end{aligned}$$ which is indeed of a single-mode thermal state with average thermal photon number $n_{a/b_1}(0)=\sinh^2r_s$. Based on the same Fourier transformation, it follows that the initial system-reservoir correlations are then given by
\[tpc\] $$\begin{aligned}
&&\langle a(0)b_k(0)\rangle=\sqrt{\frac{1}{2\pi}}\sinh2r_se^{i\theta_s}\sin k,\\
&&\langle a(0)b_k^\dag(0)\rangle=0,\end{aligned}$$
namely, the initial Gaussian state only has initial two-photon correlation between the system and the reservoir.
With the initial system-reservoir correlations in Eq. (\[tpc\]), we obtain from Eq. (\[core\]) that $\nu_1(t)=0=\upsilon_0(t)=\upsilon_2(t)$ and
\[core1s\] $$\begin{aligned}
&\nu_2(t)=-i\frac{\sinh2r_se^{i\theta_s}}{\sqrt{2\pi}}\mathcal{F}(t),
\\ &\upsilon_1(t)=\frac{2}{\pi}\sinh^2r_s|\mathcal{F}(t)|^2,\end{aligned}$$
where $$\begin{aligned}
\mathcal{F}(t)&=\int_0^td\tau\sum_kg_k\sin(k)e^{-i\omega_k\tau}u(t-\tau)\notag
\\
&=\frac{\eta}{\sqrt{2\pi}}\int_0^td\tau\int_0^\infty d\omega
\sin[k(\omega)]e^{-i\omega \tau}u(t-\tau).\end{aligned}$$ The last line of the above equation has been applied to the waveguide band structure given in Eq. (\[strength\]), so that $\eta=\frac{\lambda}{\lambda_0}$ and $\sin[k(\omega)]=\frac{1}{2\lambda_0}\sqrt{4\lambda_0^2-(\omega-\omega_0)^2}$ with $\omega_0-2\lambda_0\leq\omega\leq \omega_0+2\lambda_0$.
After obtaining the time-dependent functions $\nu_j(t)$ and $\upsilon_j(t)$ given above, Eq. (\[e\]) becomes
$$\begin{aligned}
&\langle a(t)\rangle =0 , \\
&n(t)=|u(t)|^2n_a(0)+\upsilon_1(t),\\
&s(t)=2u(t)\nu_2(t).\label{pjgz}\end{aligned}$$
This solution indicates that for the given initial thermal state $\rho_a(0)$ in Eq.(\[rstate\]), the cavity field at time $t$ is in a squeezed thermal state [@marian], which can be expressed as $$\begin{aligned}
\rho(t)=S_a[r(t)]\rho_{\rm th}(t) S_a^\dag[r(t)],\end{aligned}$$ where the single-mode squeezing operator $$\begin{aligned}
S_a[r(t)]=\exp[-\frac{r(t)}{2}e^{-i\theta(t)}a^2+\frac{r(t)}{2}e^{i\theta(t)}a^{\dag2}],\end{aligned}$$ with the squeezing parameters $$\begin{aligned}
r(t)=\frac{1}{4}\ln\frac{n(t)+|s(t)|+1/2}{n(t)-|s(t)|+1/2},\end{aligned}$$ and $\theta(t)=\arg[s(t)]$. The thermal state $$\begin{aligned}
\rho_{\rm th}(t)=\sum_{k}\frac{[\bar{n}(t)]^n}{[\bar{n}(t)+1]^{k+1}}|n_a\rangle\langle
n_a|,\end{aligned}$$ where the average thermal photon number $\bar{n}(t)=\sqrt{(n(t)+1/2)^2-|s(t)|^2}-1/2$. By defining the quadrature operators $X=(a + a^\dag )/\sqrt{2}$ and $Y=(a - a^\dag
)/\sqrt{2i}$, the covariance matrix are given by $$\begin{aligned}
\begin{pmatrix} \Delta X^2 & \Delta\{XY\} \\
\Delta \{YX\}& \Delta Y^2 \end{pmatrix}
=\Big[\bar{n}(t) + \frac{1}{2}\Big]\Big[\frac{\cosh 2r(t)}{2} I ~~\notag \\
+ \frac{\sinh 2r(t)}{2} \begin{pmatrix} \cos\theta(t) & \sin\theta(t) \\
\sin\theta(t) & -\cos\theta(t) \end{pmatrix} \Big].\end{aligned}$$ If $\bar{n}(t)=0$, the above covariance matrix is reduced to the standard form for a pure squeezed state [@Zhang90867]. Obviously, the squeezed thermal state squeezes the thermal-state fluctuation $\bar{n}(t) +1/2$. Thus, the squeezing in the squeezed thermal state can be described by the squeezing parameter $r(t)$. If there is no initial system-reservoir correlation, then we have $\nu_2(t)=0$ so that $s(t)=0$ which leads to the squeezing parameter $r(t)=0$.
In Fig. \[fig2\], the time evolution of the cavity intensity $n(t)$ and the squeezing parameter $r(t)$ are plotted for the different coupling strengths $\eta=\lambda/\lambda_0$. As shown in Fig. \[fig2\] (a), for a weak coupling ($\eta=0.4$), the cavity intensity decays monotonically and eventually approaches to zero, as a result of the Markovian damping dynamics at zero temperature. Also, the small but non-zero squeezing parameter $r(t)$ indicates that the initial two-photon correlation $\langle a(0)b_k(0)\rangle$ between the system and reservoir induces a small squeezing effect to the cavity field in the beginning. However, the long-time behavior of the squeezing parameter shows that the effect of the initial system-reservoir correlation is washed out in the long-time limit, which is also consistent with the Markovian dynamics. In contrast, by increasing the coupling strength, as depicted in Fig. \[fig2\] (b), the cavity intensity decays rather fast in the beginning and then it revives and damps with oscillation in which some non-Markovian memory effect appears. Interestingly, the squeezing parameter $r(t)$ shows a similar behavior of the non-Markiovan effect, except for the beginning where the initial two-photon correlation $\langle a(0)b_1(0)\rangle$ generates a stronger squeezing effect to the cavity field, in comparison with the weak coupling case. When the coupling strength continues increasing to $\eta=2.0$ (the strong non-Markovian regime [@Wu1018407]), the cavity intensity decays faster in the very beginning and then revives and keeps oscillating without damping from then on, see Fig. \[fig2\] (c). In this situation, we find that the initial-correlation-induced squeezing dynamics also oscillates over all the time. Therefore, we can conclude that the initial two-photon correlations $\langle a(0)b_k(0)\rangle$ can lead to a nontrivial squeezing dynamics of the cavity field, as a consequence of strong non-Markovian memory dynamics, but it is negligible in the steady-state limit in the Markovian regime.
Initially system-reservoir correlated mixed thermal states
----------------------------------------------------------
Next, we investigate the effect of the initially system-reservoir correlated state with the correlation $\langle
a(0)b_k^\dag(0)\rangle \neq 0$. To this end, we consider an initially mixed state $$\begin{aligned}
\rho_{ab_1}(0)=B(\vartheta)\rho_a\otimes\rho_{b_1}B^\dag(\vartheta),
\label{ics}\end{aligned}$$ where the operator $B(\vartheta)=\exp[\frac{\vartheta}{2}(ab_1^\dag-a^\dag b_1)]$ and the density operators $\rho_{a/b_1}$ represent the thermal states with average thermal photon numbers $\bar{n}_{a/b_1}$. This initially correlated state can be formed via the bilinear coupling between the cavity field $a$ and the resonator mode $b_1$ in the thermal states, and note that nonclassical entanglement are not present in this initially correlated state [@kim]. A direct calculation shows that the initial system-reservoir correlations
$$\begin{aligned}
&\langle a(0)b_k(0)\rangle=0, \\
& \langle
a(0)b_k^\dag(0)\rangle=\frac{\sin\vartheta}{\sqrt{2\pi}}(\bar{n}_a-\bar{n}_{b_1})\sin
k.\end{aligned}$$
For the initially correlated state of Eq.(\[ics\]), it is easy to find that the reduced density matrices $\rho_a(0)$ and $\rho_{b_1}(0)$ of the cavity field and the resonator mode $b_1$ are also the thermal states with the average thermal photon numbers $n_{a}(0)=\frac{1}{2}[\bar{n}_a+\bar{n}_{b_1}+(\bar{n}_a-\bar{n}_{b_1})\cos\vartheta]$ and $n_{b_1}(0)=\frac{1}{2}[\bar{n}_a+\bar{n}_{b_1}-(\bar{n}_a-\bar{n}_{b_1})\cos\vartheta]$, respectively. From Eq.(\[core\]), we obtain $$\begin{aligned}
\nu_1(t)&=&-i
\frac{(\bar{n}_a-\bar{n}_{b_1})\sin \vartheta}{\sqrt{2\pi}}\mathcal{F}(t),\\
\upsilon_1(t)&=&\frac{2n_{b_1}(0)}{\pi}|\mathcal{F}(t)|^2,\end{aligned}$$ and $\nu_2(t)=0$, and $\upsilon_0(t)=0$, $\upsilon_2(t)=0$. Thus, the corresponding physical observables of the cavity system for the above initially correlated state of Eq.(\[ics\]) are given by $$\begin{aligned}
n(t)
=|u(t)|^2n_a(0)+2Re[u^*(t)\nu_1(t)]+\upsilon_1(t),\end{aligned}$$ and $\langle a(t)\rangle=0$, $ s(t) =0$. It indicates that the cavity state remains in a thermal state over all the time with the cavity field intensity $\sim n(t)$.
In Fig. \[fig3\], the time evolution of the cavity field intensity $n(t)$ is plotted for different coupling strengths $\eta$ between the nanocavity and the waveguide. Fig. \[fig3\] (a) shows the the average photon number for a weak coupling $\eta=0.4$. It reveals that the intensity of the cavity field decays monotonically to a steady-state value, as a character of the Markovian dynamics. The initial system-reservoir correlation $\langle
a(0)b_k^\dag(0)\rangle$ leads to the intensity oscillating around the decay line of the case of the initially uncorrelated state. The amplitude of the local oscillations increases in the beginning and then decreases to a unnoticeable effect as time develops. In other words, the effect of the initial correlation $\langle
a(0)b_k^\dag(0)\rangle$ will be washed out in the steady limit. With the increasing of the coupling strength, the intensity no longer monotonically decays and some revival phenomena occur as a character of the non-Markovian memory dynamics [@Wu1018407], as depicted in Fig. \[fig3\] (b). When the coupling is increased to $\eta=2.0$ as a strong coupling value, we see from Fig. \[fig3\] (c) that the intensity and the initial system-reservoir induced oscillation keep oscillating in the whole time regime. In other words, the effect resulted from the initial system-reservoir correlation in the non-Markovian regime will not be washed out by the interaction between the system and the reservoir.
Summary
=======
In summary, we investigate the dynamics of open quantum systems in the presence of initial system-reservoir correlations. We take the photonic cavity system coupled to a non-Markovian reservoir as a specific open quantum system. By solving the exact dynamics of the cavity system, the effects of the initial correlations are explicitly built into the solution of the cavity field intensity and the two-photon correlation function. We also derive the time-convolutionless exact master equation which incorporates with the initial system-reservoir correlations. The non-Markovian memory effects are fully embedded into the time-dependent coefficients in the master equation. The fluctuation coefficient $\gamma_2(t)$ is modified by the initial system-reservoir photonic scattering correlation but the frequency shift $\Delta (t)$ of the cavity and the dissipation coefficient $\gamma_1(t)$ remain unchanged. However, the initial two-photon correlation between the system and the reservoir induces two-photon decoherence terms in the master equation, which can lead to photon squeezing in the cavity. We also take a nanocavity coupled to a coupled resonator optical waveguide (serving as a structured reservoir) as an experimentally realizable system, from which we find that the effects of the initial correlations are fragile for a Markovian reservoir but play an important role in the non-Markovian regime. In fact, in the strong non-Markovian regime, the initial two-photon correlation between the cavity and the reservoir can induce oscillating squeezing dynamics in the cavity. But in Markovian regime, the effects of the initial system-reservoir correlations will be washed out in the steady-state limit.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is supported by the National Science Council of ROC under Contract No. NSC-99-2112-M-006-008-MY3, the National Center for Theoretical Science of Taiwan, National Natural Science Foundation of China (Grant No.10804035), and SDRF of CCNU (Grant No. CCNU 09A01023).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We have investigated the quantum capacitance ($C_Q$) in functionalized graphene, modified with ad-atoms from different groups in the periodic table. Changes in the electronic band structure of graphene upon functionalization and subsequently the quantum capacitance ($C_Q$) of the modified graphene were systematically analyzed using density functional theory(DFT) calculations. We observed that the quantum capacitance can be enhanced significantly by means of controlled doping of N, Cl and P ad-atoms in the pristine graphene surface. These ad-atoms are behaving as magnetic impurities in the system, generates a localized density of states near the Fermi energy, which intern increases charge(electron/hole) carrier density in the system. As a result, a very high quantum capacitance was observed. Finally, the temperature dependent study of $C_Q$ for Cl and N functionalized graphene shows that the $C_Q$ remains very high in a wide range of temperature near the room temperature.'
address: 'Department of Physics, National Institute of Technology, Srinivasnagar, Surathkal, Mangalore Karnataka-575025, India'
author:
- 'Sruthi T & Kartick Tarafder'
title: Route to Achieving Enhanced Quantum Capacitance in Functionalized Graphene based Supercapacitor Electrodes
---
Introduction
============
Large scale generation of green energy from renewable energy sources is utmost necessary in the current scenario. Sunlight is the most viable renewable energy source present in our planet. However energy cannot be produced from the sun uniformly all the time through out the year in many parts of the glob. Therefore, an efficient storage for generated energy and its cost effective transportation is very essential. Hence the designing of efficient energy storage devices is one of the active area of research in green energy production. Supercapacitors based on two-dimensional materials would be a promising technology may provide conceivable alternative for the energy storage [@1]. As basic requirements, supercapacitor should have a very large ion-density, fast charging and discharging capacity with long life time. Two dimensional (2D) materials could play an important role to design an efficient supercapacitor electrodes. With a very high surface area, conductivity and mechanical robustness in 2D materials, specially functionalized graphenes could be the best choice for supercapacitor electrodes [@2; @3; @4]. The total capacitance ($C_{T}$) of a supercapacitor depends on two component [@5], namely the electrical double layer capacitance ($C_D$) and the quantum capacitance (${C_Q}$) such that .
$$\frac{1}{C_{T}}=\frac{1}{C_D} +\frac{1}{C_Q}$$
Insufficiency in either of them will reduce the total capacitance of the device. Thus electrode materials with sufficiently large quantum capacitance is an obligatory factor to obtain high energy density. The quantum capacitance part of electrodes depend on the electronic structure of the electrode materials[@6; @7]. In case of pristine graphene, the quantum capacitance is very small[@8]. However the capacitance can be enhance in graphene based electrode by introducing vacancy defect as well as doped with impurities in a control manner[@9]. Nitrogenation and chlorination of graphene could be an effective way to improve the quantum capacitance in the system [@10; @11]. Recently, Hirunsit [*et. al*]{}.[@12] studied the influence of Al, B, N and P doping on graphene electronic structures and change in quantum capacitance by using DFT calculations. Their report indicates that the $C_Q$ in the monolayer graphene changes substantially when doped with N and in presence of vacancy defect. Later, Song [*et al*]{}[@13] studied the quantum capacitance in ad-atom functionalized reduced graphene oxide and found a significant enhancement in $C_Q$. Therefore it is not difficult to realize from the recent studies that the quantum capacitance in graphene based electrodes can be improve significantly by means of an adequate functionalization. Several study of $C_Q$ on functionalized graphene have been recently reported, however, a systematic investigation of quantum capacitance in functionalized graphene considering various type of ad- atoms with a variable concentration and the basic theoretical understanding of their effect on the $C_Q$ is still lacking. In this present study we have used density functional theory calculations to investigate the quantum capacitance of different functionalized graphene in a systematic way. The functionalization has been done using ad-atoms from different groups in periodic table. The role of vacancy defects on electronic structure and its effect on quantum capacitance in functionalized graphene(FG) has also been carefully investigated.
Computational Method
====================
To accomplish our theoretical investigation of $C_Q$, we first obtain the accurate electronic structure of the doped graphene using plane wave based density functional theory calculations implemented in Vienna Ab-initio Simulation Package(VASP)[@14; @15; @16]. Projected augmented wave method[@17] was used to optimize the geometric structure of the functionalized graphene. The exchange correlation energy functionals were approximated using generalized gradient approximation with PBE parametrization[@18; @19]. A very high kinetic energy cut-off (>400eV) was used in all our calculations for the accurate results. In order to explore the effect of different functionalization on the quantum capacitance, calculations were done using 3$\times$3$\times$1 supercells of graphene unit cell, having 18 carbon atoms of graphene sheet (G18) with one functional group. The vacancy defected configurations were realized on a 5$\times$5$\times$1 supercell of graphene unit cell (50 C atoms of graphene, G50) with a variable concentration of vacancy in the range from 2 to 8 percent. A sufficiently large vacuum has been considered along the out of plane direction of graphene sheet (height$\textgreater$10Å) to avoid the interaction with periodic images. We used a 6$\times$6$\times$1 $\Gamma$ point pack of k-point mesh to sample the Brillouin zone for geometry optimization with 10$^{-6}$H tolerance in total energy for convergence. A denser 24$\times$24$\times$1 k-point grid was used for the precise extraction of electron density of states D(E) and atom projected density of states(PDOS).
The quantum capacitance of materials can be seen as the rate of change of excessive charges(ions) with respect to the change in applied potential[@20]. Therefore, it is directly related to the electronic energy configuration of the electrode materials and can defined as the derivative of the net charge on the substrate/electrode with respect to electrostatic potential. i.e. $$C{_Q}$ = $\frac{dQ}{d\phi}$$
where Q is the excessive charge on the electrode and $\phi$ is the chemical potential. The total charge is proportional to the weighted sum of the electronic density of states upto the Fermi level $E_F$. Due to an applied potential, the chemical potential will be shifted, the excessive charge on the electrode (Q) then can be expressed by an integral term associated to the electronic density of state D(E) and the Fermi$-$Dirac distribution function f(E) as
$$Q = e\int_{-\infty}^{+\infty} D(E)[f(E) - f(E - \phi)] dE$$
Therefore, when the density of states (DOS) is known, the quantum capacitance $C_Q$ of a channel at a finite temperature T can be calculated as
$$C{_Q} =\frac{dQ}{d\phi} = \frac{e^2}{4kT}\int_{-\infty}^{+\infty} D(E) Sech^{2}\left(\frac{E - {e\phi}}{2kT}\right) dE
\label{qc}$$
Here [*e*]{} is the electrons charge, $\phi$ is the chemical potential, [*D(E)*]{} is the DOS and $k$ is the Boltzmann constant. We therefore have estimated the $C{_Q}$ for all the system directly form the density of states.
Results and discussion
======================
It is evident from the expression of $C_Q$ in equation (\[qc\]) that the quantum capacitance is directly proportional to the density of state present near the Fermi energy. Since $(E-e\phi)$ represents the energy with respect to Fermi level and $Sech^{2} (x)$ rapidly goes to zero for $|x| > 0$, the states which are energetically far from the $E_F$ are not contribute much on $C_Q$ . The density of state near the Fermi level for a given material can be tune by means of an efficient chemical modification of the system using external ad-atoms or creating defects. This is also an effective way to control the type and concentration of charge carriers in the system. The electronic Energy levels of the parent material may also be modified/shift in these process. The change in electronic structure depends on dopant type, concentration and doped position in the sublattice. In this study we have considered atoms from different groups in periodic table with an increasing order of electronegativity such as K$<$ Na$<$ Al$<$ P$<$ N$<$ Cl, to functionalize the graphene. The stable adsorption position on graphene was estimated by placing ad-atoms in different possible adsorption sites and comparing the optimized total energies. The hollow position was found to be the most favourable position for ad-atoms like K, Na, Al, bridge position for P, N and top position for Cl ad-atoms respectively. The optimized structure of functionalized graphene with different adsorption positions are shown in Fig. \[FG-Adatom Optimized structure\]. The stability of the functionalized structure was investigated by estimating average adsorption energy $E_{ad}$ for ad-atoms using $$E_{ad} = \frac{1}{n}[E_{tot} - E_{gr} -nE_{at}]$$ where $E_{tot}$ is the total energy of the functionalized graphene unit cell, $E_{gr}$ is the total energy of pristine graphene in the same unit cell, $E_{at}$ is the per atom energy of the ad-atom and $n$ represents the number of ad-atoms present in the unit cell. The adsorption energies for different ad-atoms are listed in the Table. \[table1\]. Our calculation shows that the adsorption of N on pristine graphene surface is most favourable compare to other ad-atoms. However relatively large adsorption energies for P, Cl and Al clearly indicate that these can be easily adsorb on the pristine graphene surface. The ad-atoms are adsorbs on graphene at a distances varying from 1.5 Å to 3.3 Å from the surface. Interestingly, the planar structure of graphene is not much disturbed with these adsorptions. However a very small change in the C-C bond lengths (order of 0.002nm) compare to C-C bond length of graphene were observed near the doped site.
[@llll]{} **ad-atom** & **adsorption**& **ad-atom** & **adsorption**\
& **energy(in eV)**& & **energy(in eV)**\
Na & -0.525 & Cl & -1.142\
K & -0.790 & P & -1.234\
Al & -1.117 & N & -2.372\
![(colour online) Energy-optimized geometry of functionalized graphene with different ad-atoms. Black magenta, blue and green ball represents C, Al, N and Cl atoms respectively a) Preferred adsorption position for Al at hollow site b)Nitrogen at bridge site and (c) Chlorine at the top site. []{data-label="FG-Adatom Optimized structure"}](Structute-FG-Adatom.eps){width="0.5\linewidth"}
The electronic structure and subsequently the quantum capacitance of functionalized graphene were calculated in the optimized adsorbed geometry. We observed a slight shifts of band energies in graphene due to functionalization. Our calculation shows that quantum capacitance in functionalized graphene varies with type, concentration and distributions of ad-atoms. The calculated $C_Q$ values in functionalized graphene with ad-atoms from different groups with a 5.5% doping concentration are shown in Table. \[table2\], Our calculated result clearly indicates a significant enhancement in quantum capacitance of graphene due to functionalization in comparison with the pristine graphene which is $\sim 1.3 \mu F/cm^2 $. A similar trend is reported in previous studies[@20]. Notice that enhancement of $C_Q$ is proportional to the increase of electronegativity of ad-atoms.
[@lll]{} **Configuration** & **Electronegativity** & **C$_Q$**\
& **of ad-atom** & **($\mu$F/cm$^2)$**\
Pristine Graphene & - & 1.2947\
FG - K & 0.82 & 26.7905\
FG - Na & 0.93 & 75.9058\
FG - Al & 1.61 & 57.4976\
FG - Sn & 1.96 & 249.4770\
FG - P & 2.19 & 346.1162\
FG - N & 3.04 & 256.4227\
FG - Cl & 3.16 & 553.6849\
The doped atoms in our study can be broadly categorized in two different groups. Group 1-3 metals such as K, Na and Al ad-atoms are behave as electron donors for graphene. They are donating electrons from their outer shell to the graphene and shift the Fermi level of graphene into the conduction band. As a result the system shows a n-type behaviour. On the other hand group 15 and 17 elements such as N and Cl are electron accepting ad-atoms (p-type doping).
In case of n-type functionalization, the Dirac cone structure of the graphene bands are preserved as shown in Fig. \[n-type doping-PDOS\]. The Fermi level is located at the conduction cone, indicating a large amount of ad-atom induced free electron density between the Dirac point energy (E$_D$) and E$_F$, which can be controlled by tuning the n-type ad-atom concentration.
The density of states contributed from alkali metal ad-atoms, are far from Fermi energy and shows a weak dispersion. K doped graphene band structure is shown in Fig. \[BS-FG-K\], where coloured circles represent band contributed from the doped K atoms. Since the conduction in pristine graphene is mainly due to the de-localized $\pi$ cloud from p$_z$ orbitals of carbon, only p-states are expected to be present around the Fermi-energy. A very similar change in electronic structure were observed for Na and Al doped graphene shown in Fig. \[n-type doping-PDOS\](b)&(c). The overall change in density of state near the Fermi energy very small. As a result, the change in quantum capacitance compared to pristine graphene is expected to be very small. The maximum value of C$_Q$ among all alkali metal doped graphene is found to be $\sim 76 \mu F/cm^2$ for Na doping.
![(colour online) Atom projected density of states for functionalized graphene with (a)K (b)Na and (c)Al atoms. The shaded curve represents dos from the doped atom. The vertical blue dashed line is the Fermi energy set at E=0.[]{data-label="n-type doping-PDOS"}](n-type-FG-Adatoms-PDOS.eps){width="0.7\linewidth"}
![(colour online)Electronic band structure and DOS for graphene functionalized with K. Contribution from doped atoms are represented by the coloured curve in the DOS and coloured circle in the band structure. Horizontal green dashed line is the Fermi energy set at E=0.[]{data-label="BS-FG-K"}](BS-FG-K.eps){width="0.7\linewidth"}
The electronic behaviour of p-doped graphenes are entirely different from the alkali metal doped graphene. In all of p-doped graphene system, Dirac cone structure got distorted and the Fermi level is located at the valence Dirac cone. The atom projected density of states for N, P and Cl doped graphene are shown in Fig. \[p-type doping-PDOS\]. Free hole density appear between E$_D$ and E$_F$. The energy states near the Fermi energy are mainly contributed from the dopant atoms. We observed a maximum change is for Cl doped system. It shows strong peaks near $E_F$ which are $3p_z$ states from the doped chlorine ad-atom as shown in Fig. \[p-type doping-PDOS\]. N doped graphene band structure is shown in Fig. \[BS-FG-N\], where coloured circles represent band contributed from the doped N atoms.
Accumulation of large density of state near the Fermi energy was found in all p-doped graphene. Since the quantum capacitance is directly proportional to the measure of electron density near Fermi level, there is a significant enhancement observed in case of p-type doping system shown in Table \[table2\]. We found the maximum value of $C_Q = 553 \mu F/cm^2$ for Cl-functionalized graphene with a 5.5% doping concentration.
![(colour online)Atom projected density of states for functionalized graphene with (a)N (b)P and (c)Cl ad-atoms. The shaded curve represents dos from the doped atom. The vertical blue dashed line is the Fermi energy set at E=0. []{data-label="p-type doping-PDOS"}](p-type-FG-Adatom-PDOS.eps){width="0.7\linewidth"}
![(colour online)Electronic band structure and DOS for graphene functionalized with N. Contribution from doped atoms are represented by the coloured curve in the DOS and coloured circle in the band structure. Horizontal red dashed line is the Fermi energy set at E=0.[]{data-label="BS-FG-N"}](BS-FG-N.eps){width="0.85\linewidth"}
![(colour online) Contour plots for electron density associated with (a)Pristine Graphene, Functionalized graphene with (b)Al, (c)N and (d)Cl. Relative electron density is indicated by the colour bar.[]{data-label="Isosurface-FG-Adatom"}](Isosurface-FG-Adatom.eps){width="0.6\linewidth"}
Next we investigated the origin of large enhancement in quantum capacitance in p-type doping on graphene. Charge redistribution in presence of ad-atom could be one of the main reason. Therefore, we performed Bader charge analysis on functionalized graphene and observed that there are significant charge transfer between graphene and the ad-atoms. In case of Cl and N doping, the charge transfer is remarkably high. We found 0.6e and 0.5e charge par atom transfer to Cl and N atoms respectively from the graphene sheet. In case of P functionalized graphene, 0.3e charge has been transferred from P to the graphene sheet. A small change in the sublattice structures have also been observed in all case. The introduction of the electron accepting(donating) ad-atoms, disrupts the homogeneity of the charge distribution due to the strong correlation effects. Fig. \[Isosurface-FG-Adatom\] shows the charge redistribution upon functionalization of graphene with Al, N and Cl ad-atoms. A uniform charge distribution can be seen in the case of pristine graphene monolayer (See Fig. \[Isosurface-FG-Adatom\](a)). We notice that the charge inhomogeneity systematically increases with increasing atomic number of the ad-atom. The localization of charges near the dopant site increases systematically as we move from the Na to Cl. This can be ascribed to the fact that the strong on-site Coulombic interaction dominate in case of heavier p-block elements. We found maximum charge re-distribution on graphene caused by Cl. Further charge transfer leads to a change in the average DOS near Fermi level, which in-turn affect the quantum capacitance. Interestingly the density of states from $3p_z$ orbitals of Cl, N and P are localized very near to Fermi level as shown in Fig. \[p-type doping-PDOS\], which provides the maximum contribution to the quantum capacitance. To understand the localization of states near the Fermi energy from the doped atom we investigate the temperature dependent behaviour of the $C_Q$. We vary the temperature from 10K to 400K and calculated the $C_Q$ of the system. In case of pristine graphene monolayer the value of $C_Q$ remains nearly unchanged and it is close to $\sim 1.3 \mu$F/cm$^2$. However, $C_Q$ changes dramatically with temperature when Cl, P and N atoms are added to the graphene. Our calculation shows a sharp increase and then a gradual decrease of $C_Q$, when we increase the temperature. With a 5.5% doping concentration the maximum value of $C_Q= 648 \mu F/cm^2$ appears at 200K in Cl doped graphene and 263 $\mu F/cm^2$ arises at 234K in N-doped graphene. A significantly large value of $C_Q=1992 \mu F/cm^2 $ was observed for P doped system at 30K with a same doping concentration. The variation of $C_Q$ with temperature for N, P and Cl doped graphene are shown in Fig. \[QC-T\]. One possibility of such variation of $C_Q$ with temperature could be due to the Kondo behaviour of the doped graphene where ad-atoms may behave as the magnetic impurities. To confirm this, we perform spin polarized DFT calculations for these systems and observed that 0.371$\mu B$, 0.75$\mu B$, and 0.45 $\mu B$ magnetic moment lies on the doped Cl, N, P atoms respectively, which confirms the magnetic behaviour of the doped ad-atoms and localization of DOS near the Fermi energy.
![(colour online) Variation of Quantum Capacitance with temperature in the range of 10K to 400K for (a)FG-N (b) FG-P and (c)FG-Cl.[]{data-label="QC-T"}](QC-T.eps){width="0.7\linewidth"}
Dependence of ad-atom concentrations on $C_Q$:
----------------------------------------------
Next we investigated the impact of doping concentration on quantum capacitance. We choose Cl, N and P and their combination as ad-atoms to dope graphene with a variable concentrations, as we observed that doping with these atoms with a similar concentration shows comparatively large $C_Q$ value at the room temperature.
We observed a steady increment of C$_Q$ for N and Cl doped system upto a certain concentration and then it decrease. However, in case of P doped system the C$_Q$ decreases significantly with increase of doping concentration also we found that the P adsorption is not stable when the doping concentration is more than 8%. The maximum value of C$_Q$ appears to be 284, 280 and 1142 $\mu F/cm^2 $ for P, N and Cl doped systems with doping concentration 2%, 6.25% and 12% respectively at room temperature. The concentration dependence of $C_Q$ for different systems are summarized in the Table \[table3\]. We also observed that the $C_Q$ is sensitive to the adsorption site of the graphene lattice and it reduces drastically when ad-atoms are very close to each other. Moreover the calculated value of $C_Q$ is very small, when more than one ad-atom adsorbs on a single ring unit of the graphene or in equivalent sites of the adjacent ring. Since the N is prefer to adsorb on the bridge position and Cl prefers to adsorb on top position of the graphene lattice, therefore maximum concentration with such restriction is possible only for 6.25% and 12% doping concentration for N and Cl adsorption respectively. When the doping concentration is more than the critical value, impurity centres are interacting each other that affects on the localized states near the Fermi energy, which inturn reduces the $C_Q$ value.
[@llll]{} **Concentration of ad-atom** & **FG-N:C$_Q$**& **FG-Cl:C$_Q$** & **FG-P:C$_Q$**\
& **($\mu$F/cm$^2$ )**& **($\mu$F/cm$^2$ )**& **($\mu$F/cm$^2$ )**\
2% & 254.344 & 315.7274 & 284.9665\
4% & 269.484 & 1066.193 & 279.3714\
6.25% & 279.770 & 976.1142 & 281.5697\
8% & 249.099 & 980.3039 & 108.6805\
10% & 195.3198 & 279.7531 &\
12% & 269.7866 & 1141.6553 &\
This can be understand by comparing the projected DOS of the impurity N atom in N-doped graphene with critical doping concentration(red curve) and higher than the critical doping concentration (blue curve) as shown in Fig. \[Max-QC-Doping for FG-N\], where the impurity band width is extended for the concentration which is higher than the critical value. Dense decoration of Cl on a graphene surface leads to the desorption of Cl from the surface in the form of Cl$_2$ due to a stronger Cl$-$Cl interaction.
![(colour online) N atom projected DOS for nitrogenated graphene with two different N concentrations. Red line is for the 6.25% doping concentration, where the $C_Q$ is maximum. Blue line represent the same for higher 8% doping concentration. []{data-label="Max-QC-Doping for FG-N"}](Max-QC-Doping-FG-N.eps){width="10cm"}
A very similar situation appears in case of co-doping of Cl, N and P ad-atoms. In this case the interaction between ad-atoms are much more stronger and instabilizes the absorption of ad-atoms on the graphene surfaces. The $C_Q$ also reduces significantly as shown in the Table \[table4\].
[@ll]{} **Co-doped ad-atoms** & **C$_Q$($\mu$F/cm$^2$)**\
N,Cl & 110.05\
N,Sn & 118.22\
P,Sn & 118.45\
Sn,Cl & 102.57\
Impact of vacancy defects on $C_Q$
----------------------------------
Dislocation or defect on pristine graphene also causes charge localization, which inturn may improve the quantum capacitance. Therefore we have investigated the quantum capacitance of graphene in presence of vacancy defects. The vacancies were created by simply removing C atom from the graphene lattice. Our calculation shows that $C_Q$ increase with defect concentration up to 4% and then it decrease. The enhancement of the quantum capacitance is due to the formation of localized states near Fermi energy, induced by defects, which is very similar to the effect of ad-atom doping on graphene. It was also observed that the localized states are spin polarized induces $\sim 0.3 \mu B$ magnetic moment for each vacancy created. The value of $C_Q$ in different vacancy concentration are listed in the Table. \[table5\]. We were unable to study the vacancy defected graphene with more than 8% defect as the structure itself is unstable. Interestingly the position of the created vacancy site also has an impact on $C_Q$, very similar to the ad-atom’s positions in doped graphene. We observed a significant reduction of $C_Q$ in 4% defect configuration when the distance between two defect sites is less than 5 Å.
[@ll]{} **Vacancy concentration(%)** & **C$_Q$($\mu$F/cm$^2$)**\
0 & 1.2947\
2 & 99.0641\
4 & 260.3285\
6 & 244.5882\
8 & 127.8805\
In the final step, we have incorporated vacancy defects in graphene functionalized with N, P, Cl ad-atoms. In presence of vacancy defect, we observed a slight variation in C$_Q$ only for Cl doped system. In 2% and 4% chlorine concentration, when one vacancy was created in a 50 C atom unit cell of graphene, the $C_Q$ increases to 486 $\mu$F/cm$^2$ and 1091 $\mu$F/cm$^2$ respectively. Further increment in Cl doping concentration, the $C_Q$ reduces drastically. This is due to the strong interaction between the localized charge in the doped atom and in the defected site. We have not fund any considerable change in C$_Q$ for N and P doped graphene with vacancy defects.
Conclusion
==========
In conclusion, our theoretical study shows that the quantum capacitance of graphene-based electrodes can be enhance significantly by introducing ad-atoms and vacancy defects in the graphene sheet. Our density functional theory based calculations indicate that the quantum capacitance in functionalized graphene is mainly depends on the sublattice positions and the concentration of ad-atoms. We observed that the enhancement is significant when the graphene was functionalized with N, Cl and P ad-atoms with a specific concentrations. These ad-atoms are behaving as magnetic impurities in the system, generates localized density of states near the Fermi energy. Atom projected density of states for these systems show that new DOS are appearing near the Fermi-energy which are mainly from p-orbitals of ad-atoms, produces high charge(electron/hole) density near the Fermi level results to a very high quantum capacitance in the system. Our calculation also predicts that the Cl functionalization is most feasible for higher $C_Q$ when defect present in the graphene. Finally the temperature variation calculation shows that $C_Q$ remains large in case of Cl and N functionalized graphene in a wide range of temperature. Our study propose that the chemically modified graphene could be a promising electrode materials for super-capacitors, in which a very large quantum capacitance (($>$600 $\mu$F/c$m^2$)) can be achieve near the room temperature.
Acknowledgement
===============
KT would like to acknowledge NITK-high performance computing facility and also would like thank DST-SERB(project no. SB/FTP/PS-032/2014 ) for the financial support.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present photometry with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) of stars in the Magellanic starburst galaxy NGC 4449. The galaxy has been imaged in the F435W (B), F555W (V) and F814W (I) broad-band filters, and in the F658N (H$\alpha$) narrow-band filter. Our photometry includes $\approx$ 300,000 objects in the (B, V) color-magnitude diagram (CMD) down to V $\la$ 28, and $\approx$ 400,000 objects in the (V, I) CMD, down to I $\la$ 27 . A subsample of $\approx$ 200,000 stars has been photometrized in all the three bands simultaneously. The features observed in the CMDs imply a variety of stellar ages up to at least 1 Gyr, and possibly as old as a Hubble time. The spatial variation of the CMD morphology and of the red giant branch colors point toward the presence of an age gradient: young and intermediate-age stars tend to be concentrated toward the galactic center, while old stars are present everywhere. The spatial variation in the average luminosity of carbon stars suggests that there is not a strong metallicity gradient ($\lesssim 0.2$ dex). Also, we detect an interesting resolved star cluster on the West side of the galaxy, surrounded by a symmetric tidal or spiral feature consisting of young stars. The positions of the stars in NGC 4449 younger than 10 Myr are strongly correlated with the H$\alpha$ emission. We derive the distance of NGC 4449 from the tip of the red giant branch to be ${\rm D=3.82 \pm 0.27}$ Mpc. This result is in agreement with the distance that we derive from the luminosity of the carbon stars.'
author:
- 'F. Annibali , A. Aloisi , J. Mack, M. Tosi , R.P. van der Marel, L. Angeretti, C. Leitherer, M. Sirianni'
title: 'Starbursts in the Local Universe: new HST/ACS observations of the irregular galaxy NGC 4449[^1]'
---
Introduction
============
Starbursts are short and intense episodes of star formation (SF) that usually occur in the central regions of galaxies and dominate their integrated light. The associated star-formation rates (SFR) are so high that the existing gas supply can sustain the stellar production only on timescales much shorter than a cosmic time ($\lesssim 1$ Gyr).
The importance of the starburst phenomenon in the context of cosmology and galaxy evolution has been dramatically boosted in recent years by deep imaging and spectroscopic surveys which have discovered star-forming galaxies at high redshift: a population of dusty and massive starbursts, with SFRs as high as $\sim$ 100 – 1000 M$_{\odot}$ yr$^{-1}$, has been unveiled in the submillimeter and millimeter wavelengths at z$>$2 [@blain02; @scott02] and star-forming galaxies at $z >$ 3 have been discovered with the Lyman break selection technique [@steidel96; @pet01] and through Lyman-$\alpha$ emission surveys (@rhoads, see also @lefevre05 for a more recent independent approach).
In the local Universe, starbursts are mostly found in dwarf irregular galaxies, and contribute $\sim$ 25% of the whole massive SF [@heck98]. Both observations and theoretical models [@larson78; @genz98; @ni86] show that strong starbursts are usually triggered by processes such as interaction or merging of galaxies, or by accretion of gas, which probably played an important role in the formation and evolution of galaxies at high redshift. Thus, nearby starbursts can serve as local analogs to primeval galaxies to test our ideas about SF, evolution of massive stars, and physics of the interstellar medium (ISM) in “extreme” environments. The high spatial resolution and high sensitivity of Hubble Space Telescope offer the possibility to study the evolution of nearby starbursts in details. This is fundamental in order to address many of the still open questions in cosmological astrophysics: What are the main characteristics of primeval galaxies? What is the nature of star-forming galaxies at high redshift? How important are accretion and merging processes in the formation and evolution of galaxies?
The Magellanic irregular galaxy NGC 4449 ($\alpha_{2000} =Ê12^h 28^m 11^{s}.9$, $\delta_{2000} =Ê+ 44^{\circ} 05^{'} 40^{"}$, $l=136.84$ and $b=72.4$), at a distance of $3.82 \pm 0.27$ Mpc (see Section 5), is one of the best studied and spectacular nearby starbursts. It has been observed across the whole electromagnetic spectrum and displays both interesting and uncommon properties. It is one of the most luminous and active irregular galaxies. Its integrated magnitude $M_B
= -18.2$ makes it $\approx$ 1.4 times as luminous as the Large Magellanic Cloud (LMC) [@hunter97]. @th87 estimated a current SFR of $\sim 1.5$ M$_{\odot}$ yr$^{-1}$. NGC 4449 is also the only local example of a global starburst, in the sense that the current SF is occurring throughout the galaxy [@hunter97]. This makes NGC 4449 more similar to Lyman break Galaxies (LBGs) at high redshift ($z \simeq3$), where the brightest regions of SF are embedded in a more diffuse nebulosity and dominate the integrated light also at optical wavelengths [@gi02].
Abundance estimates in NGC 4449 were derived in the HII regions by [@talent], [@hgr82] and [@mar97], and for NGC 4449 nucleus by @bok01. The published values are in good agreement with each other, and provide 12 + log(O/H) $\approx 8.31$. Adopting the oxygen solar abundance from @sun98, 12 + log(O/H)$_{\odot} =$ 8.83, we obtain \[O/H\] $=$ -0.52, i.e. NGC 4449 oxygen content is almost one third solar, as in the LMC. New solar abundance estimates, based on 3D hydrodynamic models of the solar atmosphere, accounting for departures from LTE, and on improved atomic and molecular data, provide 12 + log(O/H)$_{\odot} =$ 8.66 [@sun07]. However, the new lower abundances seem to be inconsistent with helioseismology data, unless the majority of the inputs needed to make the solar model are changed [@basu07]. Thus, we will adopt the old abundances from @sun98 throughout the paper. Radio observations of NGC 4449 have shown a very extended HI halo ($\sim 90$ kpc in diameter) which is a factor of $\sim 10$ larger than the optical diameter of the galaxy and appears to rotate in the opposite direction to the gas in the center [@baj94]. Hunter et al. (1998, 1999) have resolved this halo into a central disk-like feature and large gas streamers that wrap around the galaxy. Both the morphology and the dynamics of the HI gas suggest that NGC 4449 has undergone some interaction in the past. A gas-rich companion galaxy, DDO 125, at the projected distance of $\sim 40$ kpc, could have been involved [@theis].
NGC 4449 has numerous ($\sim 60$) star clusters with ages up to 1 Gyr [@gel01] and a young ($\sim$ 6-10 Myr) central cluster [@bok01], a prominent stellar bar which covers a large fraction of the optical body [@hun99], and a spherical distribution of older (3-5 Gyr) stars [@both96]. The galaxy has also been demonstrated to contain molecular clouds from CO observations [@ht96] and to have an infrared (10-150 $\micron$) luminosity of $2 \times
10^{43}$ erg s$^{-1}$ [@th87]. The ionized gas shows a very turbulent morphology with filaments, shells and bubbles which extend for several kpc (Hunter & Gallagher 1990, 1997). The kinematics of the HII regions within the galaxy is chaotic, again suggesting the possibility of a collision or merger [@va02]. Some 40% of the X-ray emission in NGC 4449 comes from hot gas with a complex morphology similar to that observed in H$\alpha$, implying an expanding super-bubble with a velocity of $\sim 220$ kms$^{-1}$ [@sum03].
All these observational data suggest that the late-type galaxy NGC 4449 may be changing as a result of an external perturbation, i.e., interaction or merger with another galaxy, or accretion of a gas cloud. A detailed study of the star-formation history (SFH) of this galaxy is fundamental in order to derive a coherent picture for its evolution, and understand the connection between possible merging/accretion processes and the global starburst. With the aim of inferring its SFH, we have observed NGC 4449 with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) in the F435W, F555W, F814W and F658N filters. In this paper we present the new data and the resulting color–magnitude diagrams (CMDs) (Sections 2, 3 and 4). We derive a new estimate of the distance modulus from the magnitude of the tip of the red giant branch (TRGB) and the average magnitude of the carbon stars in Section 5. In Section 6, the empirical CMDs are compared with stellar evolutionary tracks. With the use of the tracks, we are able to derive the spatial distribution of stars of different age in the field of NGC 4449. The Conclusions are presented in Section 7. The detailed SFH of NGC 4449 will be derived through synthetic CMDs in a forthcoming paper.
Observations and data reduction
===============================
The observations were performed in November 2005 with the ACS Wide Field Camera (WFC) using the F435W (B), F555W (V) and F814W (I) broad-band filters, and the F658N (H$\alpha$) narrow-band filter (GO program 10585, PI Aloisi). We had two different pointings in a rectangular shape along the major axis of the galaxy. Each pointing was organized with a 4 - exposure half $+$ integer pixel dither pattern with the following offsets in arcseconds: (0,0) for exposure 1, (0.12, 0.08) for exposure 2, (0.25, 2.98) for exposure 3, and (0.37, 3.07) for exposure 4. This dither pattern is suitable to remove cosmic rays and hot/bad pixels, fill the gap between the two CCDs of the ACS/WFC, and improve the PSF sampling. The exposure times in the different broad-band filters were chosen to reach at least one magnitude fainter than the TRGB, at I $\approx $24, with a photometric error below 0.1 mag (S/N $>$ 10). Eight exposures of $\sim$ 900 s, 600 s, 500 s and 90 s were acquired for each of the B, V, I and H$\alpha$ filters, respectively.
For each filter, the eight dithered frames, calibrated through the most up-to-date version of the ACS calibration pipeline (CALACS), were co-added into a single mosaicked image using the software package MULTIDRIZZLE [@Koe02]. During the image combination, we fine-tuned the image alignment, accounting for shifts, rotations, and scale variations between images. The MULTIDRIZZLE procedure also corrects the ACS images for geometric distortion and provides removal of cosmic rays and bad pixels. The total field of view of the resampled mosaicked image is $\sim$ 380 $\times$ 200 arcsec$^2$, with a pixel size of 0.035 (0.7 times the original ACS/WFC pixel size).
To choose the optimal drizzle parameters, we experimented with different combinations of the MULTIDRIZZLE parameters [*pixfrac*]{} (the linear size of the “drop" in the input pixels) and [*pixscale*]{} (the size of output pixels). One must choose a [*pixfrac*]{} value that is small enough to avoid degrading the final image, but large enough that, when all images are dropped on the final frame, the flux coverage of the output image is fairly uniform. Statistics performed on the final drizzled weight image should yield an rms value which is less than 20% of the median value. In general, the [*pixfrac*]{} should be slightly larger than the scale value to allow some of the ’drop’ to spill over to adjacent pixels. Following these guidelines, we find that [*pixfrac*]{}$=$0.8 and [ *pixscale*]{}$=$0.7 provide the best resolution and PSF sampling for our dithered images.
The total integration times are $\sim$ 3600 s, 2400 s, 2000 s and 360 s for the B, V, I and H$\alpha$ images, respectively. Only in a small region of overlap between the two pointings ($\sim$ 30 $\times$ 200 arcsec$^2$) the integration times are twice as those listed above. Figure \[image\] shows the mosaicked true-color image created by combining the data in the four filters.
The photometric reduction of the images was performed with the DAOPHOT package [@daophot] in the IRAF environment[^2]. The instrumental magnitudes were estimated via a PSF-fitting technique. We constructed a PSF template for each of the four ACS chips (2 CCDs at 2 pointings) contributing to the final mosaicked image. To derive the PSF, we selected the most isolated and clean stars, uniformly distributed within each chip. The PSF is modeled with an analytic moffat function plus additive corrections derived from the residuals of the fit to the PSF stars. The additive corrections include the first and second order derivatives of the PSF with respect to the X and Y positions in the image. This procedure allows us to properly model the spatial variation of the PSF in the ACS/WFC field of view.
The stars were detected independently in the three bands, without forcing in the shallowest frames the detection of the objects found in the deepest one. For comparison, we also ran the photometry on a list of stars detected on the sum of the B, V and I images. The luminosity functions (LFs) obtained in the various bands with the two different approaches are presented in Fig. \[lfs\]. We notice that the [*forced*]{} search pushes the detection of stars $\sim$ 0.5 magnitude deeper than the [*independent*]{} search. On the other hand, upon closer inspection the majority of the “gained" objects turn out to be spurious detections or stars with large photometric errors. Furthermore, the deeper photometry is not deep enough to detect the next features of interest in the CMD (the horizontal branch, the red clump or the asymptotic giant branch (AGB) bump, see Section 3), and thus it does not provide any additional information for our study. In the following we thus use the photometry obtained with the independent search on the three bands.
Aperture photometry with PHOT, and then PSF-fitting photometry with the ALLSTAR package, were performed at the position of the objects detected in the B, V and I images. The instrumental magnitudes were measured adopting the appropriate PSF model to fit the stars according to their position in the frame. The B,V and I catalogs were then cross-correlated with the requirement of a spatial offset smaller than 1 pixel between the positions of the stars in the different frames. This led to 299,115 objects having a measured magnitude in both B and V, 402,136 objects in V and I, and 213,187 objects photometrized in all the three bands simultaneously.
The conversion of the instrumental magnitudes $m_i$ to the HST VEGAMAG system was performed by following the prescriptions in @sir05. The HST VEGAMAG magnitudes are derived according to the equation:
$$m = m_i + C_{ap} + C_{\inf} + ZP + C_{CTE},
\label{eq1}$$
where $m_i$ is the DAOPHOT magnitude (${\rm -2.5 \times \log(counts/exptime)}$) within a circular aperture of 2–pixel radius, and with the sky value computed in an annulus from 8 to 10 pixels; $C_{ap}$ is the aperture correction to convert the photometry from the 2 pixel to the conventional $0.5\arcsec$ radius aperture, and with the sky computed at “infinite"; we computed $C_{ap}$ from isolated stars selected in our images; $C_{\inf}$ is taken from @sir05 and is an offset to convert the magnitude from the 0.5 radius into a nominal infinite aperture; ZP is the HST VEGAMAG zeropoint for the given filter. Corrections for imperfect charge transfer efficiency (CTE) were calculated from each single expousure, and then averaged, following the formulation of [@cte] (Eq. \[eq2\]), which accounts for the time dependence of the photometric losses:
$$C_{CTE}=10^A \times SKY^B \times FLUX^C \times \frac{Y}{2048} \times \frac{MJD - 52333}{365},
\label{eq2}$$
where SKY is the sky counts per pixel per exposure, FLUX is the star counts per exposure within our adopted photometry aperture (r$=$2 pixel in the resampled drizzled image, corresponding to 1.4 pixel in the original scale), Y is the number of charge transfers, and MJD is the Modified Julian Date. The coefficients of equation (\[eq2\]) were extrapolated from a r$=$3 pixel aperture to a r$=$1.4 pixel aperture (in the original scale), and are $A=1.08$, $B=-0.309$, and $C =-0.976$. The computed CTE corrections are negligible for the brightest stars, but can be as high as $\sim$ 0.1 mag for the faintest stars. We did not transform the final magnitudes to the Johnson-Cousins B, V, I system, since this would introduce additional uncertainties. However, such transformations can be done in straightforward manner using the prescriptions of @sir05. The ACS VEGAMAG magnitudes were not corrected for Galactic foreground extinction (${\rm E(B-V) = 0.019}$, [@schlegel]) and internal reddening. Concerning the internal reddening, @hill98 derived ${\rm E(B-V) \approx}$ 0.18 from the $H\alpha$/H$\beta$ ratio measured in NGC 4449 HII regions. This value can be considered an upper limit to the average internal extinction, since the nebular gas is usually associated with young star forming regions, which tend to be inherently more dusty than the regions in which older stars reside (as demonstrated explicitly for the case of the LMC; @zari).
Because of both the large number of dithered exposures and the conservative approach of the independent search on the three images, our catalog is essentially free of instrumental artifacts such as cosmic rays or hot pixels. The distribution of the DAOPHOT parameters $\sigma$, $\chi^2$ and [*sharpness*]{} is shown in Fig. \[fig1I\]. The $\sigma$ parameter measures the uncertainty on the magnitude, the $\chi^2$ is the residual per degree of freedom of the PSF-fitting procedure, and the [*sharpness*]{} provides a measure of the intrinsic size of the object with respect to the PSF. Notice that the DAOPHOT/ALLSTAR package automatically rejects the objects with $\sigma > 0.55$. The obtained distributions for $\sigma$, $\chi^2$ and [*sharpness*]{} suggest that the vast majority of the detected sources are stars in the galaxy with a small contamination from stellar blends and background galaxies. There are some objects with very bright magnitudes (${\rm m_{F814W} }\la 22$) and [*sharpness*]{} $>$0.5 in Fig. \[fig1I\] (this is also observed in the F435W and F555W filters, for which we do not show the $\chi^2$ and [*sharpness*]{} distributions). Their [*sharpness*]{} values imply that they have a larger intrinsic size than the PSF, and thus may not be individual stars. By visually inspecting these objects in all the images, we recognized several candidate star clusters and background galaxies. Some of the candidate star clusters look like fairly round but extended objects; some others present a central core, and are partially resolved into individual stars in the outskirts. We detect at least 42 clusters in our data, some of which look like very massive globular clusters. The candidate clusters and the galaxies that were identified by eye were rejected from the photometric catalog. We are left with 299,014 objects in the (B, V) catalog, 402,045 objects in the (V, I) catalog, and 213,099 objects photometrized in all the three bands. We experimented with many other cuts in $\sigma$, $\chi^2$ and [*sharpness*]{}, but none of them affected the global appearance of the CMDs and the detected evolutionary features. A detailed study of the cluster properties will be presented in a forthcoming paper (Aloisi et al., in preparation).
Incompleteness and blending
===========================
To evaluate the role of incompleteness and blending in our data, we performed artificial star experiments on the drizzle-combined frames, following the procedure described by [@tosi01]. These tests serve to probe observational effects associated with the data reduction process, such as the accuracy of the photometric measurements, the crowding conditions, and the ability of the PSF-fitting procedure in resolving partially overlapped sources. We performed the tests using to the following procedure. We divided the frames into grids of cells of chosen width (50 pixels) and randomly added one artificial star per cell at each run. This procedure prevents the artificial stars to interfere with each other, and avoids to bias the experiments towards an artificial crowding not really present in the original frames. The position of the grid is randomly changed at each run, and after a large number of experiments the stars are uniformly distributed over the frame. In each filter, we assign to the artificial star a random input magnitude between $m_1$ and $m_2$, with $m_1$ $\approx$ 3 mag brighter than the brightest star in the CMD, and $m_2$ $\approx$ 3 mag fainter then the faintest star in the CMD. At each run, the frame is re-reduced following exactly the same procedure as for the real data. The output photometric catalog is cross-correlated with a sum of the original photometric catalog of real stars and the list of the artificial stars added into the frame. This prevents cross-correlation of artificial stars in the input list with real stars recovered in the output photometric catalog. We simulated about half a million stars for each filter. At each magnitude level, the completeness of our photometry is computed as the ratio of the number of recovered artificial stars over the number of added ones. The completeness levels in the color magnitude diagrams (see Section 4) are the product of the completeness factors in the two involved passbands.
We show in Fig. \[dm\] the $\Delta m$ difference between the input and output magnitudes of the artificial stars as a function of the input magnitude, for the F435W, F555W and F814W filters. The solid lines superimposed on the artificial star distributions correspond to the mean $\Delta m$ (central line), and the $\pm 1 \sigma_m$ values around the mean. The plotted $\Delta m$ distributions provide a complete and statistically robust characterization of the photometric error as a function of magnitude, for each filter. By comparing the $\sigma_m$ with the DAOPHOT errors in Fig. \[fig1I\], it is apparent that the DAOPHOT package increasingly underestimates the actual errors toward fainter magnitudes. For instance, for a star with V$\sim$ 25.5 and I$\sim$ 24, (tip of the red giant branch, see Section 5), the mean DAOPHOT error is $\sim$ 0.05 mag in both bands, while the $\sigma_m$ from the artificial star tests is $\sim$ 0.15 and $\sim$ 0.1 in V and I, respectively. The systematic deviation from 0 of the mean $\Delta m$ indicates the increasing effect of blending, i.e. faint artificial stars recovered brighter than in input because they happen to overlap other faint objects.
Color-magnitude diagrams
========================
The CMDs are shown in Figures \[cmd1\] and \[cmd2\]. We plot the ${\rm m_{F555W}}$ versus ${\rm m_{F435W}- m_{F555W}}$ CMD of the 299,014 stars matched between the B and V catalogs in Fig. \[cmd1\], and we plot the ${\rm m_{F814W}}$ versus ${\rm m_{F555W}-m_{F814W}}$ CMD of the 402,045 stars matched between the V and I catalogs in Fig. \[cmd2\]. We indicate the 90 % (solid line) and 50 % (dashed line) completeness levels as derived from the artificial star experiments on the two CMDs. The average size of the photometric errors at different magnitudes, as derived from artificial star tests, is indicated as well.
The two CMDs show all the evolutionary features expected at the magnitudes sampled by our data: a well defined blue plume and red plume, the red horizontal tail of the carbon stars in the ${\rm m_{F814W}}$ versus ${\rm m_{F555W}-m_{F814W}}$ CMD, and a prominent red giant branch (RGB). The blue plume is located at ${\rm m_{F435W}-m_{F555W}}$ and ${\rm m_{F555W}-m_{F814W}}$ $\simeq$ $-$0.1 in the two diagrams, with the brightest stars detected at ${\rm m_{F555W}}$, ${\rm m_{F814W}}$ $\sim$ 18. It samples both stars in the main-sequence (MS) evolutionary phase and evolved stars at the hot edge of the core helium burning phase. The blue plume extends down to the faintest magnitudes in our data, at ${\rm m_{F555W} \sim 28}$. The red plume is slightly inclined with respect to the blue plume, with [$\rm m_{F555W} \la 25$]{} and colors extending from [$\rm m_{F435W}-m_{F555W} \sim 1.4$]{} to $\sim$ 1.8 in the ${\rm m_{F555W}}$, ${\rm m_{F435W}- m_{F555W}}$ CMD, and [$\rm m_{F814W} \la 23.5$]{} and colors extending from [$\rm m_{F555W}-m_{F814W} \sim 1.4$]{} to $\sim$ 2.2 in the ${\rm m_{F814W}}$ versus ${\rm m_{F555W}-m_{F814W}}$ CMD. It is populated by red supergiants (RSGs) at the brighter magnitudes, and AGB stars at fainter luminosities. At intermediate colors, below ${\rm m_{F555W} \sim 25}$ and ${\rm m_{F814W} \sim 23.5}$, we recognize the [*blue loops*]{} of intermediate-mass stars in the core helium burning phase. The concentration of red stars at ${\rm m_{F555W} \ga 25.5}$ and ${\rm m_{F814W} \ga 24}$, corresponds to low-mass old stars in the RGB evolutionary phase. Finally, a pronounced horizontal feature, at ${\rm m_{F814W} \sim 23.5}$, and with colors extending from ${\rm m_{F555W}-m_{F814W} \sim 1.8}$ to as much as ${\rm m_{F555W}-m_{F814W} \sim 4}$, is observed in the ${\rm m_{F814W}}$, ${\rm m_{F555W}-m_{F814W}}$ CMD. This red tail is produced by carbon stars in the thermally pulsing asymptotic giant branch (TP-AGB) phase.
In order to reveal spatial differences in the stellar population of NGC 4449, we have divided the galaxy’s field of view into 28 (7 $\times$ 4) rectangular regions, as shown in Fig. \[imagegrid\]. The size of the regions ($\approx$ 55 $\times$ 55 ${\rm arcsec^2}$, corresponding to ${\rm \approx 1 \times 1 \ kpc^2}$ at the distance of NGC 4449) allows us to follow spatial variations at the kpc scale, being at the same time large enough to provide a good sampling. The ${\rm m_{F555W}}$, ${\rm m_{F435W}- m_{F555W}}$, and ${\rm m_{F814W}}$, ${\rm m_{F555W}- m_{F814W}}$ CMDs derived for the different regions are shown in Fig. \[hessregionbv\] and \[hessregionvi\], respectively. The completeness levels plotted on the CMDs of the central column show that the photometry is deeper in the external regions than in the galaxy center, where the high crowding level makes the detection of faint objects more difficult. The errors, as estimated from the artificial star experiments, increase toward the galaxy center, as an effect of the higher crowding level and the higher background. Bright stars are mostly concentrated toward the galaxy center, and only a few of them are present at large galactocentric distances, in agreement with what was already observed in other dwarf irregular galaxies, (e.g., [@tosi01]). The external region (6,4) makes the exception to this observed global trend, showing a prominent blue plume in both the CMDs. The luminous blue stars observed in these CMDs correspond in the image of Fig. \[image\] to a symmetric structure. This structure is more clearly visible in the top right of Fig. \[spatial\], which will be discussed in Section 6 below. This structure could be due to tidal tails or spiral–like feature associated with a dwarf galaxy that is currently being disrupted. The structure is centered on a resolved cluster-like object that could be the remnant nucleus of this galaxy. Fig. \[imagegrid\] shows a blow-up of this object.
Carbon stars
------------
In the ${\rm m_{F814W}}$, ${\rm m_{F555W}- m_{F814W}}$ CMD of Fig. \[cmd2\], the horizontal red tail, at magnitudes brighter than the TRGB, is due to carbon-rich stars in the TP-AGB phase. Since AGB stars trace the stellar populations fromÊ$\sim$ 0.1 to several Gyrs, this well defined feature is suitable to investigate the SFH from old to intermediate ages [@cioni06]. Recently, theoretical models of TP-AGB stars have been presented by [@marigo07] for initial masses between 0.5 and 5.0 $M_{\odot}$ and for different metallicities. Their Fig. 20 shows that the position of the carbon–star tracks in the $\log L/L_{\odot}$ vs $\log T_{eff}$ plane depends on both mass and metallicity. Higher-mass stars exhibit larger luminosities for a given metallicity. On the other hand, lower metallicities imply both a wider range of masses undergoing the C-rich phase, and higher luminosities for the more massive carbon stars.
From an empirical aspect, we can get the dependence of the carbon–star luminosity on age and metallicity from the work of [@batti05] (hereafter BD05). The authors provide the following relation for the dependence of the carbon–star I band magnitude on metallicity:
$${\rm <M_{I,carbon}>=-4.33 +0.28 \times [Fe/H]},
\label{eq3}$$
which was derived through a least-square fit of the mean absolute I band magnitude of carbon stars in nearby galaxies with metallicities ${\rm -2<[Fe/H]<-0.5}$. According to (\[eq3\]), a drop of 1 dex in metallicity results in a decrease of $\approx$ 0.3 in the average magnitude of carbon stars. The age dependence of the carbon–star luminosity is more difficult to quantify since it requires a detailed knowledge of the SFH in galaxies. BD05 do not study such a dependence, but some qualitative considerations can be obtained from examination of their Fig. 4. The scatter of the ${\rm <M_{I,carbon}>}$ versus ${\rm [Fe/H]}$ relation is of the order of 0.1 mag (excluding AndII and AndVII from the fit), i.e. $\approx$ 20 % of the variation in ${\rm <M_{I,carbon}>}$ spanned by the data. We also notice that at fixed metallicity, galaxies with current star formation (empty dots in Fig. 1 of BD05) tend to have brighter ${\rm <M_{I,carbon}>}$ values, while galaxies with no current star formation preferentially lie below the relation. This suggests that at least part of the scatter of relation (\[eq3\]) is due to a spread in age, with younger stellar populations having brighter carbon stars than older stellar populations. This age dependence is consistent with the [@marigo07] models, where more massive (younger) carbon stars tend to be more luminous.
Carbon stars were selected in NGC 4449 at ${\rm 23< m_{F814W} <24, m_{555W}-m_{F814W}>2.4}$, for each of the 28 regions shown in Fig. \[imagegrid\]. The color limit was chosen to avoid a significant contribution of RGB and oxygen-rich AGB stars. For each region, we fitted the ${\rm m_{F814W}}$-band LF with a Gaussian curve, and adopted the peak of the best-fitting Gaussian as the average ${\rm <m_{I,carbon}>}$ in that region. The errors on the results of the Gaussian fits can be approximated as $\Delta m \approx \sigma / \sqrt{N}$, where $\sigma$ is the width of the Gaussian and N is the number of stars. We experimented with different cuts in color (up to ${\rm m_{555W}-m_{F814W} >2.8}$) and different binnings of the data, and found that none of them significantly affects the final results of the analysis. The results presented in Fig. \[Clum\] were obtained by binning the stellar magnitudes in bins of 0.2 mag. The quantity ${\rm \Delta m_{I,C}}$ along the ordinate is the difference between the carbon–star magnitude measured in a specific region, and the carbon–star magnitude averaged over the whole field of view of NGC 4449. Along the abscissa is the X coordinate in pixels. From top to bottom, the panels refer to regions of decreasing Y coordinate. Fig. \[Clum\] shows that the observed carbon–star luminosity is reasonably constant over almost the whole galaxy, within the errors. The only significant variation is observed in the central regions ($ 3000 \la X \la 8000$, panel c), where the carbon stars appear to be up to $\approx$ 0.25 mag brighter than the average value.
We performed Monte Carlo simulations to understand if the observed variation of the C star LF is intrinsic, i.e. due to stellar population gradients present in NGC 4449 field, or if it is an effect of the photometric error and completeness level variations across the field. We adopted an estimated [*intrinsic*]{} LF for the C stars, that we assumed to be constant all over the field. Then we investigated how the [*intrinsic*]{} LF is transformed into the [*observed*]{} one after completeness and photometric errors were applied at different galacto-centric distances. For simplicity, we assumed that the intrinsic ${\rm m_{F814}}$ distribution of the C stars is Gaussian, with parameters (${\rm m_{F814W,0}=23.64}$, $\sigma$ $\sim 0.3$) derived from the Gaussian fit to the observed LF in the most external regions ((1,1:4), (1:7,4), (7,1:4); where this notation will indicate the some of these regions). We also assumed that the C stars are not homogeneously distributed in color, but follow a power law in ${\rm m_{F555W} - m_{F814W}}$. The power law’s parameters were derived by fitting the observed C star color distribution in the same external regions. Our assumption that the observed outer LF is a good description of the intrinsic one in those regions is reasonable, since the most external ${\rm m_{F814}}$, ${\rm m_{F555W} -m_{F814W}}$ CMDs are more than 90 % complete at the average C star luminosity, and the errors $\sigma_{F814W}$ are only a few hundreds of magnitudes there (see Fig. \[hessregionvi\]). Monte Carlo extractions of (${\rm m_{F814}}$, ${\rm m_{F555W} - m_{F814W}}$) pairs were drawn from the assumed magnitude and color distributions. A completeness and a photometric error were then applied to each extracted star, for both an external and an internal region of NGC 4449. The effect of incompleteness and photometric errors on the distribution is shown in the bottom panel of Fig. \[Clf\]. In the top panel we show instead the observed LFs for an external region ((1,1:4), (1:7,4), (7,1:4)), and for the most internal one (4,3), which displays the largest shift of the distribution peak with respect to the external region. Our simulations in the bottom panel show that the C star LF is mostly unaffected by incompleteness and photometric errors in the most external regions. This result testifies that the observed LF is very close to the intrinsic one in the periphery, and that we adopted a reasonable input distribution for the Monte Carlo simulations. In the most internal region, instead, the LF is shifted toward brighter magnitudes by an amount (${\rm \Delta m_{F814W} \approx 0.25}$) comparable to the shift between the observed LFs. The resulting internal distribution also has a larger width ($\sigma \approx 0.4$) than the intrinsic one, due to the larger photometric errors.
Our results show that the detected change in the C star brightness over NGC 4449 field can be largely attributed to differences in completeness between the center and the most external regions. Accounting for this effect, the average magnitude of the C stars is constant within the errors ($\approx 0.05$ mags). From the BD05 relation [^3], a change in magnitude of 0.05 corresponds to $\Delta$\[Fe/H\] $\approx 0.2$. We interpret this as an upper limit to the metallicity variation over the field of view. This is consistent with studies of metallicity gradients in other magellanic irregulars. For example, [@cole04] derive a metallicity gradient of $\approx$ $-$0.05 dex $\times$ kpc$^{-1}$ in the LMC by comparing the abundances in the inner disk and in the outer disk/spheroid [@ols], while @gro06 find no metallicity gradients from spectroscopic studies of cluster stars in the LMC.
RGB stars
---------
In the CMD of Fig. \[cmd2\], the morphology of the RGB, at ${\rm m_{F814W}} \ga 24$ and ${\rm m_{F555W}- m_{F814W} \ga 1}$, is connected to the properties of the stellar content older than $\sim$ 1 Gyr. Despite the poorer time resolution with increasing look-back time, and the well known age-metallicity degeneracy, some constraints on the properties of the old stellar population can be inferred from an analysis of the RGB morphology.
We derived the average RGB color as a function of ${\rm m_{F814W}}$ by selecting stars with ${\rm m_{F814W}} \ga 24$, and then performing a Gaussian fit to the ${\rm m_{F555W}-m_{F814W}}$ color distribution at different magnitude bins. The peak of the Gaussian fit is as red as ${\rm m_{F555W}-m_{F814W} \approx 1.7}$ at the RGB tip, and it is ${\rm m_{F555W}-m_{F814W} \approx 1.45}$ at ${\rm m_{F814W}=25}$, one magnitude below the tip. As expected, the RGB is significantly redder than in more metal-poor star-forming galaxies that we have previously studied with HST/ACS [@alo05; @alo07].
In order to reveal the presence of age/metallicity gradients in NGC 4449, we performed a spatial analysis of the RGB morphology, following the same procedure as in Section 4.1. For each of the 28 rectangular regions identified in Fig. \[imagegrid\], we performed a Gaussian fit to the RGB ${\rm m_{F555W}- m_{F814W}}$ color distribution for bins of ${\rm \Delta m_{F814W}=0.25}$. Then we averaged for each region the colors derived in the four brightest bins, at ${\rm 24 \le m_{F555W}- m_{F814W} \le 25}$, i.e. one magnitude below the TRGB. The results of our analysis are presented in Fig. \[deltargb\]. We plot along the ordinate the difference between the RGB color in each region and the average RGB color in the total field of view of NGC 4449; along the abscissa is the X coordinate in pixels. From top to bottom, the panels refer to regions of decreasing Y coordinate. Fig. \[deltargb\] shows that the RGB is bluer in the center than in the periphery of NGC 4449, with variations up to $\approx$ 0.3 mag in the ${\rm m_{F555W}-m_{F814W}}$ color.
This effect is also shown in the top panel of Fig. \[rgbcolor\], where we plotted the observed color distributions of stars with ${\rm 24 \le m_{F814} \le 25}$, for an external region ((1,1:4), (1:7,4), (7,1:4)), and for an internal region (3:5,2) of NGC 4449. The peaks of the Gaussian fits to the external and internal distributions differ by $\approx$ 0.26 mag, and their errors are very small ($\sigma / \sqrt{N} \approx$ 0.001 mag) due to the large number of stars in each distribution ($\approx 20,000$). The Gaussian fit is broader for the internal region ($\sigma \approx 0.4$) than for the external region ($\sigma \approx 0.2$). Also, the central region has a much broader tail of stars towards blue colors, due to the contamination from younger blue-loop and MS stars. The contribution from the MS $+$ blue–loop stars at the hot edge of the core He-burning phase is recognizable as a bump at V$-$I $\approx 0$.
As was done for the C stars in Section 4.1, we performed Monte Carlo simulations to understand if the observed difference in the color distributions is intrinsic, or if it can be attributed to the larger crowding in the central regions of NGC 4449. We assumed an initial distribution, and drew Monte Carlo extractions from it with application of photometric errors and incompleteness. As a first guess for the initial distribution, we adopted a Gaussian with parameters ${\rm m_{F555W}-m_{F814W}=1.65}$ and $\sigma=0.15$. This has the same mean but somewhat smaller $\sigma$ than the observed color distribution in the most external regions. The simulated color distributions for the external and internal regions were generated by applying the photometric errors and the completeness levels derived from artificial star experiments in the considered regions of NGC 4449. The results are presented in the central panel of Fig. \[rgbcolor\]. As observed, the width of the simulated distribution in the internal region is larger than in the external one because of the higher photometric errors. While the peak of the external simulated distribution is the same as that of the initial distribution, the peak of the internal distribution is blueshifted by an amount of $\approx$ 0.06 mags. However, this is much less than the observed shift of $\approx$ 0.3 mags. The simulations therefore show that the completeness and the photometric error variations over NGC 4449 field of view can account only in part ($\approx$ 20 % ) for the shift between the observed internal and external distributions. Thus the observed shift must be mainly due to an intrinsic variation of the stellar population properties.
As a test, we performed a new simulation starting from a Gaussian with a peak as blue as ${\rm m_{F555W} - m_{F814}=1.45}$ and with $\sigma=0.15$. The result for the internal region is shown in the bottom panel of Fig. \[rgbcolor\]. Here we plot also the observed color distribution, for comparison. Both the peaks and the widths of the simulated and observed distributions are in good agreement. The only discrepancy is observed at the bluest color, where of course we do not reproduce the tail of the MS and post-MS stars. Our simulations suggest that the peak of the color distribution toward the center is intrinsically bluer than in the periphery. Assuming that the bluer peak in the center is due to a bluer RGB, the possible interpretations are 1) younger ages; 2) lower metallicities; or 3) lower reddening. If differential reddening is present within NGC 4449, we expect the central regions to be more affected by dust extinction than the periphery, and this would cause an even redder RGB in the center. Lower metallicities in NGC 4449 center are also very unlikely, since abundance determinations in galaxies show that metallicity tends to decrease from the center outwards or to remain flat (see Section 4.1). Thus, a bluer RGB would most likely indicate a younger population in the center of NGC 4449. Alternatively, the bluer peak observed in the center could be due to “contamination" by intermediate-age blue loop stars at the red edge of their evolutionary phase. But this too would imply the presence of a younger stellar population in the center than in the periphery of NGC 4449. More quantitative results on age/metallicity in NGC 4449 gradients will be derived through fitting of synthetic CMDs, and presented in a forthcoming paper (Annibali et al. 2008, in preparation).
A new distance determination
============================
The magnitude of the TRGB can be used to determine the distance of NGC 4449. The top panel of Fig. \[trgb\] shows the I-band LF of those stars in our final catalog that have V$-$I in the range $1.0$–$2.0$. Here V and I are Johnson-Cousins magnitudes, obtained from our magnitudes in the ACS filter system using the transformations of [@sir05] and applying a foreground extinction correction of ${\rm E(B-V) = 0.019}$ [@schlegel]. The TRGB is visually identifiable as the steep increase towards fainter magnitudes at ${\rm I \approx 24}$. At this magnitude, RGB stars start to contribute with a LF that increases roughly as a power law towards faint magnitudes. By contrast, the stars in the LF at brighter magnitudes are exclusively red supergiants and AGB stars. The drop at magnitudes fainter than ${\rm I \approx 25.5}$ is due to incompleteness.
To determine the TRGB magnitude we used the software and methodology developed by one of us (R.P.v.d.M) and described in detail in [@cioni00]. A discontinuity produces a peak in all of the higher-order derivatives of the LF. We use a so-called Savitzky-Golay filter on the binned LF to obtain the first- and second-order derivatives. These are shown in the middle and bottom panel of Fig. \[trgb\], respectively. Peaks are indeed visible at the expected position of the TRGB. We fit these with Gaussians and find that the first derivative has a peak at ${\rm I_1 = 24.09}$, while the second derivative has a peak at ${\rm I_2 = 23.88}$. The reason for the difference between these magnitudes is the presence of photometric errors and binning in the analysis. This smooths out the underlying discontinuity. As shown in Fig. A.1 of [@cioni00], this causes the first derivative to overestimate the magnitude of the TRGB, and the second derivative to underestimate the magnitude of the TRGB. These biases can be explicitly corrected for as in Fig. A.2 of [@cioni00], using a simple model for the true underlying LF and the measured width of the Gaussian peaks in the first- and second-order derivatives of the LF. After application of this correction we obtain the final estimates ${\rm I_{TRGB,1} = 23.99}$ and ${\rm I_{TRGB,2} =24.00}$ for the underlying TRGB magnitude, based on the first and second derivatives, respectively. The good agreement between these independent estimates shows that the systematic errors in the method are small, in agreement with [@cioni00] who adopted a systematic error ${\rm \Delta I_{TRGB} = \pm 0.02}$. The additional systematic error introduced by the uncertainties in photometric zeropoints, transformations, and aperture corrections [@sir05] is ${\rm \Delta I_{TRGB} = \pm 0.03}$. The random error on ${\rm \Delta I_{TRGB}}$ is very small due to the large number of stars detected in NGC 4449. It can be estimated using bootstrap techniques to be ${\rm \Delta I_{TRGB} = \pm 0.01}$.
Our estimate of the TRGB magnitude, ${\rm I_{TRGB} = 24.00 \pm 0.01}$ (random) $\pm 0.04$ (systematic), can be compared to the absolute magnitude of the TRGB, which was calibrated as a function of metallicity by, e.g., [@bella04]. Adopting ${\rm [M/H] =
-0.52}$ for NGC 4449 (based on the oxygen abundance given in Section 1) their calibration (top panel of their Fig. 5) predicts ${\rm M_{I, TRGB} =
-3.91}$. Comparison of different studies suggests that the systematic uncertainty in this prediction is $\sim 0.15$. This takes into account also the possibility that the RGB star metallicity is actually lower than that of the HII regions (see discussion in Section 6). The implied distance modulus for NGC 4449 is therefore ${\rm (m-M)_0 = 27.91 \pm 0.15}$, where we have added all sources of uncertainty in quadrature. This corresponds to ${\rm D = 3.82 \pm 0.27}$ Mpc.
An alternative method for estimating the distance of NGC 4449 is through the average I-band magnitude of the carbon stars. Fig. \[agbdist\] shows the I-band luminosity function of those stars in our final catalog that have V$-$I in the range $2.2$–$3.0$. There is a well-defined peak, due to the horizontal “finger” of carbon stars seen in the CMD of Fig. \[cmd2\]. A Gaussian fit to the peak yields ${\rm I_{carbon} = 23.59 \pm 0.01}$, corrected for foreground extinction. Adopting ${\rm [Fe/H] = -0.52}$ for NGC 4449 (based on the oxygen abundance given in Section 1, and assuming for simplicity that oxygen traces the iron content), the BD05 calibration in (\[eq3\]) predicts $M_{\rm I,
carbon} =-4.48$. The systematic uncertainty in this prediction is difficult to quantify. This is because carbon–star magnitudes are not well understood on the basis of stellar evolution theory (by contrast to the TRGB), because the dependence on stellar age or star formation history is poorly quantified, and because only a few empirical studies exist. We therefore adopt an uncertainty of $\sim 0.2$ mag, consistent with the discussion of Section 4.1 The implied distance modulus for NGC 4449 is then ${\rm (m-M)_0 = 28.07 \pm 0.20}$, where we have added all sources of uncertainty in quadrature. This corresponds to ${\rm D = 4.11 \pm 0.38}$ Mpc. This is in agreement with the TRGB result, given the uncertainties.
We can compare our result of ${\rm D=3.82 \pm 0.27}$ Mpc to the previous estimate of NGC 4449 distance by [@ka03], who inferred ${\rm D=4.2 \pm 0.5}$ Mpc. This result was derived by applying the TRGB-luminosity method to HST/WFPC2 data. The extinction-corrected magnitude at which they detected the TRGB is ${\rm I_{TRGB} = 24.07 \pm 0.26}$, which is consistent with our somewhat lower value of ${\rm I_{TRGB} = 24.00 \pm 0.01}$ (random) $\pm 0.04$ (systematic). Their somewhat larger distance is also due to the different value adopted for the absolute magnitude of the TRGB. They adopt ${\rm M_{I, TRGB} =-4.05}$ (as appropriate for metal-poor systems), while we adopt ${\rm M_{I, TRGB} = -3.91}$ from the more recent [@bella04] calibration (which takes into account the metallicity dependence of the TRGB luminosity).
Comparison with models
======================
For a direct interpretation of the CMDs in terms of the stellar evolutionary phases, we have superimposed stellar evolutionary tracks for different metallicities on the ${\rm m_{F555W}}$, ${\rm m_{F435W}- m_{F555W}}$, and ${\rm m_{F814W}}$, ${\rm m_{F555W}- m_{F814W}}$ CMDs (Figs. \[cmdtracks\] and \[rgbtracks\]). The tracks at Z$=$0.008, Z$=$0.004, and Z$=$0.0004 are the Padua stellar evolutionary tracks (Fagotto et al. 1994a, 1994b) transformed into the ACS Vegamag system by applying the @origlia code, and corrected for Galactic extinction (${\rm E(B-V)=0.019}$, @schlegel) and distance modulus (${\rm (m-M)_0 = 27.91}$, see Section 5). The Z$=$0.001 tracks were obtained from the Padua tracks through interpolation in metallicity [@ang06].
The metallicity range covered by the plotted tracks can account for the different populations potentially present in NGC 4449. The Z$=$0.008 and Z$=$0.004 tracks are suitable to account for the more metal rich population, given that the abundances derived in the nucleus and disk of NGC 4449 are almost one third of the solar value (see Section 1). The plotted tracks are for masses in the range 0.9–40 ${\rm M_{\odot}}$. Lower mass stars would not have the time to reach visible phases within a Hubble time at the distance of NGC4449 (in these sets a 0.8 ${\rm M_{\odot}}$ star reaches the TRGB in 19 Gyr). The tracks are divided into three groups, namely [*low-mass*]{} stars (${\rm M \le M_{HeF}}$), [*intermediate-mass*]{} stars (${\rm M_{HeF} < M \le M_{up}}$) and [*high-mass*]{} stars (${\rm M > M_{up}}$). The subdivision is made according to the critical mass at which the ignition of the central fuel (either helium or carbon) starts quietly depending on the level of core electron-degeneracy. In the adopted tracks, the value of ${\rm M_{HeF}}$ depends slightly on metallicity, being equal to 1.7, 1.8 and 1.9 ${\rm M_{\odot}}$ for Z$=$0.0004, Z$=$0.004 and Z$=$0.008, respectively. The value of ${\rm M_{up}}$ is between 5 and 6 ${\rm M_{\odot}}$. For low-mass stars, we have displayed in Figs. \[cmdtracks\] and \[rgbtracks\] only the phases up to the TRGB in order to avoid excessive confusion.
At the distance of NGC 4449, the MS is sampled only for stars with ${\rm M \ga 3 M_{\odot}}$, and corresponds to the vertical lines whose ${\rm m_{F435W}- m_{F555W}}$, ${\rm m_{F555W}- m_{F814W}}$ colors go from $\sim$ $-$0.3 to $\sim$ 0 from the 40 ${\rm M_{\odot}}$ to the 3 ${\rm M_{\odot}}$ track. The turnoff is recognizable as a small blue hook on each evolutionary track of Fig. \[cmdtracks\]. The almost horizontal blue loops correspond to the later core helium-burning phase for intermediate- and high- mass stars, while the bright vertical lines in the red portion of the CMD describe the AGB phase. For low-mass stars, our data sample only the brighter red sequences of the RGB, all terminating at the TRGB with approximately the same luminosity. The models considered here do not include the horizontal feature of the carbon stars in the TP-AGB phase, recognizable in the ${\rm m_{F814W}, m_{F555W}- m_{F814W}}$ CMD at ${\rm m_{F555W}- m_{F814W} \ga 2}$, but see @marigo07 for new appropriate models.
The comparison of the CMDs with the tracks in Fig. \[cmdtracks\] shows that the blue plume is populated by high- and intermediate-mass stars on the MS, and high-mass stars at the hot edge of the core helium-burning phase. The red plume samples bright red SGs and AGB stars. The faint red sequence at ${\rm m_{F814W} \ga 24}$, featured in Fig. \[rgbtracks\], is due to low-mass stars in the RGB phase.
The presence of high-, intermediate- and low-mass stars in the CMD testifies that young, intermediate-age and old stars (several Gyrs) are present at the same time in NGC 4449. In particular, the fact that we sample stars as massive as 40 ${\rm M_{\odot}}$ implies that the SF was active 5 Myr ago. Furthermore, the absence of significant gaps in the CMD suggests that the SF has been mostly continuous over the last 1 Gyr. The time resolution gets significantly poorer for higher look-back times because of both the intrinsic degeneracy of the tracks and the large photometric errors at faint magnitudes; thus small interruptions in the SF can be easily hidden in the CMD for ages older than $\ga 1$ Gyr.
The metallicity ${\rm [M/H] =-0.52}$ derived in Section 1 for NGC 4449 corresponds to a metal fraction Z$=$0.005, adopting Z$_{\odot}=$ 0.017 [@sun98]. However this value refers to abundance determinations in HII regions, thus it is likely to reflect the metallicity of the youngest stars. Fig. \[cmdtracks\] and \[rgbtracks\] show that the Z$=$0.004 tracks are in good agreement with all the phases of the empirical CMDs. The Z$=$0.0004 and Z$=$0.001 tracks are too blue to account for the observed RGB feature. So, if there is no significant extinction intrinsic to NGC 4449, then stars with such low metallicities do not account for a significant fraction of the stellar population in NGC 4449. Allowing for an age spread from 1 Gyr to a Hubble time, the Z$=$0.004 tracks are in good agreement with both the color and the width of the observed RGB. @hill98 derived an internal reddening ${\rm E(B-V)
\approx}$ 0.18 from the $H\alpha$/H$\beta$ ratio measured in NGC 4449 HII regions. Adopting this value, the RGB is consistent with metallicities as low as Z$=$0.001. However, this extinction value is appropriate for the young star forming regions, and we expect a lower extinction in the regions where older stars reside (as demonstrated for the LMC by @zari). Either way, the Z$=$0.008 models seem slightly too red, especially in the AGB phase. Although there are some uncertainties in the models of this phase, we do believe that this discrepancy indicates that metallicities higher than Z$=$0.008 are ruled out for this galaxy.
By superimposing the stellar tracks on the empirical CMDs, we can attempt a selection of the photometrized stars according to their age. In Fig. \[cmdsel\] we show the four regions selected on the ${\rm m_{F814W}, m_{F555W}- m_{F814W}}$ CMD that correspond to different stellar masses, namely M${\rm \ge 20 \ M_{\odot}}$, ${\rm 5 M_{\odot} \le M < 20 M_{\odot}}$, ${\rm 1.8 M_{\odot} < M < 5 M_{\odot}}$, and ${\rm M \le 1.8 M_{\odot}}$. These regions roughly define the loci of [*very young*]{} stars, with ages $\la$ 10 Myr, [*young*]{} stars, with ${\rm 10 \ Myr \la age \la 100 \ Myr}$, [*intermediate-age*]{} stars, with ages between $\sim$ 100 Myr and 1 Gyr, and [*old*]{} stars, with ages $>$ 1 Gyr. The spatial distribution of the four groups of stars is shown in Fig. \[spatial\]. We notice that old stars are homogeneously distributed over the galaxy, except for the central regions, where the high crowding level makes their detection more difficult. As we approach younger ages, the distribution becomes more and more concentrated. Stars with ages between 10 and 100 Myr are highly clustered in an [*S shaped*]{} structure centered on the galaxy nucleus. This could be a bar, which is a common feature among Magellanic irregular galaxies. Stars with ages between 10 and 100 Myr also clearly outline the symmetric structure around the resolved cluster-like object (see Fig. \[imagegrid\]) that was already discussed in Section 4. The same structure is visible to a lesser extent in the spatial distribution of stars with ages 100 Myr – 1 Gyr. The resolved cluster-like object itself has a clear RGB (not shown here), and it is therefore not a young super star cluster. Very young stars, with ages younger than 10 Myr, are strongly clustered and detected only in the very central regions of NGC 4449. Such young stars should be able to ionize the surrounding interstellar medium and produce HII regions. Indeed, Fig. \[halpha\], where we plotted the F658N (H$\alpha$) image together with the positions of the stars younger than 10 Myr, shows that this is the case, with a strong correlation between the position of the stars younger than 10 Myr and the HII regions. This indicates that most of the emission is due to photoionization rather than to shocks, due to, e.g., supernovae explosions.
Conclusions
===========
We have acquired HST/ACS imaging in the F435W (B), F555W (V), F814W (I) and F658N (${\rm H\alpha}$) filters of the Magellanic starburst galaxy NGC 4449 in order to infer its star formation history and understand the properties of the observed global starburst. In this paper we present the B, V and I photometry of the resolved stars. We detect 299,014 objects in the (B,V) CMD, 402,045 objects in the (V,I) CMD, and 213,099 objects with a measured magnitude in all the three bands. The derived CMDs span a magnitude range of $\approx$ 10 mag, and sample both the young and the old resolved stellar population in NGC 4449. We also detected several candidate clusters (at least $\approx$ 40, some of which look like very massive globular clusters) and background galaxies in our images.
We derived a new distance from the TRGB method. The TRGB is detected at a Johnson-Cousins I magnitude of ${\rm I_{\rm TRGB} = 24.00 \pm 0.04}$. At the metallicity of NGC 4449, the TRGB is expected at an absolute magnitude of ${\rm M_{\rm I, TRGB} =-3.91}$, with a systematic error of $\sim 0.15$ mag. This provides a distance modulus of ${\rm (m-M)_0 = 27.91 \pm 0.15}$, i.e. a distance of $3.82 \pm 0.27$ Mpc. We used also the alternative method of the carbon–star luminosity, and found a distance of $4.11 \pm 0.38$ Mpc, which is consistent with the result from the TRGB method. Our distance determinations are consistent within the errors with the value of D$=4.2 \pm 0.5$ Mpc previously provided by Karachentsev et al. (2003).
In the CMDs of NGC 4449 we observe a well defined blue plume (MS and post-MS stars) and red plume (red SGs and AGB stars), the horizontal tail of the carbon stars in the TP-AGB phase (in the I, V$-$I CMD), and a prominent RGB. The presence of all these evolutionary features implies ages up to at least 1 Gyr and possibly as old as a Hubble time. The comparison of the observed CMDs with the Padua stellar evolutionary tracks, corrected for the derived distance modulus and foreground extinction, shows that stars as massive as 40 ${\rm M_{\odot}}$ are present in NGC 4449. Such high masses imply that the star formation was active 5 Myr ago, and possibly it is still ongoing. The absence of significant gaps in the CMDs suggests also that the star formation has been mostly continuous over the last 1 Gyr. However, interruptions in the star formation can be easily hidden in the CMD for ages older than $\ga 1$ Gyr, because of the intrinsic degeneracy of the tracks and the large photometric errors at faint magnitudes. The presence of a prominent RGB testifies that NGC 4449 hosts a population possibly as old as several Gyrs or more. However, the color-age degeneracy of the tracks with increasing look-back time (at a given metallicity, large age differences correspond to small color variation in the RGB), and the well known age-metallicity degeneracy, prevent us from establishing an exact age for the galaxy. We will derive a better age estimate with the synthetic CMD method, when we’ll study the detailed SFH (Annibali et al. 2008, in preparation).
Abundance estimates in NGC 4449 HII regions provide 12 + log(O/H)$ \approx 8.31$ [@talent; @mar97], which corresponds to \[O/H\] $= -0.52$ if we assume a solar abundance of 12 + log(O/H)$_{\odot} =$ 8.83 [@sun98]. These measures are biased toward regions where the interstellar medium has been significantly reprocessed, and thus are likely to reflect the metallicity of the youngest generation of stars. We expect lower metallicity for the oldest stars in NGC 4449. Interestingly, though, the Z$=$0.004 stellar evolutionary tracks (the closest in the Padua set to the metallicity of NGC 4449), corresponding to ${\rm \log(Z/Z_{\odot}) = -0.63}$ if we adopt ${\rm Z_{\odot}=0.017}$ [@sun98], seem to be in very good agreement with all the features observed in the empirical CMDs, if we assume that there is not significant extinction intrinsic to NGC 4449. In particular, the next lower metallicity tracks (at ${\rm Z=0.001}$) are definitively too blue to account for the observed RGB colors, implying that the bulk of the stellar population older than 1 Gyr was already enriched in metals.
We investigated the presence of age and metallicity gradients in NGC 4449. To this purpose, we divided the total galaxy’s field of view into 28 rectangular regions of $\approx$ 1 kpc$^2$ area, and derived the CMDs for the different regions. The CMD morphology presents a significant spatial dependence: while the RGB is detected over the whole field of view of the galaxy, the blue plume, red plume and blue-loop stars are present only in the more central regions, indicating that the stellar population is younger in the center than in the periphery of NGC 4449.
We also studied the spatial behavior of the carbon–star luminosity and of the RGB color. Once the effect of incompleteness and photometric errors is taken into account, the average magnitude of the C stars turns out to be constant within the errors ($\approx$ 0.05 mags). This gives an upper limit of $\approx$ 0.2 dex on the metallicity variation over the field of view of NGC 4449. On the other hand, we find that the RGB is intrinsically bluer in the center than in the periphery of the galaxy. Bluer RGB colors can be due to younger and/or more metal poor stellar populations. However, as spectroscopic-based abundance determinations in galaxies show that metallicity tends to decrease from the center outwards, or to remain constant, we interpret this as the result of a younger, and not more metal poor, stellar population in the center of NGC 4449.
With the help of the Padua tracks, we identified in the observed ${\rm m_{F814W}, m_{F555W}- m_{F814W}}$ CMD four different zones corresponding to different stellar masses, namely M${\rm \ge 20 M_{\odot}}$, ${\rm 5 M_{\odot} \le M < 20 M_{\odot}}$, ${\rm 1.8 M_{\odot} \le M < 5 M_{\odot}}$, and ${\rm M < 1.8 M_{\odot}}$. These regions roughly define the loci of stars with age $\la$ 10 Myr, ${\rm 10 \ Myr \la age \la 100 \ Myr}$, 100 Myr $<$ age $<$ 1 Gyr, and age $>$ 1 Gyr. Low-mass old stars are homogeneously distributed over the galaxy’s field of view, with the exception of the central regions, where the high crowding level makes their detection more difficult. As we approach younger ages, the spatial distribution of the stars becomes more and more clustered. Intermediate-age stars (100 Myr $<$ age $<$ 1 Gyr) are mostly found within $\approx$ 1 kpc from the center. Stars with ages between 100 Myr and 10 Myr are found in an [*S–shaped*]{} structure centered on the galaxy nucleus, and extending in the North-South direction up to 1 kpc away from the center. This could be a bar, which is a common feature among Magellanic irregular galaxies. Stars younger than 10 Myr are very rare, and found only in the galaxy nucleus and in the North arm of the [*S shape*]{} structure. The comparison with the H$\alpha$ image shows a tight correlation between the position of the stars younger than 10 Myr and the HII regions, indicating that we have identified the very massive and luminous stars that ionize the surrounding interstellar medium.
One of the many star clusters visible in our image is of particular interest. This cluster on the West side of the galaxy is surrounded by a symmetric structure that is particularly well outlined by stars with ages in the range of 10–100 Myr (see Fig. \[spatial\]). This structure could be due to tidal tails or spiral–like feature associated with a dwarf galaxy that is currently being disrupted by NGC 4449. The cluster could be the remnant nucleus of this galaxy. It is resolved into red stars, and it has a significant ellipticity (see Fig. \[imagegrid\]). This is reminiscent of the star cluster $\omega$ Cen in our own Milky Way, which has also been suggested to be the remnant nucleus of a disrupted dwarf galaxy [@omegacen]. More details about this object will be presented in a forthcoming paper (Aloisi et al. 2008, in preparation).
Quantitative information on the star formation history of NGC 4449 is fundamental in order to understand the connection between the global starburst observed and processes such as merging, accretion and interaction. The star formation history of NGC 4449 will be derived in a forthcoming paper through the synthetic CMD method, which is based on stellar evolutionary tracks, and is able to fully account for the effect of observational uncertainties, such as photometric errors, blending and incompleteness of the observations.
Support for proposal \#10585 was provided by NASA through a grant from STScI, which is operated by AURA, Inc., under NASA contract NAS 5-26555. We thank Livia Origlia for providing the photometric conversion tables to the ACS Vegamag system.
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[^1]: Based on observations with the NASA/ESA [*Hubble Space Telescope*]{}, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., for NASA under contract NAS5-26555.
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by AURA, Inc., under cooperative agreement with the National Science Foundation
[^3]: BD05 use the Johnson-Cousins I-band. This is very similar to ${\rm m_{F814W}}$ [@sir05], and the small difference can be ignored for the purpose of the differential argument presented here.
| {
"pile_set_name": "ArXiv"
} |
[**Where and How Do They Happen ?**]{}\
\
[**Elemér E Rosinger**]{}\
\
Department of Mathematics\
and Applied Mathematics\
University of Pretoria\
Pretoria\
0002 South Africa\
eerosinger@hotmail.com\
\
[**Abstract**]{}\
This is a two part paper. The first part, written somewhat earlier, presented standard processes which cannot so easily be accommodated within what are presently considered as physical type realms. The second part further elaborates on that fact. In particular, it is argued that quantum superposition and entanglement may better be understood in extensions of what we usually consider as physical type realms, realms which, as it happens, have so far never been defined precisely enough.\
\
[**Part I**]{}\
\
[**Abstract**]{}\
It has for ages been a rather constant feature of thinking in science to take it for granted that the respective thinking happens in realms which are totally outside and independent of all the other phenomena that constitute the objects of such thinking. The imposition of this divide on two levels may conflict with basic assumptions of Newtonian and Einsteinian mechanics, as well as with those in Quantum Mechanics. It also raises the question whether the realms in which thinking happens have no any other connection with the realms science deals with, except to host and allow scientific thinking.\
\
[**0. The Yet Undefined Physical Realms ...**]{}\
In the sequel, based on rather obvious and simple, even if so far seldom considered facts within, or related to Physics, we shall argue that what are usually assumed to be the Physical Realms may have to be extended. Such possible additional realms, however, are not along those infinitely many of Everett’s “many-worlds” view of Quantum Mechanics. Instead, they are suggesting a finite number of further physical type realms, thus they can be seen as a development of the classical Cartesian realm of “res extensa”.\
As for what Physical Realms may actually mean, or rather, Physics itself, here is a recent and quite appropriate view on that never yet clarified issue, \[8, pp. 153,154\] :
> “Physics is the study of those phenomena that are successfully treatable with well-specified and testable models.\
> For example, Physics treats atoms and simple molecules. Chemistry, on the other hand, deals with all molecules, most of whose electron distributions cannot be well specified. A physicist might study a well specified biological system, but the functioning of a complex organism lies in the domain of biologists.\
> Anything not successfully treatable with a well-specified and testable model is rather quickly defined out of Physics.”
It is quite clear in this spirit that, even if no one seems to care much about a more precise definition of Physics, and thus, of Physical Realms, phenomena such as human thinking, let alone, human consciousness or awareness, are not expected to concern Physics any time soon. Consequently, what for Descartes constituted “res cogitans”, that is, the realms of thinking, are supposed to remain in the splendour of their undisturbed solitude, as far as Physics is concerned. And then, anything that may be seen as remotely acceptable from a physical point of view, may be but a refinement, or rather, a structural enrichment of the Cartesian “res extensa”, that is, of the realms which, at least intuitively, are supposed to have to do with Physics.\
And yet, as seen in the sequel, the story is not quite that simple, not even from a strictly physical point of view ...\
\
[**1. Conflict with Newtonian Mechanics**]{}\
Instant action at arbitrary distance, such as in the case of gravitation, is one of the basic assumptions of Newtonian mechanics. This certainly does not appear to conflict with the fact that we can think instantly and simultaneously about phenomena which are no matter how far apart from one another in space or in time. However, absolute space is also a basic assumption of Newtonian Mechanics. And it is supposed to contain absolutely everything that may exist in Creation, be it in the past, present or future. Consequently, it is supposed to contain, among others, the physical body of the thinking scientist as well.\
Yet it is not equally clear whether it also contains scientific thinking itself which, traditionally, is assumed to be totally outside and independent of all phenomena under its consideration, therefore in particular, of the Newtonian absolute space, and also, of absolute time.\
And then the question arises :
> Where and how does such a scientific thinking take place or happen ?
[ ]{}\
[**2. A difference with Mathematics**]{}\
Mathematical thinking, especially in its modern and abstract variants, does not appear to need the assumption of any absolute space, or for that matter, absolute time. Such thinking may appear to unfold during appropriate local time intervals. However, when seen all in itself, and unrelated to the physical body of the respective mathematician, it is quite likely that such thinking has no location in any space, be it relative or absolute.\
\
[**3. Conflict with Einsteinian Mechanics**]{}\
In Einsteinian Mechanics a basic assumption is that there cannot be any propagation of action faster than light.\
Yet just like in the case we happen to think in terms of Newtonian Mechanics, our thinking in terms of Einsteinian Mechanics can again instantly and simultaneously be about phenomena no matter how far apart from one another in space or time.\
Consequently, the question arises :
> Given the mentioned relativistic limitation, how and where does such a thinking happen ?
[**4. Conflict with Quantum Mechanics**]{}\
Let us consider the classical EPR, or Einstein-Podolsky-Rosen entanglement phenomenon, and for simplicity, do so in the terms of quantum computation. For that purpose it suffices to consider double qubits, that is, elements of $\mathbb{C}^2 \bigotimes \mathbb{C}^2$, such as for instance the EPR pair\
(4.1) $ \begin{array}{l}
|~ \omega_{00} > ~=~ |~ 0, 0 > ~+~ |~ 1, 1 > ~=~ \\ \\
~~~=~ |~ 0 > \bigotimes |~ 0 > ~+~ |~ 1 > \bigotimes |~ 1 >
\, \in \mathbb{C}^2 \bigotimes \mathbb{C}^2
\end{array} $\
which is well known to be [*entangled*]{}, in other words, $|~ \omega_{00} > $ is [*not*]{} of the form\
$~~~~~~ ( \alpha |~ 0 > ~+~ \beta |~ 1 > ) \bigotimes ( \gamma |~ 0 > ~+~ \delta |~ 1 > )
\, \in \mathbb{C}^2 \bigotimes \mathbb{C}^2 $\
for any $\alpha, \beta, \gamma,\delta \in \mathbb{C}^2$.\
Here we can turn to the usual and rather picturesque description used in quantum computation, where two fictitious personages, Alice and Bob, are supposed to exchange information, be it of classical or quantum type.\
Alice and Bob can each take their respective qubit from the entangled, or EPR pair of qubits $|~ \omega_{00} >$, and then go away with it no matter how far apart from one another. And the two qubits thus separated in space will remain entangled, unless of course one or both of them get involved in further classical or quantum interactions. For clarity, however, we should note that the single qubits which, respectively, Alice and Bob take away with them from the EPR pair $|~ \omega_{00} >$ are neither one of the terms $|~ 0, 0 >$ or $|~ 1, 1 >$ in (4.1), since both these are themselves already pairs of qubits, thus they cannot be taken away as mere single qubits, either by Alice, or by Bob. Consequently, the single qubits which Alice and Bob take away with them cannot be described in any other form, except that which is implicit in (4.1).\
Now, after that short detour into the language of quantum computation, we can note that, according to Quantum Mechanics, the entanglement in the EPR double qubit $|~ \omega_{00} >$ implies that the states of the two qubits which compose it are correlated, no matter how far from one another Alice and Bob would be with them. Consequently, knowing the state of one of these two qubits can give information about the state of the other qubit. On the other hand, in view of General, or even Special Relativity, such a knowledge, say by Alice, cannot be communicated to Bob faster than the velocity of light.\
And yet, anybody who is familiar enough with Quantum Mechanics, can instantly know and understand all of the above, no matter how far away from one another Alice and Bob may be with their respective single but entangled qubits.\
So that, again, the question arises :
> How and where does such a thinking happen ?
[ ]{}\
[**5. Two, Among Other Possible Alternatives**]{}\
Let us first assume that scientific thinking does indeed happen in realms outside and independent of all the realms in which the variety of phenomena studied by scientific thinking takes place. Then the very existence of scientific thinking proves the existence of realms transcendental to those which at present are customarily the object of that scientific thinking.\
In this case, one may ask whether the realms in which scientific thinking happens have, indeed, no any other connection whatsoever with the realms which are the object of study of science, except to host and allow such scientific thinking.\
A cautious answer is of course not one of categorical negation. Furthermore, any answer, including a categorically negative one, may need some supporting evidence, and possibly of experimental or empirical kind as well.\
If alternatively, we assume that, after all, there is only one overall realm in which everything happens, then quite likely we may have to extend rather significantly, if not in fact dramatically, the list of entities, phenomena, or processes which are, or can be relevant in Physics, Chemistry, Biology, and so on. Certainly, in such a case it can no longer be taken for granted - and done so without any supporting evidence - that the whole range of entities and their interactions which form the object of science are isolated in some subdomain of that unique overall realm. And very much isolated they appear to be, since usual scientific thinking itself is assumed to be outside and independent of them, plus we deal with all those entities and their interactions as if they were perfectly self-contained.\
\
[**6. Conclusions**]{}\
It may be useful to ask the following four questions :
> 1\. Do we believe that whatever in Creation which may be relevant to science is already accessible to our awareness ?\
> 2. And if not - which is most likely the case - then do we believe that it may become accessible during the lifetime of our own generation ?\
> 3. And if not - which again is most likely the case - then do we believe that we should nevertheless try some sort of two way interactions with all that which may never ever become accessible to the awareness of our generation, yet may nevertheless be relevant to science even in our own days ?\
> 4. And if yes - which most likely is the minimally wise approach - then how do we intend to get into a two way interaction with all those realms about which our only awareness can be that they shall never ever be within our awareness, or perhaps, not even of human awareness as such, no matter how long our species may live ?
[ ]{}\
[**Part II**]{}\
\
[**Abstract**]{}\
It is further argued that quantum superposition and entanglement may better be understood in extensions of what we usually consider to be physical type realms, realms which in fact have never been defined precisely enough.\
\
[**1. Superposition : a Typically Quantum\
Fundamental Phenomenon**]{}\
In the non-relativistic Quantum Mechanics of a finite system $S$ described by states in a Hilbert space $H$, if for instance $\psi_1,~ \psi_2 \in H$ are two possible orthogonal states of the system $S$, then further states of $S$ are given by the arbitrary linear combinations\
(1.1) $ \psi = c_1 \psi_1 + c_2 \psi_2,~~~ c_1, c_2 \in \mathbb{C} $\
where the usual normalizing conditions are assumed\
(1.2) $ || \psi_1 || = || \psi_2 || = 1,~~~ | c_1 |^2 + | c_2 |^2 = 1 $\
hence resulting as well in\
(1.3) $ || \psi || = 1 $\
So far, in no other theory of Physics is such a property present. As for the importance of that property in Quantum Mechanics it suffices to recall two facts : it leads to yet unsolved foundational controversies, as in the celebrated argument in Schrödinger’s Cat, and it is considered to be one of the basic resources of quantum computers, a resource which allows them unprecedented computational power, a power not possible to attain with usual electronic computers.\
\
[**2. Realms Physical, and Other Ones Less So ?**]{}\
It is nowadays a fundamental assumption that the Physical Realms do surely contain all there is, or at least, all there is of interest to Physics. And as with many a fundamental assumption, this one is so deeply ingrained that hardly anyone finds any reason at all to make it explicit to any extent.\
One of the amusing aspects of such an approach is the convenient [*circularity*]{} of the argument, a circularity which, however, does not seem to concern in the least its proponents ...\
Another amusing aspect is the recently emerging credo, according to which “information is physical” ...\
This credo does, of course, reflect an awareness that what earlier were perceived, mostly tacitly, as the possible boundaries of the Physical Realms should now be extended in order not to leave out such an entity of fast growing importance like information.\
And needless to say, such a move to encompass information within the Physical Realms is rather easy to accomplish, since the latter remains as undefined as it has always been ...\
Indeed, there is here a significant mismatch between the rather clear definition of information in present day science and technology, and on the other hand, the actual, and quite convenient vagueness of what the Physical Realms are supposed to be about. Not to mention that, in spite of the insistent propagation of that newly emerged credo, the concept of information is in fact treated as a second class one at best in most of the present day fundamental theories of Physics, including in Quantum Mechanics.\
On the other hand, and despite of the above, it is quite clear that the so called Physical Realms, even in their ever vague and latest extended sense, do [*not*]{} contain all that is of interest. And on top of it, they happen to [*fail*]{} to do so precisely on their own terms.\
Several such instances were discussed in Part I, and here we recall one of them :
> Anybody, and even more so a physicist, can at the same time think about two arbitrarily far away places in the universe, for instance, two galaxies.\
> On the other hand, according to Relativity Theory, no physical interaction can take place with arbitrary velocity.\
> Thus such a thinking, so easily and so commonly available to quite everybody, [*cannot*]{} be of a physical nature.
And then, the question arises :
> Where and how does such a thinking happen, if not within the Physical Realms, and definitely not there, in view of Physics itself ?
And while such a question remains unanswered, and in fact, not even considered by present day Physics, perhaps one may as well compound the issue with the following.\
It was Descartes in the early 1600s, who suggested the existence of two distinct realms. His “res extensa” was more or less what has been meant by the Physical Realms. On the other hand, his “res cogitans” was a realm beyond and outside of “res extensa”, and it encompassed thinking.\
As a consequence, Descartes has for long been ridiculed as being a dualist ...\
Such a judgment misses, however, the fundamental fact that, as so many major European scientists of his time, Descartes himself was a deeply religious person in the Christian tradition. Consequently, he could not possibly be less removed from dualism than anybody else, since he saw God as underlying all Creation, and thus in particular, both “res extensa” and “res cogitans”.\
Now of course, Descartes himself did not advocate the study of “res cogitans” by the means of Physics, whatever the latter may mean under reasonable conditions.\
And Classical Physics, that is, prior to the 20th century, did not in any way seem to require a more direct involvement of “res cogitans” than it would usually happen in the customary thinking process of normal humans, among them, physicists.\
Relativity Theory, in spite of the above question, has not changed that classical situation, and it did not appear to need to do so. What it does instead, and even if not yet seriously considered, is to point quite sharply to the existence of at least two very different realms. And for the lack of better terms, as well as a homage to Descartes, we can still call those two realms as “res extensa” and “res cogitans”, respectively.\
\
[**3. Does Superposition Need a Third Realm ?**]{}\
\
This may not be such an easy to answer question as one would like it. Indeed, Schrödinger’s Cat already shows that it is not trivial. Therefore, let us consider it with some care.\
What is obvious from (1.1) - (1.3) is that superposition takes place in the Hilbert space $H$, that is, within the mathematical model of the quantum system $S$. And as mathematical models go, they may hopefully reflect their respective system which, of course, is supposed to be situated in “res extensa”, but on the other hand, as mere models, are [*not*]{} supposed to be identical with such a system.\
This failure to distinguish between a physical system and its model is precisely one of the reasons one ends up with the controversy about Schrödinger’s Cat.\
And then, the question arises :
> Are superpositions (1.1) - (1.3) bona fide physical phenomena, or on the contrary, they are merely convenient features of the respective mathematical model ?
Well, as far as one can understand, this question does not have a clear enough answer in present day Quantum Mechanics.\
However, as it may happen with not a few physicists, in case one tends to consider superpositions as genuine physical phenomena, then the foundational controversy around Schrödinger’s Cat may simply be set aside by considering a [*third*]{} realm which we may call “res super-extensa”, and in which such superpositions take place. This realm contains the usual “res extensa”, in the sense that $\psi$ in (1.1), as a superposition of $\psi_1$ and $\psi_2$, belongs to it, without however belonging to “res extensa”, while $\psi_1$ and $\psi_2$ belong to the latter. Clearly, just as with “res extensa”, there is no need for any overlapping between “res super-extensa” and “res cogitans”.\
Here, however, one should note that the mathematical model (1.1) - (1.3), assumed to be in “res cogitans”, need not always distinguish between “res extensa” and “res super-extensa”. Indeed, $\psi$ in (1.1), as an element of the Hilbert space $H$, can in itself belong to “res extensa”, as long as it is not seen as being constituted as a superposition.\
The point to note with the above is that it is precisely the preference to see superpositions as physically real, that is, as having genuine physical existence, and not merely being representations in a mathematical model, which, when considered together with conundrums such as Schrödinger’s Cat, can suggest the consideration of a third realm, such as that of “res super-extensa”.\
\
[**4. And How About Entanglements ?**]{}\
As seen in Part I, entanglement also raises a question as to where and how it happens, given what appears to be its instantaneous nonlocal manifestation.\
And yet, it may appear that entanglement, even more than superposition, is seen by physicists as a genuine physical phenomenon, and not merely as some occurrence in the mathematical model.\
In this regard, no less than superpositions, entanglements are typical quantum phenomena, as well as unprecedented resources in quantum computation. As for their foundational importance, it suffices to recall the celebrated EPR paper, with all the related subsequent developments.\
Thus a fundamental and still controversial issue which entanglements bring up is that of [*nonlocality*]{}. This fact, as is well known, was brought forward most starkly with the celebrated Bell Inequalities.\
Here however, once one may consider the possibility of a third realm, like for instance, the above “res super-extensa”, which is in fact but a [*larger*]{} instance of the customary “res extensa”, the very issue of nonlocality may benefit from a new view and understanding.\
Indeed, it may simply happen that in “res super-extensa” the dichotomy “local - nonlocal” is meaningless.\
And here we should recall that such a possibility is not at all strange, since in a bounded system modelled mathematically by a compact space, the very concept of “nonlocal” loses much of its usual difficulties, if not in fact, its meaning. And in this regard we can recall that, so far, the very question whether the whole of the universe itself is in fact bounded is still open.\
But then, and as if to complicate the issues, entanglements need [*not*]{} necessarily happen in the same extension of “res extensa” in which superposition may happen. Consequently, we may yet have to consider another, namely, third physical type realm as well.\
A further possible consequence of considering physical extensions of the usual “res extensa” is that the foundational controversy related to the so called “hidden variables” in Quantum Mechanics may give way in favour of whole physical realms which, so far, were themselves hidden. In other words, it may well happen that what has been missing were not some hidden variables within this or that quantum entity, but rather whole physical type [*realms*]{} within which the very quantum processes as a whole may actually take place.\
And with the acceptance in String Theory of the fact that the so called Physical Realms may have highly counterintuitive large finite dimensions, some of them so contracted as to make the dichotomy “local - nonlocal” quite meaningless, there is no longer any particular reason to be so parsimonious when considering the possible realms, beyond the usual “res extensa”, that may be relevant to Physics.\
\
[**5. Conclusions**]{}\
Several extensions of what usually is meant by the otherwise undefined concept of Physical Realms were argued, based on rather obvious, simple, as well as fundamental physical considerations. In section 2, such an extension is motivated by the limitation of velocity of physical interactions, as follows from Relativity. In section 3, it was argued that, precisely to the extent that quantum superposition is not a mere feature of a mathematical model, but a genuine physical phenomenon, an extension of the customary concept of Physical Realms may be needed. In section 4, it was argued that quantum entanglement may need yet another such extension.\
And as suggested, such possible extensions of the concept of Physical Realms need not necessarily be given by one and the same additional realm.\
In case such a multiplicity of realms, beyond the two classical Cartesian ones, may raise certain concerns, one can always remember that, as thinking humans, thus in particular, physicists, our basic realm is in fact the “res cogitans”. No wonder that Descartes insisted on what he considered as the fundamental ontological fact for us humans, namely, “cogito, ergo sum” ...\
And therefore, without much further intellectual effort, we may at a certain stage subsume all other possible realms to that one. In other words, we may as well consider that everything is but a model, including what for so long we considered as having “objective” existence, whatever “objective” may happen to mean, namely, the Physical Realms.\
The only major difference such a subsummation may imply is that we should redefine accordingly what we mean by “experimental evidence”, and in particular, by “falsifiability”.\
[99]{}
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Auletta, G : Foundations and Interpretation of Qunatum Mechanics. World Scientific, Singapore, 2000
Dirac, P A M : Lectures on Quantum Mechanics. Dover, New York, 2001
Einstein, A : Relativity. Routledge, London, 2003
Greenstein G, Zajonc A G : The Quantum Challenge, Modern Research on the Foundations of Quantum Mechanics (second edition). Jones & Bartlett, Boston, 2006
Hirvensalo, M : Quantum Computing. Springer, New York, 2001
Isham, C J : Quantum Theory, Mathematical and Structural Foundations. Imperial College Press, London, 1997
Rosenblum B, Kuttner F : Quantum Enigma, Physics Encounters Consciuousness. Oxford Univ. Press, 2006
Rosinger E E : Mathematics and “The Trouble with Physics”, How Deep We Have to Go ? arXiv:0707.1163
Silagadze Z K : Realtivity without Tears. arXiv:0708.0929
| {
"pile_set_name": "ArXiv"
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---
abstract: 'We construct finite mass, asymptotically flat black hole solutions in $d=4$ Einstein–Yang-Mills theory augmented with higher order curvature terms of the gauge field. They possess non-Abelian hair in addition to Coulomb electric charge, and, below some non-zero critical temperature, they are thermodynamically preferred over the Reissner-Nordström solution. Our results indicate the existence of hairy non-Abelian black holes which are stable under linear, spherically symmetric perturbations.'
author:
- |
[Eugen Radu]{}$^{\dagger}$ and [D. H. Tchrakian]{}$^{\star \diamond }$\
$^{\dagger}$[Institut für Physik, Universität Oldenburg, D-26111 Oldenburg, Germany]{}\
$^{\star}$[Department of Computer Science, National University of Ireland Maynooth, Maynooth, Ireland]{}\
$^{\diamond}$[School of Theoretical Physics – DIAS, 10 Burlington Road, Dublin 4, Ireland ]{}
title: ' Stable black hole solutions with non-Abelian fields'
---
[** Introduction.– **]{} In recent years it has been realized that the electrically charged Reissner-Nordström (RN) black hole, when considered as solution of a more general theory, may become unstable to forming hair at low temperatures. This has lead to the discovery of some holographic models for condensed matter systems, and, in particular, to a gravitational description of superconductivity (see [@Horowitz:2010gk] for a review).
The case of Einstein-Yang-Mills (EYM) model with negative cosmological constant $\Lambda$ in $d=4$ spacetime dimensions provides an interesting illustration of these aspects. As shown in [@Gubser:2008zu], there is a second order phase transition between the RN–anti-de Sitter solutions, which are preferred at high temperatures, and symmetry breaking non-Abelian black holes, which are preferred at low temperatures. In [@Gubser:2008zu], $\Lambda$ plays an essential role; although electrically charged hairy black holes do exist also in a Minkowski spacetime background [@Galtsov:1991au], they have rather different properties as compared to the anti-de Sitter (AdS) solutions in [@Gubser:2008zu]. In particular they do not emerge as perturbation of the RN black holes, and, similar to the well-known $d=4$ asymptotically flat, purely magnetic EYM solutions [@Volkov:1998cc], are also perturbatively unstable.
However, one might take the view that in the strong coupling regime the ($\Lambda=0$) EYM theory is incomplete. Perhaps the simplest possibility to describe this situation is to supplement the action of the EYM model with higher order curvature terms, for both gravitational and gauge field sectors. As discussed $e.g.$ in [@Donets:1995ya], the inclusion of (string theory inspired-) corrections to the gravity action does not lead to qualitatively new features. By contrast, we will here argue that the situation is different [*vis a vis*]{} the inclusion of higher order Yang-Mills (YM) curvature terms. This possibility has been overlooked so far in the literature. The first relevant order in this case is the fourth, in which case the most general such density added to the Lagrangian consists of the four terms, $$\begin{aligned}
\label{Ls}
&&\mathcal{L}_s=
c_1 {\rm Tr}\left\{ F_{\mu\nu}F_{\rho\sigma} F^{\mu\nu}F^{\rho\sigma} \right \}
+
c_2 {\rm Tr}\left\{ F_{\mu\nu}F^{\mu\nu} F_{\rho\sigma}F^{\rho\sigma} \right \}
\\
&&{~~~~~~~~~}+
c_3 {\rm Tr}\left\{ F_{\tau\nu}F^{\mu\tau} F_{\mu\lambda}F^{\lambda\nu} \right \}
+c_4 {\rm Tr}\left\{ F_{\mu\nu}F^{\nu\rho} F_{\rho\lambda}F^{\lambda\mu} \right \},
\nonumber\end{aligned}$$ with some constant coefficients $c_i$. A particularly priviledged such combination, which we adopted here, is that with $c_1=c_2=-4 c_3,$ $c_4=0$. In that case, $\mathcal{L}_s$ features only the second power of any “velocity field” and is a causal density just like the Gauss-Bonnet term in gravity [@Zwiebach:1985uq] or the Skyrme [@Skyrme:1962vh] term of the $O(4)$ sigma model. With this specific choice of the constants $c_i$, the Lagrange density [(\[Ls\])]{} is nothing else than the trace of the square of the $4-$form curvature $F_{\mu\nu \rho \sigma}= \{F_{\mu [\nu},F_{\rho\sigma]} \}$. This is the second member of the YM hierarchy [@Tchrakian:1984gq], providing a natural generalization of the usual YM model. A convenient way to express this system is $ {\rm Tr}\left\{ (F_{\mu \nu} {}\tilde F^{\mu \nu})^2 \right \}$, where a tilde denotes the Hodge dual.
Notwithstanding our specific choice for the constants $c_i$ in (\[Ls\]), we have verified that for certain other choices, some salient features of the solutions discussed in this work, in particular the instability of the RN black hole, persist.
[** The model.– **]{} Ignoring for simplicity other possible corrections, we consider the following action for the model $$\begin{aligned}
\label{action}
S=\int d^4 x
\sqrt{-g}
\bigg [
\frac{1}{4}R-\frac{1}{2 }{\rm Tr}\left \{ F_{\mu\nu}F^{\mu\nu} \right\}
+\frac{3\tau}{2 }
{\rm Tr}\left\{ (F_{\mu \nu} {}\tilde F^{\mu \nu})^2 \right \}\bigg ],\end{aligned}$$ (here we have set $4\pi G/e^2=1$, such that the only parameter of the theory is $\tau$).
In what follows, we shall prove that the presence of the last term in (\[action\]) leads to an instability of the RN black hole, together with the occurance of stable black holes with non-Abelian hair outside the horizon. We shall restrict attention to the following spherically symmetric Ansatz: $$\begin{aligned}
\label{metric}
ds^2=\frac{dr^2}{N}+r^2(d\theta^2+\sin^2 \theta d\phi^2)-N\sigma^2dt^2,~\end{aligned}$$ where $N,\sigma$ are functions of $ r$ and $t$ in general. The minimal gauge group for which the superposition of a Coulomb field and a non-Abelian hair is not forbidden by the ’baldness’ theorems [@bald] is $SU(3)$. Then, as in the $\tau=0$ case in [@Galtsov:1991au], we shall restrict to an $SU(2)\times U(1)$ truncation of the $SU(3)$ group, the general spherically symmetric ansatz for the gauge potential being $$\begin{aligned}
\label{YMansatz}
A=
\bigg \{
(\nu T_3+U T_8) dr+
(w T_1+\tilde w T_2) d\theta
+\left( (w T_2-\tilde w T_1)\sin \theta +\cos \theta T_3 \right) d\phi
+(v T_3 +V T_8) dt
\bigg \}
,\end{aligned}$$ where $\nu,w,\tilde w,v$ and $U,V$ are functions of $(r,t)$ and $T_i$ are the standard generators of the $SU(3)$ Lie algebra.
For static solutions, one can set the functions $\nu,\tilde w, v$ and $U$ to zero without any loss of generality, resulting in the equations $$\begin{aligned}
\nonumber
&&m'=Nw'^2+\frac{(1-w^2)^2}{2r^2}+\frac{r^2 V'^2}{2\sigma^2}+\tau \frac{(1-w^2)^2V'^2}{r^2 \sigma^2},
~~~
\sigma'=\frac{2\sigma}{r}w'^2,
\\
\label{eqs}
&&w''+(\frac{N'}{N}+\frac{\sigma'}{\sigma})w'
+\frac{w(1-w^2)}{r^2 N}+\frac{2\tau (w^2-1)V'^2}{r^2N\sigma^2}=0,
\end{aligned}$$ together with the first integral for the electric potential, $$\begin{aligned}
\label{int-V}
V'=Q\frac{\sigma}{r^2}\left(1+\frac{2\tau(1-w^2)^2}{r^4}\right)^{-1},\end{aligned}$$ with $Q$ an arbitrary constant.
The RN solution corresponds to a vanishing SU(2) field, $w(r)=\pm 1$, and $m(r)=M-\frac{Q^2}{2r}$, $\sigma=1$, $V(r)=\Phi-Q/r$. This solution has an outer event horizon at $r_h=M+\sqrt{M^2-Q^2}$, which becomes extremal for $Q= M$.
Solutions with nonzero magnetic gauge fields should also exist. However, one can see that, for $Q\neq 0$, the first integral (\[int-V\]) excludes the existence of particle-like configurations with a regular origin. Thus, the only physically interesting solutions of this model describe black holes, with an event horizon at $r=r_h>0$, located at the largest root of $N(r_h)=0$. The regularity conditions at the horizon imply the following series expansion there $$\begin{aligned}
\nonumber
&&m(r)=\frac{r_h}{2}+m_1(r-r_h)+\dots,
~
\sigma(r)=\sigma_h+\frac{2 \sigma_h w_1^2}{r_h}(r-r_h)+\dots,
\\
\label{exp-eh}
&&w(r)=w_h+w_1(r-r_h)+\dots,~
V(r)=v_1(r-r_h)+\dots,\end{aligned}$$ where $
v_1=\frac{Q r_h^2 \sigma_h}{r_h^4+2\tau (1-w_h^2)^2},
$ $
m_1=\frac{ \sigma_h^2(1-w_h^2)^2+v_1^2(r_h^4+2\tau (1-w_h^2)^2)}{2r_h \sigma_h^2}
$, $
w_1= \frac{(\sigma_h^2-2\tau v_1^2)w_h(w_h^2-1)}
{(1-2 m_1)r_h\sigma_h^2}.
$ It is also straightforward to show that the requirement of finite energy implies the following asymptotic behavior at large $r$ $$\begin{aligned}
\label{exp-inf}
m(r)=M-\frac{Q^2}{2r} +\dots,
~~
\sigma(r)=1-\frac{ J^2}{2r^4}+\dots,
~~ w(r)=\pm 1+\frac{ J}{r}+\dots,~~V(r)=\Phi-\frac{Q}{r}+\dots~.\end{aligned}$$ Once the parameters $\sigma_h,~w_h$ and $J,~M,~Q$ are specified, all other coefficients in (\[exp-eh\]), (\[exp-inf\]) can be computed order by order. $M$ and $Q$ correspond to the mass and electric charge of the solutions; other quantities of interest are the Hawking temperature $T_H= \frac{1}{4 \pi} \sigma(r_h) N'(r_h)$, the entropy $S=\frac{A_H}{4}={\pi r_h^2}$ and the chemical potential $\Phi$. $J$ here is an order parameter describing the deviation from the Abelian solution.
[** The results.– **]{} The solutions of the equations (\[eqs\]), (\[int-V\]) interpolating between the asymptotics (\[exp-eh\]), (\[exp-inf\]) were constructed numerically, by employing a shooting strategy. Finite mass, non-Abelian black holes exist for any $\tau \geq 0$. For given $Q,r_h,\tau$, the solutions are found for discrete values of the parameter $w_h$, labeled by the number of nodes, $n$, of the magnetic YM potential $w(r)$. To simplify the picture, in this work we have restricted attention to solutions with a monotonic behavior of the magnetic gauge potential $w(r)$ (and thus $n=0,1$ only).
In characterizing the non-Abelian configurations, it is convenient to introduce the quantity $$\begin{aligned}
\label{kappa}
\kappa=\frac{\tau}{Q^2}.\end{aligned}$$ For $0\leq \kappa< 1/2$, the solutions can be thought of as nonlinear superpositions of the RN and the purely magnetic SU(2) black holes in [@89]. In particular, they are unstable since the magnetic gauge potential has always $n\geq 1$ nodes. Moreover, their free energy $F=M-T_H S$ is always greater than that of the RN configuration with the same $T_H$ and $Q$.
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The picture is very different for $\kappa\geq 1/2$. In this case, for given $\kappa$, we notice the existence of a set of non-Abelian solutions emerging as perturbations of the RN black holes, for a critical value of the charge to mass ratio. This instability of the Abelian configuration is found within the Ansatz (\[YMansatz\]), for values of the magnetic gauge potential $w(r)$ close to the vacuum values $ \pm 1$ everywhere, $w(r)= \pm 1+ \epsilon W(r)$. The perturbation $W(r)$ starts from some nonzero value at the horizon and vanishes at infinity, being a solution of the linear equation $$\begin{aligned}
\label{stab1}
(N W')'-(1-\frac{2\tau Q^2}{r^4})\frac{2W}{r^2}=0~,\end{aligned}$$ where $N=(1-\frac{r_h}{r})(1-\frac{Q^2}{r_h r})$. The second term in this equation can be seen as an effective mass term $\mu^2$ for $W$ near the horizon, with $\mu^2\sim 1-2\kappa (Q/r_h)^4$. We have found that for any $\kappa\geq 1/2$ ($i.e.$ $\mu^2<0$), an instability occurs for a critical value of the mass to charge ratio of the RN solution. An approximate value of this ratio is found by using an asymptotic matching expansion for the approximate solutions of (\[stab1\]) at the horizon and at infinity, the result $$\begin{aligned}
\label{QM}
\frac{Q}{M}=\frac{2\sqrt{6}\sqrt{1+\sqrt{1+48\kappa}}}{7+\sqrt{1+48\kappa}} \end{aligned}$$ providing good agreement with the numerical data.
This unstability signals the emergence of a symmetry breaking branch of the non-Abelian solution bifurcating from the RN black holes. In contrast to the solutions with $\kappa <1/2$, here we notice the existence of a fundamental branch of solutions without nodes in the magnetic gauge function $w(r)$, see $e.g.$ the typical profile shown in Fig. 1.
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=6.5cm
Among the solutions bifurcating from RN black holes, those with $1/2 \leq \kappa \leq 2$ are of special interest, sharing some striking similarity with the picture found for the EYM-AdS system in [@Gubser:2008zu]. A plot of the horizon area as a function of the temperature reveals the existence of a single branch of non-Abelian solutions, which exist below a critical temperature $T_H^{(c)}$ only (the range of the scaled temperature $t_H=T_H Q$ being $0<t_H<0.0203$). In a canonical ensemble, the non-Abelian black holes within this range of $\kappa$ exist for a finite interval of $r_h$ ($i.e.$ of the entropy) only. These solutions always possess a positive specific heat, the Hawking temperature vanishing for a minimal value $r_h^{(min)}= \sqrt{2\tau}\sqrt{\sqrt{2\kappa}-1}$ of the event horizon radius. As $r\to r_h^{(min)}$, an extremal non-Abelian black hole solution with a regular horizon is approached, the charge to mass ratio of this configurations being always greater than one. Furthermore, it turns out that the free energy of a RN solution is always larger than the free energy of a non-Abelian solution with the same temperature and electric charge. Therefore, for $1/2 \leq \kappa \leq 2$ these non-Abelian black holes are preferred. The difference of free energies scales like $(T_H^{(c)}-T_H)^2$ near the transition point, signaling that this is a second order phase transition, while $J \sim \sqrt{1-T_H/T_H^{(c)}}$. Moreover, for the same values of the mass and electric charge, the RN solution has a smaller event horizon radius (and thus a smaller entropy), than the non-Abelian black hole.
The picture is somehow different for $\kappa>2$. Again, one finds a single branch of solutions emerging as a perturbation of the RN black hole, nodeless configurations existing also in this case. However, these solutions exist for values of the temperature greater than the critical value $T_H^{(c)}$ and are thermally unstable.
Some of these features are shown in Fig. 2 where we plot the scaled horizon area $a_H=A_H/Q^2$ as a function of the scaled temperature $t_H= T_H Q$ for several values of the ratio $\tau/Q^2$. The branch of RN solutions is also shown there. In Fig. 3 the scaled free energy $f=F/Q$ is plotted as a function of the scaled temperature $t_H$ for several values of $\kappa$. The inlet there shows the behavior of the parameter $J$ which enters the asymptotic behavior of the magnetic potential, as a function of the ratio $T_H/T_H^{(c)}$.
[** Existence of stable solutions.– **]{} An outstanding question now is whether the $F^4$ term leads also to stable non-Abelian black holes. The fact that, for a range of $\kappa$, we have found nodeless solutions which are thermodynamically favored over the RN black holes suggests a positive answer to this question.
For simplicity, we consider linear, spherically symmetric perturbations only. Even in this case, the analysis is highly involved and the details will be presented elsewhere. Here we briefly outline just the key features.
As usual, all field variables are written as the sum of the static equilibrium solution whose stability we are investigating and a time dependent perturbation. Choosing a gauge such that $U=v=0$, one finds that the fluctuations decouple in two groups. $\delta w(r,t)$, $\delta V(r,t)$, $\delta \sigma(r,t)$ and $\delta m(r,t)$ form even-parity perturbations, whereas $\delta \tilde w(r,t)$ and $\delta \nu(r,t)$ form odd-parity perturbations. The linearized equations imply that $\delta \sigma(r,t)$, $\delta V(r,t)$ and $\delta m(r,t)$ are determined by $\delta w(r,t)=w_1(r)e^{-i\Omega t}$, leading to a single Schrödinger equation $$\begin{aligned}
\label{schr-even}
-\frac{d^2 w_1}{d \rho^2}+U_{even}(\rho)w_1=\Omega^2 w_1,\end{aligned}$$ (where $dr/d \rho=N \sigma $) with a potential $$\begin{aligned}
\label{pot-even}
&&
U_{even}= \frac{N\sigma }{r^2}
\Bigg [
\left (1-\frac{2\tau V'^2}{\sigma^2} \right)
\left (3w^2-1+\frac{8ww'}{r}(w^2-1)-\frac{4w'^2(1-w^2)^2}{r^2} \right)
\\
\nonumber
&&
+4w'^2\left(\frac{2(1-w^2)^2}{r^2}+\frac{r^2V'^2}{\sigma^2}-1\right)
+\frac{32\tau^2w^2(1-w^2)^2V'^2}{\sigma^2 r^4(1+\frac{2\tau}{r^4}(1-w^2)^2)}
\Bigg ],\end{aligned}$$ which is a regular function in the entire range $-\infty<\rho<\infty$. The corresponding analysis for the odd sector is much more evolved. After much algebra, the perturbation equations for $\delta \tilde w(r,t)=\tilde w_1(r) e^{-i\Omega t}$ and $\delta \nu(r,t)= \nu_1(r) e^{-i\Omega t}$ can be cast in the form $$\begin{aligned}
\label{schr-odd}
-\frac{d^2 \Psi}{d \rho^{*2}}+U_{odd}(\rho^*)\Psi= \Omega^2 \Psi,\end{aligned}$$ where $\Psi=[(1+6\tau (1-w^2)^2/r^4)\nu_1-12 \tau(w^2-1)w'/r^4 \tilde w_1] F_1$, $dr/d \rho^*=N \sigma/(1+12 \tau Nw'^2/(r^2(1+6\tau (1-w^2)^2/r^4)))^{1/2} $ and $\nu_1(r)=F_2 \Psi+F_3 \Psi'$. The potential $U_{odd}$ and the functions $F_i$ have complicated expressions depending on the equilibrium functions $m,\sigma,w$ and $V$, with $F_1>0$. For nodeless solutions, $U_{odd}$ and $F_i$ are regular in the entire range $-\infty<\rho^*<\infty$.
For both equations (\[schr-even\]) and (\[schr-odd\]), the potential vanishes near the horizon and at infinity. Then standard results [@Messiah] imply that there are no negative eigenvalues for $\Omega^2$ (and hence no unstable modes) if the potentials $U_{even}$ and $U_{odd}$ are everywhere positive.
Our results indicate that this is indeed the case for some of the solutions with $\kappa >1/2$, see $e.g.$ the inlet in Fig. 1. Interestingly, approaching the extremality appears to imply generically positive values for the potentials in (\[schr-even\]), (\[schr-odd\]).
Therefore we conclude that, in contrast to all other known $d=4$ asymptotically flat hairy black holes with non-Abelian gauge fields only [@Volkov:1998cc], at least some of our solutions here are linearly stable.
[** Further remarks.– **]{} We close with some remarks on the generality of the results in this work. First, we remark that the $F^2$ and $F^4$ terms in (\[action\]) are the pieces which also enter the Lagrangian of the non-Abelian Born-Infeld theory [@Tseytlin:1997csa] describing the low energy dynamics of $D-$branes. Although the equations of motion of that model coupled with gravity are more complicated and do not admit the RN black hole as a solution, it is natural to expect that the picture we have found here will share some similarities with the results in that case.
Non-Abelian fields featuring both the $F^2$ and $F^4$ terms appear also in the higher loop corrections to the action of the $d=10$ heterotic string [@Polchinski:1998rr]. However, a generalization in that framework of the solutions considered here is not an easy task, due to the occurrence there of a variety of other fields. Previous work in this direction [@Donets:1995ya] indicates that the solutions with a standard $F^2$ term only, share the basic features of the EYM black holes in [@89], in particular being unstable. It appears however, that inclusion of higher order gauge field curvature terms could lead to a very different situation (for example, we have found rather similar results to those discussed above, for a generalization of (\[action\]) with an extra dilaton field, a Gauss-Bonnet term and gauge group $SO(5)$). Thus we expect the following picture to be generic: the $F^4$ term introduces a supplementary interaction between electric and magnetic fields, which, for some range of the parameters, implies a tachyonic mass for the vacuum perturbations of the non-Abelian magnetic fields around the Abelian solutions. Then the Abelian gauge symmetry is spontaneously broken near a black hole horizon for some critical value of the charge to mass ratio, with the appearance of a condensate of magnetic non-Abelian gauge fields there. The possible relevance of these aspects in providing analogies to phenomena observed in condensed matter physics is yet to be explored.\
\
[*[ Acknowledgements.]{}*]{} This work is carried out in the framework of Science Foundation Ireland (SFI) project RFP07-330PHY. E.R. gratefully acknowledges support by the DFG.
[99]{}
G. T. Horowitz, arXiv:1002.1722 \[hep-th\]. S. S. Gubser, Phys. Rev. Lett. [**101**]{} (2008) 191601 \[arXiv:0803.3483 \[hep-th\]\]. D. V. Galtsov and M. S. Volkov, Phys. Lett. B [**274**]{} (1992) 173. M. S. Volkov and D. V. Gal’tsov, Phys. Rept. [**319**]{} (1999) 1 \[arXiv:hep-th/9810070\]. E. E. Donets and D. V. Gal’tsov, Phys. Lett. B [**352**]{} (1995) 261 \[arXiv:hep-th/9503092\];\
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P. Bizon, Phys. Rev. Lett. [**64**]{} (1990) 2844. A. Messiah, [*Quantum Mechanics*]{}, North-Holland, Amsterdam (1962). A. A. Tseytlin, Nucl. Phys. B [**501**]{} (1997) 41 \[arXiv:hep-th/9701125\]. J. Polchinski, [*‘String theory. Vol. 2: Superstring theory and beyond’*]{}, Cambridge, Cambridge University Press, (1998).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the ionic Hubbard model on a triangular lattice at three-quarters filling. This model displays a subtle interplay between metallic and insulating phases and between charge and magnetic order. We find crossovers between Mott, charge transfer and covalent insulators and magnetic order with large moments that persist even when the charge transfer is weak. We discuss our findings in the context of recent experiments on the layered cobaltates A$_{0.5}$CoO$_2$ (A=K, Na).'
author:
- Jaime Merino
- 'B. J. Powell'
- 'Ross H. McKenzie'
title: ' Interplay of frustration, magnetism, charge ordering, and covalency in the ionic Hubbard model on the triangular lattice at three-quarters filling'
---
The competition between metallic and insulating states in strongly correlated materials leads to many novel behaviours. The Mott insulator occurs when a single band is half-filled and the on-site Coulomb repulsion, $U$, is much larger than hopping integral, $t$. A menagerie of strongly correlated states is found when a system is driven away from the Mott insulating state, either by doping, as in the cuprates [@Anderson-RVB], or reducing $U/t$, as in the organics [@organics-review]. Geometric frustration causes yet more novel physics in Mott systems [@organics-review]. Therefore the observation of strongly correlated phases in the triangular lattice compounds A$_{0.5}$CoO$_2$, where $A$ is K or Na [@ong-cava-science], has created intense interest.
An important model for investigating insulating states in correlated materials is the ionic Hubbard model. On a half-filled square lattice this model displays a crossover between Mott and band insulating states which has been analyzed with quantum Monte Carlo (QMC) [@bouadim], dynamical mean field theory (DMFT) and its cluster extensions [@kancharla]. However, except for the case of one dimension [@penc2], this model has not been studied away from half-filling [@penc1] and/or on geometrically frustrated lattices.
In this Letter we study the ionic Hubbard model on a triangular lattice at three-quarter filling. This Hamiltonian displays a subtle interplay between metallic and insulating phases and charge and magnetic order. It has regimes analogous to Mott, charge transfer [@Zaanen], and covalent insulators [@Sarma]. The study of this model is motivated in part by our recent proposal [@MPM] that it is an effective low-energy Hamiltonian for , at values of $x$ at which ordering of the sodium ions occurs.
The Hamiltonian for the ionic Hubbard model is $$H=-t\sum_{\langle ij\rangle\sigma} c^\dagger_{i\sigma} c_{j\sigma} +U \sum_i n_{i \uparrow} n_{i \downarrow}
+\sum_{i\sigma} \epsilon_i n_{i \sigma}, \label{ham}
\label{model}$$ where $c^{(\dagger)}_{i\sigma}$ anihilates (creates an electron with spin $\sigma$ at site $i$, $t$ is the hopping integral, $U$ is the effective Coulomb repulsion between electrons on the same site, and $\epsilon_i$ is a the site energy. We specialise to the case with two sublattices, A ($\epsilon_i=\Delta/2$) and B ($\epsilon_i=-\Delta/2$), consisting of alternating rows, with different site energies on the two sublattices (c.f., Fig. 15 of Ref. ). This is the lattice relevant to where the difference in site energies results from the ordering of the A-atoms [@williams-argyriou; @Na_ordering; @Na-expt].
Two limits of model (\[model\]) at $3/4$-filling may be easily understood. For non-interacting electrons, $U=0$, a metallic state occurs for all $\Delta$ as at least one-band crosses the Fermi energy. In the atomic limit $t=0$, and $U>\Delta$ one expects a charge transfer insulator with a charge gap of about $\Delta$ whereas for $U<\Delta$ a Mott insulator with charge gap of $U$ occurs. However, realistic parametrization of materials imply $U\gg\Delta$ and $\Delta \sim |t|$ [@foot-parms]; we will show below that in this parameter regime the model show very different behaviour from either of the limits discussed above. This interesting regime needs to be analyzed using non-perturbative and/or numerical techniques. Thus, we have performed Lanczos diagonalization calculations on 18 site clusters with periodic boundary conditions.
In Fig. \[fig:nanb\] we plot the charge transfer, $n_B-n_A$ as a function of $\Delta/|t|$ for several values of $U$. We also plot $n_B-n_A$ in two analytically tractable limits: the non-interacting limit, $U=0$ [@footnon]; and the strong coupling limit $U\gg\Delta\gg|t|$ [@footstrong]. Several interesting effects can be observed in this calculation. Firstly, the sign of $t$ strongly effects the degree of charge transfer on the triangular lattice. Secondly, charge transfer depends only weakly on $U$. Thirdly, regardless of the sign of $t$ or the magnitude of $U$, the charge transfer increases rather slowly as $\Delta$ increases.
\
The charge gap, i.e., the difference in the chemical potentials for electrons and holes, is $\Delta_c \equiv
E_0(N+1)+E_0(N-1)-2E_0(N)$, where $E_0(N)$ is the ground state energy for $N$ electrons. We plot the variation of $\Delta_c$ with $\Delta$ for various values of $U$ in Fig. \[fig:gap\]. $\Delta_c$ vanishes for $U=0$, however finite size effects mean that we cannot accurately calculate $\Delta_c$ for small $\Delta$. $\Delta_c =\Delta$ for $t=0$ and $U \gg
\Delta$; this result is reminiscent of a charge transfer insulator [@Zaanen]. Both perturbative [@footstronggap] and numerical results show that the charge gap depends on the sign of $t$ due to the different magnetic and electronic properties arising from the geometrical frustration of the triangular lattice. In contrast, on a square lattice, $\Delta_c$, does not depend on the sign of $t$.
\
In the limit, $\Delta\gg U\gg|t|$, the A and B sublattices are well separated in energy; the B sites are doubly occupied (i.e., the B-sublattice is a band insulator) and the A sublattice is half-filled and hence becomes a Mott insulator. If there were no hybridisation between that chains, one would find a metallic state for any finite charge transfer from the B-sites to the A-sites (self doping), even for $U\gg |t|$ as the A-chains are now electron-doped Mott insulators and the B-chains are hole-doped band insulators. However, Fig. \[fig:gap\] shows that the insulating regime of the model extends far beyond the well understood $n_B-n_A=1$ regime. This is because the real space interpretation is incorrect as hybridization between A and B chains is substantial. For $|t|\sim\Delta\ll U$ the system can remain insulating with a small gap \[${\cal O}(t)$\]. This state is analogous to a covalent insulator [@Sarma].
One expects that for $\Delta=0$ the ground state is metallic as there the system is $3/4$-filled. However, a small but finite $\Delta=0^+$ leads to a strongly nested Fermi surface for $t>0$ whereas for $t<0$ the Fermi surface rather featureless. Thus, rather different behaviors might be expected for different signs of $t$ even at weak coupling. At large $U$ our exact diagonalization results suggest that a gap may be present even for a small value of $\Delta/t$. However finite-size effects, inherent to the method, mean that it is not possible to resolve whether a gap opens at $\Delta=0$ or at some finite value of $\Delta$.
To test this covalent insulator interpretation in the $\Delta\sim|t|$ and large $U$ regime we have also calculated the spectral density, $A(\omega)$, c.f., Fig. \[fig:dos\]. There are three distinct contributions to the $A(\omega)$: at low energies there is a lower Hubbard band; just below the chemical potential ($\omega=\mu$) is a weakly correlated band; and just above $\omega=\mu$ is the upper Hubbard band. Furthermore, the large energy separation, much larger than the expected $U=15|t|$, between the lower and upper Hubbard bands is due to an upward (downward) shift of the upper (lower) Hubbard bands due to the strong hybridization. In contrast, in the strong coupling limit $A(\omega)$ has a much larger gap, ${\cal O}(\Delta)$, between the contributions from the weakly correlated band and the upper Hubbard band.
The magnetic moment associated with the possible antiferromagnetism, $m_ \nu =(3 \langle S^z_{i} S^z_{j} \rangle)^{1/2}$, where $\nu =A$ or $B$ and $S^z_{i}={1 \over 2} (n_{i \uparrow} -n_{i
\downarrow})$, is evaluated between two next-nearest neighbors on the $\nu$ sublattice at the center of cluster (to reduce finite size effects [@sandvik]). Fig. \[fig4\] shows that $m_A$ increases with $\Delta$ and is substantially enhanced by $U$, whereas $m_B$ is always small. This is in marked contrast to a spin density wave, as predicted by Hartree-Fock calculations where the magnetic moment is far smaller than that experimentally observed [@RPA].
We now turn to discuss the consequences of our results for understanding experiments; for simplicity and concreteness we focus on . The $x=0.5$ materials have remarkably different properties from those on other values of $x$ [@foo; @wang]. Above 51 K the intralayer resistivity of is weakly temperature dependent with values of a few m$\Omega$cm [@foo] characteristic of a bad metal [@MPM]. Below 51 K the resistivity increases, consistent with a small gap opening ($\sim$10 meV) [@foo]. Thus a (bad) metal-insulator transition occurs at 51 K. The insulating state of has a number of counterintuitive properties, not the least of which is the absence of strong charge ordering. NMR observes no charge ordering up to a resolution of $n_B-n_A<0.4$ [@bobroff; @nmr], while neutron crystallography suggest $n_B-n_A\simeq0.12$ [@williams-argyriou]. Thus the insulating state is not the simple charge-transfer-like state predicted by (\[model\]) in the strong coupling limit. develops a commensurate magnetic order below 88 K [@bobroff; @gasparovic-yokoi]. A large magnetic moment \[$m=0.26(2)\mu_B$ per magnetic Co ion\] is observed in spite of the weak charge order \[note that classically $m<(n_B-n_A)\mu_B/2$\]. Above 100 K the optical conductivity [@wang] shows no evidence of a Drude peak, consistent with a bad metal. In the insulating phase spectral weight is lost below $\sim$10 meV, consistent with the gap seen in the dc conductivity and a peak emerges at $\sim$20 meV, which is too sharp and too low energy to correspond to a Hubbard band [@wang]. ARPES shows that the highest energy occupied states are $\sim$10 meV below the Fermi energy [@qian]. No equivalent insulating state is seen in the misfit cobaltates [@bobroff], which supports the contention that Na-ordering is vital for understanding the insulating state.
Various theories have been proposed to explain these intriguing experiments. Lee [[*et al*]{}. ]{}[@Lee] have performed LDA+U calculations, which include Na-ordering, but not strong correlations. Other groups [@theory] have studied strongly correlated models that include the Coulomb interaction with neighbouring sites, but neglect the effects of Na-ordering. Marianetti and Kotliar [@Marianetti] have also studied the Hamiltonians proposed in [@MPM] for $x=0.3$ and 0.7.
In order to compare our results with experiments on we need to restrict ourselves to the relevant parameter values: $t<0$ and $|t|\sim\Delta\ll U$ [@foot-parms]. This corresponds with the regime of the three quarters filled ionic Hubbard model that is both the most interesting and the most difficult to study via exact diagonalisation because of the deleterious finite size effects. Nevertheless we propose that in the insulating state is analogous to a covalent insulator. This explains a wide range of experiments. The peak observed at $\omega \sim30$ meV in the optical conductivity [@wang], is interpreted as the transfer of an electron from the weakly correlated band to form a doublon in the strongly correlated band. The weak charge transfer ($n_B-n_A=0.1-0.3$; c.f., Fig. \[fig:nanb\]) is caused by the strong hybridisation between the A and B sublattices and is consistent with the value extracted from crystallographic experiments (0.12 [@williams-argyriou]) and the bounds from NMR ($<$0.4 [@bobroff]). The large moment (0.1-0.2$\mu_B$; c.f., Fig. \[fig4\]) is comparable to the moment found by neutron scattering ($0.26\mu_B$ [@gasparovic-yokoi]) and results from the electrons in the strongly correlated band, i.e., the single spin hybridised between the A and B sublattices. Finite size effects mean that we cannot accurately calculate the charge gap in this regime. However, we propose that the experimental system corresponds to a parameter range where the gap is small, $\Delta_c<{\cal O}(|t|)$, consistent with the gap, $\sim$7-10 meV [@foo; @qian], seen experimentally in ARPES and resistivity. This is consistent with the expectation that $\Delta_c\rightarrow0$ as $\Delta/|t|\rightarrow0$. Accurately calculating $\Delta_c$ for small $\Delta/|t|$ and large $U$, and hence further testing our hypothesis, therefore remains an important theoretical challenge.
We thank H. Alloul, Y.S. Lee, and R. Singh for helpful discussions. J.M. acknowledges financial support from the Ramón y Cajal program, MEC (CTQ2005-09385-C03-03). B.J.P. was the recipient of an ARC Queen Elizabeth II Fellowship (DP0878523). R.H.M. was the recipient of an ARC Professorial Fellowship (DP0877875). Some of the numerics were performed on the APAC national facility.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss the underlying relativistic physics which causes neutron stars to compress and collapse in close binary systems as has recently been observed in numerical (3+1) dimensional general relativistic hydrodynamic simulations. We show that compression is driven by velocity-dependent relativistic hydrodynamic terms which increase the self gravity of the stars. They also produce fluid motion with respect to the corotating frame of the binary. We present numerical and analytic results which confirm that such terms are insignificant for uniform translation or when the hydrodynamics is constrained to rigid corotation. However, when the hydrodynamics is unconstrained, the neutron star fluid relaxes to a compressed nonsynchronized state of almost no net intrinsic spin with respect to a distant observer. We also show that tidal decompression effects are much less than the velocity-dependent compression terms. We discuss why several recent attempts to analyze this effect with constrained hydrodynamics or an analysis of tidal forces do not observe compression. We argue that an independent test of this must include unconstrained relativistic hydrodynamics to sufficiently high order that all relevant velocity-dependent terms and their possible cancellations are included.'
address:
- ' University of Notre Dame, Department of Physics, Notre Dame, IN 46556'
- ' University of California, Lawrence Livermore National Laboratory, Livermore, CA 94550'
- ' University of Notre Dame, Department of Physics, Notre Dame, IN 46556'
author:
- 'G. J. Mathews and P. Marronetti'
- 'J. R. Wilson'
title: 'RELATIVISTIC HYDRODYNAMICS IN CLOSE BINARY SYSTEMS: ANALYSIS OF NEUTRON-STAR COLLAPSE'
---
INTRODUCTION {#sec:level1}
============
The physical processes occurring during the last orbits of a neutron-star binary are currently a subject of intense interest [@wm95]-[@sbs98]. In part, this recent surge in interest stems from relativistic numerical hydrodynamic simulations in which it has been noted [@wm95; @wmm96; @mw97] that as the stars approach each other their interior density increases. Indeed, for an appropriate equation of state, our numerical simulations indicate that binary neutron stars collapse individually toward black holes many seconds prior to merger. This compression effect would have a significant impact on the anticipated gravity-wave signal from merging neutron stars. It could also provide an energy source for cosmological gamma-ray bursts [@mw97].
In view of the unexpected nature of this neutron star compression effect and its possible repercussions, as well as the extreme complexity of strong field general relativistic hydrodynamics, it is of course imperative that there be an independent confirmation of the existence of neutron star compression before one can be convinced of its operation in binary systems. In view of this it is of concern that the initial numerical results reported in [@wm95; @wmm96; @mw97] have been called into question. A number of recent papers [@lai; @rs96; @wiseman; @shibata; @lombardi; @lw96; @brady; @flanagan; @thorne97; @baumgarte; @sbs98] have not observed this effect in Newtonian tidal forces [@lai], first post-Newtonian (1PN) dynamics [@rs96; @wiseman; @shibata; @lombardi; @lw96; @sbs98], tidal expansions [@brady; @flanagan; @thorne97], or in binaries in which rigid corotation has been imposed [@baumgarte]. The purpose of this paper is to point out that none of these recent studies could or should have observed the compression effect which we observe in our calculations.
Moreover, this flurry of activity has caused some confusion as to the physics to which we attribute the effects observed in the numerical calculations. The present paper, therefore, summarizes our derivation of the physics which drives the collapse. We illustrate how such terms have been absent in some Newtonian or post-Newtonian approximations to the dynamics of the binary system. We also present numerical results and analytic expressions which demonstrate how the compression forces result in an orbiting dynamical system from the presence of fluid motion with respect to the corotating frame. As such, they could not appear in an an analysis of relativistic external tidal forces no matter how many orders are included in the tidal expansion parameter (e.g. [@flanagan; @thorne97]) unless self gravity from internal hydrodynamic motion is explicitly accounted for. The effect could also not arise in systems with uniform translation or rigid corotation.
The implication of the present study is that any attempt to confirm or deny the compression driving force requires an unconstrained, untruncated relativistic hydrodynamic treatment. At present, ours is still the only existing such calculation. Hence, despite claims to the contrary [@lai; @rs96; @wiseman; @shibata; @lombardi; @lw96; @brady; @flanagan; @thorne97; @baumgarte; @sbs98], the neutron star compression effect has not yet been independently tested.
Another confusing aspect surrounding the numerical results has been our choice of a conformally flat spatial three-metric for the solution of the field equations. Indeed, it has been speculated that this approximate gauge choice (in which the gravitational radiation is not explicitly manifest) may have somehow led to spurious results. A second purpose of this paper, therefore, is to emphasize that the compression driving terms are a completely general result from the relativistic hydrodynamic equations of motion. The advantages of the conformally flat condition are that the algebraic form of the compression driving terms is easier to identify and that the solutions to the field equations obtain a simple form. It does not appear to be the case, however, that the imposition of a conformally flat metric drives the compression. It has been nicely demonstrated in the work of Baumgarte et al. [@baumgarte] that conformal flatness does not necessarily lead to neutron-star compression.
The Spatially Conformally Flat Condition
========================================
There has been some confusion in the literature as to the uncertainties introduced by imposing a conformally flat condition (henceforth [*CFC*]{}) on the spatial three-metric. Therefore we summarize here some attempts which we and others have made to quantify the nature of this approximation.
The only existing strong field numerical relativistic hydrodynamics results in three unrestricted spatial dimensions to date have been derived in the context of the [*CFC*]{} as described in detail in [@wm95; @wmm96; @mw97].
We begin with the usual ADM (3+1) metric [@adm62; @york79] in which there is a slicing of the spacetime into a one-parameter family of three-dimensional hypersurfaces separated by differential displacements in a timelike coordinate, $$ds^2 = -(\alpha^2 - \beta_i\beta^i) dt^2 +
2 \beta_i dx^i dt + \gamma_{ij}dx^i dx^j~~,
\label{metric}$$ where we take Latin indices to run over spatial coordinates and Greek indices to run over four coordinates. We also utilize geometrized units ($G = c = 1$) unless otherwise noted. The scalar $\alpha$ is called the lapse function, $\beta_i$ is the shift vector, and $\gamma_{i j}$ is the spatial three metric.
In what follows, we make use of the general relation between the determinant of the four metric $g_{\alpha \beta}$ and the ADM metric coefficients $$det(g_{\alpha \beta} ) = - \alpha^2 det({\gamma_{i j}}) \equiv \alpha^2 \gamma^2~~,$$ where $\gamma \equiv \sqrt{-det(\gamma_{i j})}$.
The conformally flat metric condition simply expresses the three metric of Eq. (\[metric\]) as a position dependent conformal factor $\phi^4$ times a flat-space Kronecker delta $$\gamma_{i j} = \phi^4 \delta_{ij}~~.$$
It is common practice (e.g. [@evans85; @cook93; @brugmann97]) to impose this condition when solving the initial value problem in numerical relativity. It is the natural choice for our three-dimensional quasiequilibrium orbit calculations [@wmm96] which in essence seek to identify a sequence of initial data configurations for neutron-star binaries.
The reason conformal flatness is chosen most frequently for the initial value problem is that it simplifies the solution of the hydrodynamics and field equations. The six independent components of the three metric are reduced to a single position dependent conformal factor.
Since conformal flatness implies no transverse traceless part of $\gamma_{i j}$ it can minimize the amount of initial gravitational radiation apparent in the initial configuration. However, in general the physical data still contain a small amount of preexisting gravitational radiation. This has been clearly demonstrated in numerical calculations of axisymmetric black-hole collisions [@smarr]. In exact numerical simulations, the gravitational radiation appears as the time derivatives of the spatial three metric ($\dot \gamma_{i j}$) and its conjugate (the extrinsic curvature $\dot K_{i j}$) are evolved. The immediate evolution of the fields from conformally flat initial data is characterized by the development of a weak gravity wave exiting the system.
An estimate of the radiation content of initial data slices for axisymmetric black hole collisions has been made by Abrahams [@abrahams]. Even for high values of momentum, the initial slice radiation is always less than about 10% of the maximum possible radiation energy (as estimated from the area theorem).
Two questions then are relevant to our application of the [*CFC*]{}. One is the validity of this metric choice for the initial value problem, and the other is the effect on the system of the “hidden” gravitational radiation in the physical data.
Regarding the validity of the [*CFC*]{} one has a great deal of freedom in choosing coordinates and initial conditions as long as the initial space is Riemannian and the metric coefficients satisfy the constraint equations of general relativity [@mtw]. Indeed, we have shown in [@wmm96] that exact solutions for the [*CFC*]{} metric coefficients can be obtained by imposing the ADM Hamiltonian and momentum constraint conditions. Nevertheless, in three dimensions a physical space is conformally flat if and only if the Cotton-York tensor vanishes [@kramer80; @eisenhart], $$C^{i j} = 2 \epsilon^{i k l}\biggl( R^j_{~k}
- {1 \over 4} \delta^j_{~k}R \biggr)_{;l}~~,
\label{cy}$$ where $R^j_{~k}$ is the Ricci tensor and $R$ is the Ricci scalar for the three space.
Equation (\[cy\]) vanishes by fiat for the three-space metric we have chosen. However, conformally flat solutions for physical problems have only been proven [@kramer80; @eisenhart] for spaces of special symmetry (e.g. constant curvature, spherical symmetry, time symmetry, Robertson-Walker, etc. [@kramer80]). Hence, the invocation of the CFC here and in other applications is an assumption. That is, it is a valid solution to the Einstein constraint equations, but does not necessarily describe a physical configuration to which two neutron stars will evolve. Nevertheless, this is a valid approximation as long as the nonconformal contributions from the $\dot \gamma_{i j}$ and $\dot K_{i j}$ equations in the exact two-neutron star problem remain small. Indeed, numerical tests for an axisymmetric rotating neutron star [@cookcfa] and a comparison of the [*CFC*]{} vs. an exact metric expansion for an equal-mass binary [@rs96] have indicated that conformal flatness is a good approximation when it can be tested.
As a related illustration, consider the Kerr solution for a rotating black hole. It is well known that the Kerr metric is not conformally flat. The close binaries we study have specific angular momentum only slightly greater than that of an extreme Kerr black hole. Also, they ultimately merge and collapse to a single Kerr black hole. Hence, an analysis of the Cotton-York tensor for a Kerr black hole is another indicator of the degree to which conformal flatness is a valid approximation for neutron-star binaries.
Figure \[fig1\]. gives the dimensionless scaled Cotton-York parameter $C^{\theta \phi} m^3$ for a maximally rotating Kerr black hole as a function of proper distance. For illustration, consider the decrease of this quantity as one moves away from the horizon at $m = r$ as a measure of the rate at which the metric becomes conformally flat. The maximally rotating ($a = m$) black hole of this example, however, is an extreme example of compactness and angular velocity relative to any of the neutron stars in our simulations.
It can be seen in figure \[fig1\] that, even for this extreme case, the dimensionless tensor coefficient $C^{\theta \phi} m^3$ diminishes rapidly away from the black hole. At the separation of interest for binary neutron stars approaching their final orbits ($r/m \sim 25$ where $m$ is the total binary mass and $r$ the separation between stars), this coefficient has already diminished to $\sim 10^{-3}$ of the value at the event horizon, ($r/m \sim 1$). Thus, the effect of either star on its companion is probably well approximated by conformal flatness. Regarding the interior of the neutron stars themselves, in our studies the stars are rotating so slowly (even when corotating) that the deviation from conformal flatness is probably negligible. Thus, it seems plausible that conformal flatness is a reasonable approximation for most physical aspects involving the spatial three-metric of binary neutron-star systems.
The next issue concerns the “hidden” radiation in the physical data. To address this we decompose the extrinsic curvature into longitudinal $K^{i j}_L$ and transverse $K^{i j}_T$ components as proposed by York [@york73], $$K^{ij} = K^{i j}_L + K^{i j}_T~~.
\label{kdecomp}$$ By definition the transverse part obeys $$D_i K^{i j}_T = 0~~,
\label{kt}$$ where $D_i$ are covariant derivatives. The longitudinal part can be derived from a properly symmetrized vector potential. We find $$D_i K^{i j}_L = 8 \pi S^i~~,
\label{kl}$$ where $S^i$ are spatial components of the contravariant four-momentum density.
The product $K^{i j}_T K_{Tij}$ is a measure of the hidden radiation energy density. To find $K^{i j}_T$ then from our numerical calculations, we first find $K_{i j}$ by choosing maximal slicing \[$Tr(K_{i j}) = 0$\] and requiring that the trace free part of the $\dot \gamma_{i j}$ equation vanish. This gives [@wmm96] $$2\alpha K_{ij} = (D_i \beta_j+D_j \beta_i -{2\over 3} \phi^{-4}
\delta_{ij} D_k \beta^k)~~.
\label{detweiler}$$ We then determine $K^{i j}_L$ from the equilibrium momentum density \[Eq. (\[kl\])\] and subtract $K^{i j}_L$ from $K^{i j}$.
We find that this measure of the “hidden” gravitational radiation energy density is a small fraction of the total gravitational mass energy of the system, $$\int K^{i j}_T K_{Tij} {dV\over 8 \pi} \approx 2 \times 10^{-5} ~
{\rm M_G}~~.$$ Hence, we conclude that the CFC is probably a good approximation to the initial data.
This should be an excellent approximation for the determination of stellar structure and stability. However, an unknown uncertainty enters if one attempts to reconstruct the time evolution of the system (e.g. the gravitational waveform) from this sequence of quasistatic initial conditions. At present we make this connection approximately via a multipole expansion [@thorne] for the gravitational radiation as described in [@wmm96].
An Electromagnetic Analogy
--------------------------
The meaning of imposing a conformally flat spatial metric can, perhaps, be qualitatively understood in an electromagnetic analogy. Both the ADM formulation of relativity and Maxwell’s equations can be written as two constraint equations plus two dynamical equations. In electromagnetism the constraint equations for electric and magnetic fields are embodied in the $\nabla \cdot E$ and $\nabla \cdot B$ equations, while the dynamical equations are contained in Ampere’s law and Faraday‘s law. In relativity the analogous constraint equations are the ADM momentum and Hamiltonian constraints. The dynamical equations are the ADM $\dot K_{i j}$ and $\dot \gamma_{i j}$ equations. In either electromagnetism or gravity, any field configuration which satisfies the constraint equations alone represents a valid initial value solution. However, one must analyze its physical meaning.
For example, consider two orbiting charges. One could construct an electric field which satisfies the constraint by simply summing over the electrostatic field from two point charges. Similarly, one can construct a static magnetic field from the charge current by imposing $\dot E = \dot B = 0$ in the dynamical equations. However, by forcing the dynamical equations to vanish, one has precluded the existence of electromagnetic radiation. In this field configuration, therefore one has unknowingly imposed ingoing radiation to cancel the outgoing electromagnetic waves.
Similarly, enforcing $\dot K_{i j}$ = $\dot \gamma_{i j} = 0$ might in part be thought of as implying the existence of ingoing gravitational radiation to cancel the outgoing gravity waves. Nevertheless, in both cases, this remains a good approximation to the physical system (with no ingoing wave) as long as the energy density contained in the radiation is small compared to the energy in orbital motion. Gravity waves enter in two ways: as estimated above there is an insignificant amount of “hidden” radiation induced by our choice of the CFC; there is also the emission of gravitational radiation by the orbiting binary system. The binary gravity-wave emission is estimated in our calculations by evaluating the multipole moments and using the appropriate formulas [@wmm96]. The fractional energy and angular momentum loss rate as determined by the multipole expansion method is quite small, e.g. $\dot J/\omega J \sim 10^{-4}$ in all of our calculations [@wmm96; @mw97]. Hence, it can be concluded that the energy in gravitational radiation is indeed small compared to the energy in orbital motion.
The emission of gravity waves also induces a reaction force which we have incorporated into our hydrodynamic equations by the quadrupole formula. The radiation reaction force is so small, however, that it is difficult to discern it in the numerical results. In most of our calculations we simply neglect the back reaction terms and thereby obtain quasistatic orbit solutions.
Solutions to Field Equations
----------------------------
With a conformally flat metric, the constraint equations for the field variables $\phi$, $\alpha$, and $\beta^i$ reduce to simple Poisson-like equations in flat space. The Hamiltonian constraint equation [@york79] for the conformal factor $\phi$ becomes [@wmm96; @evans85], $$\nabla^2{\phi} = -2\pi
\phi^5 \biggl[ (1+U^2) \sigma - P
+ {1 \over 16\pi} K_{ij}K^{ij}\biggr]~.
\label{phieq}$$ where $\sigma$ is the inertial mass-energy density $$\sigma \equiv \rho (1 + \epsilon) + P ~~,$$ and $\rho$ is the local proper baryon density which is simply related to the baryon number density $n$, $\rho = \mu m_\mu
n/N_A$, where $\mu$ is the mean molecular weight, $m_\mu$ the atomic mass unit, and $N_A$ is Avogadro’s number. [$\epsilon$]{} denotes the internal energy per unit mass of the fluid, and $P$ is the pressure. In analogy with special relativity we have also introduced a Lorentz-like variable $$\begin{aligned}
\biggl[ 1 + U^2\biggr]^{1/2} \equiv \alpha U^t &= &\biggl[ 1 + U^j U_j \biggr]^{1/2}
\nonumber \\
&&= \biggl[ 1 + \gamma^{i j} U_i U_j \biggr]^{1/2}~~,
\label{weq}\end{aligned}$$ where $U_i$ is the spatial part of the covariant four velocity. Here we explicitly write $U^2$ (in place of $W^2 - 1 $ used in [@wm95; @wmm96; @mw97]) because it emphasizes the extra velocity dependence here and in the equations of motion.
In the Newtonian limit, the r.h.s. of Eq. (\[phieq\]) is dominated [@wmm96] by the proper matter density $\rho$, but in relativistic neutron stars there are also contributions from the internal energy density $\epsilon$, pressure $P$, and extrinsic curvature. This Poisson source is also enhanced by the generalized curved-space Lorentz factor $(1+U^2)$. This velocity factor becomes important as the orbit decays deeper into the gravitational potential and the orbital kinetic energy of the binary increases.
It was pointed out in the appendix of [@mw97] that in analogy to the velocity-dependent enhancement of the source for Eq. (\[phieq\]), the Poisson source for the $v^4$ post-Newtonian correction to the effective potential also exhibits velocity-dependence. This appendix has been misinterpreted as a statement that we attribute the compression to a first post-Newtonian effect. We therefore wish to state clearly that the appendix in that paper was merely an illustration of how the effective gravity begins to show velocity dependence even in a post-Newtonian expansion. The velocity dependence of the post-Newtonian source is not the main compression driving force. The compression derives mostly from the hydrodynamic terms described herein. It is not obvious, however, at what post-Newtonian order the compression effect should be counted, since different authors have treated these terms differently. We return to this point below.
In a similar manner [@wmm96], the Hamiltonian constraint, together with the maximal slicing condition, provides an equation for the lapse function, $$\begin{aligned}
\nabla^2(\alpha\phi) = && 2 \pi
\alpha \phi^5 \biggl[ (3(U^2+1) \sigma \nonumber \\
&& - 2 \rho(1 + \epsilon) + 3 P
+ {7 \over 16\pi} K_{ij}K^{ij}\biggr]~~.
\label{alphaeq}\end{aligned}$$ Here again, the source is strengthened when the fluid is in motion through the presence of a $U^2+1$ factor and the $K_{ij}K^{ij}$ term.
The momentum constraints [@york79] provide an elliptic equation [@wmm96] for the shift vector, $$\nabla^2 \beta^i = {\partial \over \partial x^i} \biggl({1 \over 3}
\nabla \cdot \beta\biggr) + 4 \pi \rho_3^i~,
\label{wilson3}$$ $$\begin{aligned}
\rho_3^i & =& \biggl(4\alpha \phi^4 S_i - 4 \beta^i(U^2+1)\sigma \biggr)
\nonumber \\
&& {1 \over 4\pi} {\partial ln(\alpha/\phi^6) \over \partial x^j}
({\partial \over \partial x^j}\beta^i + {\partial \over \partial x^i} \beta^j
-{2\over 3}\delta_{ij} {\partial \over \partial x^k} \beta^k)~,
\label{betaeq}\end{aligned}$$ where we have introduced [@w79] the Lorentz contracted coordinate covariant momentum density, $$S_i = \sigma W U_i~~.
\label{momdef}$$
As noted previously and in Ref. [@wmm96], we only solve equation (\[wilson3\]) for the small residual frame drag after the dominant $\vec \omega \times \vec r$ contribution to $\vec \beta$ has been subtracted.
Relativistic Hydrodynamics
==========================
The techniques of general relativistic hydrodynamics have been in place and well studied for over 25 years [@w79]. The basic physical processes which induce compression can be traced to completely general terms in the hydrodynamic equations of motion. To illustrate this we first summarize the completely general derivation of the relativistic covariant momentum equation in Eulerian form and identify the terms which we believe to be the dominant contributors to the relativistic compression effect.
For hydrodynamic simulations it is convenient to explicitly consider two different spatial velocity fields. One is $U_i$, the spatial components of the covariant four velocity. The other is $V^i$, the contravariant coordinate matter three velocity, which is related to the four velocity $$V^i = { U^i \over U^t} =
{\gamma^{i j} U_j \over U^t} - \beta^i~~.
\label{threevel}$$ It is convenient to select the shift vector $\beta^i$ such that the coordinate three velocity vanishes when averaged over the star, $\langle V^i\rangle = 0$. This minimizes coordinate fluid motion with respect to the shifting ADM grid.
The perfect fluid energy-momentum tensor is $$T_{\mu \sigma} = \sigma U_\mu U_\sigma
+ P g_{\mu \sigma}~~.$$ However, it is convenient to derive the hydrodynamic equations of motion using the mixed form, $$T_\mu^{~\nu} =
g^{\sigma \nu} T_{\mu \sigma} = \sigma U_\mu U^\nu + P \delta_{\mu}^{~\nu}~~,$$
The vanishing of the spatial components of the divergence of the energy momentum tensor $$\biggr(T_i^{~\mu}\biggl)_{;\mu} = 0$$ leads to an evolution equation for the spatial components of the covariant four momentum, $$\begin{aligned}
{1\over\alpha \gamma}{\partial (S_i \gamma )\over\partial t} +&&
{1\over\alpha \gamma}{\partial (S_i V^j\gamma )\over\partial x^j}
+{\partial P\over \partial x^i} \nonumber \\
&& + {1 \over 2} {\partial g^{\alpha \beta}\over\partial
x^i} { S_\alpha S_\beta \over S^t} = 0
~~.
\label{divmom}\end{aligned}$$ The covariant momentum equation is used because of its close similarity with Newtonian hydrodynamics. The first two terms are advection terms familiar from Newtonian fluid mechanics. The latter two terms are the pressure and gravitational forces, respectively.
Expanding the gravitational acceleration into individual terms we have $$\begin{aligned}
{\dot S_i}& + & S_i{\dot \gamma \over\gamma}
+{1\over\gamma}{\partial\over\partial x^j}(S_iV^j\gamma)
+ {\alpha \partial P\over \partial x^i}
- S_j {\partial \beta^j \over \partial x^i}
\nonumber \\
& + & \sigma {\partial \alpha \over \partial x^i}
+ \sigma \alpha \biggl( U^2 {\partial \ln{\alpha} \over \partial x^i}
+ {U_j U_k \over 2 } {\partial \gamma^{j k}
\over \partial x^i}\biggr) = 0~~.
% - 2 {\partial \ln{\phi}\over\partial x^i} \biggr) = 0
%&+ & \sigma\biggl((1+U^2){\partial \alpha \over \partial x^i}
%+ \alpha {U_j U_k \over 2 } {\partial \gamma^{j k}
%\over \partial x^i}\biggr) = 0~~.
\label{hydromom}\end{aligned}$$
Similar forms can be derived for the condition of baryon conservation and the evolution of internal energy [@wmm96; @w79]. However, the above momentum equation is sufficient for the present discussion.
It is now worthwhile to consider the ”gravitational” forces embedded in the expanded terms of Equation (\[hydromom\]). These result from the affine connection terms $\Gamma^\mu_{\mu \lambda} T^{\mu \lambda}$ in the covariant differentiation of $ T^{\mu \nu}$.
The term containing $\partial \alpha/\partial x^i$ comes from the time-time component of the covariant derivative. It is of course the well known analog of the Newtonian gravitational force as can easily be seen in the Newtonian limit $\alpha \rightarrow 1 - Gm/r$.
The term $S_j (\partial \beta^j /\partial x^i)$ comes from the space-time covariant derivative. In an orbiting system it is convenient to allow $\beta^j$ to follow the orbital motion of the stars. In our specific application [@wmm96] we let $\vec \beta = \vec \omega \times
\vec R + \vec \beta_{resid}^{drag}$ where $\omega$ is chosen to minimize matter motion on the grid. Hence, $\vec \omega \times
\vec R$ includes the major part of rotation plus frame drag. The quantity $\vec \beta_{resid}^{drag}$ is the residual frame drag after subtraction of rotation and is very small for the almost nonrotating stars which result from our calculations. With $\beta^j$ dominated by $ \vec \omega \times
\vec R $, the term $S_j (\partial \beta^j /\partial x^i)$ is predominantly a centrifugal force.
The $U^2 \partial \ln{\alpha} / \partial x^i$ term arises from the time-time component of the affine connection piece of the covariant derivative. The $(U_j U_k /2 ) \partial \gamma^{j k}
/\partial x^i$ term similarly arises from the space-space components. They do not have a Newtonian analog. As we shall see, these terms cancel when a frame can be chosen such that the whole fluid is at rest with respect to the observer (or in the flat space limit). However, for a star with fluid motion in curved space, they describe additional velocity-dependent forces.
We identify the nonvanishing combination of these $U^2$-dependent force terms and the $S_j (\partial \beta^j /\partial x^i)$ term as the major contributors to the net compression driving force.
This suggests some useful test problems for our hydrodynamic simulations. For example, in simple uniform translation the effects of these terms must cancel to leave the stellar structure unchanged. Similarly, as discussed below, any fluid motion such that the four velocity can be taken as proportional to a simple Killing vector (e.g. rigid corotation) these force terms must cancel [@baumgarte; @kramer80]. However, for more general states of motion, e.g. noncorotating stars, differential rotation, meridional circulation, turbulent flow, etc., these forces do not obviously cancel, but must be evaluated numerically.
Indeed, as discussed below, the sign of these terms is such that a lower energy configuration for the stars than that of rigid corotation can be obtained by allowing the fluid to respond to these forces. As we shall see, the numerical relaxation of binary stars from corotation (or any other initial spin configuration) produces a nonsynchronous (approximately irrotational) state of almost no intrinsic neutron-star spin in which the central density and gravitational binding energy increase.
Conformally Flat Relativistic Hydrodynamics {#hydro}
-------------------------------------------
The practical implementation of conformal flatness means that, given a distribution of mass and momentum on some manifold, we first solve the constraint equations of general relativity at each time for a given distribution of mass-energy. We then evolve the hydrodynamic equations to the next time step. Thus, at each time slice we obtain a solution to the relativistic field equations and then can study the hydrodynamic response of the matter to these fields [@wmm96].
For the [*CFC*]{} metric, the relativistic momentum equation is derived by simply replacing $\gamma^{j k} \rightarrow \phi^{-4}
\delta^{j k}$ in Eq. (\[hydromom\]). $$\begin{aligned}
{\partial S_i\over\partial t}& +& 6 S_i{\partial \ln\phi\over\partial t}
+{1\over\phi^6}{\partial\over\partial x^j}\biggl(\phi^6S_iV^j\biggr)
+\alpha{\partial P\over \partial x^i}
- S_j {\partial \beta^j \over \partial x^i}
\nonumber \\
& + & \sigma {\partial \alpha \over \partial x^i}
+ \sigma \alpha U^2 \biggl( {\partial \ln{\alpha} \over \partial x^i}
- 2 {\partial \ln\phi\over\partial x^i} \biggr) = 0~~.
\label{hydromomcfa}\end{aligned}$$ Here as in Eq. (\[hydromom\]), the first term with ${\partial \alpha/\partial x^i}$ is the relativistic analog of the Newtonian gravitational force.
In Eq. (\[hydromomcfa\]) there are two ways in which the effective gravitational force might increase for finite $U^2$. One is that the matter contribution to the source densities for $\alpha$ or $\phi$ are increased by factors of $\sim 1 + U^2$ \[cf. Eqs. (\[phieq\]) and (\[alphaeq\])\]. The more dominant effect, however, is from the combination of the $S_j {\partial \beta^j /\partial x^i}$ term and the $U^2 [\partial \ln{\alpha} / \partial x^i -
2\partial \ln{\phi}/\partial x^i]$ terms in Eq. (\[hydromomcfa\]).
As noted previously, these compression driving terms result from the affine connection part $\Gamma^\mu_{\mu \lambda} T^{\mu \lambda}$ of the covariant differentiation of $ T^{\mu \nu}$. These terms have no Newtonian analog but describe a general relativistic increase in the gravitational force as $U^2$ increases. As noted in [@wmm96; @mw97] (see also Fig \[fig2\] below) for a binary, $U^2$ is approximately uniform over the stars, and the increase in central density due to these additional forces scales as $\approx U^4$. This scaling, however, is the net result from a nontrivial cancellation of terms and must be treated carefully. We shall return to this point below.
The proper way to determine the post-Newtonian order at which the compression driving terms enter would be to count the powers of $c^2$ which appear in the denominator of a term. For example, if we divide the last two terms in Eq. (\[hydromomcfa\]) by the gradient of the $\alpha$ term (the analog of the Newtonian gravitational force) we would obtain a ratio of order $U^2/c^2$ which would be manifestly first post Newtonian. However, in the first post-Newtonian treatment of Wiseman [@wiseman], these velocity terms were explicitly disregarded. Thus, the effects of these terms could not have been present in that calculation. It is no surprise, therefore that no effect was observed in Ref. [@wiseman].
Also note that the $2{\partial \ln{\phi}/\partial x^i}$ term in Eq. (\[hydromomcfa\]) enters with a sign such that the total $U^2$-dependent contribution is further increased by about twice that from the ${\partial \ln{\alpha}/\partial x^i}$ contribution alone. (The factor of 2 in front of the derivative comes from the requirement that $\phi^2 \sim (1/\alpha)$ in the Newtonian limit [@wmm96].)
A further increase of binding arises from the $K^{ij}K_{ij}$ terms in the field sources, but these terms are much smaller than the $U^2$ contributions for a binary system.
Comment on the Relativistic Bernoulli Equation
----------------------------------------------
For comparison with other work in the literature it is instructive to discuss the derivation of the relativistic Bernoulli equation from equation (\[hydromom\]). It has been pointed out (e.g. [@baumgarte]) that the hydrodynamics reduces to a simple equation for a fluid in which the velocity field can be represented by a Killing vector. In our notation this equation can be written, $$d \ln{(U^t)} = {dP \over \sigma}~~.
\label{bernoulli}$$
The demonstration that the relativistic Bernoulli equation (\[bernoulli\]) is exactly reproduced from Eq. (\[hydromom\]) and indeed for any case in which a Killing vector can be imposed, was recently brought to our attention by T. Nakamura [@nakamura]. We summarize the derivation here in the conformally flat metric both for clarity and to show that conformal flatness does not violate this important constraint.
To begin with, note that in the ADM formalism, the existence of a Killing vector is equivalent to being able to choose a shifted ADM grid such that $V^i = 0$ everywhere for the fluid. Next use Eq. (\[threevel\]) to solve for $\beta^i$ and divide by $\sigma$. The resulting equation for stationary motion is $${1 \over \sigma} {\partial P\over \partial x^i} = U^t U_i {\partial \over \partial x^i}
\biggl({U_i \over \phi^4 U^t}\biggr)
- (\alpha U^t)^2 {\partial ln{\alpha} \over \partial x^i}
+ 2 U^2 {\partial \ln{\phi} \over \partial x^i} ~~.
\label{bernoulli1}$$ The recovery of the relativistic Bernoulli equation requires that the r.h.s. $= \partial \ln{U^t}/\partial x^i$. With some straightforward algebraic manipulation it is possible to show that all of the terms on the r.h.s. cancel except for one term from the $\beta$ derivative, $-\phi^{-4} U^2 \partial \ln{U^t}/\partial x^i$. The completion of the proof is simply to note that this term is equal to $\partial \ln{U^t}/\partial x$ by Eq. (\[weq\]). The result is Eq. (\[bernoulli\]).
It is instructive to consider the change in the relativistic Bernoulli equation when there is no Killing vector, i.e. $V_i \ne 0$. Along the same lines of the derivation of Eq. (\[bernoulli1\]), It can be shown [@nakamura] that the momentum equation can be rewritten in our notation as, $$\begin{aligned}
{1 \over \alpha \sigma}\biggl[{\dot S_i}& + & S_i{\dot \gamma \over\gamma}
+{1\over\gamma}{\partial\over\partial x^j}(S_iV^j\gamma)\biggr]
+ U^t U_j {\partial V^j \over \partial x^i} \nonumber \\
& =&
-{1 \over \sigma} {\partial P\over \partial x^i}
+ {\partial ln U^t \over \partial x^i}~~.
\label{hydromomb}\end{aligned}$$ The r.h.s. is just the relativistic Bernoulli equation in the limit that the l.h.s. vanishes. In general fluid flow, however, the l.h.s. contains not only the advection terms (in brackets), but also an additional surviving part of the $\beta^j$ derivative.
It can be seen from this that imposing a Killing vector ($V^i = 0$) means that only simple hydrostatic equilibrium is obtained for stationary systems. However, when nontrivial hydrodynamic motion is allowed, the extra forces embodied in the l.h.s. of Eq. (\[hydromomb\]) are manifest. The presence of fluid motion not represented by a simple Killing vector, thus leads to a deviation from the simple relativistic Bernoulli solution. Any attempt to model this deviation requires a careful treatment of the dynamical properties of the fluid described by the l.h.s. of Eq. (\[hydromomb\]).
CONSTRAINED HYDRODYNAMICS
=========================
Further insight into the complexity of the physics contained in the relativistic equations of motion can be gained by considering some simple examples of constrained hydrodynamics for which the answer is known. These pose useful tests of our numerical scheme. Since some have proposed that the effect we observe may be an artifact of numerical resolution or approximation, we present here a summary of various test problems designed to illustrate the stability of the numerics and also to compare with some of the calculations in the literature. These calculations also demonstrate that the compression effect vanishes in the limiting cases which have been studied by others. Hence, they could not have been observed. They highlight the fact that the effect we observe only appears in a strong field dynamic treatment which accounts for internal motion of stellar material in response to the binary and its effect on the star’s self gravity. At present, ours may be the only existing result. This is consistent with the conclusion of [@shapiro] based upon test particle dynamics.
Bench Mark Calculations
-----------------------
To test for the presence of the compression driving forces we consider two bench-mark initial calculations. The bench mark of no compression is that of an isolated star. In our three dimensional hydrodynamic calculations, the single star structure is derived from Eq. (\[hydromom\]) in the limit $$S_i = U_i = V^i = \beta^i = 0~~.$$ The condition of hydrostatic equilibrium in isotropic coordinates is then trivially derived from Eq. (\[hydromom\]) $${\partial P\over \partial x^i} = - \sigma {\partial \ln{
\tilde \alpha} \over \partial x^i} ~~,
\label{static}$$ where the tilde denotes that the metric coefficients are evaluated in the fluid rest frame. The Newtonian limit of the right hand side is recovered as $\tilde \alpha \rightarrow 1 - G m/r$. Hence, we again identify the ${\partial \ln{\tilde \alpha} / \partial x^i}$ term with the relativistic analog of the Newtonian gravitational force. Eq. (\[static\]) also trivially reproduces to relativistic Bernoulli equation (\[bernoulli\]).
We have of course tested our three dimensional calculations for single isolated stars. A single star remains stable on the grid indefinitely, except when baryon mass exceeds the maximum stable mass allowed by the TOV equations. Above the maximum TOV mass the stars begin to collapse on a dynamical timescale as they should. We have also checked that the grid resolution used in our binary calculations is adequate to produce the correct central density, stellar radius, and gravitational mass of a single isolated star [@mw97]. Hence, it seems unlikely that inadequate grid resolution is the source of the compression effect as some have proposed.
In order to facilitate comparisons with the literature, and to avoid confusion over equation of state (EOS) issues, we have employed a simplistic $\Gamma=2$ polytropic EOS, $P = K\rho^\Gamma$, where $K= 1.8 \times 10^5$ erg cm$^3$ g$^{-2}$. This gives a maximum neutron-star mass of 1.82 $M_\odot$. The gravitational mass of a single $m_B = 1.625$ $M_\odot$ star in isolation is 1.51 $M_\odot$ and the central density is $\rho_c =5.84\times 10^{14}$ g cm$^{-3}$. The compaction ratio is $m/R = 0.15$, similar to one of the stars considered in [@baumgarte]. Note that this EOS leads to stars with a lower compaction ratio than the stars we considered in [@wmm96; @mw97] for which $m/R \approx 0.2$. Hence, the effects of tidal forces in the present calculations should be more evident.
The bench mark in which compression is present is that of two equal mass stars in a binary computed with unconstrained hydrodynamics. The binary stars have the same baryon mass ($m_B = 1.625$ $M_\odot$ each), the same EOS, and a fixed angular momentum $J = 2.5 \times 10^{11}$ cm$^2$ ($J/M_B^2 = 1.09$ where M$_B = 2 m_B$). For these conditions the binary stars have $U^2 = 0.025$ and are at a coordinate separation of $\approx 100$ km. The stars are stable but close to the collapse instability. Hence, they have experienced some compression which has increased their central density by 14% up to $\rho_c = 6.68\times 10^{14}$ g cm$^{-3}$.
The central densities of these two bench marks are summarized in the first and last entries of Table \[table1\]. To compare with these bench-mark calculations we have computed equilibrium configurations for stars under the various conditions outlined below. The test for the presence or absence of compression inducing forces will be the comparison of the numerically computed central density with that of a single isolated star or stars in a binary.
Stars in Uniform Translation {#linear}
----------------------------
As a first nontrivial test, now consider a star as seen from an observer in an inertial frame which is in uniform translation with respect to the fluid. Choosing motion along the $x$ coordinate, the fluid three velocity is, $${ U^x \over U^t} = V^x = {\rm Constant} ~~.$$ However, the observer is still free to choose the ADM shift vector such that the computational grid remains centered on the star. That is, although $S_i, U_i, \ne 0$, we can still choose $V^i = 0$. This gives a restriction on $\beta^i$ from Eq. (\[threevel\]), $$\beta^x = {\gamma^{x x} \alpha U_x \over W} ~~.
\label{betadef}$$ Note, that this is an ADM coordinate freedom. It is not equivalent to a coordinate boost. It is in fact a Killing vector which is convenient for numerical hydrodynamics. It allows the matter to remain centered on the grid even though the equations of motion are being solved for fluid which is not at rest with respect to the observer.
With $V^i = 0$, the $x$ component of the momentum equation (in equilibrium) becomes, $${\partial P\over \partial x} =
{S_x \over \alpha} {\partial \beta^x \over \partial x}
- \sigma \biggl[(U^2 + 1)
{\partial ln{\alpha} \over \partial x}
+ {U_j U_k \over 2 } {\partial {\gamma^{j k}} \over \partial x} \biggr]~~.$$ With a [*CFC*]{} metric this becomes, $${\partial P\over \partial x} = {S_x \over \alpha} {\partial \beta^x \over \partial x}
- \sigma \biggl[(U^2 + 1)
{\partial ln{\alpha} \over \partial x}
- 2 U^2 {\partial \ln{\phi} \over \partial x} \biggr]~~.
\label{hydrostatcfa}$$
There are now several differences between this expression and that for an observer in the fluid rest frame. For one, there is the shift vector derivative $\partial \beta^x /\partial x$. Even in uniform translation this derivative is nonzero due to the variations of the metric coefficients over the star (cf. Eq. \[betadef\]). Also, the effective gravity is enhanced by the $(U^2 + 1)$ velocity factor. The ${U_j U_k / 2 } {\partial {\gamma^{j k}} / \partial x^i}$ term also appears. In addition, the effective source terms (\[phieq\]) and (\[alphaeq\]) for the [*CFC*]{} metric coefficients are enhanced both by $(U^2 + 1)$ factors and the $K_{i j} K^{i j}$ term.
In spite of these differences, we nevertheless know that the locally determined pressure and inertial density must be the same as those determined for a star at rest. Indeed, since we can choose a Killing vector ($V^i = 0$) these equations must reduce to the relativistic Bernoulli equation (\[bernoulli\]).
Thus, this is an important numerical test problem. We solve the full hydrodynamic equations explicitly, e.g. Eq. (\[hydromomcfa\]), under the initial condition of nonzero $U_x$ for a single star. The cancellations embedded in the hydrodynamics are not obvious. Nevertheless, in the end, all of these effects must cancel to leave the stellar central density unchanged (except for a Lorentz contraction factor).
To solve the uniform translation problem numerically we have applied the Hamiltonian and momentum constraints to determine the metric coefficients. We then evolved the full hydrodynamic equations to equilibrium. Figure \[fig2\] shows the numerically evaluated central density for such translating stars as a function of $U^2$. These stars were calculated with the EOS of [@wmm96]. This is compared with the central density for binary stars evolved at the same $U^2$ value using the same EOS. One can see that the translating stars maintain a constant central density (within numerical error) as they should. In contrast, the central density of binary stars grows as $\approx U^4$. This growth is the nontrivial net result from the velocity dependent terms in Eq. (\[hydromomcfa\]). It is not obvious, however, to what post-Newtonian order this dependence corresponds.
As summarized in Table \[table1\], the central density for a uniformly translating $\Gamma=2$ star with $U^2 = 0.025$ (for comparison with the binary bench-mark calculation) is $5.90 \times 10^{14}$ g cm$^{-3}$. Within numerical accuracy, this central density is identical with that of an isolated star at rest.
This is at least indicative that our observed growth in central density may not be numerical error as some have suggested (e.g. [@brady; @thorne97; @shapiro]). Such an error would likely be apparent in this test case. We argue that the difference between simple translation and binary orbits relates to the physics of the binary system itself, in particular physics which is not apparent in uniform translation, an analysis of tidal forces, or a truncated expansion which does not contain sufficient terms to adequately describe the dynamical response of the fluid.
Tidal Forces
------------
It has been pointed out [@lai; @flanagan; @thorne97] that tidal forces are in the opposite sense to the compression driving forces discussed here. That is, tidal forces distort the stars and decrease the central density and therefore render the stars less susceptible to collapse. We have argued [@mw97] that although such stabilizing forces are present in our calculations they are much smaller in magnitude than the velocity-dependent compression driving terms. Nevertheless, the evolution of the matter fields in a calculation in which only tidal forces are present still represents a useful test of our numerical results. Stars in which only tidal forces act, should be stable and the central density should decrease rather than increase as the stars approach.
To test the effects of tidal forces alone we have constructed an artificial test calculation in which we place stars on the grid in a binary, but with no initial angular or linear momentum, i.e. $J = 0$ and $U^2 = 0$. This initial condition would normally evolve to an axisymmetric collision between the stars. However, after updating the matter fields, we artificially return the center of mass of the stars to the same fixed separation after each time step. We also reset to zero the mean velocity component directed along the line between centers. This sequence is repeated until the matter fields come to equilibrium. Since the velocity dependent forces eventually vanish, the only remaining forces are the pressure and static gravitational (including tidal) forces.
Results as a function of separation distance are shown in Table \[table2\] for the $\Gamma=2$ EOS and Table \[table3\] for the realistic EOS used in [@wmm96]. For the realistic EOS the central density indeed decreases as the stars approach consistent with the expectations from Newtonian and relativistic tidal analyses [@lai; @thorne97]. For the $\Gamma=2$ polytropic EOS, the central density also decreases as the stars approach and remain below the central density of an isolated star. The fact that this table is not monotonic at the innermost point, however, may be due to a limitation of this numerical approximation for tidal forces as the ratio of separation to neutron-star radius diminishes.
Although the tidal forces do indeed stabilize the stars, their effect on the central density is quite small ($\sim 0.2\%$ decrease) compared to the net increase in density caused by the compression forces present for the binary. This is consistent with the relative order-of-magnitude estimates for these effects described in [@mw97].
Stars in Rigid Corotation
-------------------------
As a next nontrivial example, consider stars in a binary system which are restricted to rigid corotation. In a recent series of papers, Baumgarte et al. [@baumgarte] have studied neutron-star binaries using the same conformally flat metric. Their work differs from ours in that rather than solving the hydrodynamic equations, they describe the four velocity field by a Killing vector whereby the stars are forced to corotate rigidly. They also impose spatial symmetry in the three Cartesian coordinate planes so that they can solve the problem in only one octant. One should keep in mind, however, that rigid corotation is not necessarily the lowest energy configuration or the most natural [@bc92] final state for two neutron stars near their final orbits. This assumption, though artificial, is nevertheless a means to constrain and simplify the fluid motion degrees of freedom. It is much easier to implement and therefore becomes an interesting test problem for codes seeking to explore the true hydrodynamic evolution of close binaries.
Indeed, it is possible to show [@kramer80] that in this limit, the neutron star hydrostatic equilibrium can be described by a simple Bernoulli equation in which the compression driving force terms are absent except for a weak velocity dependence. Analytically, the reason for this is trivially obvious from Eq. (\[hydromomb\]). The existence of a Killing vector is equivalent to setting $V^i = 0$ globally. Choosing the ADM coordinates to remain centered on the stars, in steady state the time derivatives vanish along with the rest of the l.h.s of Eq. (\[hydromomb\]). Only the relativistic Bernoulli equation (\[bernoulli\]) survives.
It is not surprising, therefore that in [@baumgarte] it has been demonstrated that in this special symmetry, the central density of the stars does not increase (within numerical error) as the stars approach the inner most stable circular orbit. In very close orbits the density actually decreases relative to the central density of stars at large separation. They also find that the orbit frequency remains close to the Newtonian frequency. Both of these results are interesting in that they confirm that the compression effect does not occur (as it should not) in this special symmetry. They also demonstrate that conformal flatness is not the source of the compression.
Accepting the results of [@baumgarte] as correct, this then becomes another important test of our calculations. That is, if we artificially impose rigid corotation, then the central density should remain nearly constant until the stars are close enough that tidal effects cause the central density to decrease rather than increase.
Imposing rigid corotation, however, is not a trivial test problem to implement without completely replacing the hydrodynamic equations with the corresponding Bernoulli solution of [@baumgarte; @kramer80]. (Indeed, we have done this [@marronetti] and reproduce the results of [@baumgarte] quite well). Moreover, we have found that directly modifying the hydrodynamic equations in an attempt to mimic a dynamically unstable configuration is difficult. One might think that the simplest way to implement corotation would be to impose a high fluid viscosity. Indeed high viscosity would resist the hydrodynamic forces described herein. However, a high fluid viscosity also resists the much weaker tidal forces and prevents the numerical relaxation to quasistatic equilibrium. It is thus difficult to achieve tidal locking by simply increasing the viscosity.
Instead, we introduce artificial forces on the fluid which continually drive the system toward a state of rigid corotation while allowing the system to at least somewhat respond hydrodynamically. To do this we define accelerations $(\dot U_i)_{\rm Rigid}$ necessary to achieve rigid rotation by $$(\dot U_i)_{\rm Rigid} \equiv {(\tilde U_i - U_i) \over \Delta t}$$ where $\tilde U_i$ are components of the rigidly corotating covariant four velocity in the $ x- y$ orbit plane. These are determined by requiring that $\beta^i = (\omega \times r)^i$ and setting $V^i = 0$ in Eq. (\[threevel\]). $$\tilde U_y = {\omega x \phi^4 \over \alpha
\sqrt{1 - \omega^2 R^2 \phi^4/\alpha^2 }}~~,$$ $$\tilde U_x = {-\omega y \phi^4 \over \alpha
\sqrt{1 - \omega^2 R^2 \phi^4/\alpha^2 }}~~,$$ where $R$ is the coordinate distance from the center of mass of the binary.
At each time step we then update the momentum density using a combination of the hydrodynamic and corotating acceleration terms, $$\dot U_i = f (\dot U_i)_{\rm Rigid} + (1 - f)(\dot U_i)_{\rm Hydro}$$ where $(\dot U_i)_{\rm Hydro}$ is the acceleration from the full hydrodynamic equation of motion \[Eq. (\[hydromomcfa\])\].
Numerically, we find that if $f$ is small ($< 0.2$) the hydrodynamic forces dominate and corotation is not obtained. On the other hand, for $f > 0.2$ the system is not stable, i.e. the stars deform and the velocities become erratic. We have therefore run with $f = 0.2$ which temporarily produces a velocity field which is close to rigid corotation. That is, the residual three velocities are damped to a fraction of the orbit speed. This is, perhaps, good enough to make qualitative comparisons with the expectations from a truly corotating system.
Starting from the unconstrained initial configuration, we find that when the stars have achieved approximate corotation the central density has decreased from $6.68 \times 10^{14}$ g cm$^{-3}$ to $5.90 \times 10^{14}$ g cm$^{-3}$ which is close to the value for stars in isolation ($5.84 \times 10^{14}$ g cm$^{-3}$). The calculated gravitational mass is slightly greater than that of the unconstrained binary. However, with the large artificial force terms needed to approximate corotation, gravitational mass is not a well defined quantity in this simulation. Also, the orbit frequency was not sufficiently converged for a meaningful comparison.
The Spin of Binary Stars
------------------------
As noted above our simulations indicate that neutron stars relax to a state of almost no intrinsic spin. In a separate paper [@wm98] we analyze the nature and formation of this state in more detail. For the present discussion, however, we summarize in Figure \[fig3\] a study of the relaxation to this state from states of arbitrary initial rigid rotation (including corotation).
As a means to distinguish the intrinsic spin motion of the fluid with respect to a non-orbiting distant observer, we define a quantity which is analogous to volume averaged intrinsic stellar spin in the orbit plane, $$J_S = \sum_{i=1,2}\int \biggl[(x - \tilde x_i)S_y - (y-\tilde y_i)S_x \biggr]
{\phi^2 \over \alpha} d V_i~~,$$ where $(\tilde x_i,\tilde y_i,\tilde z_i = 0)$ is the coordinate center of mass of each star.
In this study we have imposed an initial angular velocity $\omega_S$ in the corotating frame to obtain various initial rigidly rotating spin angular momenta (including corotation, $\omega_S = 0$), but for fixed total $J/M_B^2 = 1.4$. We have considered spin angular frequencies in the range $-900 < \omega_S < 900$ rad sec$^{-1}$, corresponding to $-0.03 < J_S/m_B^2 < 0.17$. We then let the system evolve hydrodynamically with the stars maintained at zero temperature.
In Ref. [@mw97] we showed that neutrino emission is sufficient to radiate away the released gravitational energy and keep the stars at near zero temperature until just before collapse. This is the reason that we have treated this as a relaxation problem. That is, unlike a true hydrodynamic calculation, the relaxation calculation presented here, assumes that the stars radiate efficiently and stay at zero temperature. Therefore, this evolution does not need to conserve energy or circulation. This relaxation assumption is the reason the stars can evolve to a different spin (lower energy) state without violating the circulation theorem.
Figure \[fig3\] shows the spin $J_S/m_B^2$ as a function of time for each initial condition. In each case, the system relaxed to a state of almost no net spin within about three sound crossing times ($t \sim 0.6$ msec). These calculations suggest that rapidly spinning neutron stars in close orbits are unstable. The true evolution time, however, would be much longer.
We also note that the quantity $\int \sigma [\sqrt{1+ U^2}
-1] dV$ decreased as the system evolved from rigid rotation to hydrodynamic equilibrium. Since this quantity is related to the kinetic energy of the binary, this indicates that the hydrodynamic lowest energy state is one of lower kinetic energy (for fixed total angular momentum) than that of rigid rotation.
As far as the compression effect is concerned, one wishes to know whether the response of the stars is simply due to that fact that they have no spin (and therefore no internal centrifugal force to support them against the compression forces), or whether more complex fluid motion within the star itself affects the stability. To test this, we have constructed stars of no spin ($J_S = 0$) by simply damping the residual motion to that of $J_{S} = 0$ after each update of the velocity fields.
Since this no-spin state is so close to the true hydrodynamic equilibrium, this produced stable $J_{S} = 0$ equilibrium stars for the binary. For this case, the central density converges to $\rho_c = 6.56 \times 10^{14}$ g cm$^{-3}$ which is very close to the high value for the unconstrained hydrodynamics. This result would seem to indicate that most of the increase in density can be attributed to the velocity with respect to the corotating frame generated by the fact that the stars have almost no spin.
DISCUSSION
==========
For clarity, we summarize in this section our conclusions regarding why the neutron-star compression effect was not observed in some other recent works.
First consider post-Newtonian expansions. In the work of Wiseman [@wiseman] the force terms containing $U^2$ were explicitly deleted from the computation of the stellar structure \[cf. Eq. (8) in that paper\]. Only the $dln{\alpha}/dx$ term was included. The recovery of simple hydrostatic equilibrium was thus unavoidable.
The PN orbiting ellipsoids of Shibata et al. [@shibata] included more terms. Indeed, it was noted that there are two effects at 1PN order. One is the self gravity of each star of the binary and the other is the gravity acting between the stars. In their calculations the self gravity dominates causing the stars to become more compact. This is consistent with the compression effect described here in the sense that relativistic corrections can dominate over Newtonian tidal forces. However, the self gravity terms in [@shibata] appear to only include the usual 1PN terms which would equally apply to stars in isolation. Hence, the velocity-dependent compression driving terms are probably not present.
Their results for stars in corotation are consistent with ours under the same constraint. They also note that approaches in which PN corrections to the gravity between the stars is included without also including the corrections to the self gravity (as in [@lw96]) can be misleading.
In the work of Lombardi et al. [@lombardi] both corotating and irrotational equilibria were computed. However, in their calculations it appears that the stars become less compact as they approach contrary to our results and the results of [@shibata]. It may be that the reason for this is that in Lombardi et al. the post-Newtonian corrections to self gravity were only computed for stars “instantaneously at rest”. The authors chose to “exclude the spin kinetic energy contribution to the self energy”. It is such terms, however, which we identify with the compression effect.
The conformally flat corotating equilibria computed by Baumgarte et al. [@baumgarte] are consistent with our results. Since their stars were restricted to rigid corotation, only the hydrostatic Bernoulli solution would result. They could not have observed the compression forces which result from fluid motion with respect to the corotating frame.
We have argued in this paper that if one wishes to explore this effect, it would be best to apply a complete unconstrained strong-field relativistic hydrodynamic treatment for stars which are not in corotation. In this regard, a recent paper [@sbs98] has come to our attention in which hydrodynamic simulations of both corotating and irrotational binaries have been studied in a first post-newtonian approximation to conformally-flat gravity but using the full relativistic hydrodynamics equation (\[hydromom\]). For both corotational and irrotational stars the central density is observed to oscillate about a value which is less than that of isolated stars. Hence, the authors conclude that no compression effect is present.
Since this calculation contains many of the higher order terms to which we attribute the compression effect, it is not immediately obvious why the compression effect was not observed. This may indeed be a real contradiction. We suggest, however, that this simulation did not observe the effect because of their use of an unrealistically soft $\Gamma = 1.4$ EOS. The authors chose this EOS because the stars become so extended that one can compute arbitrarily close binaries without encountering the relativistic inner orbit instability. For the irrotational stars (model $Bc$ in [@sbs98]), which is the only simulation that might have observed the compression effect, the compaction ratio is only $M/R = 0.023$. Hence, a 1.45 M$_\odot$ neutron star would have an unrealistic radius of 93 km.
However, since they have simulated very extended stars at very close separation, the tidal forces are much stronger relative to the relativistic compression driving terms than in any of the simulations which we have done.
The ratio of the stabilizing tidal correction $\Delta E_{tidal}$ to the destabilizing energy from compression $\Delta E_{comp}$ should scale [@mw97; @lai] as $${\Delta E_{tidal} \over \Delta E_{comp}} \propto \biggl({R \over r}\biggr)^6~~,$$ where $R$ is the neutron star radius and $r$ is the orbital separation. For model Bc in [@sbs98] we estimate that this ratio is $^>_\sim 200$ times greater than any of the binary stars we have considered. Hence, it is quite likely that the authors have simply chosen an unrealisticly soft equation of state for which the tidal forces dominate over compression. It might be very interesting to see the results from a similar study for stars with a realistic compaction ratio and several radii apart.
Concerning tidal expansions, in Brady & Hughes [@brady] an attempt was made to analyze the stability of a central star perturbed by an orbiting point particle. The metric and stress-energy were perturbed in terms of order $\epsilon = \mu/R$ where $\mu$ is the point particle mass and $R$ its coordinate distance from the central star. The Einstein equation was then linearized to terms of order $\epsilon$. The result of this linearization was that the only possible correction to the central density was a single monopole term of order $\mu/R \sim v^2$. However, in our numerical results as shown in Figure \[fig3\]. the central density is observed to increase as $v^4$. Hence, it may be that the expansion of Ref. [@brady] was truncated at too low order to observe the compression effect described here. The main reason that they could not observe the effect, however, is that the terms involving motion of the central star were discarded. We attribute the compression effect to an enhancement of the self gravity due to motion of the stars with respect to the corotating frame. Hence, the neglect of terms involving motion of the central star precludes the possibility of observing the effect.
We believe that the same conclusion is true in the treatments by Refs. [@flanagan; @thorne97]. The analysis of Flanagan [@flanagan] is based upon the method of matched asymptotic expansion. The metric is approximated $$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}^{NS} + h_{\mu \nu}^B~~,$$ where the superscript $NS$ refers to the self contribution from one star and $B$ refers to the contribution from a distant companion. The internal gravity of a static neutron star $h_{\mu \nu}^{NS}$ is expanded to all orders. The binary tidal contribution $h_{\mu \nu}^B$ is expanded in powers of the ratio of stellar radius to orbital separation.
First we suggest that such a decomposition may be questionable for a close neutron-star binary. In our metric one can write the metric perturbation as $$h_{i j} = (\phi^4 -1) \delta_{i j}~~.$$ The conformal factor $\phi$ is a solution to a Poisson equation involving source terms from the two stars. Between the stars, the only source of the fields arises from the $K_{i j} K^{i j}$ terms which are quite small. Hence, neglecting $K_{i j} K^{i j}$ terms, $\phi$ is additive in the “vacuum” between the stars, $$\phi = \phi_1 + \phi_2 = 1 +
{m_1 \over 2 \vert r-r_1 \vert}
+ {m_2 \over 2 \vert r-r_2 \vert} ~~.$$ Expanding $h_{i j}$ around star $1$ in the presence of a distant companion $2$ we have $$\begin{aligned}
h_{i j}& =& {4 \over 2}\biggl({m_1 \over \vert r-r_1 \vert} +
{m_2 \over \vert r-r_2 \vert}\biggr)\nonumber \\
& & + {6 \over 4}\biggl({m_1 \over \vert r-r_1 \vert} +
{m_2 \over \vert r-r_2 \vert}\biggr)^2 + \cdot \cdot \cdot \nonumber \\
&&\nonumber \\
&& = h_{\mu \nu}^{NS} + h_{\mu \nu}^B + {\rm cross~terms} ~~.\end{aligned}$$ However, for the binary systems we have considered, the cross terms are $\sim$ 15% to 20% of the sum $h_{\mu \nu}^{NS} + h_{\mu \nu}^B$. Hence, they can not be neglected. The errors associated with this decomposition may be part of the reason that the compression effects are not apparent in this work.
A related concern is with the expansion of the stress-energy tensor in [@flanagan]. We have noted that most of the compression arises from the net effect of velocity dependent terms in the covariant derivative of the stress-energy tensor. In [@flanagan] the stress energy is expanded is powers of the curvature $R^{-m}$. The author states [@flanagan] “We assume initial conditions of vanishing $T_{\mu \nu}^{(2)}$, so that the only source for perturbations is the external tidal field.” An analysis which only considers perturbations from the external tidal field (and not motions of the fluid) will not observe the compression effect. The result of [@flanagan] is that the central density is unchanged until tidal forces enter at $O(R^6)$. This is consistent with our results in the limit of only tidal perturbations acting on the stars. It is not clearto us, however, to what degree the velocity dependent terms are included or excluded by this expansion. A more careful recent revision (E. Flanagan, Priv. Comm.) shows an effect coming in a lower order, but not necessarily as strong as we have noted.
In the paper of Thorne [@thorne97], a similar tidal expansion is applied. In that work only the stabilizing effect of tidal forces was considered along with the stabilizing effect of rotation. However, the increased self gravity from velocity-dependent forces was not included. Hence, the conclusions of [@thorne97] are consistent with our results based upon tidal forces. So are the Newtonian tidal effects computed in [@lai].
CONCLUSIONS
===========
The results of this study (cf. Table \[table1\]) are that we see almost no difference between the central density of an isolated star and a binary star in which rigid corotation has been artificially imposed, or one in which only tidal effects are included. Indeed, in the case of tidal forces alone, the central density in our simulations actually decreases as stars approach, consistent with other works.
An increase in the central density is only apparent in our binary simulations for stars with fluid motion with respect to the corotating frame. (Specifically we considered stars of low intrinsic spin in a binary.) In such cases there is no simple Killing vector which can be imposed to cancel the compression driving forces. We have argued here and in [@mw97] that the main compression effect arises from the net result of velocity-dependent hydrodynamic terms [@fn1]. These terms arise from the affine connection part of the covariant differentiation of the stress-energy tensor.
We show here that the compression effect would not have been observed in a study of tidal forces or any model which artificially imposes rigid corotation of the fluid. A proper treatment must consider all of the force terms apparent in the momentum equation (\[hydromom\]) to sufficient order that their effects on the fluid self gravity survive. A similar conclusion has been reached in [@shapiro] based on test particle dynamics near a Schwarzschild black hole. In that work it is concluded that at least 2.5 post-Newtonian particle dynamics is necessary before a dynamical collapse instability is manifest.
We argue that the results of this study are thus consistent with results in a number of recent papers [@lai; @rs96; @wiseman; @shibata; @lombardi; @lw96; @brady; @flanagan; @thorne97; @baumgarte] which have analyzed the stability of binary stars in various approximations and limits and see no effect. Since we do not disagree with the lack of a compression effect in the limits which they have imposed, we conclude that the existence or absence of the neutron-star compression effect has not yet been independently tested.
Therefore, if one wishes to explore this effect, it would be best to apply a complete unconstrained strong-field relativistic hydrodynamic treatment employing an EOS which produces realistically compact neutron stars. Another alternative, however, might be to study the quasi-equilibrium structure of nonspinning irrotational binary stars at sufficiently high order. In this regard a recently proposed formalism [@bonazzola] to compute quasi-equilibria for nonsynchronous binaries may be of some use. We have begun calculations in this independent formalism. The results will be reported in a forthcoming paper.
Regarding the existence of this low spin state, we find that such a state represents the unconstrained hydrodynamic equilibrium for a close binary. In Newtonian theory, stars are driven to corotation by tidal forces. However in [@bc92] it has been shown that Newtonian tidal forces are insufficient to produce corotation before neutron-star merger unless the viscosity is unrealistically high. Nevertheless, in the absence of strong tidal forces, neutron stars stars gradually spin down. Therefore, even apart from the hydrodynamic effects described here, stars of low spin are likely to be members of close binaries. The hydrodynamic effects described herein, however, could hasten the spin down as stars approach their final orbits and cause the stars to become more compact.
Work at University of Notre Dame supported in part by DOE Nuclear Theory grant DE-FG02-95ER40934, NSF grant PHY-97-22086, and by NASA CGRO grant NAG5-3818. Work performed in part under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405-ENG-48 and NSF grant PHY-9401636.
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------------- --------------------- ----------------------------------
Environment Constraints $\rho_c$ ($10^{14}$ g cm$^{-3})$
Single Star Hydrostatic $5.84$
Single Star Uniform Translation $5.90$
Binary Tidal Only $5.82$
Binary Rigid Corotation $5.90$
Binary Rigid No Spin $6.56$
Binary Full Hydrodynamics $6.68$
\[table1\]
------------- --------------------- ----------------------------------
: Central density for m$_B = 1.625$ M$_\odot$ stars in various conditions using a $\Gamma = 2$ EOS.
----------------- ---------------------------------- --
Separation (km) $\rho_c$ ($10^{14}$ g cm$^{-3})$
41.8 $5.821$
50.6 $5.816$
81.8 $5.821$
$\infty$ $5.837$
\[table2\]
----------------- ---------------------------------- --
: Central density vs. coordinate separation between centers for m$_B = 1.625$ M$_\odot$ ($\Gamma=2$) stars in which only tidal forces are included. The neutron star radius (in isotropic coordinates) is 12 km.
----------------- ---------------------------------- --
Separation (km) $\rho_c$ ($10^{14}$ g cm$^{-3})$
31.2 $14.15$
37.4 $14.16$
64.8 $14.20$
103.8 $14.25$
$\infty$ $14.30$
\[table3\]
----------------- ---------------------------------- --
: Same as Table II but for the EOS of ref. \[2\]. The neutron star radius (in isotropic coordinates) is 6 km.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The traditional difficulty about stochastic singular control is to characterize the regularities of the value function and the optimal control policy. In this paper, a multi-dimensional singular control problem is considered. We found the optimal value function and the optimal control policy of this problem via Dynkin game, whose solution is given by the saddle point of the cost function. The existence and uniqueness of the solution to this Dynkin game are proved through an associated variational inequality problem involving Dirichlet form. As a consequence, the properties of the value function of this Dynkin game implies the smoothness of the value function of the stochastic singular control problem. In this way, we are able to show the existence of a classical solution to this multi-dimensional singular control problem, which was traditionally solved in the sense of viscosity solutions, and this enables the application of the verification theorem to prove optimality. [^1]'
author:
- 'Yipeng Yang [^2]'
title: 'A Multi-dimensional Stochastic Singular Control Problem Via Dynkin Game and Dirichlet Form'
---
Dynkin game, Dirichlet form, Multi-dimensional diffusion, Stochastic singular control
49J40, 60G40, 60H30, 93E20
Introduction and Problem Formulation
====================================
The characterization of the regularities of value function and optimal policy in stochastic singular control remains a big challenge in stochastic control theory, especially the higher dimensional case, see, e.g., [@Soner89]. The traditional approach is to use the viscosity solution technique, see [@Fleming06] [@Bayraktar12] [@Bassan02], which usually yields a less regular solution. Another approach to solve singular control problems and characterize the regularity of value functions is through variational inequalities and optimal stopping or Dynkin game, see, e.g., Karatzas and Zamfirescu [@Karatzas05], Guo and Tomecek [@Guo08]. In [@Karatzas85] Karatzas and Shreve studied the connection between optimal stopping and singular stochastic control of one dimensional Brownian motion, and showed that the region of inaction in the control problem is the optimal continuation region for the stopping problem. In [@Baldursson97], the authors established and exploited the duality between the myopic investor’s problem (optimal stopping) and the social planning problem (stochastic singular control), where an integral form and change of variable formula were also presented on this connection. Ma [@ma92] dealt with a one dimensional stochastic singular control problem where the drift term is assumed to be linear and the diffusion term is assumed to be smooth, and he showed that the value function is convex and $C^2$ and the controlled process is a reflected diffusion over an interval. Guo and Tomecek [@Guo09] solved a one dimensional singular control problem via a switching problem [@Guo08], and showed, using the smooth fit property [@Pham07], that under some conditions the value function is continuously differentiable ($C^1$).
It is found that [@Fuku02] through the approach via game theory and optimal stopping, it is possible to show the existence of a smooth solution. The connection is the following: given a symmetric Markov process on a locally compact separable metric space, it is well known that the solution of an optimal stopping problem admits its quasi continuous version of the solution to a variational inequality problem involving Dirichlet form, e.g., see Nagai [@Nagai78]. Zabczyk [@Zab84] extended this result to a zero-sum game (Dynkin game). In the one dimensional case, the integrated form of the value function of the Dynkin game was identified to be the solution of an associated stochastic singular control problem, e.g., see Taksar [@Taksar85], Fukushima and Taksar [@Fuku02] where a more general one dimensional diffusion is assumed. As a result, the classical smooth solution ($C^2$) can be obtained for this singular control problem.
This paper extends the work by Fukushima and Taksar [@Fuku02] to multi-dimensional stochastic singular control problem. There are many difficulties in this extension. In the one dimensional singular control problem, each point in the space has a positive capacity [@Fuku02], hence the nonexistence of the proper exceptional set. However, this is no longer the case in multi-dimensional singular control problem. We overcome this difficulty using the absolute continuity of the transition function of the underlying process [@Fuku06]. Under some conditions, the optimal control policy of the one dimensional case is proved to be the reflection of the diffusion at two boundary points, but the form of the optimal control policy and the conditions on the regularity of the value function in multi-dimensional case are much more complicated. For instance, in the two dimensional case, the boundary of the continuation region can have various formats, e.g., bounded curves, unbounded curves, singular points, disconnected curves, line segments, etc. The difficulty in characterizing the continuation region is due to the fact that its boundary is a free boundary, and this paper investigates such issues.
In this paper, we are concerned with a multi-dimensional diffusion on ${\mathbb{R}}^n$: $$\label{omodel}
d{\bf X}_t={\bf \mu}({\bf X}_t)dt+{\bf\sigma}({\bf X}_t)d{\bf B}_t,$$ where $${\bf X}_t=\left(\begin{array}{c}X_{1t}\\
\vdots\\
X_{nt}
\end{array}\right), \mu=\left(\begin{array}{c}\mu_1\\
\vdots\\
\mu_n
\end{array}\right), \sigma=\left(\begin{array}{ccc} \sigma_{11} &\ \cdots & \sigma_{1m}\\
\vdots & & \vdots\ \\
\sigma_{n1} &\ \cdots & \sigma_{nm}\end{array}\right), {\bf
B}_t=\left(\begin{array}{c}B_{1t}\\
\vdots\\
B_{mt}\end{array}\right),$$ in which $\mu_i=\mu_i({\bf X}_t)$ and $\sigma_{i,j}=\sigma_{i,j}({\bf X}_t)$ ($1\leqslant i\leqslant n,1\leqslant j\leqslant
m$) are continuous functions of $X_{1t},X_{2t},...,X_{(n-1)t}$, and ${\bf B}_t$ is $m$-dimensional Brownian motion with $m\geqslant n$. Thus we are given a system $(\Omega, \mathcal{F},\mathcal{F}_t, {\bf
X},\theta_t,P_{\bf x})$, where $(\Omega,\mathcal{F})$ is a measurable space, ${\bf X}={\bf X}(\omega)$ is a mapping of $\Omega$ into $C({\mathbb{R}}^n)$, $\mathcal{F}_t=\sigma({\bf X}_s,s\leqslant
t)$, and $\theta_t$ is a shift operator in $\Omega$ such that ${\bf
X}_s(\theta_t\omega)={\bf X}_{s+t}(\omega)$. Here $P_{\bf x}$(${\bf
x}\in{\mathbb{R}}^n$) is a family of measures under which $\{{\bf
X}_t,t\geqslant 0\}$ is an $n$-dimensional diffusion with initial state ${\bf x}$. We assume that $\mu$ and $\sigma$ satisfy the usual Lipschitz growth condition.
A control policy is defined as a pair $(A_t^{(1)},A_t^{(2)})=\mathcal{S}$ of $\mathcal{F}_t$ adapted processes which are right continuous and nondecreasing in $t$ and we assume $A_0^{(1)},A_0^{(2)}$ are nonnegative. Denote $\mathbb{S}$ the set of all admissible policies, whose detailed definition will be given in Section \[mdssc\].
Given a policy $\mathcal{S}=(A_t^{(1)},A_t^{(2)})\in\mathbb{S}$ we define the following controlled process: $$\begin{array}{l}
dX_{1t}=\mu_1dt+\sigma_{11}dB_{1t}+\cdots+\sigma_{1m}dB_{mt},\\
\vdots \quad\quad \vdots \quad\quad\quad \vdots\\
dX_{nt}=\mu_ndt+\sigma_{n1}dB_{1t}+\cdots+\sigma_{nm}dB_{mt}+dA_t^{(1)}-dA_t^{(2)},\\
{\bf X}_0={\bf x},
\end{array}$$ with the cost function $$\begin{aligned}
\label{scost}
k_{\mathcal{S}}({\bf x})=E_{\bf x}\left(\int_0^\infty e^{-\alpha
t}h({\bf X_t})dt+\int_0^\infty e^{-\alpha t}\left(f_1({\bf
X}_t)dA_t^{(1)}+f_2({\bf X}_t)dA_t^{(2)}\right)\right),&&\\
f_1({\bf x}),f_2({\bf x})>0,\ \forall {\bf
x}\in{\mathbb{R}}^n.&&\nonumber\end{aligned}$$ Here we assume that $A_t^{(1)}-A_t^{(2)}$ is the minimal decomposition of a bounded variation process into a difference of two increasing processes.
A natural question is that why the control only applies on one dimension. The difficulty arises in the step where the value function of the zero-sum game is integrated (in one dimension) to obtain the value function of the singular control problem. If the control were applied to multi dimensions, no result so far is know on the choice of the direction of integration. This represents a traditional difficulty in multi-dimensional singular control problem. Interested readers are referred to [@Soner89] for a result on two dimensional singular control problem.
There are two types of costs associated with the process ${\bf X_t}$ for each policy $\mathcal{S}$. The first one is the holding cost $h({\bf X}_t)$ accumulated along time. The second one is the control cost associated with the processes $(A_t^{(1)},A_t^{(2)})$, and this cost increases only when $(A_t^{(1)},A_t^{(2)})$ increase.
One looks for a control policy $\mathcal{S}$ that minimizes $k_{\mathcal{S}}({\bf x})$, i.e., $$\label{mincostw}
W({\bf x})=\min_{\mathcal{S}\in\mathbb{S}} k_{\mathcal{S}}({\bf x}).$$
As an application of this model, a decision maker observes the expenses of a company under a multi-factor situation but only has control over one factor, yet she still wants to minimize the total expected cost. Analogously, by studying the associated maximization problem, i.e., taking the negative of $\min$, this model can be used to find the optimal investment policy where an investor observes the prices of several assets in a portfolio and manages the portfolio by adjusting one of them. Notice that every time there is a control action, it yields a certain associated cost, e.g., the transaction cost.
The rest of this paper is organized as follows: we first introduce some preliminaries on Dirichlet form and a variational inequality problem in Section \[DformDgame\]. In Section \[DgameFBP\] we identify conditions for the value function as well as the optimal policy of the associated Dynkin game. The integrated form of the value function of this Dynkin game is shown in Section \[mdssc\] to be the value of a multi-dimensional singular control problem, and the optimal control policy is also determined consequently. In the appendix we shall correct an error found in the paper by Fukushima and Taksar [@Fuku02].
Dirichlet Form and a Variational Inequality Problem {#DformDgame}
===================================================
Let $\mathbb{D}$ be a locally compact separable metric space, ${\mathbb m}$ be an everywhere dense positive Radon measure on $\mathbb{D}$, and $L^2(\mathbb{D},m)$ denotes the $L^2$ space on $\mathbb{D}$. We assume that the Dirichlet form $(\mathcal{E},\mathscr{F})$ on $L^2(\mathbb{D},m)$ is regular in the sense that $\mathscr{F}\cap C_0(\mathbb{D})$ is $\mathcal{E}_1$ dense in $\mathscr{F}$ and is uniformly dense in $C_0(\mathbb{D})$, where the $\mathcal{E}_1$ norm is defined as follows: $$\|u\|_{\mathcal{E}_1}=\left(\mathcal{E}(u,u)+\int_{\mathbb{D}}u({\bf
x})^2{\mathbb m}(d{\bf x})\right)^{1/2}.$$ Analogously we define $\mathcal{E}_\alpha(u,v)$ as $\mathcal{E}_\alpha(u,v)=\mathcal{E}(u,v)+\alpha(u,v)\ (\alpha>0)$, where $$(u,v)=\int_{\mathbb{D}}u({\bf x})v({\bf x}){\mathbb m}(d{\bf x}).$$
For this Dirichlet form, there exists an associated Hunt process ${\bf M}=({\bf X}_t,P_{\bf x})$ on $\mathbb{D}$, see [@Fuku11], such that
$$p_tf({\bf x}):=E_{\bf x}f({\bf X}_t),\quad {\bf x}\in\mathbb{D}$$
is a version of $T_tf$ for all $f\in C_0(\mathbb{D})$, where $T_t$ is the $L^2$ semigroup associated with the Dirichlet form $(\mathcal{E},\mathscr{F})$. Furthermore, the $L^2$-resolvent $\{G_\alpha,\ \alpha>0\}$ associated with this Dirichlet form satisfies $$\label{resol}
G_\alpha f\in\mathscr{F},\quad \mathcal{E}_\alpha(G_\alpha
f,u)=(f,u),\quad \forall f\in L^2(\mathbb{D};m),\ \forall
u\in\mathscr{F},$$ and the resolvent $\{R_\alpha,\ \alpha>0\}$ of the Hunt process ${\bf M}$ given by $$R_\alpha f({\bf x})=E_{\bf x}\left(\int_0^\infty e^{-\alpha t}f({\bf
X}_t)dt\right),\quad {\bf x}\in\mathbb{D},$$ is a quasi-continuous modification of $G_\alpha f$ for any Borel function $f\in
L^2(\mathbb{D};m)$.
For $\alpha>0$, a measurable function $f$ on $\mathbb{D}$ is called $\alpha$-excessive if $f({\bf x})\geqslant 0$ and $e^{-\alpha t}p_t
f({\bf x})\uparrow f({\bf x})$ as $t\downarrow 0$ for any ${\bf
x}\in\mathbb{D}$. A function $f\in\mathscr{F}$ is said to be an $\alpha$-potential if $\mathcal{E}_\alpha(f,g)\geqslant 0$ for any $g\in\mathscr{F}$ with $g\geqslant 0$. For any $\alpha$-potential $f\in\mathscr{F}$, define $\hat{f}({\bf x})=\lim_{t\downarrow 0}p_t
f({\bf x})$, then $f=\hat{f}\ m$-a.e. and $\hat{f}$ is $\alpha$-excessive (see Section 3 in [@Fuku06]). $\hat{f}$ is called the $\alpha$-excessive regularization of $f$. Furthermore, any $\alpha$-excessive function is finely continuous (see Theorem A.2.7 in [@Fuku11]).
As related literature, Nagai [@Nagai78] considered an optimal stopping problem and showed that there exist a quasi continuous function $w\in\mathscr{F}$ which solves the variational inequality $$w\geqslant g,\quad \mathcal{E}_\alpha(w,u-w)\geqslant 0,\quad
\forall u\in\mathscr{F} {\rm \ with\ }u\geqslant g,$$ and a properly exceptional set $\mathbb{N}$ such that for all ${\bf x}\in\mathbb{D}/\mathbb{N}$, $$w({\bf x})=\sup_\sigma E_{\bf x}\left(e^{-\alpha\sigma}[g({\bf
X}_\sigma)]\right)=E_{\bf x}\left(e^{-\alpha\hat{\sigma}}[g({\bf
X}_{\hat{\sigma}})]\right),$$ where $g$ is a quasi continuous function in $\mathscr{F}$ and $$\hat{\sigma}=\inf\{t\geqslant 0;w({\bf X}_t)=g({\bf
X}_t)\}.$$ Moreover, $w$ is the smallest $\alpha$-potential dominating the function $g$ ${\mathbb m}$-a.e.
Zabczyk [@Zab84] then extended this result to the solution of the zero-sum game (Dynkin game) by showing that there exist a quasi continuous function $V({\bf x})\in\mathscr{F}$ which solves the variational inequality $$\label{vineqH0}
g\leqslant V\leqslant h\ \ {\mathbb m}\ {\rm a.e.},\
\mathcal{E}_\alpha(V,u-V)\geqslant 0,\quad \forall u\in\mathscr{F},\
g\leqslant u\leqslant h\ \ {\mathbb m}\ {\rm a.e.},$$ and a properly exceptional set $\mathbb{N}$ such that for all ${\bf
x}\in\mathbb{D}/\mathbb{N}$, $$\label{saddleH0}
V({\bf x})=\sup_\sigma \inf_\tau J_{\bf x}(\tau,\sigma)=\inf_\tau
\sup_\sigma J_{\bf x}(\tau,\sigma)$$ for any stopping times $\tau$ and $\sigma$, where $$\label{JH0}
J_{\bf x}(\tau,\sigma)=E_{\bf
x}\left(e^{-\alpha(\tau\wedge\sigma)}\left(I_{\sigma\leqslant\tau}g({\bf
X}_\sigma)+I_{\tau<\sigma}h({\bf X}_\tau)\right)\right),$$ and $g\leqslant h$ $\mathbb m$-a.e. are quasi-continuous functions in $\mathscr{F}$.
In these works, there always existed an exceptional set $\mathbb{N}$. Fukushima and Menda [@Fuku06] showed that, if the transition function of ${\bf M}$ satisfies an absolute continuity condition, i.e., $$\label{abscont} p_t({\bf x},\cdot)\ll {\mathbb m}(\cdot),$$ for all $t>0$ and ${\bf x}\in\mathbb{D}$, and $g,h$ satisfy the following separability condition:\
*There exist finite $\alpha$-excessive functions $v_1,v_2\in\mathscr{F}$ such that, for all ${\bf x}\in\mathbb{D}$, $$\label{sepcond}
g({\bf x})\leqslant v_1({\bf x})-v_2({\bf x})\leqslant h({\bf x}),$$*then Zabczyk’s result still holds and there does not exist the exceptional set $\mathbb{N}$. In what follows we shall introduce a version of Theorem 2 in [@Fuku06], where we used $-f_1,f_2$ in places of $g,h$ respectively for the convenience of later use.
Let $f_1,f_2\in\mathscr{F}$ be finely continuous functions such that for all ${\bf x}\in\mathbb{D}$ $$\label{bdcond}
-f_1({\bf x})\leqslant f_2({\bf x}),\ |f_1({\bf
x})|\leqslant\phi({\bf x}),\ |f_2({\bf x})|\leqslant \psi({\bf x}),$$ where $\phi,\psi$ are some finite $\alpha$-excessive functions, and $f_1,f_2$ are assumed to satisfy the following separability condition $$\label{sepcondf}
-f_1({\bf x})\leqslant v_1({\bf x})-v_2({\bf x})\leqslant f_2({\bf
x}).$$
We further define the set $$\label{dk}
K=\{u\in\mathscr{F}:\ -f_1\leqslant u\leqslant f_2,\ {\mathbb m}
{\rm -a.e.}\}.$$ Considering the variational inequality problem $$\label{vineqH0f}
V\in K,\ \mathcal{E}_\alpha(V,u-V)\geqslant 0,\quad \forall u\in K,$$ we have:
\[fuku2\] Assume conditions (\[abscont\]), (\[bdcond\]) and (\[sepcondf\]). There exists a finite finely continuous function $V$ satisfying the variational inequality (\[vineqH0f\]) and the identity $$V({\bf x})=\sup_\sigma\inf_\tau J_{\bf
x}(\tau,\sigma)=\inf_\tau\sup_\sigma J_{\bf x}(\tau,\sigma),\quad
\forall {\bf x}\in\mathbb{D},$$ where $\sigma,\tau$ range over all stopping times and $$\label{JH0f}
J_{\bf x}(\tau,\sigma)=E_{\bf
x}\left(e^{-\alpha(\tau\wedge\sigma)}\left(I_{\sigma\leqslant\tau}(-f_1({\bf
X}_\sigma))+I_{\tau<\sigma}f_2({\bf X}_\tau)\right)\right),$$
Moreover, the pair $(\hat{\tau},\hat{\sigma})$ defined by $$\hat{\tau}=\inf\{t>0:V({\bf X}_t)=f_2({\bf X}_t)\},\quad
\hat{\sigma}=\inf\{t>0:V({\bf X}_t)=-f_1({\bf X}_t)\},$$ is the saddle point of the game in the sense that $$J_{\bf x}(\hat{\tau},\sigma)\leqslant J_{\bf
x}(\hat{\tau},\hat{\sigma})\leqslant J_{\bf
x}(\tau,\hat{\sigma}),\quad \forall {\bf x}\in\mathbb{D},$$ for all stopping times $\tau,\sigma$.
For a given function $H\in L^2(\mathbb{D};m)$ one looks for a solution $V\in K$ to the following variational inequality problem $$\label{vi} \mathcal{E}_\alpha(V,u-V)\geqslant(H,u-V),\ \ \forall
u\in K.$$ Then we have the following proposition:
\[solVIeq\] There exists a unique finite finely continuous function $V\in K$ which solves (\[vi\]).
The proof is essentially identical to the proof of Proposition 2.1 in [@Fuku02] and is omitted here.
We assume further the following separability condition:
There exist finite $\alpha$-excessive functions $v_1,v_2\in\mathscr{F}$ such that, for all ${\bf x}\in\mathbb{D}$, $$\label{sepcondHneq0}
-f_1({\bf x})-G_\alpha H({\bf x})\leqslant v_1({\bf x})-v_2({\bf
x})\leqslant f_2({\bf x})-G_\alpha H({\bf x}),$$
then the following result holds:
\[DgameHneq0\] For any function $H\in L^2({\mathbb D};{\mathbb m})$ ) and any $f_1,f_2\in\mathscr{F}$ such that $f_1({\bf x})+G_\alpha H({\bf x})$ and $f_2({\bf x})-G_\alpha H({\bf x})$ are finely continuous and bounded by some finite $\alpha$-excessive functions, respectively. Assuming (\[abscont\])(\[sepcondHneq0\]), we put $$\begin{aligned}
\label{Jcost}
J_{\bf x}(\tau,\sigma)&=&E_{\bf x}\left(\int_0^{\tau\wedge\sigma}
e^{-\alpha t}H({\bf X}_t)dt\right)\\
&&+E_{\bf x}\left( e^{-\alpha
(\tau\wedge\sigma)}\left(-I_{\sigma\leqslant\tau}f_1({\bf
X}_\sigma)+I_{\tau<\sigma}f_2({\bf X}_\tau)\right)\right)\nonumber\end{aligned}$$ for any stopping times $\tau,\sigma$. Then the solution of (\[vi\]) admits a finite finely continuous value function of the game $$V({\bf x})=\inf_\tau\sup_\sigma J_{\bf
x}(\tau,\sigma)=\sup_\sigma\inf_\tau J_{\bf x}(\tau,\sigma),\quad
\forall {\bf x}\in\mathbb{D}.$$
Furthermore if we let $$E_1=\{{\bf x}\in
{\mathbb{D}}: V({\bf x})=-f_1({\bf x})\}, \quad E_2=\{{\bf x}\in
{\mathbb{D}}: V({\bf x})=f_2({\bf x})\},$$ then the hitting times $\hat{\tau}=\tau_{E_2}$, $\hat{\sigma}=\tau_{E_1}$ is the saddle point of the game $$\label{saddleHn0}
J_{\bf x}(\hat{\tau},\sigma)\leqslant J_{\bf
x}(\hat{\tau},\hat{\sigma})\leqslant J_{\bf x}(\tau,\hat{\sigma})$$ for any ${\bf x}\in \mathbb{D}$ and any stopping times $\tau,\sigma$. In particular, $$V({\bf x})=J_{{\bf x}}(\hat{\tau},\hat{\sigma}),\quad \forall {\bf
x}\in \mathbb{D}.$$
$E_1$ is the set of points where $V=-f_1$ and $E_2$ is the set of points where $V=f_2$. So $\hat{\tau}$ and $\hat{\sigma}$ in Theorem \[DgameHneq0\] can be defined in the same way as in Theorem \[fuku2\]. The proof of Theorem \[DgameHneq0\] is identical to Theorem 2.1 in [@Fuku02].
The Dynkin Game and Its Value Function {#DgameFBP}
======================================
Two players $P_1$ and $P_2$ observe a multi-dimensional underlying process ${\bf X}_t$ in (\[omodel\]) with accumulated income, discounted at present time, equalling $\int_0^\sigma e^{-\alpha
t}H({\bf X}_t)dt$ for any stopping time $\sigma$. If $P_1$ stops the game at time $\sigma$, he pays $P_2$ the amount of the accumulated income plus the amount $f_2({\bf X}_\sigma)$, which after been discounted equals $e^{-\alpha \sigma}f_2({\bf X}_\sigma)$. If the process is stopped by $P_2$ at time $\sigma$, he receives from $P_1$ the accumulated income less the amount $f_1({\bf X}_\sigma)$, which after been discounted equals $e^{-\alpha \sigma}f_1({\bf
X}_\sigma)$. $P_1$ tries to minimize his payment while $P_2$ tries to maximize his income. Let $\tau,\sigma$ be two stopping times, the value of this game is thus given by $$\label{vgame}
V({\bf x})=\inf_\tau\sup_\sigma J_{{\bf x}}(\tau,\sigma),\quad
\forall {\bf x}\in {\mathbb{R}}^n,$$ where $J_{\bf x}$ is given by (\[Jcost\]) on ${\mathbb{R}}^n$.
For the diffusion (\[omodel\]), define its infinitesimal generator $\mathcal{L}$ as $$\label{infgen}
\mathcal{L}:=\sum_{i=1}^n\mu_i\frac{\partial}{\partial
x_i}+\sum_{i,j=1}^n A_{ij}\frac{\partial^2}{\partial x_i\partial
x_j},$$ where $[A_{ij}]={\bf A}=\frac{1}{2}\sigma
\sigma^T$. We assume that ${\bf A}$ is non-degenerate.
Define the measure ${\mathbb m}(d{\bf x})=\rho({\bf x})d{\bf x}$, where $\rho({\bf x})$ satisfies the following condition: $$\label{rhocond}
{\mathbb A}\nabla\rho=\rho\cdot(\mu-{\bf b}),$$ where $b_i=\nabla\cdot{\bf A}_i,i=1,2,...,n$ in ${\bf
b}$. (Notice that when $\mu$ and ${\mathbb A}$ are constants, $\rho({\bf x})$ reduces to $\rho({\bf x})=\exp(({\mathbb
A}^{-1}\mu)\cdot {\bf x})$.) It can be seen that the absolute continuity condition (\[abscont\]) is satisfied.
We are unable to solve the case with a general multidimensional diffusion. Even in the case of one dimensional diffusion, conditions on $\mu$ and $\sigma$ should be made (see Appendix).
For the generator $\mathcal{L}$, its associated Dirichlet form $(\mathcal{E},\mathscr{F})$ densely embedded in $L^2({\mathbb{R}^n};{\mathbb m})$ is then given by $$\label{NDform}
\mathcal{E}(u,v)=\int_{\mathbb{R}^n}\nabla u({\bf x})\cdot{\bf
A}\nabla v({\bf x}){\mathbb m}(d{\bf x}), \quad u,v\in\mathscr{F},$$ where $$\mathscr{F}=\{u\in L^2({\mathbb{R}^n};{\mathbb m}):\ u {\rm\ is\
continuous},\ \int_{{\mathbb{R}^n}}\nabla u({\bf
x})^T\nabla u({\bf x}){\mathbb m}(d{\bf x})<\infty\}.$$
For given functions $H,f_1,f_2$ satisfying the conditions of Theorem \[DgameHneq0\], and noticing that ${\bf X}_t$ is a non-degenerate Ito diffusion, we can conclude that $V({\bf x})$ in Eq.(\[vgame\]) is finite and continuous, and it solves (\[vi\]). Furthermore if we let $$\label{e12R} E_1=\{{\bf x}\in
{\mathbb{R}}^n: V({\bf x})=-f_1({\bf x})\}, \quad E_2=\{{\bf x}\in
{\mathbb{R}}^n: V({\bf x})=f_2({\bf x})\},$$ then the hitting times $\hat{\tau}=\tau_{E_2}$, $\hat{\sigma}=\tau_{E_1}$ is the saddle point of the game $$\label{saddle_Cont}
J_{\bf x}(\hat{\tau},\sigma)\leqslant J_{\bf
x}(\hat{\tau},\hat{\sigma})=V({\bf x})\leqslant J_{\bf
x}(\tau,\hat{\sigma})$$ for any ${\bf x}\in {\mathbb{R}}^n$ and any stopping times $\tau,\sigma$.
In the next section we shall give conditions on $H,f_1,f_2$ and characterize the regularities of $V({\bf x})$ and the form of the optimal control policy.
Optimal Stopping Regions
------------------------
In the one dimensional case, if the functions are defined over a bounded interval, a lot of properties are automatically satisfied [@Fuku02]. But in multi-dimensional case, this is much harder.
It is obvious that the conditions on $H,f_1,f_2$ are critical on the form of optimal control policy. For example, if $H\equiv 0$ and $-f_1({\bf x})<0<f_2({\bf x}),\ \forall {\bf x}$, then no party would ever stop the game and there is no optimal control.
\[assumpHf\] $f_1,f_2\in\mathscr{F}$ are smooth functions, $-M<-f_1({\bf
x})<0<f_2({\bf x})<M,\ \forall {\bf x}\in\mathbb{R}^n$ where $M$ is a constant, and $H\in L^2(\mathbb{R}^n;\mathbb{m})$ is everywhere continuous, and the separability condition (\[sepcondHneq0\]) holds. $H(\bar{\bf x},x_n)$ is strictly increasing in $x_n$, $f_1(\bar{\bf x},x_n)$ is nondecreasing in $x_n$, $f_2(\bar{\bf
x},x_n)$ is nonincreasing in $x_n$. Further more, $(\alpha-\mathcal{L})f_1(\bar{\bf x},x_n)+H(\bar{\bf x},x_n)$ is strictly increasing in $x_n$ and $(\alpha-\mathcal{L})f_2(\bar{\bf
x},x_n)-H(\bar{\bf x},x_n)$ is strictly decreasing in $x_n$. The (hyper)curves $a(\bar{\bf x})$, $b(\bar{\bf x})$ such that $$\begin{aligned}
(\alpha-\mathcal{L})f_1(\bar{\bf x},a(\bar{\bf x}))+H(\bar{\bf
x},a(\bar{\bf x}))&=&0,\\
(\alpha-\mathcal{L})f_2(\bar{\bf x},b(\bar{\bf x}))-H(\bar{\bf
x},b(\bar{\bf x}))&=&0,\end{aligned}$$ with $a(\bar{\bf x})<b(\bar{\bf
x})$, $\forall \bar{\bf x}\in{\mathbb{R}}^{n-1}$, are assumed to be bounded and uniformly Lipschitz continuous.
Then it is easy to see that
\[f1f2Hsign\] Assume Assumption \[assumpHf\]. For any $(\bar{\bf x},x_n)$ with $x_n<a(\bar{\bf
x})$, $$(\alpha-\mathcal{L})f_1(\bar{\bf x},x_n)+H(\bar{\bf x},x_n)<0,$$ and for any $(\bar{\bf x},x_n)$ with $x_n>a(\bar{\bf
x})$, $$(\alpha-\mathcal{L})f_1(\bar{\bf x},x_n)+H(\bar{\bf x},x_n)>0.$$ Similarly, for any $(\bar{\bf x},x_n)$ with $x_n<b(\bar{\bf x})$, $$(\alpha-\mathcal{L})f_2(\bar{\bf x},x_n)-H(\bar{\bf x},x_n)>0,$$ and for any $(\bar{\bf x},x_n)$ with $x_n>b(\bar{\bf
x})$, $$(\alpha-\mathcal{L})f_2(\bar{\bf x},x_n)-H(\bar{\bf x},x_n)<0.$$
Define the set $$\label{cregion}
E=\{{\bf x}\in{\mathbb{R}}^n:-f_1({\bf x})<V({\bf x})<f_2({\bf
x})\}.$$Since $P_2$ would stop the game once $V({\bf x})\leqslant -f_1({\bf x})$ and the instant payoff is $-f_1({\bf x})$, while $P_1$ would stop the game once $V({\bf
x})\geqslant f_2({\bf x})$ and the instant payoff is $f_2({\bf x})$, we could write ${\mathbb{R}}^n$ as a partition: $${\mathbb{R}}^n=E_1\cup E\cup E_2,$$ where $E_1,E_2$ were given in (\[e12R\]).
\[Jandf\] Assume Assumption \[assumpHf\]. For each ${\bf x}\in E_1$, $$(\alpha-\mathcal{L})f_1({\bf x})+H({\bf x})\leqslant0,$$ and for each ${\bf x}\in E_2$, $$(\alpha-\mathcal{L})f_2({\bf
x})-H({\bf x})\leqslant 0.$$
We only give proof to the first half. We know at the point ${\bf
x}\in E_1$ it must be true that $V({\bf x})\leqslant-f_1({\bf x})$, and it is optimal for $P_2$ to stop the game immediately. Suppose $$(\alpha-\mathcal{L})f_1({\bf x})+H({\bf x})>0,\ {\bf x}\in E_1,$$ then by the smoothness of $f_1$ and the continuity of $H$, we can find a small ball $B_r({\bf x})$ containing the point ${\bf x}$, such that for each ${\bf y}\in B_r({\bf x})$, $$(\alpha-\mathcal{L})f_1({\bf y})+H({\bf y})>0.$$ Consider a policy for $P_2$ to stop the game at the first exit time of $B_r({\bf x})$, denoted $\tau_{B_r}$. Then by Dynkin’s formula, the payoff would be $$\begin{aligned}
J_{{\bf x}}&=&E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha t}H({\bf
X}_t)dt+E_{{\bf
x}}(e^{-\alpha \tau_{B_r}}(-f_1({\bf X}_{\tau_{B_r}})))\\
&=&E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha t}H({\bf
X}_t)dt-f_1({\bf x})+E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha
t}(\alpha-\mathcal{L})f_1({\bf X}_t)dt\\
&=&-f_1({\bf x})+E_{{\bf x}}\int_0^{\tau_{B_r}}e^{-\alpha
t}[(\alpha-\mathcal{L})f_1({\bf
X}_t)+H({\bf X}_t)]dt\\
&>&-f_1({\bf x}).\end{aligned}$$ This is a contradiction since $P_2$ tries to maximize his payoff but we assumed that the optimal policy at ${\bf
x}$ was to stop the game immediately.
Assume Assumption \[assumpHf\]. If ${\bf x}=(\bar{\bf x},x_n)\in E_1$, then for any point $(\bar{\bf x},y)$ with $y< x_n$, $$(\alpha-\mathcal{L})f_1(\bar{\bf x},y)+H(\bar{\bf x},y)<0.$$ If ${\bf x}=(\bar{\bf x},x_n)\in E_2$, then for any point $(\bar{\bf
x},y)$ with $y> x_n$, $$(\alpha-\mathcal{L})f_2(\bar{\bf x},y)-H(\bar{\bf x},y)<0.$$ Furthermore, $E_1\subseteq{\mathbb R}^{n-1}\times(-\infty,a]$ and $E_2\subseteq{\mathbb R}^{n-1}\times[b,\infty)$.
This can be easily seen from Proposition \[Jandf\] and the conditions on $f_1,f_2, H$ given in Assumption \[assumpHf\].
Further, noticing the conditions on the curves $a({\bar{\bf x}})$ and $b({\bar{\bf x}})$, $\bar{\bf x}\in{\mathbb R}^{n-1}$, we have the following:
Assume Assumption \[assumpHf\]. $E\supseteq {\mathbb R}^{n-1}\times(a,b)$ and hence $E$ is not empty. Furthermore, the value of this game $V$ is bounded by $M$, where $M$ is given in Assumption \[assumpHf\].
Take any point ${\bf x}=({\bar{\bf x}},x_n)\in E_1$, and denote $\sigma_a$ the hitting time to the curve $a(\cdot)$. Notice that the diffusion (\[omodel\]) is a conservative process by the given conditions, and also by noticing the conditions given on $a(\cdot)$, it can be concluded that $E_{\bf x}(e^{-\alpha \sigma_a})$ goes to zero as $x_n$ goes to $-\infty$. Similarly $E_{\bf x}(e^{-\alpha
\sigma_b})$, ${\bf x}=({\bar{\bf x}},x_n)\in E_2$, goes to zero as $x_n$ goes to $\infty$.
\[assumAB\] There exist functions $A(\bar{\bf x}), B(\bar{\bf x})$, $\bar{\bf
x}\in{\mathbb{R}}^{n-1}$ that are uniformly bounded and such that for any ${\bf x}\in{\mathbb R}^{n-1}\times(-\infty,A]$, $$E_{\bf x}\left(\int_0^{\sigma_a} e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha\sigma_a}M\right)<-f_1({\bf x}),$$ and for any ${\bf x}\in{\mathbb
R}^{n-1}\times[B,\infty)$, $$E_{\bf x}\left(\int_0^{\sigma_b} e^{-\alpha t}H({\bf
X}_t)dt-e^{-\alpha\sigma_b}M\right)>f_2({\bf x}).$$
Assume Assumptions \[assumpHf\] and \[assumAB\], then $A(\bar{\bf x})<a(\bar{\bf x})$ and $B(\bar{\bf
x})>b(\bar{\bf x})$, $\forall\bar{\bf x}\in{\mathbb{R}}^{n-1}$. Furthermore, on ${\mathbb R}^{n-1}\times(-\infty,A]$ player $P_2$ would stop the game immediately and $V=-f_1$, and on ${\mathbb
R}^{n-1}\times[B,\infty)$ player $P_1$ would stop the game immediately and $V=f_2$.
Suppose there is a point ${\bf x}=(\bar{\bf x},x_n)$ with $a(\bar{\bf x})\leq x_n\leq A(\bar{\bf x})$. Then by Dynkin’s formula, $$\begin{aligned}
&&E_{\bf x}\left(\int_0^{\sigma_a\wedge T} e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha(\sigma_a\wedge T)}M\right)\\
&& > E_{\bf x}\left(\int_0^{\sigma_a\wedge T} e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha(\sigma_a\wedge T)}(-f_1({\bf X}_{\sigma_a\wedge
T}))\right)\\
&&=-f_1({\bf x})+E_{\bf x}\left(\int_0^{\sigma_a\wedge T}e^{-\alpha
t}((\alpha-\mathcal{L})f_1({\bf X}_t)+H({\bf
X}_t))dt\right)>-f_1({\bf x})\end{aligned}$$ by Proposition \[f1f2Hsign\]. Taking $T\to\infty$ we get a contradiction.
Now suppose ${\bf x}\in{\mathbb R}^{n-1}\times(-\infty,A]$. For any stopping time $\sigma$ for player $P_2$ the payoff will be $$\begin{aligned}
&&E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]\\
&&=\left(E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]\right)P_{\bf
x}(\sigma\leq \sigma_a)\\
&&+\left(E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]\right)P_{\bf
x}(\sigma>\sigma_a).\end{aligned}$$ When $\sigma\leq\sigma_a$, the following quantity $$\begin{aligned}
&&=E_{\bf x}\int_0^{\sigma\wedge T\wedge\sigma_a}e^{-\alpha t}H({\bf
X}_t)dt+E_{\bf
x}[e^{-\alpha (\sigma\wedge T\wedge \sigma_a)}(-f_1({\bf X}_{\sigma\wedge T\wedge\sigma_a}))]\\
&&=-f_1({\bf x})+E_{\bf x}\int_0^{\sigma\wedge
T\wedge\sigma_a}e^{-\alpha t}((\alpha-\mathcal{L})f_1({\bf
X}_t)+H({\bf X}_t))dt\end{aligned}$$ is less than $-f_1({\bf x})$, $\forall T>0$ by Proposition \[f1f2Hsign\].
When $\sigma>\sigma_a$, $$\begin{aligned}
&&E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf x}[e^{-\alpha \sigma}(-f_1({\bf
X}_{\sigma}))]\\
&&\leq E_{\bf x}\left(\int_0^{\sigma_a}e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha\sigma_a}M\right)\end{aligned}$$ because $M$ is the bound of the payoff of each player. Hence by Assumption \[assumAB\], $$E_{\bf x}\int_0^{\sigma}e^{-\alpha t}H({\bf X}_t)dt+E_{\bf
x}[e^{-\alpha \sigma}(-f_1({\bf X}_{\sigma}))]<-f_1({\bf x}).$$ As a summary, if ${\bf x}\in{\mathbb R}^{n-1}\times(-\infty,A]$, then for any stopping policy of $P_2$, the expected payoff is less than $-f_1({\bf x})$, and the optimal strategy is to stop the game immediately. The other half of this proposition can be proved in a similar way.
By the properties of $A(\cdot)$ (or $B(\cdot)$), we can choose a bounded and continuous curve below $A(\cdot)$ (or a bounded and continuous curve above $B(\cdot)$) which also has the properties as given in Assumption \[assumAB\]. Without loss of generality, we assume $A(\cdot)$ and $B(\cdot)$ are bounded and continuous.
Now it is easy to see that ${\mathbb
R}^{n-1}\times(-\infty,A]\subseteq E_1$ and ${\mathbb
R}^{n-1}\times[B,\infty)\subseteq E_2$. Since the functions $V,f_1,f_2$ are all continuous, the boundary of $E$ consists of continuous curves.
Let $\tilde{E}_1$ be the largest connected region in $E_1$ containing the set ${\mathbb R}^{n-1}\times(-\infty,A]$, and $\tilde{E}_2$ be the largest connected region in $E_2$ containing the set ${\mathbb R}^{n-1}\times[B,-\infty)$, then obviously $V({\bf
x})=-f_1({\bf x}),\forall {\bf x}\in\tilde{E}_1$, and $V({\bf
x})=f_2({\bf x}),\forall {\bf x}\in\tilde{E}_2$. Furthermore $\tilde{E}_1$ has a continuous boundary curve $\tilde{a}$ that is bounded by $A$ and $a$, $\tilde{E}_2$ has a continuous boundary curve $\tilde{b}$ that is bounded by $b$ and $B$.
$\tilde{E}_1,\tilde{E}_2$ are simply connected regions.
It suffices to prove that $\tilde{E}_1$ is simply connected. Suppose there is a point ${\bf x}_0$ such that $V({\bf x}_0)>-f_1({\bf
x}_0)$, and the boundary $\partial D_{\bf x_0}$ of the largest connected region containing ${\bf x}_0$ as well as the points ${\bf
x}$ with $V({\bf x})>-f_1({\bf x})$ belongs to $\tilde{E}_1$, i.e., $\partial D_{\bf x_0}\subset\tilde{E}_1$. Then for any ${\bf x}\in
\partial D_{\bf x_0}$, $V({\bf x})=-f_1({\bf x})$, and for any ${\bf x}\in D_{\bf
x_0}$, $(\alpha-\mathcal{L})f_1({\bf x})+H({\bf x})<0$. Consider any stopping strategy for player $P_2$ with the reward $$\begin{aligned}
&&E_{\bf x_0}\left(\int_0^\sigma e^{-\alpha t}H({\bf
X}_t)dt+e^{-\alpha\sigma}(-f_1({\bf X}_\sigma))\right)\nonumber\\
&&=-f_1({\bf x}_0)+E_{\bf x_0}\int_0^\sigma e^{-\alpha
t}((\alpha-\mathcal{L})f_1({\bf X}_t)+H({\bf X}_t))dt\nonumber\\
&&<-f_1({\bf x}_0),\end{aligned}$$ since $\sigma\leqslant \tau_{\partial D_{\bf x_0}}$ a.s., where $\tau_{\partial D_{\bf x_0}}$ is the first hitting time to $\partial D_{\bf x_0}$, and on $\partial D_{\bf x_0}$ it is known that the game should be stopped by $P_2$ with the payoff $-f_1$. But this is a contradiction since we have assumed that $V({\bf
x}_0)>-f_1({\bf x}_0)$. Therefore there is no *hole* in $\tilde{E}_1$.
If there is any point $(\bar{\bf x},x_n)$ with $x_n<a(\bar{\bf x})$ such that $V(\bar{\bf x},x_n)>-f_1(\bar{\bf x},x_n)$, then the connected region containing this point with $V>-f_1$ is connected to the region ${\mathbb{R}}^{n-1}\times[a,b]$. A similar result holds on the curve $b$.
Suppose not, then the boundary of the connected region containing $(\bar{\bf x},x_n)$ with $V>-f_1$ is contained in ${\mathbb
R}^{n-1}\times(-\infty,a)$. For any stopping strategy for player $P_2$, the process is stopped before it hits the curve $a$. The expected payoff is less than or equal to $-f_1(\bar{\bf x},x_n)$ because in this region $(\alpha-\mathcal{L})f_1+H<0$, and this contradicts the assumption $V(\bar{\bf x},x_n)>-f_1(\bar{\bf
x},x_n)$.
Now it is clear that the region $E$ is connected. Recall that $\hat{\sigma}=\tau_{E_1},\hat{\tau}=\tau_{E_2}$ are the first hitting times to the sets $E_1,E_2$ respectively, and they are finite a.s., we can rewrite $V$ as $$V({\bf x})=E_{\bf
x}\left(\int_0^{\hat{\tau}\wedge\hat{\sigma}}e^{-\alpha t}H({\bf
X}_t)dt+I_{\hat{\sigma}<\hat{\tau}}e^{-\alpha\hat{\sigma}}(-f_1({\bf
X}_{\hat{\sigma}}))+I_{\hat{\tau}<\hat{\sigma}}e^{-\alpha\hat{\tau}}f_2({\bf
X}_{\hat{\tau}})\right).$$
But at this point we still can not tell that $E$ is simply connected. Let ${\bf U}$ be any connected region such that ${\mathbb
R}^{n-1}\times[a,b]\subset {\bf U}$ and ${\bf U}\subset {\mathbb
R}^{n-1}\times[A,B]$. Define $\tau_{\bf U}$ the first exit time of this region, then obviously $\tau_{\bf U}$ is finite a.s. Define the function $$\label{fdef}
F_{\bf U}({\bf x})=E_{\bf x}\left(\int_0^{\tau_{\bf U}}e^{-\alpha
t}H({\bf X}_t)dt+e^{-\alpha\tau_{\bf U}}R({\bf X}_{\tau_{\bf
U}})\right),\ \ \forall {\bf x}\in{\bf U},$$ where $R({\bf X}_{\tau_{\bf U}})=-f_1({\bf X}_{\tau_{\bf
U}})$ if ${\bf X}_{\tau_{\bf U}}\in{\mathbb R}^{n-1}\times[A,a]$, and $R({\bf X}_{\tau_{\bf U}})=f_2({\bf X}_{\tau_{\bf U}})$ if ${\bf
X}_{\tau_{\bf U}}\in{\mathbb R}^{n-1}\times[b,B]$.
We put an assumption jointly on the process ${\bf X}_t$ and the functions $f_1,f_2,h$. Consider any point $(\bar{\bf
x}_0,x_n)\in\bar{\bf U}$ with $x_n<a(\bar{\bf x}_0)$. Define the cone $$C_{(\bar{\bf x}_0,x_n)}=\left\{(\bar{\bf
x},y):x_n\leqslant y\leqslant a(\bar{\bf
x}),\frac{y-x_n}{\|<\bar{\bf x}-\bar{\bf x}_0,y-x_n>\|_2}\geqslant
\zeta\right\},$$ where $\zeta\in(0,1)$ is a constant. Construct the new connected region $\tilde{\bf U}={\bf U}\cup
C_{(\bar{\bf x}_0,x_n)}$ and define the function $F_{\tilde{\bf U}}$ similarly as in (\[fdef\]).
\[unif1\] For any point $(\bar{\bf x}_0,x_n)\in\bar{\bf U}$ with $x_n<a(\bar{\bf x}_0)$, $F_{\tilde{\bf U}}(\bar{\bf
x}_0,x_n)\geqslant F_{\bf U}(\bar{\bf x}_0,x_n)$.
Similarly, if we consider any point $(\bar{\bf x}_0,x_n)\in\bar{\bf
U}$ with $x_n>b(\bar{\bf x}_0)$, we may define the cone $$C_{(\bar{\bf x}_0,x_n)}=\left\{(\bar{\bf
x},y):x_n\geqslant y\geqslant b(\bar{\bf
x}),\frac{x_n-y}{\|<\bar{\bf x}-\bar{\bf x}_0,y-x_n>\|_2}\geqslant
\zeta\right\},$$ where $\zeta\in(0,1)$ is a constant. Construct the new connected region $\tilde{\bf U}={\bf U}\cup
C_{(\bar{\bf x}_0,x_n)}$ and define the function $F_{\tilde{\bf U}}$ similarly as in (\[fdef\]), then we put the following assumption.
\[unif2\] For any point $(\bar{\bf x}_0,x_n)\in\bar{\bf U}$ with $x_n>b(\bar{\bf x}_0)$, $F_{\tilde{\bf U}}(\bar{\bf
x}_0,x_n)\leqslant F_{\bf U}(\bar{\bf x}_0,x_n)$.
In the case of one dimensional diffusion, Assumptions \[assumpHf\] and \[assumAB\] are sufficient to imply that $E$ - the continuation region - is an interval, but in the multidimensional case, more conditions are needed to guarantee a regular property.
Under Assumptions \[unif1\],\[unif2\], the continuation region $E$ is simply connected. Furthermore, the lower and upper boundaries of $E$ (in $x_n$) are uniformly Lipschitz functions of $\bar{\bf
x}\in{\mathbb R}^{n-1}$.
We have shown that $E$ is connected. For any point $(\bar{\bf
x},x_n)\in\bar{E}$ with $x_n<a(\bar{\bf x})$, we consider the new continuation region $E\cup C_{(\bar{\bf x},x_n)}$, and have $F_{E\cup C_{(\bar{\bf x},x_n)}}(\bar{\bf x},x_n)\geqslant
V(\bar{\bf x},x_n)$ by Assumption \[unif1\]. Since in the region below the curve $a$, only player $P_2$ wants to stop the game who wants to maximize his payoff, the new continuation region $E\cup
C_{(\bar{\bf x},x_n)}$ is certainly better than $E$, if not identical, and this is a contradiction since $E$ is assumed to be optimal. Thus $E\cup C_{(\bar{\bf x},x_n)}=E$. Since this holds for any point $(\bar{\bf x},x_n)\in\bar{E}$ with $x_n<a(\bar{\bf x})$, we know that the region in $E$ below the curve $a$ is simply connected, and the lower boundary of $E$ is uniformly Lipshcitz continuous. The second half is proved in a similar manner.
Now we can claim that $\tilde{E}_1=E_1,\tilde{E}_2=E_2$. In what follows we shall still use $\tilde{a}$ as the upper boundary (in $x_n$) of the set $E_1$ (also the lower boundary of $E$), and $\tilde{b}$ as the lower boundary (in $x_n$) of the set $E_2$ (also the upper boundary of $E$), and they are uniformly Lipschitz continuous and bounded curves. We notice that the curve $a$ in Assumption \[assumpHf\] is not necessarily identical to the curve $\tilde{a}$, and the curve $b$ is not necessarily identical to the curve $\tilde{b}$ either. Figure \[region\] illustrates the continuation region $E$ in a two dimensional case.
Regularities of the Value Function
----------------------------------
\[prop2\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}=\tilde{a}(\bar{\bf x})$, $\tilde{b}=\tilde{b}(\bar{\bf x})$ are smooth, then $V$ is smooth on ${\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})$, and $$\begin{aligned}
\alpha V({\bf x})-\mathcal{L}V({\bf x})&=H({\bf x}),\quad {\bf
x}\in{\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}),\label{inAB}\\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&>H({\bf x}),\quad {\bf
x}\in{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}), \\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&<H({\bf x}),\quad {\bf
x}\in{\mathbb{R}}^{n-1}\times(\tilde{b},\infty).\end{aligned}$$
It is still not clear under what conditions the free boundary curves are smooth. But it is reasonable to believe that any condition on this should involve jointly the underlying process and all the reward functions of this problem.
We notice that $V$ is $H$-$\alpha$ harmonic on ${\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})$, which by [@Fuku11] implies the validation of the following equation: $$\label{VIeq}
\mathcal{E}_\alpha(V,u)=(H,u),\quad \forall u\in
C_0^{1,\cdots,1}({\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})).$$ The continuity of $H$ implies that $V$ is smooth on the same region, and an integration by parts yields (\[inAB\]) on this region. The rest of the proof follows from the fact that $V({\bf
x})=-f_1({\bf x}),\forall {\bf
x}\in{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a})$ and $V({\bf
x})=f_2({\bf x}),\forall {\bf x}\in(\tilde{b},\infty)$.
In what follows we will characterize the regularity of $V$ on the boundary curves.
Let $\partial_{\bf u} V$ denote the one-sided directional derivative along a unit vector ${\bf u}\in{\mathbb{R}}^n$ defined in the following manner $$\partial_{\bf u} V({\bf x})=\lim_{\lambda\to 0+}\frac{V({\bf x}+\lambda{\bf u})-V({\bf
x})}{\lambda}.$$ By Proposition \[prop2\], it can be seen that $\partial_{\bf u}
V({\bf x})$ is well defined at any point ${\bf x}$ in any direction ${\bf u}$, and $\partial_{\bf u} V({\bf x})$ is continuous. If ${\bf
x}$ is not on the curves $\tilde{a}(\cdot)$ or $\tilde{b}(\cdot)$, then obviously $\partial_{\bf u} V({\bf x})=\partial_{-\bf u} V({\bf
x})$. The following proposition will characterize the property of $\partial_{\bf u} V({\bf x})$ when ${\bf x}$ is on $\tilde{a}(\cdot)$ or $\tilde{b}(\cdot)$.
Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ are smooth, then $$\partial_{\bf u} V(\bar{\bf x},\tilde{a}(\bar{\bf
x}))=-\partial_{\bf u} f_{1}(\bar{\bf x},\tilde{a}(\bar{\bf x})), \
\partial_{\bf u} V(\bar{\bf x},\tilde{b}(\bar{\bf x}))=\partial_{\bf u} f_{2}(\bar{\bf x},\tilde{b}(\bar{\bf x})),\ \forall
\bar{\bf x}\in{\mathbb{R}}^{n-1},\ \forall {\bf u}\in{\mathbb{R}}^n.$$
- Pick any point $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$ on the curve $\tilde{a}(\cdot)$ and any unit vector ${\bf u}\in{\mathbb
R}^n$ (similar result can be derived for the curve $\tilde{b}(\cdot)$), and construct a ball $B_\delta(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))$ centered at this point with radius $\delta$, and such that for any point $(\bar{\bf x},x_n)\in
B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, we have $A({\bar{\bf x}})<x_n<b(\bar{\bf x})$. This can be easily done under Assumptions \[assumpHf\] and \[assumAB\]. Also by the smoothness and Lipschitz property of the curve $\tilde{a}(\cdot)$, we can choose $\delta$ small so that all the points $(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))+\lambda {\bf u}$ ($0<\lambda\leqslant\delta$) are either on the curve $\tilde{a}(\cdot)$, below the curve $\tilde{a}(\cdot)$ or above the curve $\tilde{a}(\cdot)$. In the first two cases, since we have showed that $V=-f_1$, the result automatically holds. Now we assume $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\lambda {\bf u}$ ($0<\lambda\leqslant\delta$) are above the curve $\tilde{a}(\cdot)$, then it has been shown that $V((\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))+\lambda {\bf u})>-f_1((\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))+\lambda {\bf u})$, hence $\partial_{\bf u}V(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))\geqslant -\partial_{\bf
u}f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$. Assume the equality does not hold, then there is $\epsilon>0$ such that $\partial_{\bf
u}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))= -\partial_{\bf
u}f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\epsilon$. By the continuity of $\partial_{\bf u}V$ and the Lipschitz property of $\tilde{a}(\cdot)$, we can construct a cone $\mathcal{C}_{\bf u}$ containing ${\bf u}$ such that $\forall {\bf v}\in \mathcal{C}_{\bf
u}$, the points $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\lambda
{\bf v}$ ($0<\lambda\leqslant\delta$) are above the curve $\tilde{a}(\cdot)$ and $\partial_{\bf v}V(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))>-\partial_{\bf v}f_1(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))+\epsilon/2$.
- Define $\tau_{\tilde{a},\delta}$ the first exit time from $B_\delta(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))$. Firstly, by the fact that $\bf{X}_t$ is uniformly elliptic, we have $$E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}(\tau_{\tilde{a},\delta})=o(\delta),$$where $o(\delta)$ is a small quantity satisfying $\lim_{\delta\to 0}{o(\delta)}/{\delta}=0$, (see, e.g., [@Fuku02], [@Karatzas91]). Further, we notice that $$1-E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}\right)=\alpha
E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha
t}dt\right)\leqslant\alpha E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}(\tau_{\tilde{a},\delta})=o(\delta).$$
- Recall that player $P_2$ would stop the game at the point $(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$ immediately, and the payoff is $-f_1(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))$. Consider the region $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, it is shown that play $P_1$ would not stop the game in this region. Since $\hat{\tau}=\tau_{\tilde{a},\delta}+\hat{\tau} \circ
\theta_{\tau_{\tilde{a},\delta}}$ where $\theta_t$ is the shift operator, we get $$\hat{\tau}\wedge\sigma=\tau_{\tilde{a},\delta}+(\hat{\tau}\wedge\hat{\sigma})\circ\theta_{\tau_{\tilde{a},\delta}},$$ where $\sigma=\tau_{\tilde{a},\delta}+\hat{\sigma}\circ\theta_{\tau_{\tilde{a},\delta}}$. Therefore by (\[saddle\_Cont\]), $$\begin{aligned}
\label{ineqVx0}
&&-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))=V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\geqslant J_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(\hat{\tau},\sigma)\nonumber\\
&&\ \ \ \ \ \ \ =E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha t}H({\bf
X}_t)dt\right) +E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}V({\bf
X}_{\tau_{\tilde{a},\delta}})\right).\end{aligned}$$ By Assumption \[assumpHf\], it is obvious that $H$ is bounded on $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, i.e., there exists $U>0$ such that $|H({\bf x})|\leqslant U,\ \forall {\bf x}\in
B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$. Then we have $$\begin{aligned}
&&0\leqslant \left|E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha t}H({\bf
X}_t)dt\right)\right|\leqslant UE_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}e^{-\alpha
t}dt\right)\nonumber\\
&&= \frac{U}{\alpha}\left(1-E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}\right)\right)
=o(\delta).\nonumber\end{aligned}$$
The other part $$\begin{aligned}
&&E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}V({\bf
X}_{\tau_{{a},\delta}})\right)=E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left((1-\alpha\tau_{\tilde{a},\delta}+o(\alpha\tau_{\tilde{a},\delta}))V({\bf
X}_{\tau_{\tilde{a},\delta}})\right)\nonumber\\
&&=E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(V({\bf
X}_{\tau_{\tilde{a},\delta}})\right)+E_{(\bar{\bf
x}_0,\tilde{a}(\bar{\bf
x}_0))}\left((-\alpha\tau_{\tilde{a},\delta}+o(\alpha\tau_{\tilde{a},\delta}))V({\bf
X}_{\tau_{\tilde{a},\delta}})\right).\nonumber\end{aligned}$$ Since $V$ is also bounded and $E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}(\tau_{\tilde{a},\delta})=o(\delta)$, we get $$E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(e^{-\alpha\tau_{\tilde{a},\delta}}V({\bf
X}_{\tau_{\tilde{a},\delta}})\right)=E_{(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))}\left(V({\bf
X}_{\tau_{\tilde{a},\delta}})\right)+o(\delta).$$
- By Dynkin’s formula, $$E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(-f_1({\bf
X}_{\tau_{\tilde{a},\delta}})\right)=-f_1(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))+E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}\left(\int_0^{\tau_{\tilde{a},\delta}}\mathcal{L}(-f_1)({\bf
X}_t)dt\right).$$ Since the function $-f_1$ is bounded on $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))$, and $E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}(\tau_{\tilde{a},\delta})=o(\delta)$, we get $$\label{f_1approx}
E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(-f_1({\bf
X}_{\tau_{\tilde{a},\delta}})\right)=-f_1(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))+o(\delta).$$ On the other hand, for each point ${\bf y}\in \partial
B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, i.e., the boundary of $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, and by the smoothness of $-f_1$, $$-f_1({\bf y})=-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))+\partial_{\bf v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))\delta+o(\delta),$$ so $$\begin{aligned}
\label{f_1approx1}
&&E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(-f_1({\bf
X}_{\tau_{\tilde{a},\delta}})\right)\nonumber\\
&&=-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int\partial_{\bf
v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta p_{(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})+o(\delta),\end{aligned}$$ where $p_{(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})$ is the probability density function of the exit distribution. By comparing (\[f\_1approx\]) with (\[f\_1approx1\]), we get $$\int\partial_{\bf v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))\delta p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
y})=o(\delta).$$
- On the boundary of $B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$, $V({\bf y})\geqslant -f_1({\bf y})$. Since $V$ is smooth a.e., $$V({\bf y})=V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\partial_{\bf
v}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta+o(\delta),$$ and $$\begin{aligned}
&&E_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}\left(V({\bf
X}_{\tau_{\tilde{a},\delta}})\right)\\
&&=V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int \partial_{\bf
v}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta p_{(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})+o(\delta)\nonumber\\
&&=-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int \partial_{\bf
v}V(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta p_{(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf y})+o(\delta)\nonumber\\
&&\geqslant -f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))+\int
\partial_{\bf v}(-f_1)(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))\delta
p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
y})\nonumber\\
&& + \int_{\mathcal{C}_{\bf u}\cap\partial B_\delta(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))} \frac{\epsilon}{2}\delta
p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
y})-|o(\delta)|,\nonumber\end{aligned}$$ in view of the conclusion of Step 1, and $\mathcal{C}_{\bf u}\cap\partial B_\delta(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))$ is the part of $\partial
B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))$ in the cone $\mathcal{C}_{\bf u}$. Substitute in (\[ineqVx0\]) the results from Steps 3 and 4, we get $$\begin{aligned}
&&-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))=V(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))\nonumber\\
&& \geqslant -f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))+\int_{\mathcal{C}_{\bf u}\cap\partial B_\delta(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))} \frac{\epsilon}{2}\delta
p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}(d{\bf
y})-|o(\delta)|.\nonumber\end{aligned}$$ Since ${\bf X}_t$ is uniformly elliptic, $\int_{\mathcal{C}_{\bf
u}\cap\partial B_\delta(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))}
p_{(\bar{\bf x}_0,\tilde{a}(\bar{\bf
x}_0))}(d{\bf y})\to \eta$ as $\delta\to 0$, where $0<\eta<1$ is a constant. By choosing $\delta$ small, we get $$\begin{aligned}
-f_1(\bar{\bf x}_0,\tilde{a}(\bar{\bf x}_0))=V(\bar{\bf
x}_0,\tilde{a}(\bar{\bf x}_0))\geqslant -f_1(\bar{\bf
x}_0,\tilde{a}(\bar{\bf
x}_0))+\frac{\epsilon\delta\eta}{4}-|o(\delta)|.\end{aligned}$$ If $\delta$ is sufficiently small, the part $\frac{\epsilon\delta\eta}{4}-|o(\delta)|>0$, and we get a contradiction.
Now the proof is complete.
As a summary we have the following theorem:
\[Vcond\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ are smooth, then $$\begin{aligned}
-f_1({\bf x})<&&V({\bf x})<f_2({\bf x}),\quad \forall{\bf x}\in
{\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}),\\
V({\bf x})&&=-f_1({\bf x}), \quad \forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}],\\
V({\bf x})&&=f_2({\bf x}), \quad \forall{\bf x}\in
{\mathbb{R}}^{n-1}\times[\tilde{b},\infty),\end{aligned}$$ $$\partial_{\bf u} V(\bar{\bf x},\tilde{a}(\bar{\bf x}))=-\partial_{\bf u}
f_{1}(\bar{\bf x},\tilde{a}(\bar{\bf x}),\ \partial_{\bf u}
V(\bar{\bf x},\tilde{b}(\bar{\bf x}))=\partial_{\bf u}
f_{2}(\bar{\bf x},\tilde{b}(\bar{\bf x})),\ \forall \bar{\bf
x}\in{\mathbb{R}}^{n-1},$$ where ${\bf u}$ is any directional vector.
Furthermore $V$ is $C^{1,\cdots,1,1}$ on ${\mathbb{R}}^{n}$, $C^{2,\cdots,2}$ on ${\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})\cup{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a})\cup{\mathbb{R}}^{n-1}\times(\tilde{b},\infty)$ and $$\begin{aligned}
\alpha V({\bf x})-\mathcal{L}V({\bf x})&=H({\bf x}),\quad
&\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}), \\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&>H({\bf x}),\quad
&\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}), \\
\alpha V({\bf x})-\mathcal{L}V({\bf x})&<H({\bf x}),\quad
&\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{b},\infty),\end{aligned}$$ where $\mathcal{L}$ is given in (\[infgen\]).
The Multi-Dimensional Stochastic Singular Control Problem {#mdssc}
=========================================================
Define $h({\bf x}),W({\bf x}),\ {\bf x}\in{\mathbb{R}}^n$, as follows: $$\begin{aligned}
h(\bar{\bf x},y)&=&\int_{\tilde{a}(\bar{\bf x})}^yH(\bar{\bf x},u)du+C(\bar{\bf x}),\label{hcost}\\
W(\bar{\bf x},y)&=&\int_{\tilde{a}(\bar{\bf x})}^yV(\bar{\bf
x},u)du,\quad \bar{\bf x}\in{\mathbb{R}}^{n-1},\
y\in{\mathbb{R}},\label{wval}\end{aligned}$$ where $C(\bar{\bf x})$ is a function of $\bar{\bf x}$ such that $$\lim_{y\to \tilde{a}(\bar{\bf x})+}\alpha W(\bar{\bf
x},y)-\mathcal{L}W(\bar{\bf x},y)-h(\bar{\bf x},y)=0,$$ then $h(\bar{\bf x},y)$ and $W(\bar{\bf x},y)$ satisfy the following:
\[Wcond\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\]. If the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ are smooth, then $W$ is $C^{2,\cdots,2}$ on ${\mathbb{R}}^n$ and $$\begin{aligned}
&&\alpha W({\bf x})-\mathcal{L}W({\bf x})=h({\bf x}),\quad
\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}), \\
&&\alpha W({\bf x})-\mathcal{L}W({\bf x})<h({\bf x}),\quad
\forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}) \cup
{\mathbb{R}}^{n-1}\times(\tilde{b},\infty),\\
&&-f_1({\bf x})<\frac{\partial }{\partial x_n}W({\bf x})<f_2({\bf x}),\quad \forall{\bf x}\in {\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b}),\\
&&\frac{\partial}{\partial x_n}W({\bf x})=-f_1({\bf x}), \quad \forall{\bf x}\in {\mathbb{R}}^{n-1}\times(-\infty,\tilde{a}],\\
&&\frac{\partial}{\partial x_n}W({\bf x})=f_2({\bf x}), \quad
\forall{\bf x}\in {\mathbb{R}}^{n-1}\times[\tilde{b},\infty),\end{aligned}$$ and $$\begin{aligned}
&&\frac{\partial^2}{\partial x_n\partial x_k}W(\bar{\bf
x},\tilde{a}(\bar{\bf x}))=-\frac{\partial f_1}{\partial
x_k}(\bar{\bf
x},\tilde{a}(\bar{\bf x})),\nonumber\\
&&\frac{\partial^2}{\partial x_n\partial x_k}W(\bar{\bf
x},\tilde{b}(\bar{\bf x}))=\frac{\partial f_2}{\partial
x_k}(\bar{\bf x},\tilde{b}(\bar{\bf x})),\ \forall\bar{\bf
x}\in{\mathbb{R}}^{n-1},\ 1\leqslant k\leqslant n.\nonumber\end{aligned}$$
We first need a lemma.
\[cont\_alpha\_L\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\] and \[unif2\] and that the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ being smooth. The function $\alpha W({\bf x})-\mathcal{L}W({\bf x})$ is continuous.
This result obviously holds for ${\bf
x}\in{\mathbb{R}}^{n-1}\times(-\infty,\tilde{a})\cup{\mathbb{R}}^{n-1}\times(\tilde{a},\tilde{b})\cup{\mathbb{R}}^{n-1}\times(\tilde{b},\infty)$. On the curves $\tilde{a}(\bar{\bf x})$ and $\tilde{b}(\bar{\bf x})$, $W$ is twice continuously differentiable along the $x_n$ direction by (\[wval\]). The only term that seems not to be continuous in this function is $\mathcal{L}W({\bf x})$, which involves the first and second derivative with respect to each variable. Denote $\partial_{x_k} W$ the directional derivative along $x_k, 1\leqslant
k\leqslant n-1$, then by (\[wval\]) we have the following: $$\partial_{x_k}W(\bar{\bf x},y)=\int_{\tilde{a}(\bar{\bf
x})}^y \partial_{x_k}V(\bar{\bf x},u)du-V(\bar{\bf
x},\tilde{a}(\bar{\bf x}))\partial_{x_k}\tilde{a}(\bar{\bf x}).$$ Notice that $V$ is $C^{1,1,...,1}$ on ${\mathbb{R}}^n$, so $\partial_{x_k}W(\bar{\bf
x},y)=\partial_{x_k}W({\bf x})$ is continuous in ${\bf x}$. Now consider $$\begin{aligned}
\frac{\partial^2 W}{\partial x_k^2}(\bar{\bf
x},y)&=&\int_{\tilde{a}(\bar{\bf x})}^y\frac{\partial^2 V}{\partial
x_k^2}(\bar{\bf x},u)du-2\partial_{x_k}V(\bar{\bf
x},\tilde{a}(\bar{\bf x}))\cdot\partial_{x_k}\tilde{a}(\bar{\bf
x})\\
\quad && -\partial_{x_n}V(\bar{\bf x},\tilde{a}(\bar{\bf
x}))\cdot(\partial_{x_k}\tilde{a}(\bar{\bf x}))^2-V(\bar{\bf
x},\tilde{a}(\bar{\bf x}))\frac{\partial^2 \tilde{a}}{\partial
x_k^2}(\bar{\bf x}).\end{aligned}$$ Because the two curves $\tilde{a}(\bar{\bf x})$ and $\tilde{b}(\bar{\bf
x})$ have zero Lebesgue measure, and the functions $\partial_{x_k}V$, $\partial_{x_k}\tilde{a}$ and $\frac{\partial^2
\tilde{a}}{\partial x_k^2}$ are all continuous, we conclude that $\frac{\partial^2 W}{\partial x_k^2}$ is continuous, $1\leqslant
k\leqslant n-1$. The continuity of $\frac{\partial^2 W}{\partial
x_i\partial x_j},i\neq j,$ can be proved in a similar manner. Combined with previous argument that $\frac{\partial^2 W}{\partial
x_n^2}$ is continuous, this lemma is proved.
Since $(\alpha-\mathcal{L})W$ is continuous, and the functions $V$ and $H$ are continuous too, we know that the function $h({\bf x})$ in (\[hcost\]) is continuous, hence the continuity of $C(\bar{\bf
x})$ in (\[hcost\]).
For fixed $\bar{\bf x}$, consider the function $$U(y)=\alpha W(\bar{\bf x},y)-\mathcal{L}W(\bar{\bf x},y)-h(\bar{\bf
x},y)$$ with $$U'(y)=\alpha V(\bar{\bf x},y)-\mathcal{L}V(\bar{\bf x},y)-H(\bar{\bf
x},y),$$ and we know $U(\tilde{a}(\bar{\bf x}))=0$. Notice that $U'(y)=0$ for $\tilde{a}(\bar{\bf x})< y< \tilde{b}(\bar{\bf
x})$; $U'(y)>0$ for $y<\tilde{a}(\bar{\bf x})$; $U'(y)<0$ for $y>\tilde{b}(\bar{\bf x})$, and by Lemma \[cont\_alpha\_L\] the function $U(y)$ is continuous, it can be seen that $$\alpha W(\bar{\bf x},y)-\mathcal{L}W(\bar{\bf x},y)<h(\bar{\bf
x},y), {\rm\ for\ } y<\tilde{a}(\bar{\bf x}) {\rm\ or\ }
y>\tilde{b}(\bar{\bf x}).$$ The rest of the proof is obvious.
The result of Theorem \[Wcond\] gives conditions to the solution of the stochastic singular control problem (\[scost\]) and (\[mincostw\]) (see, e.g., [@Taksar85]), where the holding cost $h(\cdot)$ is given in (\[hcost\]) and the boundary penalty costs $f_1(\cdot),f_2(\cdot)$ are given in Assumption \[assumpHf\].
We call a quadruplet $\mathcal{S}=(S,{\bf X}_t,A_t^{(1)},A_t^{(2)})$ ($\mathcal{S}=(A_t^{(1)},A_t^{(2)})$ for simplicity) admissible policy if the following conditions are satisfied:
\[assumpgen\]
1. $S$ is a compact region given in the form ${\mathbb{R}}^{n-1}\times[\beta,\gamma]$ where $\beta(\bar{\bf x}),
\gamma(\bar{\bf x})$ are continuous functions of $\bar{\bf
x}\in{\mathbb{R}}^{n-1}$ with $\beta(\bar{\bf x})< \gamma(\bar{\bf
x})$.
2. There is a filtered measurable space $(\Omega,\{\mathcal{F}_t\}_{t\geqslant
0})$ subject to usual conditions and a probability measure $\{P_{\bf
x}\}_{{\bf x}\in S}$ on it such that
- $\{{\bf X}_t\}_{t\geqslant 0}$ is an $\{\mathcal{F}_t\}$-adapted process, and
- $\{A_t^{(1)},A_t^{(2)}\}_{t\geqslant 0}$ are $\{\mathcal{F}_t\}$-adapted right continuous processes with bounded variation such that $$\label{Acond}
E_{\bf x}\left(\int_{0^-}^\infty e^{-\alpha
t}dA_t^{(1)}\right)<\infty, E_{\bf x}\left(\int_{0^-}^\infty
e^{-\alpha t}dA_t^{(2)}\right)<\infty,\forall {\bf x}\in S,$$ and $A_t^{(1)}-A_t^{(2)}$ is the minimal decomposition of a bounded variation process into a difference of two increasing processes.
3. There are $\{\mathcal{F}_t\}$-adapted independent Brownian motions $B_{1t},...,B_{mt}$ ($m\geqslant n$) starting at the origin under $P_{\bf x}$ for any ${\bf x}\in S$ such that the following controlled diffusion ${\bf X}_t=(X_{1t},...,X_{nt})$ $$\begin{aligned}
\label{cdiff}
dX_{1t}&=&\mu_1dt+\sigma_{11}dB_{1t}+\cdots+\sigma_{1m}dB_{mt},\\
\vdots && \quad\quad \vdots \quad\quad\quad \vdots\nonumber\\
dX_{nt}&=&\mu_ndt+\sigma_{n1}dB_{1t}+\cdots+\sigma_{nm}dB_{mt}+dA_t^{(1)}-dA_t^{(2)},\nonumber\\
{\bf X}_0&=&{\bf x},\nonumber\end{aligned}$$ holds $P_{\bf x}$-a.s., $\forall{\bf x}\in
S$. Furthermore we assume $$P_{\bf x}({\bf X}_t\in S,\forall t\geqslant 0)=1,\quad \forall {\bf
x}\in S.$$
The probability space $\Omega$ with the filtration $\{\mathcal{F}_t\}$ is not fixed a priori. It is part of an admissible policy. The filtration $\{\mathcal{F}_t\}$ is assumed to be right continuous and $\mathcal{F}_0$ is assumed to contain every $P_{\bf x}$-negligible set for any ${\bf x}\in S$.
Both $A_t^{(1)}$ and $A_t^{(2)}$ are nontrivial in the sense that for any $T>0$, $$P_{\bf x}(A_t^{(i)}=A_0^{(i)},\ \ \forall t\in[0,T])=0,\quad \forall
{\bf x}\in S,\ \ i=1,2.$$
If both $A_t^{(1)}$ and $A_t^{(2)}$ are trivial, ${\bf X}_t$ will hit every open region of positive Lebesgue measure in ${\mathbb{R}}^n$ with positive probability, but this is a contradiction since ${\bf X}_t$ is concentrated on $S$. If either $A_t^{(1)}$ or $A_t^{(2)}$ is trivial, ${\bf X}_t$ can not be concentrated on $S$ which again is a contradiction.
Define the following notations: $$\begin{aligned}
\Delta A_t^{(i)}&=&A_t^{(i)}-A_{t^-}^{(i)},\quad t\geqslant 0,
i=1,2,\\
\Delta {\bf X}_t&=&{\bf X}_t-{\bf X}_{t^-},\quad t\geqslant
0,\\
\Delta W({\bf X}_t)&=&W({\bf X}_t)-W({\bf X}_{t^-}),\quad
t\geqslant 0.\end{aligned}$$ Then due to the fact that $A_t^{(1)},A_t^{(2)}$ are the minimal decomposition of a bounded variation process into a difference of two increasing processes, $\Delta A_t^{(1)}\cdot\Delta
A_t^{(2)}=0,\ \forall t\geqslant 0$. By convention we let $$B_{1t}=\cdots=B_{nt}=0,\ A_{t}^{(1)}=A_{t}^{(2)}=0,\quad \forall
t<0,$$ so that $$\Delta A_0^{(i)}=A_0^{(i)},\quad i=1,2,\quad {\bf X}_0={\bf x}\quad
P_{\bf x} {\rm\ a.s.},\quad {\bf x}\in S.$$
Notice that the integrals in (\[Acond\]) involve the possible jumps at time $0$ so that they are the sum of the integrals over $(0,\infty)$ as well as $A_0^{(i)},i=1,2$. In particular, the jump only happens to the $x_n$ coordinate. In what follows, we use $A_t^{(i),c} (i=1,2)$ to denote the continuous part of the processes $A_t^{(i)},i=1,2$.
\[Veri\] Assuming Assumptions \[assumpHf\], \[assumAB\], \[unif1\], \[unif2\], \[assumpgen\] and that the curves $\tilde{a}(\bar{\bf x})$, $\tilde{b}(\bar{\bf x})$ being smooth. Let $k_{\mathcal S}({\bf x})$ be given by the following $$\begin{aligned}
\label{kcostjump}
k_{\mathcal{S}}({\bf x})=&&E_{\bf x}\left(\int_0^\infty e^{-\alpha
t}h({\bf X_t})dt\right)\\
&&+E_{\bf x}\left(\int_0^\infty e^{-\alpha t}\left(f_1({\bf
X}_t)dA_t^{(1),c}+f_2({\bf X}_t)dA_t^{(2),c}\right)\right)\nonumber\\
&&+E_{\bf x}\left(\sum_{0\leqslant t<\infty}e^{-\alpha
t}\left(\int_{X_{nt^-}}^{X_{nt^-}+\Delta A_t^{(1)}} f_1({\bf
X}_t)dy\right.\right.\nonumber\\
&&\left.\left.+\int_{X_{nt^-}-\Delta A_t^{(2)}}^{X_{nt^-}} f_2({\bf
X}_t)dy\right)\right),\nonumber\end{aligned}$$ then
1. For any admissible policy $\mathcal{S}$, $W({\bf x})\leqslant k_{\mathcal S}({\bf
x}),\ \forall {\bf x}\in{\mathbb{R}}^n$.
2. $W({\bf x})= k_{\mathcal S}({\bf
x}),\ \forall {\bf x}\in{\mathbb{R}}^n$, if and only if $S={\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$, where $\tilde{a}(\bar{\bf x}),\tilde{b}(\bar{\bf x})$ are given in Theorem \[Wcond\], the process ${\bf X}_t$ is the reflecting diffusion on $S$, and $\mathcal{S}=(A_t^{(1)},A_t^{(2)})$ where $A_t^{(1)}$ increases only when ${\bf X}_t$ is on the boundary $(\bar{\bf
x},\tilde{a}(\bar{\bf x}))$ and $A_t^{(2)}$ increases only when ${\bf X}_t$ is on the boundary $(\bar{\bf x},\tilde{b}(\bar{\bf
x})),\ \forall\bar{\bf x}\in{\mathbb{R}}^{n-1}$.
The cost function consists of several parts. The first integral in (\[kcostjump\]) is the holding cost. The second integral is a control cost associated with the increment of controls $A_t^{(i)}$ ($i=1,2$) in the continuous part. The last integral is a control cost associated with the jumps in $A_t^{(i)},i=1,2$ (or equivalently jumps in ${\bf X}_t$). We further extend $k_{\mathcal{S}}({\bf x})$ outside the region ${\mathbb{R}}^{n-1}\times[\beta,\gamma]$ for two continuous functions $\beta(\bar{\bf x})<\gamma(\bar{\bf x}),\
\forall\bar{\bf x}\in{\mathbb{R}}^{n-1}$ as the following: $$\begin{aligned}
\label{extcost}
k_{\mathcal{S}}({\bf x})&=&k_{\mathcal{S}}(\bar{\bf
x},\beta(\bar{\bf x}))+\int_{x_n}^{\beta(\bar{\bf x})} f_1(\bar{\bf
x},u)du,\quad \forall{\bf
x}=(\bar{\bf x},x_n)\in{\mathbb{R}}^{n-1}\times(-\infty,\beta),\\
k_{\mathcal{S}}({\bf x})&=&k_{\mathcal{S}}(\bar{\bf
x},\gamma(\bar{\bf x}))+\int_{\gamma(\bar{\bf x})}^{x_n}f_2(\bar{\bf
x},u)du,\quad \forall{\bf x}=(\bar{\bf
x},x_n)\in{\mathbb{R}}^{n-1}\times(\gamma,\infty),\end{aligned}$$ and we are looking for an admissible control $\mathcal{S}$ such that $$\label{winf}
W^*({\bf x})=\inf_{\mathcal{S}\in\mathbb{S}}k_{\mathcal S}({\bf
x}),\quad \forall{\bf x}\in{\mathbb{R}}^n,$$ where $\mathbb{S}$ is the set of all admissible control policies.
1. Consider the diffusion given in (\[cdiff\]) with ${\bf x}\in S$. Applying the generalized Ito formula to $e^{-\alpha t}W({\bf X}_t)$ (see [@Harrison83]) yields $$\begin{aligned}
\label{gIto}
e^{-\alpha t}W({\bf X_t})&=&W({\bf x})-\alpha\int_0^t e^{-\alpha
s}W({\bf X}_s)ds+\int_0^t e^{-\alpha s}\mathcal{L}W({\bf
X}_s)ds\nonumber\\
&&+\int_0^t e^{-\alpha s}\nabla W({\bf
X}_s) \cdot \sigma({\bf X}_s) d{\bf B}_{s}\\
&&+\int_0^te^{-\alpha s}\frac{\partial }{\partial x_n}W({\bf
X}_s)(dA_{s}^{(1),c}-dA_s^{(2),c})+\sum_{0< s\leqslant t}e^{-\alpha
s}\Delta W({\bf X}_s).\nonumber\end{aligned}$$ Using the following identity $$W({\bf x})+\sum_{0< s\leqslant t}e^{-\alpha s}\Delta W({\bf
X}_s)=W({\bf X}_{0^-})+\sum_{0\leqslant s\leqslant t}e^{-\alpha
s}\Delta W({\bf X}_s),$$ and taking expectation of both sides of (\[gIto\]) with respect to $P_{\bf x}$ and let $t\to\infty$, we get the following: $$\begin{aligned}
\label{widen}
W({\bf x})&=&E_{\bf x}\left(\int_0^\infty e^{-\alpha
t}\left(\alpha-\mathcal{L}\right)W({\bf X}_t)dt\right)\\
&&-E_{\bf x}\left(\int_0^\infty e^{-\alpha
t}\frac{\partial}{\partial x_n}W({\bf
X}_t)(dA_{t}^{(1),c}-dA_t^{(2),c})\right)\nonumber\\
&&-E_{\bf x}\left(\sum_{0\leqslant t <\infty}e^{-\alpha t}\Delta
W({\bf X}_t)\right).\nonumber\end{aligned}$$ Therefore $$\begin{aligned}
\label{kmW}
&&\quad\quad \quad k_{\mathcal S}({\bf x})-W({\bf x})\\
&&=E_{\bf x}\left(\int_0^\infty
e^{-\alpha t}\left[h({\bf X}_t)-(\alpha-\mathcal{L})W({\bf
X}_t)\right]dt\right)\nonumber\\
&&+E_{\bf x}\left(\int_0^\infty e^{-\alpha t}\left[f_1({\bf
X}_t)+\frac{\partial }{\partial x_n}W({\bf
X}_t)\right]dA_t^{(1),c}\right)\nonumber\\
&&+E_{\bf x}\left(\int_0^\infty e^{-\alpha t}\left[f_2({\bf
X}_t)-\frac{\partial }{\partial x_n}W({\bf X}_t)\right]dA_t^{(2),c}\right)\nonumber\\
&&+E_{\bf x}\left(\sum_{0\leqslant t<\infty}e^{-\alpha t}\Delta
W({\bf X}_t)\right)\nonumber\\
&&+E_{\bf x}\left(\sum_{0\leqslant t<\infty}e^{-\alpha
t}\left(\int_{X_{nt^-}}^{X_{nt^-}+\Delta A_t^{(1)}} f_1({\bf
X}_t)dy+\int_{X_{nt^-}-\Delta A_t^{(2)}}^{X_{nt^-}} f_2({\bf
X}_t)dy\right)\right).\nonumber\end{aligned}$$ By Theorem \[Wcond\], the first three integrands in (\[kmW\]) are all nonnegative for the process ${\bf X}_t$ staying in the region $S$.
Define the sets $$\Gamma_+=\{t\geqslant 0: \Delta
A_t^{(1)}>0\},\quad \Gamma_-=\{t\geqslant 0: \Delta A_t^{(2)}>0\},$$ then $\Gamma_+\cap\Gamma_-=\phi$. Rewrite the last two expectations of (\[kmW\]) as $$\begin{aligned}
&&E_{\bf x}\left(\sum_{t\in\Gamma_+}e^{-\alpha
t}\int_{X_{nt^-}}^{X_{nt^-}+\Delta A_t^{(1)}}\left[
\frac{\partial}{\partial x_n}W({\bf X}_t)+f_1({\bf
X}_t)\right]dy\right)\\
&&+E_{\bf x}\left(\sum_{t\in\Gamma_-}e^{-\alpha
t}\int_{X_{nt^-}-\Delta A_t^{(2)}}^{X_{nt^-}}\left[
-\frac{\partial}{\partial x_n}W({\bf X}_t)+f_2({\bf
X}_t)\right]dy\right).\end{aligned}$$ By Theorem \[Wcond\] this quantity is nonnegative, and this shows $k_{\mathcal{S}}({\bf x})\geqslant
W({\bf x}),\forall {\bf x}\in S$.
Due to the extension (\[extcost\]), we proved $k_{\mathcal{S}}({\bf x})\geqslant W({\bf x}),\forall {\bf
x}\in{\mathbb{R}}^n$.
2. If $S={\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$ and the process ${\bf X}_t$ is the reflecting diffusion on $S$, then by Theorem \[Wcond\], the first integral in (\[kmW\]) is obviously zero. As to the second and third integrals in (\[kmW\]), because $dA_t^{(1)},dA_t^{(2)}$ are zero whenever ${\bf X}_t$ is in ${\mathbb{R}}^{n-1}\times({a},{b})$, while at the boundary where $A_t^{(1)},A_t^{(2)}$ increases, the integrands are zero, these two integrals are zero too. The last two expectations are also zero due to this construction hence $W({\bf x})= k_{\mathcal S}({\bf
x}),\forall {\bf x}\in S$.
On the other hand, suppose $W({\bf x})= k_{\mathcal S}({\bf
x}),\forall {\bf x}\in S$, then all the expectations in (\[kmW\]) must be zero. Assume $S={\mathbb{R}}^{n-1}\times[\beta,\gamma]$ and at least one of the inequalities is true: $\beta(\bar{\bf x})\neq
g_1(\bar{\bf x}),\gamma(\bar{\bf x})\neq g_2(\bar{\bf x})$, then due to the continuity of these four functions we know that the sum of the first three integrals in (\[kmW\]) is positive by Theorem \[Wcond\]. And because the sum of the last two expectations in (\[kmW\]) is nonnegative, it can be seen that $W({\bf x})<
k_{\mathcal S}({\bf x})$. Therefore in order to have $W({\bf x})=
k_{\mathcal S}({\bf x})$, $S$ must be the region ${\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$.
Again by Theorem \[Wcond\], we see that the processes ${\bf X}_t$ and $A_t^{(i)}$ ($i=1,2$) must all be continuous in order to eliminate the last two expectations in (\[kmW\]), which implies $A_t^{(i)}=A_t^{(i)c}$ ($i=1,2$) when $\beta(\bar{\bf
x})=\tilde{a}(\bar{\bf x}),\gamma(\bar{\bf x})=\tilde{b}(\bar{\bf
x})$. Therefore $({\bf X}_t,A_t^{(1)},A_t^{(2)})$ must be the reflecting diffusion on ${\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$.
The possible jumps, $\Delta A_t^{(i)},i=1,2,$ only happen at time zero. When the process ${\bf X}_t$ starts at a point outside the region ${\mathbb{R}}^{n-1}\times[\tilde{a},\tilde{b}]$, the control brings it back to this region immediately, and after that, the process will be a continuous reflected diffusion. The confirmation of the last assertion is shown below.
If we let $\gamma=(0,0,...,0,1)^T$, then the reflected diffusion can be written as $$\label{refdif}
d{\bf X}_t=\mu({\bf X}_t) dt+\sigma({\bf X}_t) d{\bf B}_t+\gamma
dA_t^{(1)}-\gamma dA_t^{(2)},\ \ t>0,$$ where $A_t^{(1)}$ increases only at the boundary $\tilde{a}(\cdot)$ and $A_t^{(2)}$ increases only at the boundary $\tilde{b}(\cdot)$.
We notice that the reflection only happens to the last component of the process. Since the two curves $\tilde{a}(\cdot)$ and $\tilde{b}(\cdot)$ are smooth and uniformly Lipschitz, if we let $n({\bf x})$ be the inward normal for ${\bf x}$ at the boundary, then we can show that there exist positive constants $\nu_1,\nu_2$ such that $$\begin{aligned}
\forall {\bf x}&=&(\bar{\bf x},\tilde{a}({\bar{\bf x}})),\
(\gamma,n({\bf
x}))\geqslant \nu_1,\\
\forall {\bf x}&=&(\bar{\bf x},\tilde{b}({\bar{\bf x}})),\
(\gamma,n({\bf x}))\leqslant -\nu_2.\end{aligned}$$ By a localization technique and Theorem 4.3 in [@Lions84], it can be shown that there exists a solution $({\bf
X}_t,A_t^{(1)},A_t^{(2)})$ to the reflected diffusion (\[refdif\]). This problem is called the Skorohod problem.
Concluding Remarks {#concluding-remarks .unnumbered}
==================
In this paper, we studied a multi-dimensional stochastic singular control problem via Dynkin game and Dirichlet form. The value function of the Dynkin game satisfies a variational inequality problem, and the integrated form of this value function turns out to be the value function of the singular control problem. By characterizing the regularities of the value function of the Dynkin game and its integrated version, we showed the existence of a classical solution to the Hamilton-Jacobi-Bellman equation associated with this multi-dimensional singular control problem, and this kind of problems were traditionally solved through viscosity solutions. We also proved that, under some conditions, the optimal control policy is given by two curves and the controlled process is the reflected diffusion between these two curves. Unlike the one dimensional singular control problem, where under some conditions, the boundary of the optimal continuation region are given by two points [@Fuku02], it is much more difficult to characterize the boundaries of the continuation region in the multi-dimensional singular control problem. This paper investigates some conditions on the regularity of value function and the form of optimal singular control policies of multi-dimensional diffusion, and it provides a basis for the search of further conditions and further regularities in this realm.
Appendix {#appendix .unnumbered}
========
In this appendix we shall correct an error found in the paper [@Fuku02]. In the paper “Dynkin Games Via Dirichlet Forms and Singular Control of One-Dimensional Diffusion”[@Fuku02], the authors tried to show the existences of a smooth value function and an optimal policy to a one-dimensional stochastic singular control problem via Dynkin game and Dirichlet form. The value function $V(x)$ of a Dynkin game is known to exist [@Zab84], which is the solution of a variational inequality problem involving Dirichlet form. The integration of $V(x)$ turns out to be a smooth optimal return function $W(x)$ for a stochastic singular control problem. Thus the traditional technique of viscosity solution is avoided.
In their paper, the underlying process is a generalized one dimensional diffusion process given by $dX_t=\mu(X_t)dt+\sigma(X_t)dw_t$, in which $w_t$ is a Wiener process. It is found that a different diffusion process should be considered in the proofs, and as a result the main theorem of this paper should be amended.
In their paper, the infinitesimal generator is defined as (see page 693, Eq. 4.1 in [@Fuku02]) $$\label{gen}
Lu(x)=\frac{d}{dm}\frac{d}{ds}u(x)=\mu(x)u'(x)+\frac{1}{2}\sigma(x)^2
u''(x),$$ where $ds(x)=\dot{s}(x)dx,dm(x)=\dot{m}(x)dx$, and (see Eq. 4.2 in [@Fuku02]) $$\label{sam}
\dot{s}(x)
=\exp\left(-\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right),\quad
\dot{m}(x) =
\frac{2}{\sigma(x)^2}\exp\left(\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right).$$
The value function $W(x)$ of the stochastic singular control problem is assumed to satisfy the following PDE (see Eq. 3.23 on page 693 in [@Fuku02]) $$\alpha W(x) -\frac{d}{dm}\frac{d}{ds}W(x) = h(x),$$ or equivalently $$\alpha W(x) -\mu(x)W'(x)-\frac{1}{2}\sigma(x)^2 W''(x) = h(x),$$ where (see Eq. 3.21 and Eq. 3.22 in [@Fuku02]) $$\label{defh}
h(x) = \int_0^x H(y)\dot{s}(y)dy + C,$$ and $$\label{defW}
W(x) = \int_a^x
V(y)\dot{s}(y)dy+\frac{1}{\alpha}\left(-\frac{f_1'(a)}{\dot{m}(a)}+h(a)\right).$$
Then in the proof of Theorem 3.2 on page 693 in [@Fuku02], the authors constructed the function $$U(x)=\alpha W(x)-\frac{d}{dm}\frac{d}{ds}W(x)-h(x),$$ and claimed that $$\frac{1}{\dot{s}(x)}U'(x) = \alpha
V(x)-\frac{d}{ds}\frac{d}{dm}V(x)-H(x).$$ This is equivalent to $$\label{defU}
U(x)=\alpha W(x)-\mu(x)W'(x)-\frac{1}{2}\sigma(x)^2W''(x)-h(x),$$ and $$\label{hjbV}
\frac{1}{\dot{s}(x)}U'(x) = \alpha
V(x)-\mu(x)V'(x)-\frac{1}{2}\sigma(x)^2V''(x)-H(x).$$ However, by a careful examination, it can be seen that the above proposition is not true in general. The reason here is that $\mu(x)$ and $\sigma(x)$ are both functions of $x$, and when taking the derivative of $U(x)$, the product rule has to be applied. The details are shown below.
By the definition of $W(x)$ in (\[defW\]), $h(x)$ in (\[defh\]) and $\dot{s}(x)$ in (\[sam\]) in [@Fuku02], we get $$\begin{aligned}
W'(x) &&= V(x)\dot{s}(x),\nonumber\\
h'(x) &&= H(x)\dot{s}(x),\nonumber\\
\ddot{s}(x)&&=-\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2},\nonumber\end{aligned}$$ hence $$\begin{aligned}
W''(x) &&=
V'(x)\dot{s}(x)-V(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2},\nonumber\\
W'''(x)&&=V''(x)\dot{s}(x)-2V'(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}+V(x)\dot{s}\frac{4\mu(x)^2}{\sigma(x)^4}\nonumber\\
&&\quad
-V(x)\dot{s}(x)\left(2\mu'(x)\sigma(x)^{-2}-4\mu(x)\sigma(x)^{-3}\sigma'(x)\right).\nonumber\end{aligned}$$ Now if we take the derivative of $U(x)$ in (\[defU\]) we get $$\begin{aligned}
U'(x)=&&\alpha
V(x)\dot{s}(x)-\mu'(x)V(x)\dot{s}(x)-\mu(x)V'(x)\dot{s}(x)+\mu(x)V(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}\nonumber\\
&&-\sigma(x)\sigma'(x)V'(x)\dot{s}(x)+\sigma(x)\sigma'(x)V(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}\nonumber\\
&&-\frac{1}{2}\sigma(x)^2\left(V''(x)\dot{s}(x)-2V'(x)\dot{s}(x)\frac{2\mu(x)}{\sigma(x)^2}+V(x)\dot{s}\frac{4\mu(x)^2}{\sigma(x)^4}\right.\nonumber\\
&&\quad
\left.-V(x)\dot{s}(x)\left(2\mu'(x)\sigma(x)^{-2}-4\mu(x)\sigma(x)^{-3}\sigma'(x)\right)\right)-H(x)\dot{s}(x).\nonumber\end{aligned}$$ After simplifying this expression and comparing it with (\[hjbV\]) we should have the following $$0=-\sigma(x)\sigma'(x)V'(x)+2\mu(x)V'(x),$$ which does not hold in general. The following condition should be added to make it hold. $$\label{result}2\mu(x)=\sigma(x)\sigma'(x)$$
A second concern of this paper might be more profound. The Dirichlet form in this paper is defined as (see Eq. 3.3 on page 686 in [@Fuku02]) $$\label{dform}
\mathcal{E}(u,v)=\int_{-A}^Au'(x)v'(x)\frac{1}{\dot{m}(x)}dx,\quad
u,v\in\mathcal{F},$$ where $$\begin{aligned}
\mathcal{F}&&=H^1((-A,A);dx)\nonumber\\
&&=\{u\in L^2((-A,A);dx):\ u{\rm\ is\ absolutely\ continuous,\
}u'\in L^2((-A,A);dx)\}.\nonumber\end{aligned}$$
The authors claimed that this Dirichlet form $(\mathcal{E},\mathcal{F})$ is regular on $L^2([-A,A];ds)$ and the associated underlying process is a reflecting barrier diffusion on $[-A,A]$ with infinitesimal generator $\frac{d}{ds}\frac{d}{dm}$, i.e., the generator $L$ given in (\[gen\]). The correspondence is given by (see Corollary 1.3.1 on page 21 of [@Fuku11]) $$\label{da}
\mathcal{E}(u,v)=(-Lu,v),\quad u\in \mathcal{D}(L),v\in\mathcal{F},$$ where $\mathcal{D}(L)$ is the domain of $L$. Since the underlying process is a reflecting barrier diffusion on $[-A,A]$, $\mathcal{D}(L)$ is given by (see page 22 of [@Fuku11]) $$\begin{aligned}
\mathcal{D}(L)=\{&&u\in\mathcal{F}: u'{\rm\ is\ absolutely\
continuous,\ }\nonumber\\
&&u''\in L^2((-A,A);dx), u'(-A)=u'(A)=0\}.\nonumber\end{aligned}$$
Now we try the integration by parts on (\[dform\]) and get $$\begin{aligned}
\mathcal{E}(u,v)&&=\int_{-A}^A
u'(x)v'(x)\frac{\sigma(x)^2}{2}\exp\left(-\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right)dx\nonumber\\
&&=-\int_{-A}^A\left(\frac{\sigma(x)^2}{2}u''(x)+\sigma(x)\sigma'(x)u'(x)-\mu(x)u'(x)\right)v(x)\exp\left(-\int_0^x\frac{2\mu(y)}{\sigma(y)^2}dy\right)dx.\nonumber\end{aligned}$$
Once again, when the condition (\[result\]) holds, we get $$\frac{\sigma(x)^2}{2}u''(x)+\sigma(x)\sigma'(x)u'(x)-\mu(x)u'(x)=\frac{\sigma(x)^2}{2}u''(x)+\mu(x)u'(x)=Lu(x),$$ and (\[da\]) holds.
As a conclusion, if the condition (\[result\]) is added, then all the results in that paper still hold, but for a very particular Ito diffusion.
In the following we give another way to amend the results of that paper which makes the theorems more general. If we just simply consider the diffusion $$\label{ndiff}
dX_t = \gamma(X_t)dt + \sigma(X_t)dw_t,$$ where $$\gamma(x)=\sigma(x)\sigma'(x)-\mu(x),$$ and define the infinitesimal generator $$L_\gamma u(x) = \gamma(x)u'(x)+\frac{1}{2}\sigma(x)^2 u''(x),$$ while the Dirichlet form is still defined as in (\[dform\]) and $\dot{s}(x),\dot{m}(x)$ are still given in (\[sam\]), then we get $$\mathcal{E}(u,v)=(-L_\gamma u,v).$$ That means the underlying process associated with the Dirichlet form (\[dform\]) should be (\[ndiff\]). With this in mind, we can examine again the results of that paper [@Fuku02]. Results in Section 2 are classical on variational inequalities and optimal stopping. In Section 3, the part $\frac{d}{ds}\frac{d}{dm}$, whenever it appears before Theorem 3.2, should be replace by $L_\gamma$. Let the functions $h(x),W(x)$ still be defined as in Eqn. (3.21) (3.22) in that paper, respectively, then Theorem 3.2 holds intact. But in the proof of this theorem, after setting $$\begin{aligned}
U(x)&=&\alpha W(x)-\frac{d}{dm}\frac{d}{ds}W(x)-h(x)\nonumber\\
&=&\alpha
W(x)-\mu(x)W'(x)-\frac{1}{2}\sigma(x)^2W''(x)-h(x)\nonumber,\end{aligned}$$ and taking the derivative of both sides, we should get $$\begin{aligned}
\frac{1}{\dot{s}(x)}U'(x) &=& \alpha
V(x)-(\sigma(x)\sigma(x)'-\mu(x))V'(x)-\frac{1}{2}\sigma(x)^2V''(x)-H(x)\nonumber\\
&=&\alpha V(x)-L_\gamma V(x)-H(x).\nonumber\end{aligned}$$ Since Theorem 3.1 has been amended, the rest of the proof of Theorem 3.2 just follows. Section 4 of that paper is about a verification theorem, and the results there still hold.
It might be interesting to notice that when $\sigma$ is a constant, we get $\gamma(x)=-\mu(x)$, and $$\mathcal{E}(u,v)=\int_{-A}^A
u'(x)v'(x)\frac{\sigma^2}{2}\exp\left(\int_0^x\frac{2\gamma(y)}{\sigma^2}dy\right)dx.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
In memory of Dr. Michael Taksar.
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[^1]: The idea of this paper was proposed by Dr. Michael Taksar. Dr. Taksar passed away in February, 2012, however, his contributions should always be remembered.
[^2]: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri, 65211 (yangyip@missouri.edu)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The recently proposed Fully-Renormalized QRPA (FR-QRPA), which fullfils the Ikeda sum rule (ISR) exactly, is applied to the two-neutrino double beta decay of $^{76}$Ge, $^{82}$Se, $^{100}$Mo, $^{116}$Cd, $^{128}$Te and $^{130}$Xe. The results obtained are compared with those of other approaches, standard QRPA and self-consistent QRPA (SCQRPA). The similarities and the differences among the methods are discussed. The influence of the restoration of the Ikeda sum rule on the $2\nu\beta\beta$-decay amplitude is analyzed.'
address:
- |
$^1$ Institut für Theoretische Physik der Universität Tübingen,\
Auf der Morgenstelle 14, D-72076 Tübingen, Germany
- '$^2$Department of Nuclear Physics, Comenius University, Bratislava, Slovakia'
author:
- 'L. Pacearescu$^1$, V. Rodin$^1$, F. Šimkovic$^{1,2}$ and Amand Faessler$^1$'
title: 'Two-neutrino double beta decay within Fully-Renormalized QRPA: Effect of the restoration of the Ikeda sum rule'
---
Introduction
============
Observation of the neutrinoless double beta decay ($0\nu\beta\beta$-decay), violating the total lepton number by two units, would give unambiguous evidence for new physics beyond the Standard Model [@fae98; @hax84; @Vog]. For instance, at least one of the neutrinos would have to be a Majorana particle with non-zero mass [@sch82]. The current experimental upper limits on the $0\nu\beta\beta$-decay half-life impose stringent constraints, e.g., on the parameters of Grand Unification and super-symmetric extensions of the Standard Model.
Rates of the $2\nu\beta\beta$-decay, which is a second order process allowed within the Standard Model, can be calculated within the same nuclear structure models. Thus, the results of the nuclear structure calculations can be directly compared with the corresponding experimental data available for a number of nuclei [@exp]. Such a comparison provides a very useful test of the models.
The Quasiparticle Random Phase Approximation (QRPA) [@book] has been successfully exploited in nuclear physics to describe properties of the excited states of open-shell nuclei and to calculate intensities of various nuclear reactions, including the double beta decay (see reviews [@fae98]).
It was shown that the experimental data on the $2\nu\beta\beta$-decay rates can be reproduced in QRPA calculations with a sufficiently large strength of the particle-particle interaction [@vog86]. But the proximity of the value to the point of the QRPA collapse questions the reliability of the results. It is known that QRPA collapse occurs due to the use of the quasi-boson approximation (QBA) which violates the Pauli exclusion principle (PEP) and generates too many ground state correlations.
Renormalized QRPA (RQRPA) was formulated in Refs. [@rqrpa] to restore PEP in an approximate way. The main goal of the method is to use a self-consistent iteration of the QRPA equation with taking into account quasiparticle occupation numbers in the QRPA ground state. That leads to a modification of the commutation relations for bifermionic operators as compared to the ordinary quasiboson approximation (QBA). At the same time so-called scattering terms (describing transitions of the quasiparticles) are neglected in the Hamiltonian and in the phonon operators. The RQRPA does not collapse for physical values of the particle-particle interaction strength and has been extensively used to calculate the intensities of the double beta decay [@fae98; @FKSS97; @Toi97]. It has been also shown that the RQRPA provides better agreement with the exact solution of the many-body problem within schematic models, even beyond the critical point of the standard QRPA (see, e.g. [@schm] and references therein).
The self-consistent RQRPA (SCQRPA) is a more complex version of RQRPA to describe the strongly correlated Fermi systems. Within this method one goes a step further beyond the RQRPA. In the SCQRPA at the same time the quasiparticle mean field is changed by minimizing the energy and fixing the number of particles in the correlated ground state of RQRPA instead of the uncorrelated one of BCS as is done in the other versions of the RQRPA. In this way SCQRPA partially overcomes the inconsistency between RQRPA and the BCS approach and is closer to a fully variational theory.
Nevertheless, the main drawback of the modern versions of RQRPA and SCQRPA is the violation of the model-independent Ikeda sum rule (ISR) [@Toi97; @Sto01; @Bob00]. A modification of the phonon operator by including scattering terms is needed in order to restore the ISR within RQRPA. The fully-Renormalized QRPA (FR-QRPA) was formulated in Ref. [@Rod02] for even-even nuclei in such a way that it complies with restrictions imposed by the commutativity of the phonon creation operator with the total particle number operator. It was shown analytically that the Ikeda sum rule is fulfilled within the FR-QRPA [@Rod02]. Also FR-QRPA is free from the spurious low-energy solutions which would be generated by the scattering terms considered as additional degrees of freedom as suggested in [@Rad98].
The aim of the paper is twofold. First, we would like to describe the FR-QRPA equations in more details (as compared with the original paper [@Rod02]) for a simple case of a Hamiltonian with the separable residual interaction in both particle-hole and particle-particle channels. Second, the first numerical application of FR-QRPA is given to calculate $2\nu\beta\beta$-decay intensities and relevant quantities. So far, the full convergence of the FR-QRPA solution has been obtained only for a rather small model space. Nevertheless a comparison of the results obtained within FR-QRPA and SCQRPA can be provided.
Basic relationships of the Fully-Renormalized QRPA
==================================================
Within RPA an excited nuclear state, with angular momentum $J$ and projection $M$, is created by applying the phonon operator $Q^{\dagger}_{JM}$ to the vacuum state $|0^+_{RPA}\rangle$ of the initial, even-even, nucleus: $$|JM \rangle = Q^{\dagger}_{JM }|0^+_{RPA}\rangle
\qquad \mbox{with} \qquad
Q_{JM }|0^+_{RPA}\rangle=0.
%\label{eq:19}$$
As was shown in Ref. [@Rod02], the most appropriate way is to write down the phonon structure in terms of the particle creation and annihilation operators. That allows to fulfill the important principle of the commutativity of $Q^{\dagger}_{JM}$ with the total particle number operator $\hat A =\hat N + \hat Z$. The phonon operator has the following structure: $$Q^{\dagger}_{JM } = \sum\limits_{pn}
\left [x_{(pn, J )} C^\dagger(pn, JM)
-y_{(pn, J )}\tilde{C}(pn, JM)\right ],
\label{Qc}$$ with $C^\dagger(pn, JM)=\left[c^\dagger_{p}{\tilde{c}}_{n}\right]_{JM}$ and $\tilde C(pn, JM)=(-)^{J-M}C(pn, J\,-M)$, where $c^{+}_{\tau m_{\tau}}$ ($c^{}_{\tau m_\tau}$) denotes the particle creation (annihilation) operator for protons and neutrons ($\tau=p,n$). Going into the quasiparticle representation, the quasiparticle creation and annihilation operators $a^{+}_{\tau m_{\tau}}$ and $a^{}_{\tau m_{\tau}}, (\tau=p,n)$ can be defined by the Bogolyubov transformation $$\left( \matrix{ a^{+}_{\tau m_{\tau} } \cr
{\tilde{a}}_{\tau m_{\tau} }
}\right) = \left( \matrix{
u_{\tau} & v_{\tau} \cr
-v_{\tau} & u_{\tau}
}\right)
\left( \matrix{ c^{+}_{\tau m_{\tau}} \cr
{\tilde{c}}_{\tau m_{\tau}}
}\right),
\label{uv}$$ that leads to the following expression for the phonon operator $Q^{\dagger}_{JM}$: $$\begin{aligned}
&Q^{\dagger}_{JM } = \sum\limits_{{p}{n}}
\left [ X_{({p}{n}, J )} \bar A^\dagger({p}{n}, JM)
- Y_{({p}{n}, J )}\tilde{\bar A\,}({p}{n}, JM)\right ],
\label{Qa1} \\
&\nonumber\\
& \bar A^\dagger= A^\dagger+\left(u_{n}v_{n}B^\dagger-
u_{p}v_{p}\tilde B\right)\left/(v_{n}^2-v_{p}^2)\right. \nonumber\\
& A^\dagger(pn, JM)=\left[a^\dagger_{p} a^\dagger_{n}\right]_{JM}\ ; \ \
B^\dagger(pn, JM)=\left[a^\dagger_{p}{\tilde{a}}_{n}\right]_{JM}\nonumber\end{aligned}$$ where $X=u_{p}v_{n}x-v_{p}u_{n}y,\ Y=u_{p}v_{n}y-v_{p}u_{n}x$. The bifermionic operators $\bar A^\dagger,\bar A$ now being the basic building blocks of the FR-QRPA automatically contain the quasiparticle scattering terms which, however, are not associated with any additional degree of freedom. That means that there are no spurious low-lying solutions in the present theoretical scheme which would be generated by the scattering terms considered as independent constituents of the phonon operator (as proposed in [@Rad98]).
From this point we can follow the usual way to formulate the RQRPA [@rqrpa], substituting $A$ by $\bar A$ everywhere. The forward- and backward-going free variational amplitudes X and Y satisfy the equation: $$\left(
\begin{array}{cc}
{\cal A}&{\cal B}\\
{\cal B}&{\cal A}
\end{array}
\right)
\left(
\begin{array}{c}
X^m\\
Y^m
\end{array}
\right)
= {\cal E}_m
\left(
\begin{array}{cc}
{\cal U}&0\\
0&{\cal -U}
\end{array}
\right)
\left(
\begin{array}{c}
X^m\\
Y^m
\end{array}
\right),
\label{QRPA}$$ where $m$ marks different roots of the QRPA equations for a given $J^\pi$, $$\begin{aligned}
{\cal A} &=& \langle 0^+_{RPA}| \left[ \bar A, \left[ H, \bar
A^\dagger \right ]\right]
|0^+_{RPA} \rangle, \nonumber \\
{\cal B} &=& - \langle 0^+_{RPA}| \left[ \bar A, \left[ H, \bar
A \right ]\right] |0^+_{RPA}
\rangle,
\label{eq:14}\end{aligned}$$ and the renormalization matrix ${\cal U}_{pn}$ is $$\begin{aligned}
{\cal U}_{pn}&=&\langle 0^+_{RPA}|\left[ \bar A(pn, JM),
\bar A^\dagger(p'n', JM)\right]|0^+_{RPA}\rangle =
\delta_{pp'}\delta_{nn'}
{\cal D}_{pn}
.\label{Dm}\end{aligned}$$
We use a rather simple, but realistic, Hamiltonian $H$ consisting of the quasiparticle mean field $H_0$ and the residual separable particle–hole (ph) and particle–particle (pp) interactions: $$\begin{aligned}
& H = H_0 + H^{ph}_{int}+ H^{pp}_{int},\\
& H_0 = \sum\limits_{\tau=p,n} E_{\tau} a^\dagger_{\tau} a_{\tau},\\
& H_{int}^{ph} = ~\chi \sum\limits_{M} (-1)^M ( \beta^-_{1M}\beta^+_{1-M}
+ \beta^+_{1-M} \beta^-_{1M}),\\
& H_{int}^{pp}= - \kappa \sum\limits_{M} (-1)^M ( P^-_{1M}P^+_{1-M}
+ P^+_{1-M} P^-_{1M} ),\end{aligned}$$ with $\beta_{1M}^-=-\hat J^{-1}
\sum\limits_{pn}\langle p\| \sigma\| n\rangle \left[c^\dagger_{p}{\tilde{c}}_{n}\right]_{1M}$, $P_{1M}^-=\hat J^{-1}
\sum\limits_{pn}\langle p\| \sigma\| n\rangle \left[c^\dagger_{p}c^\dagger_{n}\right]_{1M}$ and $J=1$.
Taking into account the exact (fermionic) expressions for the commutators in (\[eq:14\]),(\[Dm\]), one gets the following expressions for the FR-QRPA matrices ${\cal A}$ and ${\cal B}$: $$\begin{aligned}
{\cal A} &=& \left[(E_{p}+E_{n}) {\cal D}_{pn}-
2(E_{p}-E_{n})(u_{p}^{2}v_{p}^{2}+
u_{n}^{2}v_{n}^{2}){\cal R}_{pn}
%\frac{{\cal N}_{p}-{\cal N}_{n}}{v_{n}^{2}-v_{p}^{2}}
\right] \delta_{pp'} \delta_{nn'}\nonumber\\
&&+2\chi (u_{p}v_{n}u_{p'}v_{n'}+
v_{p}u_{n}v_{p'}u_{n'}) {\cal D}_{pn} {\cal D}_{p'n'}\nonumber\\
&&-2\kappa (u_{p}u_{n}u_{p'}u_{n'} {\bar{\bar {\cal D}}}_{pn} {\bar{\bar {\cal D}}}_{p'n'}+
v_{p}v_{n}v_{p'}v_{n'} {\bar {\cal D}}_{pn} {\bar {\cal D}}_{p'n'} ) \label{A},\\
\nonumber\\
{\cal B} &=& 2(E_{p}-E_{n}) u_{p}v_{p}u_{n}v_{n}
{\cal R}_{pn}\delta_{pp'}\delta_{nn'}\nonumber\\
&&+2\chi(u_{p}v_{n}v_{p'}u_{n'}+
v_{p}u_{n}v_{p'}u_{n'}){\cal D}_{pn}{\cal D}_{p'n'}\nonumber\\
&&+2\kappa(u_{p}u_{n}v_{p'}v_{n'}{\bar{\bar {\cal D}}}_{pn}
{\bar {\cal D}}_{p'n'}+
v_{p}v_{n}u_{p'}u_{n'}{\bar {\cal D}}_{pn}{\bar{\bar {\cal D}}}_{p'n'})
.\label{B}\end{aligned}$$
The renormalization matrices ${\cal D},{\bar {\cal D}},{\bar{\bar {\cal D}}}$ entering (\[Dm\]),(\[A\]),(\[B\]) can be represented in terms of the relative quasiparticle occupation numbers ${\cal N}_{p}$ for the level $p$ in the RQRPA vacuum: $$\begin{aligned}
{\cal D}_{pn}& = &
%1+\left((u_{n}^2-v_{n}^2){\cal N}_{n}-(u_{p}^2-v_{p}^2){\cal N}_{p}\right)\left/ (v_{n}^2-v_{p}^2)\right.
1-{\cal N}_{n}-{\cal N}_{p}+\left(1-v_{p}^2-v_{n}^2\right){\cal R}_{pn}
,\label{Dpn}\\
%{\bar {\cal D}}_{pn} &=& 1-2\left(u_{p}^{2}{\cal N}_{p}-u_{n}^{2}{\cal N}_{n}\right)\left/ (v_{n}^2-v_{p}^2)\right.\nonumber\\
{\bar {\cal D}}_{pn} &=& 1-{\cal N}_{n}-{\cal N}_{p}-\left(u_{p}^2+u_{n}^2\right){\cal R}_{pn}
, \nonumber\\
%{\bar{\bar {\cal D}}}_{pn} &=& 1+2\left(v_{p}^{2}{\cal N}_{\tau}-v_{n}^{2}{\cal N}_{n}\right)\left/ (v_{n}^2-v_{p}^2)\right.
{\bar{\bar {\cal D}}}_{pn} &=& 1-{\cal N}_{n}-{\cal N}_{p}+\left(v_{p}^2+v_{n}^2\right){\cal R}_{pn}, \nonumber\end{aligned}$$ with ${\cal R}_{pn}=\frac{{\cal N}_{p}-{\cal N}_{n}}{v_{n}^{2}-v_{p}^{2}}$. In turn, the quasiparticle occupation numbers $${\cal N}_{p}=\hat j_{\tau}^{-2}\langle0^+_{RPA}|\sum\limits^{}_{m_{\tau}}
a^\dagger_{\tau m_{\tau}}{{a}}_{\tau m_{\tau}}|0^+_{RPA}\rangle;~~~\tau=p,n.
%=-\hat j_p^{-1}\langle 0^+_{RPA}|B(pp,00)|0^+_{RPA}\rangle$$ can be expressed in terms of the backgoing amplitudes $Y$ of the RQRPA solution (\[QRPA\]) [@rqrpa]. In the calculation we shall use the aproximate expression for ${\cal N}_{p}$ and ${\cal N}_{n}$:
$$\begin{aligned}
{\cal N}_{p} &\approx& \hat {j}_{p}^{-2}\sum_{n}
\left(\sum_{J,m} (2J+1)|Y^{m}_{pn,J}|^2 \right){\cal D}_{pn}\nonumber\\
{\cal N}_{n} &\approx& \hat {j}_{n}^{-2}\sum_{p}
\left(\sum_{J,m} (2J+1)|Y^{m}_{pn,J}|^2 \right){\cal D}_{pn}
\label{occup}\end{aligned}$$
where $\hat j\equiv\sqrt{2j+1}$. In the present paper we consider only $J^{\pi}=1^{+}$ contribution to the sums in (\[occup\]). Along with the modified SCQRPA and FR-QRPA equations for the chemical potential: $$\begin{aligned}
\langle 0^+_{RPA}|\hat N |0^+_{RPA}\rangle&=&\sum\limits_{n}
\hat j_n^{2}\left(v_n^2+(u_n^2-v_n^2){\cal N}_{n}\right)=N
%-\hat j_n^{-1}(u_n^2-v_n^2)\left<B(nn,00)\right>
,\nonumber\\
\langle 0^+_{RPA}|\hat Z|0^+_{RPA}\rangle&=&\sum\limits_{p}
\hat j_p^{2}\left(v_p^2+(u_p^2-v_p^2){\cal N}_{p}\right)=Z,
\label{lambdas}\end{aligned}$$ a rather complicated set of equations (\[QRPA\])-(\[lambdas\]) has to be solved.
It is noteworthy that the renormalization matrices (\[Dpn\]) become the same, ${\bar{\bar {\cal D}}}_{pn}={\bar {\cal D}}_{pn}={\cal D}_{pn}$, in the limit ${\cal R}_{pn}=0$ and coinciding with the renormalization matrix of the usual RQRPA (see,e.g., [@Toi97]). Thus, one can argue that the standard versions of RQRPA neglect effectively the differences between the quasiparticle occupation numbers whereas SCQRPA and FR-QRPA take the differences into account.
From now on we follow the usual way of solving RQRPA equations [@rqrpa]. It is useful to introduce the notation: $$\bar{X} = {\cal U}^{1/2} X, ~~~~~~\bar{Y} = {\cal U}^{1/2} Y,
\label{eq:15}$$ $$\bar{\cal A} = {\cal U}^{-1/2} {\cal A} {\cal U}^{-1/2}, ~~~~
\bar{\cal B} = {\cal U}^{-1/2} {\cal B} {\cal U}^{-1/2}.
\label{eq:16}$$ Then the amplitudes $\bar X$ and $\bar Y$ satisfy the equation of usual QRPA: $$\left(
\begin{array}{cc}
\bar {\cal A}&\bar {\cal B}\\
\bar {\cal B}&\bar {\cal A}
\end{array}
\right)
\left(
\begin{array}{c}
\bar X^m\\
\bar Y^m
\end{array}
\right)
= {\cal E}_m
\left(
\begin{array}{cc}
1&0\\
0& -1
\end{array}
\right)
\left(
\begin{array}{c}
\bar X^m\\
\bar Y^m
\end{array}
\right).
%\label{dqrpa)
%\label{eq:13}$$ Solving the FR-QRPA equations, one gets the fully renormalized amplitudes $\bar{X}$, $\bar{Y}$ with the usual normalization and closure relations: $$\begin{aligned}
\sum\limits_{pn}\bar{X}^m_{(pn, J )}\bar{X}^k_{(pn, J )}
-\bar{Y}^m_{(pn, J)}\bar{Y}^k_{(pn, J )}&=&\delta_{km},
\nonumber\\
\sum\limits_{m}\bar{X}^m_{(pn, J )}\bar{X}^m_{(p^{\phantom 1}_1n_1, J )}
-\bar{Y}^m_{(pn, J )}\bar{Y}^m_{(p^{\phantom 1}_1n_1, J )}&=&
\delta_{pp^{\phantom 1}_1}\delta_{nn_1},\nonumber\\
\sum\limits_{m}\bar{X}^m_{(pn, J )}\bar{Y}^m_{(p^{\phantom 1}_1n_1, J )}
-\bar{Y}^m_{(pn, J )}\bar{X}^m_{(p^{\phantom 1}_1n_1, J )}&=&0.
\label{closure}\end{aligned}$$
It was shown analytically that the Ikeda sum rule is fulfilled within the FR-QRPA [@Rod02], in contrast to the earlier versions of the RQRPA [@rqrpa]. The Ikeda sum rule states that the difference between the total Gamow-Teller strengths $S^{(-)}$ and $S^{(+)}$ in the $\beta^-$ and $\beta^+$ channels, respectively, is $3(N-Z)$ [@Ikeda]: $$\begin{aligned}
&ISR=S^{(-)}-S^{(+)}=3(N-Z)\label{ISR},\\
&S^{(-)}=\sum\limits_{Mm} \left|\langle 1^{+}M, m| \beta_{1M}^- | 0^+_{RPA}\rangle
\right|^2, ~~~~~ S^{(+)}=\sum\limits_{Mm'} \left|\langle 1^{+}M, m'| \beta_{JM}^+ | 0^+_{RPA}\rangle \right|^2.\nonumber\\
%&\nonumber\end{aligned}$$ With the use of the closure conditions (\[closure\]), the expressions for ${\cal D}_{pn}$ (\[Dpn\]) and the chemical potentials (\[lambdas\]), one can show [@Rod02] that $$ISR=\sum_{pn} \left|\langle p\|q_{J} \| n\rangle\right|^2
(v_n^2-v_p^2) {\cal D}_{pn}=3(N-Z).$$
The inverse half-life of the $2\nu\beta\beta$-decay can be expressed as a product of an accurately known phase-space factor $G^{2\nu}$ and the second order Gamow-Teller transition matrix element $M^{2\nu}_{GT}$: $$[ T^{2\nu}_{1/2}(0^+_{g.s.} \rightarrow 0^+_{g.s.}) ]^{-1} =
G^{2\nu} ~(g_A)^4~ | M^{2\nu}_{GT}|^2.
\label{halfl}$$ The contribution from the two successive Fermi transitions is safely neglected as they arise from isospin mixing effect [@hax84]. The double Gamow-Teller matrix element $M^{2\nu}_{GT}$ for ground state to ground state $2\nu\beta\beta$-decay transition acquires the form $$M^{2\nu}_{GT}=\sum_{{m_i m_f}}
\frac{\langle 0^+_f\parallel \beta^- \parallel 1^{+}_{m_f}\rangle
\langle 1^{+}_{m_f}|1^{+}_{m_i}\rangle
\langle 1^{+}_{m_i}\parallel \beta^- \parallel 0^+_i\rangle}
{(\omega^{m_f} + \omega^{m_i})/2}.
\label{betabeta}$$ The sum extends over all $1^+$ states of the intermediate nucleus. The index $i (f)$ indicates that the quasiparticles and the excited states of the nucleus are defined with respect to the initial (final) nuclear ground state $|0^+_i\rangle$ ($|0^+_f\rangle$). The overlap is necessary since these intermediate states are not orthogonal to each other. The two sets of intermediate nuclear states generated from the initial and final ground states are not identical within the considered approximation scheme. Therefore the overlap factor of these states is introduced in the theory as follows: $$\langle 1^{+}_{m_f}|1^{+}_{m_i} \rangle=
\sum_{pn}
[X_{pn}(1^{+}m_i)X_{pn}(1^{+}m_f)-Y_{pn}(1^{+}m_i)Y_{pn}(1^{+}m_f)]
.\label{overlap}$$
Calculation results
===================
In this section we present the $2\nu\beta\beta$-decay results obtained within the FR-QRPA for $^{76}$Ge, $^{82}$Se, $^{100}$Mo, $^{116}$Cd, $^{128}$Te and $^{130}$Xe, in comparison with the QRPA and SCQRPA ones. Rather small model bases listed in the Table I are used in order to get full convergence in the FR-QRPA method. The levels are in a vicinity of the Fermi levels and spin-orbit partners are always taken into account. FR-QRPA method is rather sensitive to the differences between occupation probabilities for protons and neutrons entering the denominator in the expression of the bifermionic operators ${\bar{A}}^{\dagger}, \bar{A}$ (\[Qa1\]) and in the expresion for $ {\cal R}_{pn}$ factor of renormalization matrices (\[Dpn\]).
For levels far from the Fermi one, the values for occupation probabilities for protons and neutrons become almost equal. Because of the denominator which appears in the expression of bifermionic operators (\[Qa1\]), that causes numerical problems, in particular the method doesn’t converge for large enough values of particle-particle strength. Therefore, the bases are fixed in order to get convergence for a larger interval of particle-particle strength, in particular up to the point of the collapse of the $2\nu\beta\beta$-decay matrix elements. The single particle energies are obtained by using a Coulomb-corrected Woods-Saxon potential with Bertsch parametrization. The proton and neutron pairing gaps are determined phenomenologically to reproduce the odd-even mass differences through a symmetric five-term formula [@waps]. Then the equations for the chemical potentials (\[lambdas\]) are solved for proton and neutron subsystems. The pairing gaps entering the BCS equations are given in the Table I.
The calculation of the QRPA energies and wave functions requires the knowledge of the particle-hole $\chi$ and particle-particle $\kappa$ strengths of the residual interaction. The value of particle-hole strength $\chi$ parameter for each nucleus is fixed in order to reproduce the experimental position of the Gamow-Teller giant resonance in odd-odd intermediate nucleus as obtained from the (p,n) reactions [@madey], [@ejiri], [@akim]. Those values are also given in the Table I. The particle-particle strength $\kappa$ is considered as a free parameter.
The numerical results are shown for two groups of nuclei, the nuclei with $A\leq 100$ and $A>100$ respectively. The calculations are done within QRPA, SCQRPA and FR-QRPA in order to show the better stability of the latter method.
The main drawback of the QRPA is the overestimation of the ground state correlations leading to the collapse of the QRPA ground state, near a certain critical interaction strength. Around this point the backward-going RPA amplitudes $Y_{pn}$ of the first $1^{+}$ states become overrated, and too many correlations in the ground state are generated with increasing strength of the particle-particle interaction. This phenomenon, as a result of the quasiboson approximation used, leads to QRPA collapse and implies an ambiguous determination of the $\beta$ and $2\nu\beta\beta$-decay matrix elements.
In Fig.1 and Fig.2 the dependence of the energy of the first excited Gamow-Teller state in daughter nuclei is plotted versus the $\kappa$ parameter. Hereafter, the dashed line corresponds to the QRPA case, the dotted line represents the SCQRPA case and the solid line describes FR-QRPA case. For all studied nuclei the collapse of the first excited state is shifted to higher values of $\kappa$ for each method and the stability increases in the FR-QRPA case. In Fig.3 and Fig.4 the $2\nu\beta\beta$-decay matrix elements as a function of the particle-particle strength $\kappa$ are shown. The calculations are done for all nuclei within the three metods. The horizontal dashed line indicates the experimental values taken from [@Vog].
For all nuclei there is a similar behaviour in the sense that QRPA and SCQRPA collapse a bit earlier than the FR-QRPA does. Although the chosen bases are rather small, the new effects we intend to emphasise as the differences between QRPA extensions are evident. The FR-QRPA method offers considerably less sensitive dependence of $M^{2\nu}_{GT}$ on $\kappa$ and shifts the collapse to larger values of particle- particle strength.
According to the definition (\[ISR\]) $S^{-}$ ($S^{+}$) is the total summed Gamow-Teller $\beta^{-} $($\beta^{+}$) transition strength from the ground state of an even-even nucleus. In Fig.5 and Fig.6 we plot the relative $\beta^{-}$ strength, $S^{-}/3(N-Z)$ for the mother nucleus (left side) and relative $\beta^{+}$ strength, $S^{+}/3(N-Z)$ for the daugther (right side), for $A\leq 100$ and $A>100$ respectively as a function of particle-particle interaction parameter in order to show the magnitude and the nature of violation of ISR.
Now we would like to discuss the conservation of the Ikeda sum rule $ISR=S_{-}-S_{+}=3(N-Z)$ in the FR-QRPA framework and to compare with the previous calculations for QRPA ans SCQRPA. We didn’t include the calculations for RQRPA because SCQRPA goes beyond and brings more improvements than RQRPA, especially for Ikeda sum rule.
Finally we combine the data of previous plots and show the ratio of SCQRPA and FR-QRPA sum $ISR/3(N-Z)$ as a function of $\kappa$. In the QRPA the Ikeda sum rule is exactely conserved as long as all spin-orbit partners of the single-particle orbitals are included. In the other extended versions of QRPA the sum rule is violated with a degree of deviation lying between $17\%$ (RQRPA) and $3\%$ (SCQRPA) [@Sto01]. In our study, following the analytical calculation of [@Rod02], we have shown numerically that Ikeda sum rule is exactly fullfiled within FR-QRPA formalism.
Conclusions
===========
In summary, the first calculation of the $2\nu\beta\beta$-decay matrix elements within the recently proposed Fully-Renormalized QRPA, which fulfills Ikeda sum rule exactly, are presented. The considered nuclear model includes the separable Gamow-Teller residual interaction. The subject of interest is the effect of the restoration of the Ikeda sum rule on the $2\nu\beta\beta$-decay observable for $A=76, 82, 100, 116, 128, 130$ systems. Within the present work we arrived to the folowing important conclusions:\
i) The SCQRPA violates the Ikeda sum rule. This phenomenon has been indicated in the previous studies [@Sto01], but the degree of violation we obtained is less than in the other calculations because we did not include all multipolarities.
ii\) In the limit when the difference between proton and neutron quasiparticle occupation numbers is neglected the FR-QRPA coincides with SCQRPA.
iii\) From a comparison of FR-QRPA with SCQRPA results we conclude that the effect of the restoration of the Ikeda sum rule is important in the range of large value of particle-particle strength beyond the point of collapse of the standard QRPA.
It is worth to mention that the FR-QRPA approach is sensitive to the precise evaluation of the proton and neutron quasiparticle occupation numbers. Due to the limitation of the approximate expression given in (\[occup\]) (motivated by a similarity to the SQRPA approach) the convergence of the FR-QRPA is achieved only for relatively small model space. However, even for such a model space the differences among the standard QRPA, SCQRPA, FR-QRPA approach are evident for $\kappa$ close to the point the standard QRPA breaks up. There is a hope that for a proper ansatz of the RPA ground state the FR-QRPA approach can work also for a large model space. This is the subject of our further study.
This work was supported in part by the Landesforschungsschwerpunktsprogramm Baden-Wuerttemberg “Low Energy Neutrino Physics”. L.P. and V.R. would like to thank the Graduiertenkolleg “Hadronen im Vakuum, in Kernen und Sternen” GRK683 and IKYDA02 project for support. The work of F. Š. was supported in part by the Deutsche Forschungsgemeinschaft (436 SLK 17/298) and by the VEGA Grant agency of the Slovac Republic under contract No. 1/0249/03.
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$^{76}$Ge $^{76}$Se $^{82}$Se $^{82}$Kr $^{100}$Mo $^{100}$ Ru
-------------------- -------------------- ------------ ---------------- ------------ ---------------- -------------
Basis 1p, 0f, 0g 1p, 0f, 0g 1p, 2s, 1d, 0g
$\Delta_p$ \[MeV\] 1.561 1.751 1.401 1.734 1.612 1.548
$\Delta_n$ \[MeV\] 1.535 1.710 1.544 1.644 1.358 1.296
$\chi$ \[MeV\] 0.21 0.18 0.17
$^{116}$Cd $^{116}$Sn $^{128}$Te $^{128}$Xe $^{130}$Te $^{130}$ Xe
Basis 1p, 2s, 1d, 0g, 0h 2s, 1d, 0g, 0h 2s, 1d, 0g, 0h
$\Delta_p$ \[MeV\] 1.493 1.763 1.127 1.177 1.299 1.043
$\Delta_n$ \[MeV\] 1.377 1.204 1.307 1.266 1.243 1.180
$\chi$ \[MeV\] 0.14 0.14 0.12
: The proton and neutron pairing gaps determined phenomenologically to reproduce the odd-even mass difference and the particle-hole strength $\chi$ chosen to reproduce the experimental position of Gamow-Teller resonance. The single particle basis for all nuclei under consideration is also shown.
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